Recursive Functionals (Studies in Logic and the Foundations of Mathematics Vol 131)

RECURSIVE FUNCTIONALS

STUDIES IN LOGIC AND

THE FOUNDATIONS OF MATHEMATICS VOLUME 131

Honorary Editor:

P. SUPPES, Stanford

Editors:

S. ABRAMSKY, London J. BARWISE, Stanford K. FINE, LosAngeles H. J. KEISLER, Madison A . S. TROELSTRA,Arnsterdarn

NORTH-HOLLAND AMSTERDAM *LONDON NEW YORK *TOKYO

RECURSIVE F"CTI0NALS

Luis E. SANCHIS School of Computer and Information Science Syracuse University Syracuse, N . Y , U . S . A .

1992

NORTH-HOLLAND AMSTERDAM - L O N D O N 0NEWYORK *TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box211,lOOOAE Amsterdam,The Netherlands

P.G. Wodehouse quoted on page vi with the permission of the Random Century Group, London, England and A.P. Watt, Ltd., London, England on behalf of The Trustees of the Wodehouse Estate. ISBN: 0 444 89447 0 1992 Elsevier Science Publishers B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. without the prior written permission of the publisher, Elsevier Science Publishers B .V., Copyright & Pcrmissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. -This publication has been registered with thc Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A. should be referred to the publisher. N o responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, instructions or ideas contained in the material herein.

This book is printed on acid-free paper. Printed in The Netherlands

To the memory of Haskell B. Curry

Too elaborate, Jeeves-that

is what you are frequently prone to become. Your methods are not simple, not straightforward. You cloud the issue with a lot of fancy stuff that is not of the essence.

P. G. Wodehouse Right Ho, Jeeves

vii

Preface This work is a self-contained elementary exposition of the theory of recursive functionals that also includes a number of advanced results. Although it was initially conceived as an extension of our previous work on numerical recursive functions, the final result is substantially different. The main reason for this departure is the role of recursion, which can be eliminated at the first order level (and eventually can be handled via reflexivity and the second recursion theorem), while it must be taken as a primitive at the second order level. In fact, recursion is the central notion in our presentation, and our extended Church’s thesis takes into account this basic situation. Although aiming basically at a theory of higher order computability, we have restricted our attention to second order functionals, where the arguments are numerical functions and the values, when defined, are natural numbers. This theory is somewhat special, for to some extent it can be reduced to the first order theory, but when properly extended and relativized it requires the full machinery of higher order computations. It appears to us that a good and sufficiently general description of the theory at this level should be in place before considering extensions to higher types. Our presentation considers not only singular functionals, which are defined only for total arguments, but general functionals with possibly undefined arguments. This approach is not standard in most of the literature, but some important precedents by Kleene, Hinman, and Tourlakis are cited in the text. T h e reasons for this extension to non-total arguments is not a simple desire for generalization. There is a price to pay here that some may consider too high. But there is a challenge, too, for the notion of computation with partially defined arguments presents a number of conceptual problems that we have tried to solve. Essentially, this form of computation appears to be ambiguous and not well defined, and this is certainly a precarious situation for any decent notion of computability. Our position is that there is still real computation there, and the ambiguity can be solved by restricting the theory to monotonic functionals. By imposing monotonicity we find that non-singular functionals come very close

...

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L.E. Sanchis

to singular functionals, for under such an assumption the values of a functional are determined in principle by the values on total arguments. At the same time, they provide extra information that is not available with singular functionals, information that is essentially related to computational properties. This being the case, we think the price we are paying is not too high, and, in fact, that there is a profit in the

transaction. In this theory of recursive monotonic functionals we are able to formulate a reasonable notion of computation which provides the right frame for what appears to be a convincing form of the extended Church’s thesis. At the same time, the theory provides sufficient room to formulate the classical results that are usually derived in terms of singular functionals. In fact, we present complete proofs of Gandy’s selector theorem, Kleene’s theorem on hyperarithmetical predicates, and Grilliot’s theorem on effectively discontinuous functionals. Since recursion is the crucial notion in this work, we have dedicated considerable space to study its foundations, mathematical and computational. The former is derived from set theory, particularly the minimal fixed point theorem, which is proved for domains. The computational aspect is formalized via recursive algorithms, which incorporate a number of features of modern higher order programming languages. As we are not satisfied with the dichotomy of set theory versus computation theory, we have included an appendix where the possibility of a far-reaching synthesis is discussed. The text is self-contained, and assumes only that the reader has some mathematical maturity. Many examples are provided for the purpose of clarifying the material. Although we do not assume familiarity with our previous work on reflexive structure, a few very general results are taken from there. Most persons cited in the references have influenced this work in one way or another. I reserve a special mention for two who are not included there. First, the late Haskell B. Curry, who taught me the special role of combinators and lambda calculus in the foundations of mathematics. Second, Dana Scott, who many years ago identified the basic mathematical ideas in a truly general theory of computation. I am very grateful to Syracuse University for the support given to this project, particularly a sabbatical leave in 1990. It is a pleasure to acknowledge the superior work of Elaine Weinman in preparing the UTEXversion of the manuscript. There is no way for me to exaggerate the patience and understanding of my

Preface

ix

wife Freide during the preparation of this work. My daughters Laura and Gabriela provided a welcome dose of good humor to alleviate the dreariness and arrogance which inevitably invade the author of a mathematical monograph. Syracuse, New York

Luis E. Sanchis

This Page Intentionally Left Blank

xi

Contents Preface.. .....................................................................

vii

1 Mappings and Domains.. ......................................................

1

CHAPTER

2 Functionals and Predicates.

CHAPTER

3 Basic Operations..

..................................................

.15

...........................................................

.29

CHAPTER

4 Primitive Recursive Operations. ..............................................

.47

5 Basic Recursion..

.67

CHAPTER

CHAPTER

CHAPTER

............................................................

6

Church’s Thesis.. ............................................................ CHAPTER

.85

7

Functional Recursion. ........................................................

.97

8 Recursive Algorithms.. ......................................................

109

9 Formalization: Structural Semantics,. .......................................

.125

10 Formalization: Reductional Semantics. ......................................

.141

CHAPTER 11 Interpreters. .................................................................

157

12 A Universal Interpreter ......................................................

171

13 Enumeration.. ..............................................................

.189

CHAPTER

CHAPTER

CHAPTER

CHAPTER

CHAPTER

L . E. Sanchis

xi1

CHAPTER 1 4 Continuous Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

15 A Selector Theorem . . . . . , . . , . . . , . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219 CHAPTER

16 Hyperenumeration . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233

CHAPTER

17 Recursion in Normal Classes

CHAPTER

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 243

APPENDIX

Recursion and Church’s Thesis.. . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259 REFERENCES..

INDEX

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265

.... . . . . . . .............................................

LIST OF S Y M B O L S . .

.. . . . . . . ........ 269

. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275

Chapter 1

Mappings and Domains This book is concerned with functionals and their computations. Formal definitions will be given later, but at this stage we can anticipate that a functional is a function or operation which can be applied to objects which are also functions or operations. Since these operations are partially defined they induce a partial ordering relation which satisfies a number of useful properties. The objective of this first chapter is to identify this type of ordering and to introduce adequate terminology which later will provide a useful tool in the study of functionals. The main result in this chapter is Theorem 1.2, where the minimal fixed point is proved for domains. Our definition of domain is to some extent idiosyncratic, but it was chosen because it provides the best approximation to the special situations where this result is applied in this book. The proof is strongly set-theoretical, but this seems to be unavoidable at the present state of mathematical practice. Let A and B be (possibly empty) sets. A mapping from set A lo set B is an operation f which can be applied to any element x E A . The application of f to x will be written fz, and it may have two possible outcomes: (i) the application fr produces a unique value y E B , in which case we write fz ~r y, which is read f z is defined with value y, or simply fx is defined; (ii) the application fx produces no To complete this value, in which case we say that fx is undefined and write fx notation we set f z 1 to indicate that fx is defined with some value which is not made explicit. Note that we introduce these fundamental notions in simple operational terms.

r.

1

2

L.E. Sanchis

This does not imply a rejection of the usual descriptions in terms of single-valued sets familiar from discrete mathematics, rather we think such an approach is unnecessary and to some extent distorts the mathematical facts. A mapping f from A to B is total if f x is defined for every x E A . There is always a totally andefined mapping from A to B , which we denote UDA. Hence UDAZis undefined for every 3: E A . If either A or B is empty, then UDA is the only possible mapping from A to B . Since we allow for partially defined mappings (partial mappings in the standard literature) we need a special notation to deal with expressions which may be defined or undefined, depending on the values of variables and parameters. The symbol ‘k” has been introduced with such a purpose, and now we generalize its use as follows. If U and V are possibly undefined expressions we write U N V to indicate that either both are undefined or both are defined with the same value. On the other hand, we write U 9 V to indicate that either both are undefined, or both are defined with different value. Note that if U and V are undefined, then both U N V and U V are true. On the other hand, if U is defined and V is undefined, then both U N V and U 9 V are false. For example, i f f and g are mappings from A to B the notation f x N gx, where x is an element in A , means that either both fx and gx are undefined, or both are defined with the same value. If we say that fx N gx holds for every x in A this means that f and g are the same mapping, hence f = g.

+

We reserve the use of the standard equality symbol “=” when we want to make assertions about totally defined expressions. For example, if x and y denote elements of a set A , then x = y is a legitimate expression with the usual meaning. Similarly, i f f and g are mappings we can write f = g, as we did in the preceding paragraph. Note that a mapping is always a defined object-only the application of a mapping to an argument can be undefined. Later we shall find it convenient to allow the notation U = V even when U or V is a partially defined expression, in which case the notation is false if either U or V is undefined. This is done in order to make possible arbitrary substitutions in predicates, but it is used in only a few instances. The introduction of a mapping f from a set A to a set B , in a particular context, requires that f be properly specified. A specification for f is any mathematical identification of f , which essentially requires it be determined for which elements

Chapter 1. Mappings and Domains

3

of A the function is defined, and which is the value produced whenever defined. We may think of this process as a definition of the relation f x N y for x E A and y E B, and it is certainly common in the literature to call this a definition of the mapping f . In the present context, since we allow for expressions where f x is defined or undefined depending on the value of x , we prefer to avoid this notation and use rather the term specification which refers to any mathematical procedure which determines the mapping f . When a total mapping f is being specified we shall find it more natural to describe the specification as a definition. This situation is very similar to the one considered above regarding the use of the symbols “=” and “E.” There are many procedures to specify mappings, and many of them will be discussed later in this book. At this stage we identify two fundamental procedures that will be referred to as explicit and relational specifications. To give an explicit specification of a mapping f from A to B we use an expression U which contains the variable z and set

fx

N

u.

This equation means that f x is defined if and only if U is defined, and in case it is defined the value of f t is the value of U . As long as the evaluation of the expression U is well understood this procedure describes unambiguously the mapping f . For this reason it is very important that rules for the evaluation of expressions be clearly determined in advance. 1.1 We set N = { 0 , 1 , 2 , . . .} = the set of all natural numbers and introduce a mapping f from N to N by explicit specification of the form

EXAMPLE

fx

N

x x (x - l),

where x is the usual multiplication, and the subtraction x - 1 is assumed to be undefined when x = 0. We shall assume that the defining expression is undefined for x = 0, hence f0 is undefined. On the other hand, for x # 0 the expression is defined, hence f x is defined with the value obtained by evaluating the expression.

A relational specification of a mapping f from A to B consists of a formal definition of the relation f x N y, hence it takes the form fx2:y iff...x...y...

L.E. Sanchis

4

where the right side must be understood as a condition for I E A and y E B . This condition must be total, i.e., defined true or false for all values of I and y, and furthermore it must be single-valued, which means that for a given I E A there is at most one y E B which satisfies the condition. If the condition is not total, or not single-valued, we do not have a proper specification, and we say that f is not well defined.

1.2 With the same set N of Example 1.1 we introduce a mapping g from N to N using a relational specification of the form

EXAMPLE

gz N y iff I = 2’. It is clear that the condition is total and single-valued, so the mapping g is well defined. EXAMPLE

1 . 3 We introduce a mapping h from N to N in the form hz

N

y iff z < y.

It is clear that the condition is total but not single-valued, hence h is not well defined. Explicit specifications are essentially substitutions-they are easy and practical, but their power is rather weak. There are stronger procedures (i.e., by cases or by recursion) which will be discussed later. Relational specifications are the most general, as any specification of any type can be reduced to a relational one. They are more difficult to use since they require checking totality and single-valuedness, which in some cases may involve a delicate argument. We denote by MAP(A, B ) , or simply by MAP, the set of all mappings from A to B. I f f and g are elements of MAP we say that g is an ertension o f f if the following relation is satisfied by all z E A and y E B: if fz N y, then gz N y. Equivalently, if for all I E A , whenever f z is defined, then f z 21 gz. If g is an extension o f f in MAP, this means, informally, that g is more defined than f , or that g is at least as defined as f . Note that the extension relation between mappings is interesting only if we allow for partially defined mappings. If this is not the case and we consider only total mappings, the relation reduces to equality. On the other hand, the relation is

C h a p t e r 1. M a p p i n g s and D o m a i n s

5

completely independent of the structure of the sets A and B ; in particular] it does not depend on any order which may be imposed on those sets. To study the properties of the extension relation we must look for operations or transformations that preserve the relation. We shall see that this is a fairly trivial property in some cases, although it requires strong assumptions in some forms of substitution. These assumptions may imply a restriction on the class of mappings involved in the relation, and may depend to a great extent on the structure of the sets A and B . The extension relation in MAP is reflexive, for it is clear that whenever f is an element in MAP, then f is an extension of f. The relation is also transitive, for if g is an extension of f, and h is an extension of g , then h is an extension of f. Finally, the extension relation is anti-symmetric] for if g is an extension of f , and f is an extension of g , then we have f x N g x for all t E A, hence f = 9. We conclude that the extension relation in MAP is a partial ordering, and MAP with this partial ordering is a partially ordered set. It is convenient now to study structures of this type in a more general setting.

If P is a non-empty set with a binary relation C p on P which is a partial ordering (i.e.] the relation E p is reflexive, transitive, and anti-symmetric), we say that P is a partially ordered s e t , or simply a posed. The set MAP(A, B ) = MAP is a poset where the partially ordering relation is defined: f SMAP g if and only if g is an extension of f . There are many examples of posets, but we are interested only in those satisfying the special requirements defined below. In particular, we note that any non-empty set A can be considered a poset under the identity relation where x 2, y if and only if t = y. This type of poset we call discrete. A poset containing only one element is necessarily discrete. On the other hand, MAP(A, B ) is discrete only if A = 0 or B = 0,in which case MAP(A, B ) = {UDA}. We shall continue to use the notational device where if P is a poset, then the partial ordering relation is denoted C p , as long as it does not introduce confusion. In some cases the name of the poset is affected by a subscript and the ordering symbol is affected by the same subscript. For example, if Po is the poset we write 20 to denote the partial ordering. The notation x cp y stands for x E p y and

# Y. We reserve the symbol 2 without subscript to denote the usual inclusion relation

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L.E. Sanchis

between sets. Hence A C B means that the set A is a subset of the set B , and A c B means that A is a subset of B and A # B . If P is a poset and the relation x C p y holds we shall say that y is an ezlension of x in P . This notation is consistent with the definition given for the poset MAP(A, B ) . The same relation will be sometimes expressed in a different language. So when y is an extension of x we may say that y is an upper bound of x , or that x is a lower bound of y.

If P is a poset and I is an element in P such that every element of P is an extension of x, we say that x is the bottom element of P . If this element exists it is clearly unique, and it is denoted in the form bp. Note that a poset P can be trivially extended to a poset PI by adding a new element as a lower bound of every element in P . This element is now a bottom element of PI. In particular, if P is a discrete poset, then P' is a trivial extension of P with a bottom element, and PI is not discrete. It has been suggested in some places that, this being the case, we should consider only posets with bottom element, disregarding altogether discrete posets. It seems to us that when these structures are considered from the point of view of computability theory we must take into account discrete posets in order to represent computation outputs. An element x in the poset P is maximal if whenever I C p y, then x = y, or equivalently when the only extension of z is z itself. If P is a discrete poset, then every element of P is maximal. In the poset MAP(A, B ) , where B is non-empty, the maximal elements are the total mappings. We do not consider here the so-called top elements in a poset P , which are upper bound of every element in P . We shall see that these elements are missing in most of the structures we discuss in this work, although in some cases they are available (see Example 1.4 below). An element x in the poset P is regular if it has a maximal extension. The poset P is regular if every element of P is regular. A discrete poset is regular. The poset MAP(A, B ) is also regular.

A is a set, let P = P(A) = the power set of A = the set of all subsets of A . Then P is a poset with the usual inclusion relation g. The empty set is the bottom element in P , and A is the only maximal element of P . Since A is an extension of every subset of A it follows that P is regular. EXAMPLE 1.4 If

Chapter 1. Mappings and Domains

7

EXAMPLE 1.5 Let Z be the set of all integers, i.e., Z = { O , l , -1,2, -2,. . .} with the standard ordering 5 (less or equal). Then Z is a poset with no bottom element and no maximal element, so this poset is not regular.

Let x and y be elements of a poset P . We say that x is consistent with y in P if they have a common extension, i.e., there is at least one element t in P such that x C p t and y c p t. This relation is reflexive, for 3: is consistent with z, and it is symmetric, for if x is consistent with y, then y is consistent with z. In general this relation is not transitive. If x is consistent with y we shall say simply that z and y are consistent. The consistency relation between mappings f and g from a set A to a set B means that whenever f z and g x are both defined for some x E A, then fz = gx. On the other hand, it is possible that fx be undefined in cases where g z is defined, and conversely. A subset M of a poset P is consistent if any two elements of M are consistent. The empty set is consistent, and a set containing only one element is consistent. In a discrete poset these are the only consistent subsets. EXAMPLE

1.6 In the poset P(A) of Example 1.4 any two elements are consistent,

so P(A) itself is consistent. On the other hand, consider the poset MAP(N,N) where

N is the set of Example 1.1. If we set fx 2: 0 for all z E N, and g z N 1 for all x E N, then f and g are not consistent. In fact, if f and g are any two different

total mappings from N to N they are not consistent. If two elements in a poset P are not consistent we say they are inconsistent. A subset M of P is inconsistent if M has at least two elements that are inconsistent. Let M be a subset of the poset P . If x is an element of P which is an upper bound of every element of M we say that x is an upper bound of M . Clearly, if M is empty, then every element of P is an upper bound of M . Furthermore, if M has an upper bound, then M is consistent. On the other hand, it is possible for a poset P to have a consistent subset with no upper bound. For example, in the poset Z of Example 1.5 the set M of the even integers is consistent but has no upper bound. Let P be a poset and M a subset of P . An element x of P is the least upper bound of M in P if it satisfies the following two conditions: (i) x is an upper bound of M ; (ii) if y is any upper bound of M , then y is an upper bound of x. It is clear that the least upper bound of a set M is unique if it exists. If M is empty, then the least upper bound must be the bottom element of P . If M is

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L.E. Sanchis

a singleton, then the least upper bound exists, and it is the unique element of M. If M has only one upper bound, then such element is necessarily the least upper bound. Note that the property of being the least upper bound of a set M in a poset P is global, i.e., depends on the whole structure of the poset P . This means the relation may disappear when P is extended to a new poset PI. Conversely, it is possible that a set without least upper bound in P may have one in the extension PI. We shall use the notation LUBp(M) to denote the least upper bound of a subset M of a poset P whenever such element exists. There is a dual notion that also plays a role in our discussion. We say that an element x of a poset P is a lower bound of a subset M if x is a lower bound of every element in M . If M is empty, then every element of P is a lower bound of M . If M = P there is at most one lower bound of M , namely the bottom element of P . Let P be a poset and M a subset of P . An element z of P is the greatest lower bound of M if it satisfies the following two conditions: (i) x is a lower bound of M ; (ii) if y is any lower bound of M , then y is a lower bound of 2. Note that if P is a poset of mappings from a set A to a set B, then every non-empty subset M has a greatest lower bound f , where f(z) N y if and only if g(z) N y for every mapping g in the class M. We shall use the notation GLBp(M) to denote the greatest lower bound of a subset M of a poset P whenever such element exists. A poset D is a domain if every consistent subset of D has a least upper bound. Since the empty set is consistent, it follows that D has a bottom element. We take the notion of domain as the bottom line of the structures that are discussed in this work. Still, this is a very general notion and to some extent is arbitrary, for in most cases we shall consider domains where the elements are mappings. On the other hand, we shall be dealing also with discrete posets which, in general, are not domains. Theorem 1.1 Let D be a domain. If M is a non-empty subset of D , then M has a greatest lower bound.

Let N be the set of all lower bounds of M . Since M is non-empty there is an upper bound of N , hence N is consistent and has a least upper bound x. If y is an element in M, then y is an upper bound of every element in N , hence y is an PROOF.

Chapter 1. Mappings and Domains

9

upper bound of x . This means that x is a lower bound of M. Since every element of N is a lower bound of x it follows that x is the greatest lower bound of M. 0 Several constructions of domains will be discussed in the next chapter. We complete this introduction by proving the minimal fixed point theorem for monotonic transformations. This result provides the foundations for the theory of recursion. Let P be a poset. A transformation on P is a total mapping T from P to P which satisfies the following monotonicity condition for all x , y in P : if x s p y, then T z G p T y , or equivalently, if y is an extension of z in P , then T y is an extension of T x in P . If T is a transformation on P , and 3: is an element of P such that T x C_p 2,we say that x is closed under T . If T x = x we say that x is a fixed point of T . Theorem 1.2 Let T be a transformation on the domain D. There is a unique element x in D such that:

(i) The element x is a fixed point of T .

(ii) If y is an element of D which i s closed under T , then y is an extension of x . To prove the existence of 3: we associate with every ordinal v an element z,,of D in such a way that the following two conditions are satisfied:

PROOF.

Condition 1:

< v then z,,G D x u ED T I , if p

2,

Condition 2: We proceed to define xu by transfinite recursion, so we assume that for every ordinal p < v the element x,, has been defined in such manner that conditions 1 and 2 are satisfied. To define I , , we consider the set Mu = {x,, : ,u < v} and note that by condition 1 this set is consistent; hence we put x < , = LUBD(M,) and define x , = T x < , , . We must show that after defining zu in this way conditions 1 and 2 are still satisfied. First we note that if p < v , then x,, z 0, then f’x is undefined if one of the applications f 0 , f l , . . . , f z - 1 is undefined. Write a specification for the minimal fixed point of T using primitive recursion.

Notes This initial chapter identifies two critical conditions, namely, domains and regularity. A domain is a structure that satisfies a closure condition, so it is complete in some sense. Regularity is a different type of condition, for it implies the existence of elements that are complete or maximal, that to some extent determine the properties of the other elements. In fact, the traditional approach has been to restrict the theory to the maximal elements, or total mappings, avoiding completely the partial elements. But this is artificial and, in the long run, distorts the theory. We have seen many times that we cannot get rid of partial functions, and it is much better to assign some explicit place to them. There is still one more condition which has not been properly discussed in this chapter. We refer to monotonicity, which is introduced in the next chapter, and it is assumed in the rest of the work. We must warn the reader that a substantial part of the theory is independent of the monotonicity assumption. In fact, only with the introduction of functional recursion in Chapter 7 does this assumption become essential. It can be avoided completely for the preceding chapters. The material introduced in this chapter is not standard in computability theory, although some elements were brought into our preceding work in Sanchis [27].

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L.E. Sanchis

Many researachers have pointed out the relevance of inductive definability for the foundations of computability theory. We follow this approach with some reservations, which are formulated in the appendix. Good introductions to this material are given in Fenstad [4] and Hinmann [lo].

Chapter 2

Functionals and Predicates In the preceding chapter we have identified a number of structures, in particular regular domains, which satisfy several closure properties. In this chapter we introduce monotonicity, which is a consistency property, and the corresponding notion of functional. Monotonicity is crucial for the theory, although it can be avoided in the initial chapters. The importance is not just mathematical, but essentially operational. This is discussed in detail in Chapter 8 where we state the extended Church’s thesis for computable functionals. To present an adequate theory of predicates with function arguments we had to confront a number of difficult choices. We have rejected the easy solution of admitting partially defined predicates, imposing at the same time the monotonicity condition. So we insist that a predicate is a total mapping which, in general, is not monotonic. With this basis we can introduce the usual boolean operations and allow for a weak form of monotonicity where a predicate is a boolean extension of another. The main difficulty in the theory of predicates is to define the associated characteristic functionals. Since these are strictly monotonic, and predicates in general are not monotonic, it follows that no faithful correspondence can be established. We adopt a compromise where the correspondence is faithful for maximal arguments, i.e., for total mappings. Let A be a poset and B a set, i.e., a discrete poset. A mapping f from A to B is monotonic if the following condition is satisfied by all c,c’ in A and y in B: if

15

L.E. Sanchis

16

f z N y and I' is an eztension of z in A, then fz' N y. Note that if A is discrete, then every mapping from A to B is monotonic. Note that we insist on requiring the set B to be discrete, while no similar requirement is imposed on the set A. There are serious reasons for this approach, for we think that the elements of B are computational outputs, hence they must be finite independent objects. A more general theory is possible (see Exercise 1.1), but it appears to run into serious difficulties. We denote by FUN(A, B ) the set of all monotonic mappings from A to B. The partial ordering in FUN, which is denoted &FUN, is the same as in MAP, hence f EFUNg if and only if whenever f z I Iy, then g z N y. Since FUN is a subset of MAP, and the order relation is the same, it follows that FUN is a sub-structure of MAP. But this is not as important as the inverse relation where MAP is a special case of FUN. In fact, if A is a discrete poset, then FUN(A, B ) = MAP(A, B ) . Hence general results about FUN will apply immediately to MAP. We know that if B is non-empty, then the maximal elements in MAP(A, B ) are the total mappings. The situation in FUN(A,B) can be very different. If A has a bottom element, and f is a total monotonic mapping from A to B , then f is a constant mapping, i.e., there is y E B such that f z N y for all z E A. On the other hand, it is generally possible that FUN contains maximal elements which are not total mappings. EXAMPLE 2.1 Consider a poset A where any two elements are consistent (see Example 1.4) and B an arbitrary set. Then FUN(A, B ) consists of all constant monotonic

mappings from A to B , although not necessarily total constant mappings. The maximal elements of FUN in this case are the total constant mappings. The elements of FUN(A, B ) , where A is a poset and B is a set, we call functionals on A with target B. This notation is consistent in the context of this work, for in most cases the set A will be a set of mappings, and operations which are applied to mappings are usually called functionals. EXAMPLE 2.2 Let f be a mapping from MAP(N, N) to N with the following relational specification, where a is an element of MAP(N,N) and y is an element of N:

fa N y iff

there is 2 E N such that a z N 0 and y = 0 or for every z E N az $0i and y = 1.

Chapter 2. Functionals and Predicates

17

More explicitly, fa N 0 if az N 0 for some z E N, and fa N 1 if az $ 0 for every z E N. Otherwise, fa is undefined. Recall that az $ 0 means that az is defined with value different from 0. This is clearly a monotonic mapping, so it is a functional on MAP(N,N) with target N. Furthermore, it is maximalin FUN(MAP(N, N), N), for if g is an extension o f f and ga N y where fa is undefined, then a has two possible extensions, a' and a'', where fa' N 0, fa'' N 1 while g a r N y and g a l r N y, which is a contradiction with the hypothesis that g is an extension o f f . The proof that FUN(A,B) is a domain is identical to the proof of the same property for MAP(A, B)in Chapter 1. First we note that Theorem 1.3 is valid also for FUN, hence two elements f and g are consistent in FUN if and only if f z N gz whenever both f z and gz are defined. Note that in proving the implication from (ii) to (i) in Theorem 1.3 the mapping h is monotonic whenever f and g are monotonic. Theorem 2.1

If A is a poset and

B i s a set, then FUN(A, B ) is a domain.

We consider a consistent subset M of FUN and define the mapping g from A to B as in the proof of Theorem 1.4:

PROOF.

gz N y iff there is a mapping f in M such that f z N y.

This specification is single-valued because M is consistent, so g is well defined. The mapping g is monotonic because every element of M is monotonic. The proof that 0 g is the least upper bound of M is exactly as in Theorem 1.4. Let A be poset and B a set. A functional f in FUN(A, B) is complete if fv N y holds whenever f z N y holds for every maximal extension z of 21. We say that f is quasi-total if f z is defined whenever z is a maximal element of A. Finally, we say that f is singular if whenever f z is defined, then z is a maximal element of A. These definitions explore the role of maximal elements in relation to functionals. If a functional f is complete, then full information about f can be derived from the values fz where z is maximal. Equivalently, if both f and g are complete, and f z N gz holds whenever z is maximal in A, then f = g. On the other hand, if f is quasi-total and A is regular, then the extensions o f f are potentially determined, in the sense that any two extensions are consistent. Finally, singularity is a kind of distortion imposed from outside. I f f is computable and singular, then f must be able to recognize the maximal elements in the set A .

18

L.E. Sanchis

Let f and g be two functionals in FUN(A, B ) . We say that f and g are similar if f x N gx whenever x is a maximal element of A.

Theorem 2.2 Let A be a regular poset, B a non-empty set, and f a functional in FUN(A, B). The following conditions are equivalent:

( i ) f is maximal in FUN(A, B ) . (ii) f quasi-total and complete. PROOF. Assume f is maximal in FUN. Then f is quasi-total, for otherwise we can extend f to g by setting gx N y where x is a maximal element of A such that f x is undefined, and y is an arbitrary element of B. It follows that f is complete, for if v is an element of A such that f v is undefined and f x N y holds for every maximal extension x of v, then we can extend f to g by setting gx' N y whenever x' is an extension (not necessarily maximal) of v. Assume now that f is quasi-total and complete, g is an extension of f, and gv N y for some v in A. Since g is monotonic we have gx N y whenever x is a maximal extension of v. Since f is quasi-total we have also f x N y whenever x is a maximal extension of v. Since f is complete this means f v N y. It follows that 0 f = g , so f is maximal in FUN.

Corollary 2.2.1 I f A is a regular poset, then FUN(A, B ) is regular. We may assume that B is non-empty, for otherwise FUN reduces to the singleton domain. Let f be an element of FUN. Without loss of generality we may assume that f is quasi-total. Let g be the functional which is specified as follows: PROOF.

gv

N

y iff there is no x which is a maximal extension of v and f x $ y.

This condition is single-valued since A is regular. Clearly, g is a monotonic extension of f, hence g is quasi-total. To show that g is complete, assume v is an element of A such that g x N y whenever c is a maximal extension of v. This means that 0 fz N y whenever x is a maximal extension of v , hence gv N y. EXAMPLE

2 . 3 Let f be the mapping from MAP(N,N) to N such that fa

N

y

iff

a is a total mapping from N t o N and there is

some x E N such that a x N 0 and y = 0 , or a x $ 0 for all x E N and y = 1.

Chapter 2. Functionals and Predicates

19

The mapping f is monotonic, so it is an element of FUN(MAP(N,N),N). The only maximal extension of f is the mapping g such that gcu

=y

iff

there is x E N such that cue 21 0 and y = 0 , or for every x E N ac $ 0 and y = 1.

We introduce now another poset construction which, although not as fundamental as the construction FUN, provides a useful tool to handle finite combinations of poset and domains. If A1,. . . , A , are sets, m 2 0, the Cartesian product denoted by A1 x . x A , is the set of all m-tuples of the form ( 2 1 , . . . , 2,) where xi is an element of Ai, i = 1,. . ., rn. If each one of the sets A1,. . . , A , is a poset, then the Cartesian product is also a poset with the partial ordering Grn such that ( 1 1 , . . . , em)Cm (ci, . . . , c&) if and only if zj Cj xi for i = 1,. . . , rn. Note that the introduction of the Cartesian product induces a subtle change in the notation to denote application. As explained in Chapter 1, i f f is a mapping from A to B , and c is an element of A , then fz denotes the application o f f to 2 . On the other hand, if A = A1 x . . . x A , and t is an element of A , then x = ( 1 1 , . . ., x,), hence the application fx can be written f ( x 1 , . . . , x,). Furthermore, if m = 1 we can identify A with A1, hence if x is an element of A1, then x is also an element of A , so x and ( 2 ) denote the same object. In this case we can use either the notation f z or the notation f(x). The properties of the Cartesian product A1 x .. x A , are easily derived from the properties of the posets A l , . . . ,A,. For example, if bj is the bottom element of the poset Ai, i = 1,. . . , m, then (bl,. . . ,bm)is the bottom element of the Cartesian product. In a similar way we can show that if the factors are regular posets, then the Cartesian product is regular. If the factor posets A l , . . . , A, are domains, then the Cartesian product is also a domain. For assume M is a consistent subset of A1 x . . . x A,, and introduce for each i = 1,.. . , m the set Mi such that z E Mi if and only if there is an element (21,. . . ,2,) in M such that t = c i . It follows that Mj is a consistent subset of Ai, hence LUBj(M;) = yi is defined. It follows that LUBm(M) = (y1, . . . , y,). In most applications of this construction all the factors will be identical, say A1 - . . . = A, = A , in which case the Cartesian product is written simply A m . Note that we allow rn = 0, and A' is the singleton poset. We shall follow the policy of identifying different product constructions which are isomorphic in an obvious way. For example, a product of the form ((A1 x . . . x A , ) x

20

L.E. Sanchis

(B1x . . . x B,,)) will be considered identical with the product (A1 x . . . x A,,, x B1 x . . . x B,,). In practice this means that pairs of the form ((ti,.. . ,z,,,), ( ~ 1 , .. . , y,,)) are written as m+n-tuples (xl,. . ., x m , yl,.. . ,y,,). While this practice is certainly convenient and harmless, in some cases we may want to keep track of the origin of the different elements and write the m+n-tuple in the form (xl,. . . , x,; y1 , . . . , y,,).

2.4 Consider the discrete poset N, and the domain MAP(N, N) = MAP. Let A = MAP x N be the Cartesian product. Then A is not a domain, since N is not a domain. Since MAP is not discrete it follows that A is not discrete. The domain FUN(A, N) contains the functionals on A with target N. For example, the following is a functional on A, where a denotes elements of MAP, and z denotes elements of N: f(a,x) N a x .

EXAMPLE

The expression ( a , z ) denotes, of course, an element of A . Note that f has been introduced by an explicit specification, so the problem of single-valuedness does not appear. An example of a functional on A given by a relational specification is the following: g(a,x) N y iff az $ 0iand x = y. Single-valuedness is clear since whenever g(a,z) N y holds, then y = x. The mapping is monotonic, for if g(a,x) 2: y, then az is defined with value different from 0, and if p is an extension of a,then p x is also defined with value different from 0, hence g(p, z) N y. It follows that g is a functional on A .

In the second part of this chapter we consider predicates over a poset A . We begin with the set BOOL = {true,false} = { t , f } . The elements of BOOL we call boolean values, so there are exactly two boolean values, t and f. To denote boolean values we shall use the capital letters X I Y , and Z,and to denote equality between Y means that the boolean values boolean values we use the symbol E. Hence X X and Y are identical, i.e., both t , or both f. We could write X = Y for the same purpose, for there is nothing fundamental in the use of a special symbol for boolean values. It is only a matter of convenience and a way to take better advantage of the fact that we can actually identify the boolean value true with the usual semantical true, and false with the semantical false. Hence, if U is a meaningful expression (say, the number 2 is odd), we can assert U , or we can say that U is true, or we

Chapter 2. Functionals and Predicates

21

can say t h a t U t. Similarly, we can negate U , or say t h a t U is false, or we can say t h a t U f. We can move also in the other direction: if X is a boolean value t, or we can say that X is true, or we can simply assert X . we can say t h a t X Similarly, we can say that X f, or say X is false, or simply assert not X. For example, the following assertion is perfectly meaningful (and, in fact, true): if X , then not (not X). is strictly equivalent to the relation if and only Under this interpretation if (iff), and from now on we shall find it convenient t o take advantage of this equivalence, particularly in writing relational specifications.

=

There are several well known operations on boolean values, and we identify a few of them in order to make clear which is our notation. We begin with negation which is a unary operation written in the form X . T h e value of X is defined as X = t iff X f. (This is sufficient since there are only two possible values for X.) T h e operation conjunction is a binary operation written in t h e form X A Y , and t h e value is given by the rule: X AY E t iff X f t and Y t. T h e operation disjunction is a binary operation written in the form X V Y , and t h e value is given by the rule X V Y E t iff either X t or Y t . It is understood t h a t these operations can be applied to assertions of any kind and will produce new assertions where the meaning is determined by t h e evaluation rules of the boolean operations.

-

-

N

=

2.5 We introduce a mapping P from N x N to BOOL by the explicit specification

EXAMPLE

P ( z ,y) We conclude that P ( z ,x)

(t

+ 1 < y) v (y + 1 < t).

f for every

t

E N, or simply not P(t,z)for every

2.

2.6 Boolean operations are very convenient to describe relational specifications. For example, we introduce a mapping f from N x N to N, in the form EXAMPLE

This is clearly a total mapping, so we can write f ( 2 , 2 ) = 0, f ( 0 , l ) = 1 , etc. Another way to produce boolean values is via quantifiers, which are introduced a t this stage only as syntactical constructions. If U is a boolean expression which

L.E. Sanchis

22

contains a variable 3: that ranges over a set B , then ( 3 x ) U means that there is some x in B such that U holds, and ( V x ) U means that for every x in B , U holds. Note that here a quantifier is not a boolean operation in the sense that it is applied to boolean values to get boolean values. Rather, it is applied to a mapping to obtain a boolean value. Such a mapping is defined informally by the expression U above, which denotes a mapping f such that f x = U . Hence f is a mapping from B to BOOL, and the quantifiers apply to f to produce a boolean value. Using the notation we introduce below we can say that the mapping f is a predicate. 2.7 We can rewrite the specification of the mapping f in Example 2.3, using the set TO of all total mappings from N to N: EXAMPLE

f a

N

y L ( a E TO) A ( ( ( 3 x ) a x

0 A y = 0) V ( ( V x ) a x 74 0 A y = 1)).

Let A be a set. A predicate on A is a total mapping P from A to BOOL. We shall use the capital letters P I Q, and R to denote predicates. Since a predicate P is total we do not need the symbol N to express application, and we write Px t or P x ~ f . Our definition is fairly general, but still it does not allow for non-total predicates. It is not uncommon in the literature to consider partially defined predicates, but we prefer to avoid this path. Our reasons are practical considerations: there are two properties that make predicates very useful-one is that they may take only two values, the other is that they are always defined. Note that the definition of a predicate on a set A ignores the possible structure of A . Essentially, this means that from the point of view of a predicate the set A is discrete. This is not true for the functionals on A , where the partial order structure of A is relevant. There are two predicates we can identify for every set A . The predicate t A such that t A x = t for all c E A , and the predicate f A such that f A x G f for all x E A . EXAMPLE

2.8 Consider the predicates P and Q on the domain MAP(N,N) such that

P a G a E TO &a E a $ TO

-

where TO is the class of all total mappings from N to N. Clearly P and Q are related &a for every o in MAP(N,N). by the condition P a

Chapter 2. Functionals and Predicates

23

Since predicates are total mappings the usual extension relation is trivial, for it reduces to the identity. We introduce another relation which is useful in many applications. If P and Q are predicates on a set A we say that Q is a boolean eztension of P if the following condition holds for every x E A: If Px E t, then

Qx t , which, of course, can be expressed simply as if Px,then Qx.This relation is clearly a partial ordering. Here again the definition of the boolean extension relation ignores the possible structure of the set A . Still, note that the usual extension relation between functionals also ignores the structure of A and, in fact, applies to arbitrary mappings with arguments in A . Actually, we are interested in some form of restriction for this extension relation. For the time being note that, as defined, we can consider predicates as subsets of A , and the extension relation becomes the usual inclusion relation between sets. We aim at a restriction where the sets, or predicates, satisfy special conditions. We assume now that A is not only a set but also a poset. As usual, the initial general situation is recaptured by assuming that A is a discrete poset. We say that a predicate P on A is boolean monotonic if whenever Px holds and x' is a n extension of I in A, then Px' also holds. The predicate P in Example 2.8 is boolean monotonic, but the predicate Q in the same example is not boolean monotonic. If A is a discrete poset, then every predicate on A is monotonic. The predicates t A and f A are always boolean monotonic, no matter the nature of the partial ordering on A. The two definitions introduced above, boolean extension and boolean monotonicity, provide a special role for the boolean value t, for they impose requirements only when the predicate holds. There are a number of reasons for this singularity, which to some extent is arbitrary. From the point of view of computability theory we can mention that the computation of a predicate is usually organized to determine when the predicate is true and assume by default that it is false when such determination fails. A typical example of this structure is a definition by induction. If A is a set we denote by PRE(A) the set of all predicates on A. This is a poset with the ordering ~ P R Ewhich is defined P CPREQ if and only if Q is a boolean extension of P . The poset PRE(A) is not only a domain but a complete lattice where every subset has a least upper bound and a greatest lower bound. In fact, PRE(A) = P ( A ) = the set of all subsets of A, and if M is a subset of PRE(A), then UM (nM)denotes the least upper (greatest lower) bound of M .

L.E. Sanchis

24

If A is a poset we denote by BMO(A) the set of all boolean monotonic predicates which is the restriction of C_PRE to on A. This is a poset with the ordering ~ B M O BMO(A). The poset BMO(A) is also a complete lattice. In fact, if M is a subset of BMO(A), then U M is also the least upper bound of M in BMO and nM is the greatest lower bound of M in BMO. We discuss now several constructions where we obtain new predicates from given predicates. They are derived from the boolean operations considered above. In some cases, but not always, they preserve monotonicity and produce boolean monotonic predicates when applied to boolean monotonic predicates. Although derived from the truth table boolean operations, these are really predicate operations which are applied to predicates to obtain new predicates. Of course, we can consider the truth values as 0-ary predicates, in which case both applications reduce to predicate operations. Let A be a poset and PRE(A) the domain of all predicates on A-. We extend the boolean operations of conjunction and disjunction to predicates on A. If Pi and P2 are predicates on A, then Q = Pl A P2 = the conjunction of Pl and P2 is specified as Q z G Plz A P ~ x If . PI and P2 are boolean monotonic, then Q is also boolean monotonic.

We extend disjunction in the same way. If P1,P2 are predicates on A, then R = Pl V P2 = the disjunction of PI and P2 is specified as Rx G Plx V P ~ z Again, . if Pl and P2 are boolean monotonic, then R is boolean monotonic. We can apply the preceding construction to any boolean operation, including negation, but in that case the monotonicity condition may fail. If P is a predicate on A, then the negation of P is the predicate Q = such that Qx 5 P x . Clearly P = Q , so P is the negation of Q. In Example 2.8 two predicates P and Q are introduced where Q is the negation of P , P is boolean monotonic, but Q is not boolean monotonic. Note that negation as an operation on predicates on A is well defined, and the result is another predicate. The only problem is that, in general, monotonicity is not preserved. Since in many cases we are interested in constructions where monotonicity is critical, this is actually a significant drawback.

-

N

2.9 Consider a boolean monotonic predicate P on MAP(N, N) and introduce a mapping f from MAP to N as follows, where we use a relational specification,

EXAMPLE

Chapter 2. Functionals and Predicates

25

a is a variable that denotes elements of MAP, and y denotes an element of N:

The condition is single-valued, so f is well defined, but f fails to be monotonic (in the sense of FUN) because p is not necessarily boolean monotonic. N

To conclude this chapter we consider a number of constructions from predicates to functionals and from functionals to predicates. Since predicates are total mappings, and the ordering relations are substantially different, we shall find that these relations are not as smooth as we might like. The translation of a predicate into a functional is intended as a device where properties of functionals are extended to predicates. It has a long tradition in the standard theory of recursive functions, usually described in terms of characteristic functions. There it was found convenient to distinguish two classes of characteristic functions, total and partial (see, for example, Sanchis [27]). When we move from functions to functional the situation is different, mainly because predicates are simply total mappings and functionals are monotonic. We still continue to distinguish between total (or dual) and partial characteristic functionals. But now we do not require that the characteristic functional of a predicate be a faithful translation. More precisely, we require consistency in the translation only for maximal arguments. Recall that in applications the maximal arguments are total functions. Let P be a predicate on a poset A. We want to identify functionals in FUN(A, N) that provide information about the predicate P , and that eventually can be used to determine the computation properties of P. A functional f in FUN(A, N) is a partial characteristic functional o f t h e predicate P on A if the following conditions are satisfied: P C F 1: If

3:

is maximal in A and Px holds, then f z is defined.

P C F 2: If fz is defined, then P z holds and fz

N

0.

A functional f in FUN(A, N) is a dual characteristic functional of the predicate

P on A if the following conditions are satisfied: DCF 1: If

2

is maximal in A , then f x is defined

DCF 2: If f z is defined, then either Pz holds and f x and fz 2: 1.

N

0 or Px does not hold

L.E. Sanchis

26 We shall use the symbol

$p

to denote an arbitrary partial characteristic func-

tional of the predicate P. There is always a minimal partial characteristic functional $ p where if $ p r is defined, then is maximal in A . There is also a maximal $ p where $ p i is defined if Px holds and there is no y which is an extension of z and Py does not hold. We shall use the symbol x p t o denote an arbitrary dual characteristic functional of the predicate P . Again, there is a minimal dual characteristic functional x p where if x p x is defined, then z is maximal. There is also a maximal dual characteristic functional x p where x p x N 0 if Px holds and there is no extension y of x where Py does not hold, and x p x N 1 if Px does not hold and there is no extension y of x where P y holds. The preceding definitions appear to be ambiguous in the sense that they do not determine a unique partial or dual characteristic functional for a given predicate P, but rather describe many functionals which satisfy some general conditions. Clearly, given one such partial or dual characteristic functional of P we cannot determine completely the predicate P. So there are limitations in the definitions, and these must be taken into account. In practice this approach works satisfactorily, so we can prove the usual closure properties familiar in the theory of recursive functions, and by using the appropriate notation we can extend the classical normal form theorems which relate computable operations to predicates. Now we consider a mapping f from a set A to a set B , and introduce Gf = the g r a p h predicate o f f , which is a predicate on B x A given by Gj(y, x) 3

fx

N

y.

If A is a poset and f is monotonic, i.e., an element of FUN(A,B), it is clear that Gf is boolean monotonic, so it is an element of BMO(B x A ) . A partial characteristic functional of G j will be denoted by $r, and a dual characteristic functional by x j . The translation from the functional f into the graph predicate G, is more satisfactory, for the latter is a faithful representation of the former. But in itself the representation is weaker for, in general, we cannot derive computation properties of the functional f from the predicate Gj. If f is a mapping from the set A to the set B we define the predicate D j = the definzlzon predicate of f , which is a predicate on A such that

Djx

f x is defined

3yGf(y,x).

Chapter 2. Functionals and Predicates

27

If A is a poset and f is monotonic, then Dj is boolean monotonic, i.e., an element of BMO(A ) . Another predicate induced by a mapping f from a set A to a set B is R j = the range predicate off, which is a predicate on B given by Rfy

31Gj(y, I).

EXERCISES

2.1 Let Z be the poset of Example 1.5. Prove that FUN(Z, N) is regular. 2.2 Let f be an element of FUN(A,N) where A is a non-regular poset. Prove that f is not complete. 2.3 Let f and g be complete functionals in FUN(A,B) such that f z

whenever

I

N

gz holds

is maximal in A . Prove that f = g .

2.4 Let A be a poset. Prove that BMO(A) is a domain where every subset has a

least upper bound and a greatest lower bound. 2.5 Let A be a poset and P a predicate on A . Prove that there is dual characteristic

functional xp such that whenever xIp is another dual characteristic functional of P , then xp is an extension of xIp. 2.6 Let f and g be the mappings of Example 2.3. Prove that g is the only maximal

extension of f .

Notes The material in this chapter, namely functionals and predicates and their relation via characteristic functionals, is traditional in computability theory. The difficulty here is that while functionals are monotonic by definition, predicates are total mappings and in general they are not monotonic. This means that a faithful correspondence is impossible, and any solution must involve some compromise. Our solution

28

L.E. Sanchis

is to require that a characteristic functional must be faithful on total arguments, but may be undefined on non-total arguments. This type of problem, where we must restrict our consideration to total arguments, will reappear several times in this work. Obviously, the problem disappears when functionals and predicates are restricted to total arguments, which is the usual approach in the literature. Another possible solution is to allow for predicates to be non-total monotonic mappings. We have decided against this approach, and we require predicates to be total mappings.

Chapter 3

Basic Operations In this work we shall be concerned with several objects derived from the natural numbers. The most important are functionals, which include as a special case numerical functions. We also consider predicates. Our task is to characterize the class of all functionals which are computable from a given class 7 of functionals. To reach this goal we must define several types of operations, starting with the basic operations that are defined in this chapter. We define basic operations by taking some initial functionals and invoking a few simple constructions, essentially substitution and definition by cases. This is a departure from the approach in Sanchis [27], where definition by cases was introduced at the level of recursive operations. From classes of functionals we move in a natural way to classes of predicates via the characteristic functionals. Since definition by cases is available, we can easily prove closure under boolean operations. Although basic operations are extremely simple, they provide a frame from which basic recursion can easily be derived. We start with the set N = (0, 1, . . .} of natural numbers. Elements of N will be called numbers, or type-0 objects. We shall use small letters from the last part of the alphabet to denote numbers: u , v , w, I , y, z . The set N is a discrete poset, so it is regular. It is not a domain, but note that every non-empty consistent subset of N has a least upper bound and a greatest lower bound. From the set N we obtain the Cartesian product N k , k 2 0. This is also a discrete

29

L.E. Sanchis

30

poset. An element of N k is a k-tuple denoted by a vector of the form (11,. . . , ~ k where 11, . . . ,i k denote elements of N. The only element of the singleton poset No is denoted by ( ). In practice we shall use a more compact notation of the form ( z ) where z stands for a list of k different numerical variables, possibly ernpty when k = 0. We identify N with N ' , so an element of N can be denoted by a letter I , but also by a vector (I). Next we introduce the domains

Fk

= MAP(Nk,N ) = F U N ( N k , N). Elements

of F k are called k-ary numerical functions or type-k objects. Usually they will be referred to simply as functions. We shall use letters f , g, h , p , q , r , s to denote functions. The application of the k-ary function f to an element (z) of N k is written in the usual form: f(z). We arrange the definitions in such a way that the numerical functions are also functionals, as defined below. This has great advantages, since we can deal uniformly with both structures. On the other hand, functions are very special and their behavior is often idiosyncratic. The crucial fact is that all numerical functions are monotonic, so this condition is trivial. Note that the elements of N are type-0, and the elements of FO = MAP(No, N) are also type-0. Clearly, we can identify every element of N , say I , with a unique element of Fo, namely f( ) N I . On the other hand, FOcontains an element UDo, where UDo( ) is undefined, and this element has no counterpart in N . To avoid ambiguities when referring to elements of N we may use the expression proper t y p e 0 object. The explicit distinction between the sets N and FO is of some importance, for it is easy to fall into the temptation of identifying the two structures. Essentially, this amounts to introducing a bottom element in the set N. As a consequence, all the other structures are changed, and the result is an elegant mathematics that is difficult to handle in terms of computability. The heart of the problem seems to be the characterization of a typical computational situation where no output is produced. To identify this situation as a value, even with the properties of the bottom element, is consistent but at the same time distorts the basic phenomenon. The set F k is a domain where the partial ordering relation is denoted by f ( I k g, although in practice we shall say simply that g is an extension o f f in Fk. Since Fk is a domain it follows that every consistent subset A4 of F k has a least upper bound denoted LUBk(A4). Furthermore, if A4 is a non-empty subset of F k the greatest lower bound exists, and it is denoted GLBk(A4).

)

Chapter 3. Basic Operations

31

T h e set F1 = MAP(", N) = FUN(", N ) will play aspecial role in our discussion. Elements of F1 are called unary numerical functions or type-1 objects. They are denoted by Greek letters: a , p, y, etc. Since we identify N with N', a type-1 object a can be applied to a proper type-0 object z, but the application is still written a(.). T h e set F i is the Cartesian product of m copies of F1, rn 1 0. Elements of FF are called type-1 m-tuples, and are denoted by vectors ( @ I , .. . , a,) where a l , . . . , a, denote elements of F1. In practice we shall use the compact notation (a)where a stands for a list of m different unary function variables. Note that FF is a domain where the consistency relation is controlled by Theorem 1.3 via the factors in the Cartesian product, all of them copies of F1. T h e partial ordering relation in

FF

will be written

Em.

We combine the domains above in the following form: we set N'J" = Nk x F T , k , m 2 0. T h e elements of Nkrm are called (k,rn)-tuples. A typical element of is a pair denoted ( ( 1 1 , . . .,zk), (a1,. . .,am)),which we identify with the N'l"

+

k rn-tuple (11, . . . , zk;a1,. . . , a,), which can be contracted to (z;a).With this notation an element of is written (z;), which can be identified with (z).In this way we identify Nkzo with N'. Similarly, we identify No,, with FF by writing ( ; afor ) an element of No>,, which can be identified with (a).Identifications of this type will be taken for granted in the future. Nkto

3.1 A typical element of N2v2 is the (2,2)-tuple ( 0 , l ; a,p),where a(.) N z + 1 and P ( z ) N z + 2. This tuple is maximal in N2,' because a and p are total mappings from N to N.

EXAMPLE

T h e partial ordering relation in N'p"' will be written ckim.Hence ( 5 ;a)Ck," ( z ' ; a ' )means that (2) = (z') and (a) (a').In particular, (2;) Ck,O (z';) means that (z)= (z') and ( ; a ) (;a') means that ( a ) (a').This is

c"' cotm

c"'

consistent with the identification of with N', and the identification of with FT. Note that since N k and F F are regular posets i t follows that Nk>" is a regular poset. O n the other hand, N'," is not a domain when k > 0. Still, it has a number of similar properties, for if M is a subset of N',", then M can be decomposed in M1 Noim

Nkio

c

which is a subset of N k and M2 which is a subset of FF, and M M1 x M z , and i t follows that the least upper bound of M exists if and only if M1 is a singleton a n d

L.E. Sanchis

32

M z is consistent in F T . I t follows also that the greatest lower bound of M exists if and only if M1 is a singleton. EXAMPLE 3.2 Let M be a subset of No,’ consisting of all (0,2)-tuples ( ; a ,B) , such t h a t a satisfies t h e condition: if a(.) N y, then y = 0, and ,B satisfies the condition: if P ( z ) N y, then y = 1. I t follows that M is consistent, and LUB0’2(M) = (; cro,,&) where c r o ( 2 ) N 0 for all 2, and ,Bo(z) N 1 for all 2. Furthermore, M is non-empty, and GLBO”(M)= (; UD1, UD1).

Finally we set Fk,m = FUN(Nkim,N), which is a regular domain. T h e elements of F L , are ~ called ( k , m ) - a y numerical functionals or type-(Ic, m ) objects. Usually they will be referred to as funclionals. We shall use letters f, g, h, p , q , r , s to denote functionals. T h e application of a (k,m)-ary functional f to a n element (z;a)of NkJ” will be written f(z;a). T h e partial ordering relation in Fk,, will b e written means that whenever f(z;a)? y, then g(z; a)N y.

ck,m.

Recall that f Ck,mg

Note that t h e extension relation between functionals is exactly the same as the extension relation between mappings in general. This means t h a t as far as the relation is concerned, the fact t h a t the arguments of a functional are functions or numbers is irrelevant. As a consequence, the relation is preserved as long as the function arguments do not change. Later we shall see t h a t function arguments can b e changed via functional substitution, which may affect the extension relation. In fact, the relation is still preserved because we assume that functionals are monotonic. T h e preceding discussion shows that we may omit monotonicity as long as functional substitution is not involved, and this takes place only at the level of recursive operations. Since we identify N k with Nksoit follows t h a t we can identify

Fk

with

Fk,o.

Hence

a Ic-ary function f (is.., a type-k object) is also a (k,0)-ary numerical functional (or

type-(k, 0) object). Still, the notation is different when we change from one situation t o the other. T h e application o f f as a k-ary function takes the form f(z),where (z)is an element of N k . When f is considered a (k,0)-ary functional the application is written f(z;).But these are purely notational differences, which we shall ignore systematically. Still, in some cases we may want t o stress one particular aspect of t h e object f , as a functional or as a function, and we shall use the notation t o make explicit which one is intended.

C h a p t e r 3. B a s i c Operations

33

+

3.3 Let f be the functional in F ~ such J that f ( r ,y; a , p) N a ( z ) p(y). Hence f(+,y; a , p) is undefined if either a(.) or p(y) is undefined. Clearly, f is quasi-total. It is also complete, for if there is some v such that f(c,y; a', ,B') N v whenever a' and p' are maximal extensions of a and p, respectively, then a ( c ) and p(y) are defined, and f ( c ,y; a,p) N v.

EXAMPLE

The case k = 1 is particularly important, for this means that we identify F1 with F1,o. Hence a unary function a is also a (l,O)-ary functional, and in an application

of the form f(c;a ) where f is a type-(1, 1)-object, both f and a are functionals. T h e identification of numerical functions with functionals plays a role in several arguments where, by assuming that a functional is, in fact, a numerical function, we are able to simplify expressions and eliminate function arguments. On the other hand, there is a natural hierarchy where we move from numbers to numerical functions and from numerical functions to functionals, and in proving some deep result we shall see that this hierarchy must be respected. In these cases an attempt to deal at the same time with functions and functionals will fail. As usual, this type of identification may create some odd notational situations. For example, if f is a (0, 1)-ary functional, and g is a (1,O) functional, then the notation f(;g) is legitimate, for we may consider g to be a type-1 object. In practice this is completely harmless, and it is even useful to stress the fundamental identities which are to some extent obscured by the formal notation.

3.4 Let f be a (1, 1)-ary functional. Then for every unary function a we can introduce a (1,O)-ary functional f,(z; ) 21 f(c;a ) . Since fa is a unary function we can introduce a ( I , 1)-ary functional g such that g(z;a ) N f(z;fa). EXAMPLE

A predicate on the set will be called a (k,m)-ary predicate or simply a predicate. We know these are actually the total mappings from NkJ" t o BOOL, so the usual ordering between mappings is trivial. We have already introduced a different ordering where a (k,m)-ary predicate Q is a boolean extension of the (k,m)-ary predicate P if whenever P ( z ;a)G t, then &(z;a)E t. We have also defined the boolean monotonic predicates where a (k,m)-ary predicate P is boolean , P ( z ' ; a ) E t. monotonic if whenever P ( z ; a ) t and (.;a) ck,m( z ' ; a ' ) then The set of all predicates on is denoted PRE(Nkim), which we abbreviate to Pk,m. This set, with the boolean extension relation, is a complete lattice which under the usual inclusion can be identified with the lattice of all subsets of Nkim

Nkim

Nkim

L.E. Sanchis

34

relation. If M is a subset of Pk,m we consider the elements of M to be subsets and denote the least upper bound of M in $he form U M = the union of of M . Similarly, we denote the greatest lower bound of M in the form n M = the intersection of M . If m = 0, then P k , O = Pk contains only total mappings from Nk to BOOL, which Nkim

we call numerical predicates. Again, this structure can be identified with the set of all subsets of N k . EXAMPLE

0.1 For every number n let Pn the (1, 1)-ary predicate such that P,(.;a)

?

a(.) < n

e (3y)(a(t)

N

y&y

< n).

If we denote by M the class of all predicates P n , n 2 0, then UM = Q where Q(z; a ) G a(.) is defined, and n M = f. The set of all boolean monotonic (6,m)-ary predicates is denoted by BMO(Nk,m) = B k , m , which is also a complete lattice under the boolean extension relation. In fact, B k , m is a sublattice of P k , m , and if M is a subset of B k , m , then M has a least upper bound in B k , m , which is the same as the least upper bound of M in P k , m , and hence is denoted U M . Similarly, n M denotes the greatest lower bound of A4 in B t , m and Pk,,. Note that if rn = 0, then = Nk is a discrete poset, and we have B k = Bk,o = Nkio

Pk,O

=

pk.

Despite the notation similarities there are important differences between the structure F k , m , which contains all (k,m)-ary functionals, and the structures Pk,, and B k , m r which contain (6, m)-ary predicates. The latter are complete lattices where every subset has a least upper bound and a greatest lower bound. The former is only a domain where the least upper bound is defined under very special restrictions. The bottom element of Pk,m and B k , m is the predicate fk,m, and the bottom element of F k , m is the totally undefined functional U D k , m . On the other hand tk,m is the top or greatest element of P k , m and B k , , , while F k , m has no top element. The structural differences between the classes F k , m and P k , m are influential in the methods of proof of some important properties. For example, the proof of Theorem 1.2, the fixed point theorem, as given in Ciapter 1 applies to domains in

Chapter 9. Basic Operations

35

general and requires transfinite recursion. In dealing with a complete lattice, which is also a domain, we can give a simpler proof where the fixed point of a monotonic transformation T is given by the intersection of all 2's that are closed under T . Such a proof would be valid for Pk,, and also for Bk,,. In fact, this is the standard proof given in the literature. On the other hand, the proof is not valid for Pk,, since in this case the intersection is defined only for non-empty classes. Subscripts and superscripts will be omitted whenever they can be derived from the context and no confusion arises. For example, t denotes an element of the set BOOL, but can also denote to,o which is an element of Po,o, hence it is a (0,O)-ary predicate and the application is written t(;). We can also consider t an element of Pk,,

and the application is written t(z; a).

I f f is a mapping from N""' to N, not necessarily a functional, we associate with f the graph predicate denoted GI. As explained in Chapter 11, this is technically a predicate on N x but here it is considered a predicate on Nk+'zm. The definition of G j is Nkim,

G ~ ( Y , za ; )

f(s;a ) N y.

Another important predicate associated with the mapping f is the (k,m)-ary predicate D j , which is defined

EXAMPLE 3.6 Let f be a mapping from N'," to N, not necessarily monotonic. T h e graph predicate G j is a well defined predicate on Nk+'tm and it is boolean monotonic only if f is monotonic. On the other hand, the negation of GI, denoted by G j , is not boolean monotonic. N

Note that in case f is a functional, i.e., an element of D j are boolean monotonic, i.e., both are elements of

Fk,,,,,

then both G j and

Bk,,.

If P is a (k,m)-ary predicate we can associate with P characteristic functionals, which are (k,m)-ary functionals and are not uniquely determined by P . Different types of characteristic functionals have been defined: partial characteristic functionals denoted by 1clp, and dual characteristic functionals denoted by x p . Note that if m = 0 and P is a numerical predicate, then P has exactly one (partial) (dual) characteristic functional.

36

L.E. S a n c h i s

The differences between predicates in general and numerical predicates parallel the differences we have already discussed between functionals in general and numerical functions. A numerical predicate, as a mapping, is always monotonic, hence is boolean monotonic. On the other hand, the extension relation between numerical predicates is essentially the same as between predicates in general. The fact that a numerical predicate has exactly one partial characteristic functional and one dual characteristic functional is not by itself of importance, since there are general predicates with the same property. More important is that the partial (or dual) characteristic function of a numerical predicate completely determines the predicate. There are many operations which can be applied to predicates to generate new predicates. We have already defined negation, conjunction, and disjunction. Note that while the class Pk,,,, is closed under all these operations, the class Bk,, is closed only under conjunction and disjunction. In this work we are concerned with classes where the elements are functionals or predicates. We shall use the letter F to denote a class of functionals, and the letter P to denote a class of predicates. Some classes have already been identified. The class Fk,,, is the class of all (k,m)-ary functionals. The classes Pk,m and Bk,,,, contain all (k,m)-ary predicates and boolean monotonic predicates, respectively. Furthermore, we set now F = the class of all functionals, P = the class of all predicates, and B = the class of all boolean monotonic predicates. If F is a class of functionals and f is an element of 3,we shall say that f is F - c o m p u t a b l e . If P is a class of predicates and Q is an element of P, we shall say that Q is 'P-definable. I f f is a functional and G f is P-definable we shall say that f is P-definable. We denote by 'Pg the class of all functionals which are 'P-definable. Let F be a class of functionals. We say that a predicate Q is p - c o m p u t a b l e in 3 if some partial characteristic functional $Q of Q is F-computable. We say that Q is d-computable in F if some dual characteristic functional xg of Q is F-computable. We denote by F p the class of all predicates that are p-computable in 3,and by F d the class of all predicates that are d-computable in F . Although the definitions of the classes Pg,F d , and FPfollow a similar pattern, there are some important differences. The functionals in the class Pg are exactly those such that the graph predicate is in P. Note that the graph predicate of a functional is completely determined by the functional. On the other hand, the

Chapter 3. Basic Operaiions

37

predicates in the classes F d and F p are those for which some dual characteristic functional, or partial characteristic functional, is in the class F.We know that the characteristic functionals of a predicate are not unique and, in general, many choices are possible. The definition does not distinguish, and any characteristic functional is sufficient for a predicate to be in F d or in F p . Still more important is the fact that the predicate is not completely determined by the characteristic functional. Subscripts applied to classes can be concatenated in the obvious way. For example, if P is a class of predicates, then Pg is a class of functionals, hence Pgp is a class of predicates and Pgpg is a class of functionals. We proceed now to introduce a number of initial functionals which express several fundamental operations on functions and numbers. C o n s t a n t Functionals. If k,m, s 2 0, then c;,, is the (k,m)-ary functional such that cl,,(z; a)N s. P r o j e c t i o n Functionals. If k 2 1 , m 2 0, and 1 5 i 5 k, then P : , ~ is the ( k ,m)-ary functional such that pi,,(z; a)N xi, where we assume that z = XI,.. . , xk. .. A p p l i c a t i o n Functionals. If k, m 2 1, 1 5 i 5 k, and 1 5 j 5 rn, then ap;:, .. is the (k,m)-ary functional such that ap>;,(z; a) N aj(z;),where we assume a = XI,.. .,xk and a = al,.. .,a,. Subscripts and superscripts will be omitted in the above notation if no confusion arises and the context is sufficient to determine the intended interpretation. The constant functionals, projection functionals, and application functionals are called basic functionals. The class of all basic functionals is denoted by BS. EXAMPLE 3 . 7 It is easy to determine the elements of the classes BSp and B s d . In fact, only the constant functionals are characteristic functionals. It follows that BSp = {tk,, : k , m 2 0) and Bsd = (tk,,,fk,,,, : k , m 2 0).

Next we introduce three operations which can be applied to functionals t o generate new functionals. They formalize well known procedures where a new functional

is obtained by substitution or definition by cases from given functionals. N u m e r i c a l Substitution. Let f , 91,. . . ,gs be given functionals where f is (s,m)-ary, s , m 2 0, and 91,.. .,gs are (k,m)-ary, k 2 0. We introduce a new (k,m)-ary functional h such that

38

L.E. Sanchis

As usual, the expression on the right is undefined if some sub-expression is undefined. Note t h a t s = 0, k > 0, are permitted. Distribution. This is a form of functional substitution where a ( E , s)-ary functional g is given, k , s 2 0, and a (k,m)-ary functional h is introduced, m > 0, in t h e form

h ( z ;a)? g ( z ; a i l 1 . . ., a i , ) , where we assume a = 0 1 , . . .,a,,, and 1 5 ij 5 m for j = 1 , .. . , s. Note t h a t s = 0 is permitted, in which case one or more (dummy) function variables are added to a given numerical k-ary function g. Definition by Cases. In this operation three ( E , m)-ary functional g , fl , and f~ are given, E , m 2 0, and a new (k,m)-ary functional h is introduced such that

h(z;a)

N

f l ( z ; a ) if

? f i ( z ; a ) if

g(z;a) NO g ( z ; a ) $0.

This specification must be understood in t h e sense t h a t h ( z ;a ) is undefined in case g ( z ; a ) is undefined. If g ( z ; a)? 0, then h ( z ;a ) 21 fl(z;a ) independently of whether fz(z;a ) is defined or not. Similarly, if g ( z ; a ) $ 0 (i.e., g ( z ; a ) is defined with value different from 0), then h ( z ;a) N fz(z;a ) independently of whether fi (2; a)is defined or not. Note that the evaluation is strictly deterministic in the sense that only one functional is being evaluated a t a given stage, and if such a functional is undefined, then t h e whole evaluation is undefined. I t is convenient t o have a more compact notation to describe the applications of definition by cases. If the functional h is given as above with given functionals g ( t h e case functional), fi and f2 (the output functionals), we shall write h ( z ;a) N Mz;a) f1(z;a ) ,f 2 ( z ; a ) ] h = [g fl,f2]’ +

+

Whenever an operation on functionals, or on predicates, has been defined we have a derived notion of closure, which in most cases is straightforward and which we assume is familiar to the reader. For example, we say that a class F of functionals is closed under definition by cases if whenever a functional h is introduced by definition by cases with given functionals g , f l , and fi which are F-computable, then h is also F-computable. In practice we need a slightly more complicated notion where several operations are involved, and also some initial functions are required.

C h a p t e r 3. B a s i c Operations

39

We shall say that a class 7 of functionals is closed u n d e r basic operations if t h e following conditions are satisfied: BS 1: T h e class

F contains the basic functionals..

BS 2: T h e class 3 is closed under numerical substitution, distribution, and definition by cases. T h e idea behind basic operations is to formalize the operations of numerical substitution and definition by cases, which are traditional in computability theory. More interesting is the restricted role that functional arguments play in these operations and, in particular, the fact that functional substitution is not allowed. T h e fact that the functional arguments are actually functions is relevant in only one operation, namely, the definition of the application functionals. And even here the application is understood simply as an unspecified relation R(y, z;a) written as a(.) N y. Whether such relation is really application is irrelevant. We shall see later that in order to make explicit the functional character of the function arguments we must introduce functional substitution, where a defined function can be substituted for a function variable. Note that this restriction is not accidental. Rather, it is an essential characteristic of the type of computation we are dealing with at this stage. In fact, the same characteristic will be a part of primitive recursive operations, basic recursion, p-recursive operations, and unrelativized recursive operations. Only in dealing with relativized recursive operations shall we need to invoke explicitly functional substit u t ion. T h e operation of distribution can be derived by functional substitution, but by itself it does not imply any functional character in t h e arguments.

3.8 Using definition by cases with the projection and constant functionals we can introduce unary functions (i.e., (1,0)-ary functionals) f1 and fz such t h a t

EXAMPLE

= [P fz = [P fl

+

+

cO,cll cl,c0I

where fl(z;)N 0 if z = 0, fl(z;)N 1 if z # 0, f i ( z ; )N 1 i f z = 0, and f i ( z ; ) N 0 if 3: # 0. If F is a class of functionals closed under basic operations, then the functions fi and f 2 are F-computable.

L.E. Sanchis

40

Theorem 3.1 Lei

F be a class of functionals closed under basic operalions. Then,

(i) FplFd are closed under conjunction. (ii) d'.

as

closed under disjunction.

(iii) Fd is closed under negation. PROOF. To prove (i) we assume R = P A Q.If $ p and $Q are partial characteristic functionals of P and Q,respectively, we can obtain a partial characteristic functional $R of R its follow^: $R=

c'].

[$P-$Q,

On the other hand, if x p and X Q are dual characteristic functionals of P and Q , respectively, then we obtain a dual characteristic functional X R in the form: XR

=

[XP

+

X Q I c'].

To prove (ii) we assume R = P V Q and take dual characteristic functionals x p and X Q . A dual characteristic functional for R is obtained in the form: XR= [XP-C

0

,XQ].

N

The proof of (iii) is similar, for if R =Q and X Q is a dual characteristic functional of Q ,then a dual characteristic functional of R is given by XR

=

[XQ

+

c l ,c O ] . 0

It is convenient to generalize the operation of definition by cases by allowing any number of cases rather than only one. General Definition by Cases. Assume (k,m)-ary functionals 91, . . . ,gs,fi , . . ., f d , f s + l are given, s 2 1. We introduce a new (k,m)-ary functional f in the form: f(z;a)21 fl(z;a) if g1(z; a)N 0

f,(z; a )

if

g,(z; a ) N O

= f , + l ( z ; a)

if

g,(z; a)9 0.

21

Chapter 9. Basic Operations

41

This specification must be understood in the sense that f(z;a)N fi(z;a) if and only if either 1 5 i 5 s , g;(z;a ) N 0 , and for all j such that 1 5 j < i g j ( z ; a ) $ 0 , or i = s 1 and for all j = l , . .. , s g,(z;a) 9 0. If this condition is not satisfied, then f ( z ;a)is undefined. Note the following features of the evaluation of f(z;a).The cases gl(z; a), . . .,gs(z; a)must be checked in the given order until the first i is found such that gi(z;a) 21 0. To move from case g j ( z ; a ) to case gj+l(z; a)it is necessary that gj(z; a)!$ 0. When such i is found we set f ( z ; a ) N fi(z;a). If no such i is found we set f(z;a)N f,+l(z; a). Note that the evaluation described above is deterministic, in the sense that at any stage only one functional is being evaluated and if this functional is not defined, then f(z;a)is undefined. The preceding characterization of deterministic computations is not intended as a formal definition. In fact, we do not intend to study non-deterministic computability in terms of functionals. An initial approach to this problem was presented in Sanchis [27] for numerical functions, where a form of Church’s thesis wits proposed. In dealing with functionals the situation is more complicated, and it is not clear to us that the previous approach can be extended. As usual, the problem disappears if we consider only singular functionals that are undefined for non-total arguments. This approach is popular in the literature. For example, see Hinman [lo]. We shall use the following notation to describe the application of general definition by cases as explained above:

+

-

f(z;a) [ d z ;a), . . . ,gs(z; a) f = [Sl, . . . , g s ---* 1 1 , . * * I f 3 f b + l . 9

I

f 1 ( . ;

a), . . .,fdz;a),fa+l(z;a)]

Theorem 3.2 If F is closed under basic operations, then 7 is closed under general definition by cases. PROOF. We prove the closure by induction on s = the number of cases in the specification. If s = 1 we have the original definition by cases and F is closed by definition. Assume the theorem is true for s, and assume a specification with s + 1 cases is given in the form:

f = [Sl,. . . ,Qb,Ss+l where 91,.. . ,gs,gs+l, f1,. . . , fs, using only s cases in the form

--+

fs+l, fs+2

. . .,f b , fs+l,

fl,

fs+2]

are F-computable. If we introduce f’

L.E. Sanchis

42

it follows by the induction hypothesis that f’ is F-computable. Now we can obtain f in the form

f = [Sl fit f’] 9

+

hence f is F-computable.

0

The operations of numerical substitution and distribution can be extended to predicates, although the former presents some difficulties due to the fact that functionals in general are not total mappings, while predicates are required to be totally defined mappings. On the other hand, the extension of distribution to predicates is straightforward. From a formal point of view there is no doubt that both distribution and numerical substitution are legitimate operations which belong to the general theory of predicates. Our interest here is rather oblique for, in general, we look at predicates via their characteristic functionals. The definitions we adopt are mainly motivated by these considerations, particularly the definition of numerical Substitution. Distribution in Predicates. Let Q be a given (k,s)-ary predicate, Ic, s 2 0. We introduce a (k,m)-ary predicate P, m > 0, in the form:

P ( z ; a)

..

Q ( ~ ; a i , ,.,a*,),

where we assume a= a1,.. .,a, and 1 5 i j 5 rn for j = 1 , . . ., s. Note that s = 0 is permitted, in which case one or more (dummy) function variables are added to a given numerical predicate. Note that i f f is a partial (or dual) characteristic functional for the predicate Q above, and f’ is obtained from f by the same distribution as P is obtained from Q, then f’ is also a partial (or dual) characteristic functional for P . It follows from this that whenever a class F of functionals is closed under basic operations, then Fp and F d are closed under distribution. The operation of distribution in predicates preserves the boolean extension relation. If Qz is a boolean extension of & I , PI is obtained from Q1 by distribution, and Pz is obtained from Qz by the same distribution, then Pz is a boolean extension of PI. The operation of numerical substitution in predicates requires special conventions for the case in which the functionals involved in the substitution are not defined. We adopt the convention that i n such a situation the defined predicate is false.

Chapter 3. Basic Operations

43

Numerical Substitution in Predicates. Let Q be a n (s,n ) - a r y predicate, s, m 2 0, and 91,. . . , g s be (k,m)-ary functionals, k 2 0. We introduce a (k,m)-ary predicate P such that

P ( z ;a )

Q(gi(z; a ) ,. . . , gs(z; a ) ; a ) -(3Y)(Gg,(Yl,z;a) A ... A G g , ( y s , z ; a ) A &(?/;a))

where we assume y = y1,. . . , ys and 3y stands for 3yl . . .3ys. In some constructions this type of substitution behaves in the way we usually associate with total substitution. For example, let Q1 and Q2 be predicates, P =

R = Q1 A

&I V Q 2 ,

Q2.

By substitution we obtain the following predicates:

Q;(z; a ) Q;(z; a ) P’(z; a ) R’(z; a )

=

Q l ( g l ( z ; a ) ,. . . , g s ( s ; a ) ;a) Qz(gi(2; a ) ,. . . , g , ( z ; a ) ;a) P ( g 1 ( s ; a ) ,. . . ,gs(z; a ) ;a ) R ( g l ( s ; a ) ,. . . ,gs(a; a ) ;a)

Now it is easy t o verify the equalities P‘ = Q; VQ; and R’ = Q; A Q2, which hold even if the functionals 91,. . . , gs are not defined. On the other hand, negation behaves in a very different way. For example Q(g(c; a ) ;a ) is true when g(z; a ) is undefined, while Q (g(c; a ) ;a ) is false in the same situation. In the evaluation of general expressions involving substitutions we adopt t h e rule that the effect of t h e substitution is determined first on the atomic components which are false whenever a functional is undefined.

-

N

3.9 Let Q b e a (1, 1)-ary predicate and g a (1, 1)-ary functional. Then the expression

EXAMPLE

Q M c ; a ) ;a ) ”

-

Q M z ; a);a )

is true when g ( c ; a ) is defined, and i t is also true when g(z; a ) is undefined, for in this case Q(g(e; a ) ;a ) holds. O n t h e other hand, the expression

-

is false when g(z; a ) is undefined.

Note t h a t we can always avoid writing predicate substitution by using the graph predicate of the functionals as we did in the definition of the substitution.

L . E . Sanchis

44

Theorem 3.3 Let

F

be a class of functionals closed under basic operations. Then,

(i) Fp is closed under distribution and numerical substitution with F-computable functionals.

(ii) Fd is closed under distribution and numerical substitution with quasi-total F computable functionals.

(iii) If Q1 is a predicate d-computable in F and Q = Q1 V Q2 is p-computable in 3.

Q2

is p-computable in F , then

Part (i) is clear from the definitions. In part (ii), note that the quasitotality of the functionals is required to make sure that condition DCF 1 is satisfied. To prove part (iii), assume x1 is a dual characteristic functional of Q1 which is F-computable, and I/J2 is a partial characteristic functional of Q2 which is also F-computable. We obtain a partial characteristic functional I/JQ which is in the class F as follows:

PROOF.

It follows that Q is p-computable in

F.

0

EXERCISES

3.1 Prove that the class BS is not closed under basic operations. 3.2 Let F be a class of functionals closed under basic operations and f a functional which is F-computable. Prove that the predicate D j is p-computable in F. 3.3 Prove there is a unique class of functionals F such that the following conditions are satisfied: (a) F is closed under basic operations.

(b) If 3' is a class closed under basic operations, then 3 5 F'. 3.4 Prove that the class

F in Exercise 3.3 contains only quasi-total functionals.

Chapter 9. Basic Operations

45

3.5 Let F be a class closed under basic operations. Give an example of predicate P which is not boolean monotonic and is d-computable in F.

Here, as in Sanchis [27], basic operations provide for a general form of numerical substitution. We include also distribution which allows for some form of functional substitution. The main difference with the treatment in [27] is the use of definition by cases, which makes it possible to obtain closure under boolean operations with a minimal of assumptions. More important is that basic operations provide a natural frame where basic recursion can be formalized. This is done in Chapter 5. Functionals with partially defined arguments have been studied before in Kleene [14] and Tourlakis [28]. The class of extended functionals introduced in Hinman [ll] is also of the same type.

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Chapter 4

Primitive Recursive Operat ions In this chapter we extend basic operations and introduce primitive recursion. This rule plays a fundamental role, as it was the case in Sanchis [27]. First, we use primitive recursion to define a basic set of functionals that are used to formulate several normal form theorems. Second, primitive recursion expresses a critical computational structure where a process is generated by a series of steps, each determined by the result of the preceding step. The theory of primitive recursive operations is preceded by a construction which completes the treatment of basic operations by means of terms where the application of several operations can be organized in a compact way. This is a natural technique derived from mathematical logic and provides a flexible machinery that can be easily extended by adding new rules as required. The reasons for this formalization go far beyond simple practicality. In fact, by using this formalism and a simple extension we can get a notion of basic recursion, which is equivalent to p-recursion, familiar from the theory of recursive functions. This extension is presented in the next chapter. We fix a class F of functionals, and given a list of variables 2; a we define basic F-terms in the variables x ; a . The definition is by induction, according t o the following rules: 47

48

L.E. Sanchis

BT 1: A variable in the list z,or a numerical constant, is a basic F-term in the list z;a. BT 2: If g is an (s, r)-ary functional in the class F , s , r 2 0, and U1,. . . , U, are basic F-terms in z;a,then g ( U 1 , . . . , Us; a i l , .. .,a*,) is a basic F-term in 5 ; a,where we assume a = al,. . .,a,, and if r > 0, then 1 5 ij 5 rn holds for all j = 1 , . . ., r . BT 3: If U1,U2, and U, are basic F-terms in the variables z;a,then [Ul -+ U z , Us] is a basic 7-term in z;a . BT 4: If U is a basic F-term in the list z;a,and a is a variable in the list a ,then a ( U ) is a basic F-term in z;a . Note that if U is a basic F-term in the variables z;a , then all the functional symbols occurring in U correspond to functionals in the class F. Furthermore, all variables occurring in U are in the list z;a,although this does not mean that every variable in the list must occur in U . Rule BT 4 allows for the direct application of a function variable to a basic term, independently of the class 3.In particular, this means that we do not require that T contain the functional ap(c;a ) N a(.). In fact, rule BT 4 can be applied even if F is empty. Our approach in dealing with basic terms is semi-formal, for we do not define explicitly which symbols are the variables. Essentially, this means that we reserve the right to consider any symbol we want to be a variable. In practice, we shall continue to use exactly the same symbols introduced in the preceding chapter. Still, there are formal elements in our treatment that to some extent may be considered irrelevant. In particular, the formal semantics introduced below is really unnecessary as long as we are interested only in strict basic terms. The situation changes later when we introduce recursive basic terms and their monotonicity properties. The use of informal terms is a technique that we use throughout this work. On the other hand, in Chapters 9 and 10 we go a step forward and present a strictly formalized system of terms. Assume the length of the list z;a is (k,rn) and U is a basic 7-term in 2 ;a.Then U determines a (k,m)-ary functional fu, which actually can be specified without N )U . The point is that the construction of any ambiguity in the form f ~ ( z ; a

Chapter

4. Primitive Recursive Operations

49

U involves operations which have been already defined. Still, we think there is some interest in writing a complete inductive definition of the functional fu. T h e definition is by induction on the construction of the term U , or simply by induction on the length of U . T h e induction has one defining rule for each construction rule which occurs in U . DBT 1: If U is a basic F-term in the list x ; a by rule BT 1, we put fu( 2; a) N U . Note that if U is a variable, then fu is a projection functional p, and if U is a numerical constant, then fu is a constant functional c. DBT 2: If U is of the form g(U1,. . . , U,; a i l , .. . , a i v ) ,we assume functionals f1 = fu,, . . . , f a = fu, have been specified, set f’(y;a) g(y; ail,. . . , a i r ) by distribution, and fu(x; a)N f ’ ( f l ( 2 ; a), . . . , fa(x;a ) ; a )by numerical substitution.

=

DBT 3: If U is of the form [Ul + Uz,U3] by rule BT 3, we assume f l = fu,, f2 = fu2, and f3 = fu, and set fu = [fi + fz, f3] by definition by cases. DBT 4: If U is of the form a(U1) by rule BT 4, we assume set fu(z; a ) N ap::j,(fl(z; a ) ; a ) .

f1

= fu,, a = a,,and

Theorem 4.1 Let U be a basic F - t e r m in the variables a;a where 3 is closed under basic operations. Then the functional fu i s 3-computable.

T h e theorem is proved by induction on the structure (or length) of the term U . All cases are trivial since in the formal definitions DBT 1 t o DBT 4 we use only basic operations. 0

PROOF.

EXAMPLE 4.1 Let 3 be a class closed under basic operations and assume the binary function add = addition (i.e., add(z,y;) N c y) is in F. It follows that the following (1,2)-ary functional f is F-computable:

+

Although the formal definition of basic terms allows only for definition by cases with just one case (rule DBT 3) in applications we shall use more general constructions involving any number of cases. Hence, if 171,.. ., U s ,V1,. . ., V,, V8+1 are basic F-terms in the variables x ; a,then [ U I , .. . , Ua + V1,. . . , V,, V,+1] is a basic

50

L.E. Sanchis

F-term in the variables z ; a . If s 2 2, we consider this expression simply as an abbreviation of the following: [Ul -+ V1, [Uz,. . .,Us V 2 , .. . , K + I ] ] . The use of definitions with several cases is just one possible construction where we allow for terms with a nested structure of definition by cases. There are infinitely many combinations available, and some of them will play a crucial role later. For example, consider a term of the form [U, + [Uz + V1,Vz],[U3 + W1, Wz]].This is actually a tree with several evaluating branches. One possible branch is U1, U z , Vz, and another is U,, U3, W1. There are two more branches, and the evaluation of the term, if defined, is always determined by exactly one of the branches. +

Let U be a basic F-term in variables a;a,and fu the functional specified by rules DBT 1 to DBT 4. We say that fu is explicitly specified from F b y the term U . It appears that the functional fu is computable in some sense, although the computation should be relative to the functionals of the class F which occur in the construction of U . This notion of computation is vague, but can be made precise in a number of equivalent ways. We discuss one example where a “program” is constructed which can be used to compute fu(z;a)for any input values (2; a ) . EXAMPLE

4.2 Let U be the following basic F-term in the variables S(X1,

[ g ( z 1 , z z ;(.2)

+

Q l ( X l ) , a2(22)1;

X I , 2 2 ; ( ~ 1 a,2

:

4,

where F is a given class, and g is a (2,l)-ary F-computable functional. An ezplicit algorithm to compute the functional fu is given by the following set of instructions, where X I , X 2 , and X3 are formal variables (or value-holding locations): 1: 2: 3: 4:

5: 6:

7: 8: The meaning and execution of these instructions is self-explanatory. As usual, it is assumed that there is a n external agent, or oracle, which provides the values requested in instructions 3, 5, 7, and 8. If such values are undefined, the computation

Chapter 4. Primitive Recursive Operations

51

is blocked (i.e., halts without output). When the computation halts, the value of fu(z1,z2;al,az) is in location X I .

Now we introduce a new operation which can be used to generate new functionals from given functionals. I t is well known from the literature and our presentation will b e rather sketchy. First, we introduce a new functional which will be taken as initial in relation to these operations. Successor Functional. T h e (1,O)-ary functional s such t h a t B(I;) N 3: 1. Since this is actually a unary function, it will be written ~(3:). T h e new operation we discuss is primitive recursion, which is introduced as a form of recursion but can be reduced immediately to an explicit evaluation. Primitive Recursion. Let g b e a (6,m)-ary functional, k,m 2 0, and h a (k 2, m)-ary functional. We introduce a new (k 1,m)-ary functional f by t h e equations:

+

+

+

f(0,z;a) f(Y + 1,z;4

= =

g(z;a)

h(f(Y, z;a), Y,z;a). These equations are understood as follows. To evaluate f(y,z; a)we use the first equation to get f(0,z; a).If this value is defined we move t o evaluate f(1,z;a), using t h e second equation with the value of f(0,z;a)given by the first equation. If f(1,z;a)is defined we use the second equation again to evaluate f(2,z; a).This procedure continues until the value of f ( y , z ; a)is obtained. If during the evaluation some expression is undefined, then f ( y , z ; a)is undefined. It should be clear t h a t this procedure is deterministic and can be easily formalized as an explicit algorithm where an external oracle provides values of g and h. T h e procedure above introduces primitive recursion in the classical way, by means of recursive equations. At t h e same time, the recursion is eliminated by describing a deterministic computation procedure that evaluates the functional. Note t h a t this procedure is not recursive, for no recursive calls are involved. O n the other hand, there is a recursive procedure available, where in order to evaluate f ( y , z ; a)with y # 0 the value f ( y - 1,z;a)is called and then used with the given functional h . T h e preceding discussion indicates t h a t , although primitive recursion is recursive and can be computed by a recursive procedure, it is, in fact, computed by a nonrecursive explicit procedure. This is actually a general phenomenon, as we shall show in the discussion of basic recursion.

L.E. Sanchis

52

We say that a class 3 of functionals is closed under primitive recursive operations if the following conditions are satisfied:

PR 1: The successor function s is F-computable. PR 2: F is closed under basic operations. PR 3: F is closed under primitive recursion. If 3 is closed under primitive recursive operations, then 7 is closed under basic operations. It follows that Theorems 3.1, 3.2,3.3,and 4.1 apply immediately to 3. In particular, it means that functionals which are specified explicitly by basic terms where all the functionals belong to class 3 are also in class F. There are several practical reasons for identifying primitive recursive operations, as is shown by the constructions introduced in the rest of this chapter, but their real significance goes beyond practical convenience. Primitive recursive operations have some of the power of general recursion and, at the same time, preserve quasitotality (see Theorem 4.4 below). Note that the latter preservation property fails very easily when recursion is introduced. By introducing primitive recursive operations we identify a computability domain with desirable completeness properties which, at the same time, avoids the ambiguities associated with non-totality. We shall show later that for some forms of general computability a kind of weak reduction to primitive recursive operations is possible. These reductions are known in the literature as normal form properties. In the presence of basic operations the rule of primitive recursion can be extended by using basic terms rather than the given functionals g and h. It is convenient to formalize this rule, for all applications of primitive recursion are given in this form. Basic Primitive Recursion. Let U be a basic F-term in the variables 5 ; a, and V a basic F-term in the variables z , g , z ; a where ( z ; a )is a ( k , m ) tuple, k , m 2 0, and y, t are variables which do not appear in the list x. We introduce a ( k 1, m)-ary functional f by the equations

+

f(0,x;a) f(Y+ 1,x;a)

N

u

=

V’

where V’ is an expression obtained by substitution of f(y,x;a)for all occurrences of z in V . This specification must be understood exactly as in the rule of primitive

Chapter 4. Primitive Recursive Operations

53

recursion. To evaluate f ( y , z ; a ) we start with f ( 0 , a ;a ) ,given by the term U , and proceed to f ( 1 , z ;a ) using V', etc., until f(y,z; a ) has been evaluated. This is a deterministic procedure, for if one of the expressions is undefined, then f (y,z; a ) is undefined. In particular, if U is undefined, then f(y,z; a)is undefined for all y, independently of the form of the term V , and even if the variable t does not occur in the term V , in which case the symbol f does not occur in V'. Note that the operation of basic primitive recursion is relative to a class F of functionals, for the terms U and V are assumed to be basic 7-terms. If the functional f introduced by the operation is 7-computable for all possible basic F-terms U and V , we say that F is closed under basic primitive recursion.

If F is a class of functionals closed under primitive recursive operations, then 3 is closed under basic primitive recursion.

Theorem 4.2

PROOF.

If we introduce functionals g and h such that g(z;a)

N

U

h(z,y,a;a)

N

v

it follows that g and h are F-computable, and f can be obtained by primitive 0 recursion from g and h.

4 . 3 Let F be a class closed under primitive recursive operations. The following (2,O)-ary functionals (i.e., binary functions) are introduced by basic primitive recursion, hence are F-computable: EXAMPLE

add(0,t) add(y 1 , t)

21

t

N

s(add(y, 2))

pr(0,z)

N

0

+

+

Pr(Y I , . ) exp(0,t) exP(Y

+ 1,

= N

add(pr(y, 2),2) 1

N

Pr(exP(Y, c),2).

In applications these functions will be denoted using the standard notation, i.e., add(y, t) = y t, pr(y, t) = y x 2. The notation exp(y, t) is preferred to 241, although in some cases the latter will be used.

+

. . .,F,,, . . . of Let F be a class of functionals. We define a sequence &,F;, classes which is determined by the class 7 as follows: we put 70= the class F plus

L.E. Sanchis

54

the basic functionals and the functional s. Assume Tnhas been defined. We set Tn+l= Tnplus all functionals which can be obtained by applying the operations of numerical substitution, distribution, definition by cases, and primitive recursion to functionals in Tn. The union of all classes Fnis denoted by PR(T), hence PR(7) = 30U TI U T 2 U

.. .

U 3 n U 3;1+1.

..

The functionals in the class PR(T) are said to be primitive recursive in T . If 7 is a singleton, say T = {g}, we write PR(g) = PR(3), and the functionals in PR(g) are primitive recursive i n g. Finally, if T is empty we write PR = PR(0). The functionals in PR are called primitive recursive functionals. The construction of the class PR(3) is essentially an elemental inductive definition, well known in the literature. The same construction will reappear a few more times, with different sets of operations. The crucial properties of the construction are given in the next theorem, which by itself can be taken as a, kind of inductive definition. The proofs of these properties will be taken by granted whenever similar definitions are introduced later. Theorem 4.3 Let T be an arbitrary class of functionals. T h e n

(i) PR(T) i s an eztension of 3 , i.e., 7 E PR(7). (ii) PR(F) is closed under primitive recursive operations. (iii) If 3' is an eztension of T , and 3' is closed under primitive recursive operation, then T' is an eztension of PR(F).

To prove (i) note that 7 E 30E PR(7). To prove (ii) we consider an application of any of the operations involved and note that the operation is applied to functionals in some class T,,,hence the result of the operation is in the class 0 Tn+l. Finally, (iii) is proved by showing by induction on n that Tn 3'. PROOF.

4.4 Consider the (2,l)-ary functional f given by basic primitive recursion in the form:

EXAMPLE

1,Ol

f(0,2; 0)

2

[a(.)

f(Y+l,+;o)

=

[42)+f(Y,C;a),

+

f(y,z;a)l.

It follows that f is primitive recursive. Note that f (y, 2; a) is defined if and only if a ( z ) is defined. In fact, f ( y , z ; a)N [a(+) + 1,0].

Chapter 4. Primitive Recursive Operations

55

Corollary 4.3.1 Let F be a class of functionals. The following conditions are equivalent:

(i) 3 is closed under primitive recursive operations.

(ii) P R ( 3 ) C F , (iii) P R ( F ) = 3 . If (i) holds, then (ii) holds by Theorem 4.3 (iii). If (ii) holds, then (iii) follows from Theorem 4.3 (i). If (iii) holds, then (i) holds by Theorem 4.3 (ii). 0 PROOF.

Theorem 4.4 If the class 3 contains only quasi-total functionals, then P R ( T ) contains only quasi-total functionals. Let F be the class of quasi-total functionals. Since T C f ,and F' is clearly closed under primitive recursive operations, it follows from Theorem 4.3 (iii) 0 that P R ( F ) & 3 ' .

PROOF.

Corollary 4.4.1 If F contains only quasi-total functionals and f is a numerical function primitive recursive in T , then f is total. PROOF.

Immediate from Theorem 4.5, since a quasi-total function is total.

0

Corollary 4.4.2 A primitave recursive numerical function is total. Immediate from Corollary 4.4.1. 0 The class PR is somewhat special. It has an easy characterization, namely, it is the smallest class that is closed under primitive recursive operations. We know that all the functionals in P R are quasi-total. This implies that the classes PRd and PRI, have natural substitution properties with the primitive recursive functionals. On the other hand, note that similar properties hold for the classes P R ( 7 ) whenever 3 contains only quasi-total functionals. Note that the numerical functions in the class PR are exactly the primitive recursive functions as defined for the example in Kleene [12]. For the proof of this relation, see Exercise 4.2 It is obvious that the primitive recursive functions in the standard sense (as defined for example in [12]), are elements of PR. The converse is also true, for every numerical function in the class PR is primitive recursive in the standard sense. PROOF.

L.E. Sanchis

56

4.5 The functionals introduced in Example 4.3 are all primitive recursive. We extend the list with the following numerical functions, which are also primitive recursive:

EXAMPLE

Closure under primitive recursive operations implies closure under several other operations which are very important for the formal development of the theory. We discuss those where some operation or search is iterated up to a given numerical upper bound. We introduce three operations of this type-bounded sum, bounded product, and bounded minimalization. We assume that the reader is familiar with the intended mechanics and we omit the details. The main purpose is to make sure that the operations can be defined using primitive recursive operations. In these definitions we depart from our usual style, where an operation is specified in terms of given functionals or predicates, as the case may be. We simply assume that a term U is given in variables y,z; a,and describe by basic primitive recursion the functional that is introduced by the rule. Bounded Sum. Let U be a basic T-term in the variables y,z; a ,where 2; a is of type-(k, m),and V a basic T-term in z;a. Using basic primitive recursion and explicit specification we obtain a (k 1,m)-ary function f’ and a (k,m)-ary functional f such that

+

f’(0,z;a)

f ’ (+ ~1

, ~a; )

N

0

N

f’(y, z;a)

f ( z ; a )N

+U

f’(V,z;a).

We say that f’ is obtained from U by bounded sum, and f is obtained from U by bounded sum with upper bound V . To denote f’ and f we shall use the following notation: f ’ ( z , z ; a ) N (Cy< z)U

f(z;a)

=

(CY < V ) U ,

Chapter 4. Primitive Recursive Operations

57

where z is a new variable not occurring in the list y , x . Note that in the specification o f f the term V may contain variables from the list x .

4.6 Let f be the (2,l)-ary functional of Example 4.4, where f(y, 2 ;a ) is defined only if a(.) is defined. Using bounded sum we can introduce a (3,l)-ary functional h’ and a (2,l)-ary functional h such that EXAMPLE

h’(z, 2; 0) h(z;a)

= =

(CY< %)f(Y, 2; a ) (CY < z ) f ( y , z ; a )

Note that h ’ ( 0 , ~a) ; N 0 and h(0;a ) N 0 for arbitrary z and a .

Bounded Product. Let U be a basic F-term in the variables y,a; a,where is of type-(k, m), and V a basic 3-term in x; a.Using basic primitive recursion and explicit specification we obtain a (k 1,m)-ary functional f’, and a (k,m)-ary functional f such that 2; a

+

N

+ 1 , z ;a)

1

N

f(z;a)

N

f’(y, 2; a)x f’(V,x;a).

f’(0,x;a) f’(y

u

We say that f’ is obtained from U by bounded product, and f is obtained from U by bounded product with upper bound V . To denote f’ and f we shall use the following notation: f ’ ( 2 , z ; a ) 21 ( n y < z)U f ( x ; a ) 21 (IIy < V)V where z is a new variable which does not occur in the list y , x . Next, we introduce bounded minimalization. Here our approach differs from the one that is found often in the literature. We do not apply the operation to a predicate, but rather to a functional. The difference is significant, for in practice the functional is a characteristic functional of some predicate, and we know this is not a faithful representation. As a consequence, we do not really evaluate the bounded minimalization of the predicate. Rather, the evaluation deals with some approximation of the predicate. Still, the approximation is faithful as long as total arguments are considered and the evaluation is meaningful with such parameters. The second difference is only technical and actually appears in many presentations. When the search up to upperbound y fails, we take as output y itself, rather than 0.

58

L.E. Sanchis

Bounded Minimalization. Let U be a basic F-term in the variables y,z;a, where z;a is of type-(k, m),and V a basic F-term in z;a . Using basic primitive recursion and explicit specification we obtain a (k + 1,m)-ary functional f‘, and a (k,m)-ary functional f such that

f’(Y

f’(0,z;a) N

0

+ 1,z;a)

[eq(f’(y,2; a ) ,Y) f/(V,z;a).

f(2;a)

1:

21

+

[U 4 Y, Y

+ 111f“Y,

2;

all

We say that f’ is obtained from U by bounded minimalization, and f is obtained from U by bounded minimilization with upper bound V . To denote f’ and f we shall use the following notation:

f’(z,.;a)

f(2;4

N

21

(PY < Z)U (cry < V)U

where z is a new variable which does not occur in the list y,z EXAMPLE 4 . 7 Consider the following (1,l)-ary functional f introduced by bounded minimalization as follows:

It follows that f(c;a ) N a(.) for all c,a.

F i s closed under bounded s u m , bounded product, and bounded minimalization with basic F -

Theorem 4.5 IfT

i s closed under primitive recursive operations, then

terms.

This is straightforward since each operation has been defined in terms of 0 basic primitive recursion and explicit specification with basic F-terms. PROOF.

Next, we list a number of primitive recursive functionals that play a technical role in several applications. They refer to properties of the prime numbers and they are used to encode tuples of numbers and also tuples of functions. Most of them are well known from the literature, and we omit details. The technique used here to encode tuples of numbers is different from the more traditional in Sanchis [27]. Still, the difference is minimal, and the idea is essentially the same.

Chapter 4. Primitive Recursive Operations

59

The encoding of functional tuples is different from those found in the literature. Since we are dealing with non-total numerical functions, we are forced to introduce a mechanism where the fact that the application of a function is undefined does not force the whole encoding to be undefined. The obvious technique to deal with this type of situation is definition by cases. The Prime Number Enumeration Function. This is a unary function pn that enumerates the prime numbers in increasing order. Hence, pn(0) N 2, and pn(y 1) 1~ the least prime number t such that t > pn(y). The Number Encoding Functions. For every k 2 0 the k-ary function nck is given explicitly by

+

nck(t1,. . ., t k )

N

(exp(2,k) x exp(3,q) x

. .. x exp(pn(k),zk)) I 1.

In applications we shall use the notation < t l , . . ., Zk> = = nck(z). The Number Decoding Function. This is a binary function nd such that nd(y, z) = the exponent of pn(y) in the prime factorization of z 1. It follows that if 1 5 i 5 k, then nd(i, < q ,... , t k > ) 1~ nd(i, )N zi. In applications we shall use the notation [.Iy N nd(y, 2). It follows that [ ] o N k, and [ < t I , .. . ,zk>]j N t i , for i = 1,.. . , h. The Function Encoding Functionals. For every m 2 0 the (1,n)-ary functional fc, is specified by cases as follows:

+

21

N

arn([z]~) if eq([l]i,m) N 0 O if eq([t11,m)$ 0 .

We shall write < & I , . . . ,a,> = a where a ( t ) N fcm(t;a1,.. .,a,). The Function Decoding Functional. This is a (2,l)-ary functional fd, which is given explicitly in the form fd(y,z;a) N a(). In applications we shall use the notation [aIy(z)N fd(y,I ;a).Hence [sly is a well defined unary function, depending on a given unary function a,and a given number y. Note that [ ] 0 = c"', and []j = Oi for i = 1 , .. . , m .

L.E. Sanchis

60

If 3 is a class of functionals closed under primitive recursive operations, then

F induces two classes of predicates, namely, Fp and F d . The closure properties of these classes has been determined in Theorem 3.1, Theorem 3.2, and Theorem 3.3 for boolean operations and numerical substitution. We introduce now two operations on predicates that involve quantification over numerical variables. Note that a general notion of quantification, both existential and universal, was introduced in Chapter 2 as a purely syntactical construction. Here we consider quantification as an operation on predicates which produces new predicates. At this stage we introduce only bounded quantification, where the range of the quantifier is finite, determined by an upper bound. This construction is closely related to bounded product and bounded sum discussed above. More general quantifications are introduced later. Unbounded numerical quantification is introduced in Chapter 13, and function quantification is introduced in Chapter 16. Universal Bounded Quantification. Let Q be a (k 1,m)-ary predicate. We introduce a ( k 1,m)-ary predicate P such that

+

+

P ( z l s;a)

(VY < ~ ) Q ( Y 5, ;a) Q(y,

x;a)holds for all y < z

If V is a basic F-term in the variables x;a we introduce a predicate R such that

R(a;a)

=

P(V,x;a) (VY < V)Q(Y,2; a).

We say that P is obtained from Q by universal bounded quantification, and R is obtained from Q by universal bounded quantification with upper bound V. Note that if V is undefined, then R ( z ;a)G f; furthermore, P ( 0 , x ;a) t. Existential Bounded Quantification. Let Q be a (k 1, m)-ary predicate. We introduce a (k 1,m)-ary predicate P such that

+

+

p ( z ,2; a) G

(3y < ~ ) Q ( Y2;, a) There is y < z such that Q(y, z;a ) holds.

If V is a basic F-term in the variables s;a we introduce a predicate R such that

R(z;a)

G

=

P(V,x;a) (31 < V)Q(Y,2; a).

Chapter 4. Primitive Recursive Operations

61

We say that P is obtained from Q by existential bounded quantification, and R is obtained from Q by existential bounded quantification with upper bound V . Note that if V is undefined, then R ( z ;a) f; furthermore, P(0,z;a) f. Theorem 4.6 Let 3 be a class of functionals closed under primitive recursive operations. Then,

(i) 3~and 3 d are closed under universal bounded quantification. (ii) 3 d is closed under existential bounded quantification. PROOF. To prove (i) assume that

P is obtained from Q in the form

P ( z ,2; a)= (Vy < z)Q(y,

2; a).

If $Q is a partial characteristic functional of Q ,then we obtain a partial characteristic functional $ p as follows:

This shows that 3~ is closed under universal bounded quantification. On the other hand, if XQ is a dual characteristic functional of Q we obtain a dual characteristic functional x p as follows:

This shows that 3 d is closed under universal bounded quantification. To prove (ii) we assume that the predicate P is obtained from Q in the form

P ( z ,z;a) (3y

< z)Q(y,

z; a).

If XQ is a dual characteristic functional of Q we obtain a dual characteristic functional x p as follows:

This proves that 3 d is closed under existential bounded quantification.

0

The functionals primitive recursive in a class 3 have been defined in terms of the given functionals in F ,the basic functionals, the numerical function s , and the operations of numerical substitution, distribution, definition by cases, and primitive

L.E. Sanchis

62

recursion. In only one of these elements do the function variables play a truly functional role, namely in the application functionals ap. In the other functionals and operations the function variables refer to arbitrary objects, different from numbers, but without any particular character. To bring out the full functional character of the function variables we need a new operation where actual functions can be substituted for function arguments. To define properly functional substitution we need a notation where functions can be specified by abstraction. We use the well known X-notation, which is defined as follows: if g is a (k + 1,m)-ary functional, then p = (Xy)g(y,z; a) is a unary function such that p(y) N g(y,z; a)for all y. Note that the function /3 depends on the variables 2; a. More generally, if U is a numerical expression depending on a number of variables, then p = (Xy)U is the unary function such that p(y) N U . Note that p depends on the variables in U different from y, and it is not assumed that y occurs in U . For example, using the X-notation we can define < c q , . . . , a,> = (Xz)fc,(z; Q1,. . . ,am). By introducing the X-notation we are able to identify functions that have been already constructed via our formalism and given functionals. If we allow that such functions be substituted for the function arguments of a previous functional, we create a powerful mechanism where the functions are executed under the control of the functional. This construction is exponential and goes beyond the simple numerical substitution which we allow in primitive recursive operations. The rules for this form of functional substitution are given below. Note that we cannot expect that closure under primitive recursive operations implies closure under functional substitution. Functional Substitution. Let f be a ( k , s)-ary functional, k 2 0, s 2 0, and 91,. . . ,gs be (k+ 1,m)-ary functionals, m 2 0. We introduce a (k,m)-ary functional h such that

h ( z ;a)= f ( z ;(XY)Sl(Y,

. . ., (Xy)ga(z; a)).

2; a),

We say that h is obtained by functional substitution in f with functionals 91,. . . ,gs. Let 3 and 3' be classes of functionals. We say that 3 is closed under functional substitution with functionals f r o m 3',if whenever a functional h is obtained by functional substitution in an 3-computable functional f with functionals 91,. . . ,ga that are F'-computable, then h is also F-computable. We say that 3is closed under functional substitution if 3 is closed under functional substitution with functionals from F.

Chapter 4. Primitive Recursive Operations

63

Theorem 4.7 Let 3 and F' be classes of functionals such that 3' EPR(3) and 3 is closed under functional substitution with functionals from F'. Then PR(F) is closed under functional substitution with functionals from F . PROOF.

A functional substitution in a basic functional with functionals from F'

will produce another basic functional, or a functional from F' in case the functional is ap. In either case the functional obtained in this way is in PR(F). It is clear that if a functional is obtained by one of the operations of numerical substitution, distribution, definition by cases, or primitive recursion, then any functional substitution will preserve the operation, hence the result of the substitution is again in PR(F). U

4 . 8 There is an alternative method to deal with bounded sum, bounded product, and bounded minimalization, which involves the use of functional substitution. For example, consider the primitive recursive (1,l)-ary functional bs such that bs(0;cr) N 0 bs(y+ I;&) N bs(y;a) a(y).

EXAMPLE

+

It follows that

f'(r,z;a) f ( z ;a )

N 2:

(Cy < z ) U (Cy < V)U

N N

bs(r;(Xy)U) bs(V; (Xy)V).

In this way we derive bounded sum using functional substitution. A similar procedure can be applied t o bounded product.

Corollary 4.7.1 If F contains only numerical functions, then PR(3) is closed under functional substitution.

This is clear from Theorem 4.7 since in this case 7 is closed under any 0 type of functional substitution. PROOF.

Corollary 4.7.2 The class PR is closed under functional substitution.

PROOF. Immediate from Theorem 4.7 with F = 0.

CI

L.E. Sanchis

64 EXERCISES

4.1 Let F be a class of functionals and U a basic F - t e r m which contains occurrences of a subterm V. Assume U’ is the result of replacing V with a new variable y in U . Prove t h e following relation holds for all values of the variables:

F be a class of total numerical functions satisfying t h e following two conditions: (i) T h e numerical projection and constant functions are in F ,and the successor function is in F. (ii) 7 is closed under numerical substitution and primitive recursion. Prove:

4.2 Let

(a) F is closed under definition by cases.

( b ) If f is a numerical function primitive recursive in T ,then f is in F , 4.3 Assume a’ = . Prove:

(4 4

2 )

(b) a(.)

= “211 N

+

1, [Pd([tIl)

+

a([~12)1011

f c ( < l , x>; a’).

4.4 Let 3 be a class of functionals closed under functional substitution with func-

tionals from BS. Prove t h a t 7 is closed under distribution.

F be a finite class of functionals closed under primitive recursive operations and functional substitution. Let g be a (k,m)-ary functional in F. Prove there

4.5 Let

is a (1,l)-ary functional f in 3 such t h a t g ( z ; a)N f(; (Xy)fc,(y; a)) 4.6 Let 3 be a class containing only quasi-total functionals. Prove that P R p ( F ) is

closed under disjunction, and P R ( F ) C_ PRdg(T). 4.7 Define a (1,l)-ary primitive recursive functional bm such t h a t whenever U is a term in variables y,z; a,and V is a term in variables a:;a,then (py

< V)U

N

b m ( V ; (Xy)U).

P be a class of predicates closed under substitution with primitiverecursive functionals, conjunction, and disjunction. Assume PRd E P . Prove t h a t if P is a given (k,m)-ary predicate the following conditions are equivalent:

4.8 Let

Chapter 4. Primitive Recursive Operations (a)

65

P is d-computable in Pg.

(b) There are (k,m)-ary boolean monotonic predicates PI and P2 in P,such that P is a boolean extension of PI, p is a boolean extension of P2,and the following equivalences hold whenever the Q are total functions: N

P(z;a) N

P(z;a)

=

PI(.;&) Pz(z;a).

Our treatment of primitive recursion operations is stronger than the traditional, as appears, for instance, in Kleene [14]. The main difference is that we allow definition by cases via basic operations. This type of definition, which we called partial definition by cases in Sanchis [27], goes beyond ordinary primitive recursion, although from the latter we can derive a weaker form that we called total definition by cases. The main application for primitive recursive operation is the definition of the encodings of numerical and functional tuples. The numerical encoding is traditional and can be found in the literature. The functional encoding is new and has to be organized in the proper way to deal with partially defined functions. We approach primitive recursion primarily as a construction that defines an operational procedure. In this way we anticipate what is intended to be a general operational approach to recursion. As a consequence, recursion becomes a central part of computability theory rather than a construction derived from more fundamental computation principles.

This Page Intentionally Left Blank

Chapter 5

Basic Recursion In this chapter we introduce a restricted form of recursion, basic recursion, defined using basic terms. We are interested in the closure properties of this operation and also in the computation of functionals that are specified by basic recursion. Essentially, basic recursion is characterized by the fact that functional substitution is not allowed. By allowing functional substitution we obtain a more general extended recursion which will be studied later. In this way we derive in a natural way two levels of computability theory, one determined by basic recursion, and the other by functional recursion that involves functional substitution. We shall show that the theory derived from basic recursion is essentially the traditional construction of recursive functions in first-order computability, relativized to non-total functions, as in Sanchis [27]. The standard Church’s thesis applies here, provided we consider only functionals that are computable relative to a numerical function. The second level derived from functional recursion requires an extended Church’s thesis which is discussed in Chapter 8. It is easy to show that primitive recursion can be reduced to basic recursion. We introduce here another operation, unbounded minimalization, which also can be reduced to basic recursion. We shall prove later that basic recursion can be reduced to primitive recursive operations with unbounded minimalization.

A fundamental property of primitive recursive operations is that the application to quasi-total functionals produces quasi-total functionals, hence the application to total functions produces total functions. Using this property we proved in the 67

68

L.E. Sanchis

preceding chapter that P R contains only quasi-total functionals. This indicates that we must extend the primitive recursive operations, for it is obvious that by simple computations we can obtain functionals that are not quasi-total. EXAMPLE

5.1 Let mu be the (0,l)-ary functional such that

4; a)

the least y such that a ( y ) 21 0

N

and a ( v ) $! 0 whenever v

< y.

It follows that mu(;c)=0 and mu(;$)is undefined, hence mu is not quasi-total. An obvious way to compute mu(;a) is to request from the oracle the values a(O), a ( l ) , a ( 2 ) ,. . . until the first y appears such that a ( y ) N 0.

+

Unbounded Minimalization. Let g be a (k 1,m)-ary functional. We introduce a new (k,m)-ary functional f such that

J ( 2 ; a)

N N

(PY)g(Y,z;a) the least y such that g(y, 2; a)N 0 and g(v, 2; a)74 0 whenever v < y .

Note that f can be obtained from g and mu in Example 5.1 using numerical and functional substitution with a projection functional, for we can set muk(z;a)

N

mu(;&)

f(2;a)

=

mud=;( x Y ) g ( Y ,2; a>).

A class T of functionals is closed under p-recursive operations if F is closed under primitive recursive operations and unbounded minimalization. In this case the functional mu is T-computable, for we can get mu in the form

We denote by M R ( T ) the smallest class of functionals which contains the class T and it is closed under p-recursive operations. The functionals in M R ( 7 ) are said to be p-recursive in 3.In case F is a singleton, say T = {g}, we write MR(g). If T is empty we write M R = MR(0). The functionals in MR(g) are p-recursive in g, and the functionals in M R are p-recursive. The class M R ( F ) is constructed essentially in the same way as P R ( T ) in Chapter 4 , and the main results proved there hold again. Hence, Theorem 4.3 can be extended, and we can prove that M R ( T ) is an extension

Chapter 5. Basic Recursion

69

of F closed under p-recursive operations. Furthermore, if F‘ is an extension of 3 closed under p-recursive operations, then 3’is an extension of MR(3). On the other hand, Theorem 4.4 fails, for MR(F) contains the functional mu which is not quasi- total. Note that Corollary 4.4.1 also fails, for MR(F) contains the totally undefined function UDo( ) N (py)s(y). This means, of course, that UDk,,,, is p-recursive for all k and m. Although we use the expression “p-recursive,” note that the only recursion that is explicitly required in the definition is primitive recursion. We shall show later that unbounded minimalization is, in fact, a kind of recursion. The treatment of the function variables remains essentially the same as in primitive recursive operations, so no functional substitution is allowed. Clearly, this definition is simply a n obvious extension of the first-order definition of recursive operations. The crucial phenomenon, which is explained in this chapter, is that p-recursive operations are sufficient to execute basic recursion. See also Sanchis [27]. We proceed now to introduce a form of recursive specification which is, in fact, very restricted, but still we shall show that it includes primitive recursion and unbounded minimalization as special cases. This form of recursion we call basic recursion, and it is obtained by a natural extension of the notion of basic F-term introduced in Chapter 4. A basic 3-term, where 3 is a given class of functionals, has been defined by four inductive rules BT 1-BT 4. The definition is relative to a given list of variables z;a of arity (k,m). We introduce now a functional symbol p of the same arity (k,rn) and define recursive basic 3-terms in p ; z ;a. Note that p is not a functional, but rather a symbol which can be used to denote a (k,m)-ary functional. Essentially, p is a functional variable as z is a list of number variables, and a is a list of unary function variables. The definition of recursive basic F-terms in p ; z ; a is given by five inductive rules RBT 1 to RBT 5. The first four rules, RBT 1 to RBT 4, are obtained from rules BT 1 to BT 4 by simply replacing every occurrence of “basic F-term” with “recursive basic 3-term,” and every occurrence of “z; a” with “ p ; z ;a.”The last rule, RBT 5, is as follows: RBT 5: If U 1 , . . . , u k are recursive basic 3-terms in p ; z ;a,then p ( U 1 , . . ., u k ; a) is also a recursive basic F-term in p;s; a.

70

L.E. Sanchis

Since rules RBT 1 to RBT 4 are reformulations of rules BT 1 to BT 4, it follows that a basic F-term is also a recursive basic F-term. Note that we do not include distribution in rule RBT 5. Technically, there is no difficulty in adopting a rule similar to RBT 2 allowing for distribution of the function variables. The discussion below in terms of monotonic transformations and minimal fixed point is not affected. Only at the operational level via the stack algorithm do we find serious difficulties due to the problem of keeping track of the variables invoked in a recursive call. The difficulty disappears by writing the rule without distribution. This restricted recursion is still sufficient for the evaluation of p-recursive operations. In dealing with recursive terms we assume in principle that the symbol p has no denotation, and as a consequence it follows that a given recursive term U has no precise meaning. Essentially, this means that the functional fu cannot be defined in the manner explained for basic terms. On the other hand, we may decide to assign to p a definite meaning, i.e., a fixed (k,m)-ary functional, in which case U becomes a basic 3‘-term in z; a,where F‘ = F U { p } . Of course, the definition of fu depends on the particular assignment of the symbol p . We are being a little ambiguous concerning the nature of the recursive symbol p . Initially it was assumed that p was an uninterpreted symbol without denotation, but now we admit the possibility of interpreting the symbol p as a functional of the proper arity. In practice there is no danger of confusion, for the context always determines whether p is an uninterpreted symbol or a constant that denotes a fixed functional. It is obvious that the choice of the symbol p is irrelevant, and we can construct a recursive term similar to U using the symbol p’ instead of p . If we denote by U’ the term with p’ and assume that p and p’ are assigned functional values, not necessarily the same, we get functional fu and ful. When p and p‘ are related in some way it are similarly related. In particular, if p’ is an follows in some cases that fu and extension of p in F k , m it follows that fu, is also an extension of fu in Fb,,. The proof of this relation involves an easy induction on the construction of U ,and it is left to the reader. The above discussion shows that a recursive basic F-term U defines a monotonic transformation T on F k , , where, if p is an arbitrary element of F k , m , then T p = fu. A fixed point of this transformation is a (k,m)S-ary functional p that satisfies the

71

Chapter 5. Basic Recursion following equation p ( z ; a)N

u.

We call this equation the recursive equation induced by U . A solution of this equation is any functional p which satisfies the equation for all values of x ;a. Since the solutions of the recursive equation are exactly the fixed points of the transformation T, it follows by Theorem 1.2 that there is a unique minimal solution which is the minimal fixed point of T. Note that the monotonicity of the transformation does not depend at all on the monotonicity of the functionals occurring in the term U . If we assume that the mappings in U (introduced by rule RBT 2) are not necessarily monotonic (so they are not required to be functionals), the transformation T is still monotonic. In fact, all the operations defined up to this point, including p-recursive operations, are well defined with arbitrary mapping, not necessarily functionals, and the same is true with basic recursion. So a more general theory is possible which we have not considered explicitly for reasons of convenience. The monotonicity condition is, in fact, essential when we introduce functional recursion. Basic Recursion. Let U be a recursive basic F-term in p , z ; a.We introduce a new (k,m)-ary functional p as the minimal solution of the recursive equation p ( z ; a)N

u.

We say that p is obtained by basic 7-recursion. We say that a class 7 of functionals is closed under basic recursion if, whenever a functional p is obtained by basic F-recursion, then p is T-computable. The idea of using a recursive specification, given a formal context of terms, as a rule where a new functional is introduced from given functionals, was used in Sanchis [27] to define the recursive operations. A similar procedure is used later with functional recursion and simultaneous functional recursion. The given functionals in this type of specification are the functionals in the recursive term U , which are identified by the application of the rule RBT 2. Closure under basic recursion means that whenever the given functionals in the specification by basic recursion of a functional p are in the class T , then p is also in the class 7 . First, we shall prove that closure under basic recursion implies closure under p-recursive operations. Later, we shall prove the converse of this relation. Assume the class F is closed under basic recursion, and U is a basic F-term in z;a,hence also a recursive basic F-term in p , z ;a.It follows that the functional p

L.E. Sanchis

72 which is specified explicitly by U satisfies the equation p ( s ; a)N

u

and is the only solution of this equation when it is considered as a recursive equation. It follows that p is F-computable, and that F is closed under basic operations. Theorem 5.1 Let F be a class closed under basic recursion and assume that F contains the numerical functions s and pd. Then F is closed under primitive recursion and unbounded minimalization. Assume the (k in the form PROOF.

+ 1,m)-ary functional f is obtained by primitive recursion f(O,=;a)

2

g(=;a)

f ( Y + 1,s;a) N h ( f ( Y , ~ ; a ) , Y , z ; a ) , where g and h are F-computable functionals. It follows that f is the minimal solution of the recursive equation

hence f is F-computable. Assume now that f is obtained by unbounded minimalization in the form

where g is F-computable. We introduce an auxiliary functional g’ such that

and note that g‘ is the minimal solution of the recursive equation

hence g’ is F-computable. Since f(y, s;a) N g’(0,z; a),it follows that f is Fcomputable. 0 A recursive specification determines completely a new functional p via the construction in the proof of Theorem 1.2. In practice, the recursive equation by itself is not very informative and additional work may be necessary to obtain more information about the functional p . In some cases it may be possible that a functional p

Chapter 5. Basic Recursion

73

is given by an alternative procedure, say, an algorithm or a different recursion, and we want to show that p is actually the functional introduced by the recursion. For this purpose it is sufficient to prove that the given functional p is a solution of the recursive equation and, furthermore, is a minimal solution, so any other solution is an extension of p . This procedure can be formalized in terms of the equation as follows. Assume a recursive equation of the form p ( z ; a ) N _ U is given, and a functional p is also given. In order to prove that the functional p is a minimal solution of the recursive equation it is sufficient to show that the following two conditions, RC 1 and RC 2, are satisfied:

RC 1: For all values of z;a : if U

N

v then p ( z ; a)? v .

RC 2: Let p’ be another (L,m)-ary functional, and let U’ be obtained from U by replacing all occurrences of p with p’. Assume that for all values of z;a : if U’ N v , then p ’ ( s ; a)=v . Then p’ is an extension of p in F k , m . In fact, since the monotonic transformation T is given by T p ( z ;a ) N U , then RC 1 means that T p p , i.e., that p is closed under T . Similarly, RC 2 means that whenever p‘ is closed under T , then p c k , m p’. Since T is monotonic, from RC 1 we get T ( T p ) c k , m T p , so T p is closed under T Iand by RC 2 it follows that p c k , m T p , i.e., p = T p is a fixed point of T . Hence p satisfies all conditions of Theorem 1.2, and p is the minimal solution of the equation p ( s ; a)? U . We apply RC 1 and RC 2 as follows. Assume a recursive equation p ( z ; a ) I IU is given and that a functional p is introduced as the computation of some algorithm. If we prove that p satisfies the two conditions we can conclude that the algorithm computes the minimal solution of the equation.

ck,m

EXAMPLE

5 . 2 Consider the recursive equation

We want to show that the minimal solution of this equation is given by

so we prove that conditions RC 1 and RC 2 are satisfied. To prove the first,

assume U

N

i where p in U is interpreted as P O . It follows that g ( z ; a ) N v for

L.E. Sanchis

74

some v 2 0. If v = 0 we have po(z;a) 2: [ a ( z )-+ 0,1] N i. If v # 0 we have p o ( z ; a ) N p o ( s ( z ) ; a )N i. Hence in either case we have p o ( z ; a )N i and RC 1 holds. To prove RC 2 we assume a functional p’ such that if U’ is obtained from U by replacing p with p‘ and U’ 1: i holds, then p ’ ( z ; a ) N i for all values of z and a. We must show that p’ is an extension of PO, so we assume p o ( z ; a ) N i. Hence (py)g(z y; a) N v, and it is clear that po(x v; a) N PO(+;a) N i . As a consequence (noting the assumption above for U’ and p’ holds for all values of z and a),it follows that p ‘ ( z v; a)N i. Using again the relation between U’ and p‘ we can prove that whenever z 5 2’ < z v, then ~ ’ ( 2 ‘ ;a)N p ’ ( s ( z ) ;a) N i . Hence p ’ ( z ; a) N i.

+

+

+

+

The algorithm we present to compute the minimal solution of a recursive equation p ( z ; a ) 2: U is an extension of the usual stack algorithm which takes into account the presence of function variables, and also of functionals from the class F. As usual, we assume an external oracle which provides values of the input functions and of the given functionals. For example, if a is an input function and the execution of the algorithm requires the value a ( z ) where z is a numerical value generated in the computation, then the oracle provides the value of a(+)if defined. If a(.) is not defined the computation is blocked and no output is produced. Similarly, the execution of the algorithm may require a value g(z; a),where g is in the class F , which is provided by the oracle if it is defined. Again, if g ( z ; a ) is undefined the computation is blocked. To refer to this peculiar behavior of the algorithm we shall say that the algorithm is deterministic. Note that the oracle is in position to answer any request without further computation. We may describe this situation by saying that, once a request is formulated to the oracle, the answer from the oracle, if defined, is independent of the algorithm itself. On the other hand, it is possible to imagine a situation where the function a is given to the oracle via a sub-routine described in the main algorithm, involving inputs or parameters, depending on the state of the computation at the time of the request. This type of situation is not contemplated here, and we assume definitely that a is completely available to the oracle in order to answer a request of the form a ( z ) or of the form g(z;a). The hypothetical situation described in the preceding paragraph corresponds to a type of algorithm allowing for functional substitution. Later we shall discuss algorithms of this type, particularly with functional substitution.

Chapter 5. Basic Recursion

75

The stack algorithm that computes the functional p is derived, of course, from the term U. The algorithm itself consists of a number of rules that operate on a stack structure which will be explained later. In order to formulate the rules we must execute several transformations until the term U is put in Polish normal form. We assume that the reader is familiar with this type of formalization and discuss only those aspects which are peculiar to the basic terms. The first transformation is to rewrite any subterm of the form g(U1,. . .,U,;a;,, . . . , a ; v )given , by rule RBT 2 in . . , U,),and from now on we consider 19,a;,,. . . , ai,l the form 19,a i l , .. . ,cqF1(U1,. as a symbol denoting an s-ary function symbol. The above transformation is not necessary with terms involving the recursive symbol p, for in this case the function variables always appear in the same order. In this case we simply omit the function variables. Here we are taking advantage of the restricted form of rule RBT 5, discussed above. Essentially, we consider that the result of a distribution in a functional must be treated as a different functional. EXAMPLE 5 . 3 Assume U is the following term in p ; el,CZ;a1,az:

g(P(%tz;

a19 az),a z ( z z ) ;a1,ad.

After the transformation it takes the following form: 19,a1,alI(P(Q,

4, ..2(4).

The Polish normal form of this term is: ~zc!P~z~219, a1,all.

Note that 19,al,all is considered a symbol exactly as are

12

and p .

In the general case the term U may contain definition by cases, and the construction of the Polish normal form must take this into account. We assume now that every definition by cases in U has been marked with a label, say a number. It does not matter which label is chosen for a particular definition by cases as long as different definitions have different labels. We may indicate the label by writing it above the initial bracket. Hence if [U,+ Uz,U3]is a subterm of U ,the label i

indication will appear in the form [ U1 + U2,U3].Assume Ui,Ui,and U i are the Polish normal forms of U1, Uz, U3,respectively. Then the Polish normal form of i

[ U1 -+ Uz,U3] is Uili,l ~ f ~ ~ ~ i ,31.2 ~Here U ~ ~li,i11,, Ii,21, and li,31 are considered new symbols added to the expression. We call these symbols markers.

76

L.E. Sanchis

EXAMPLE 5.4

Consider a non-recursive term U in z1,+2, x3; a1,( Y Z .

where each definition by cases has been assigned a label. The Polish normal form of this term is as follows (written as a column): 21 1 12

2

21

3

19, a1, all

4

P,lI

5

21

6

ff2

7

127 11

8 9

22

12>21

10

23

11 12

12,31 22

13 14

ff2

15

11, 21

13~11

16

21

17

13~21

18

23

19

13,31 1~31

20 21

There are 21 symbols in the Polish normal form, each with a numerical representative, or position, written in the column to the right. An algorithm to compute the expression depends on the values of the inputs 11,22,23; a1, a2 and will start with the first symbol in the expression (position l), evaluating some of them until the last symbol (position 21) has been evaluated. Not all the symbols in the expression must be evaluated-only those required by the definition by cases. To compute the value of U we use bracketed pairs [u,v] where u denotes a position, i.e., a value from 1 to 21, and v denotes a partial value for the symbol in

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77

that position. These pairs are organized as a sequence [ul, vl][uzl .2] . . . [.I, 4,t 2 1 which we call a stack. The last element of the stack is said to be at the t o p of the stack. The length of the stack changes during the computation. As an example, assume 2 1 = 1, 2 2 = 2 , and 2 3 = 3. Furthermore, az(1) 2: 5 and g ( l , 2 , l ; a l , a l ) 1: 0. Clearly, the value of U is then equal to 3. The stack computation proceeds as follows: 22

=1 =2

21

=1

21

=

g(1, 2 , 1 ;& 1 , Q 1 ) 0 Marker I1,lI means case 1 = 0, xl = 1 az(1) = 5 Marker 12,lI means case 2 # 0, 1 3 = 3 Marker 12,31 means ends label 2 Marker I1,21 means ends label 1 This computation can be followed using the explanation given next to every step. Note that whenever the evaluation of a definition by cases (with label i) is completed (which is detected because the next symbol is li, 21 or li, 31), the value is stored in li131. Now we consider the general case where we are given a recursive equation p ( z ; a )N U and U is a recursive basic F-term containing occurrences of the symbol p . The process described above fails precisely when we are required to evaluate occurrences of p . At this stage we start a secondary evaluation of p , but with dif-

ferent numerical inputs (the function inputs are always the same). Since this may happen several times during the evaluation, we must keep track of which inputs are involved at a particular place, and we must also keep track of the place to which the value of the secondary evaluation must be returned. To have this information available we shall use bracketed k 2-tuples (where k is the arity of z)of the form [u, v1,. . . , v k , w ] where u denotes a location, v1, . . ., vb are input values, and w is the value of the symbol in position u . We call the process of initiating a secondary evaluation a recursive call, and we need some formalism to indicate when such calls are made. For this purpose we use incomplete tuples of the form [u, v l , . . , , vt,*] where u is the location of some occurrences of the symbol p , and * is just a symbol indicating that the value of p is being evaluated.

+

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78

To describe the details of the algorithm we shall consider an example which has been discussed already in Example 5.2. In this case the recursive equation has the form

and the (1,l)-ary functional g and the unary function s are given. Technically, their values are given by an oracle, although s has been identified in advance. First, we write the Polish normal form and assign positions to the symbols. X 1 IS, 01 2

ILlI X

3

a

4

12911 0

5

12,21 1 12,31

6 7

IL21 X

8

S

9 10 11

P 1~31

Note that some markers are not assigned positions because they are not really used in the computation, although they are useful for writing the rules of the algorithm. In this case the stack algorithm involves bracketed 3-tuples of the form [u,v , v’] where u is a position, v is input, and v’ is a value. Whenever a recursive call takes place we use the incomplete bracketed triple [u, v, *]. The stack itself is a sequence of these triples. Consider, for example, the evaluation of p(1; a) where we assume g(1; a) 21 1, g(2; a) N 0, and 4 2 ) N 0. The initial stack is then [ O , l , *][1,1,1] which means we are evaluating p ( l ; a ) , and the symbol in position 1 (i.e., x) has value 1. The complete computation takes the following form:

Chapter 5. Basic Recursaon

79 Initial stack g(1;a) N 1 Case 1 # 0, go after marker I1,21 s(1) N 2 Recursive call, input = 2, go to position 1 g(2; a ) N 0 Case 1 = 0, go after marker I1,l I 4 2 ) 2! 0 Case 2 = 0, go after marker 12,lI End of case 2 End of case 1 Return recursive call End of case 1 Return recursive call, end computation.

We proceed now to describe the complete algorithm which computes the value of p ( t ; a ) whenever defined. This algorithm operates on triples which can be of two forms: complete like [u, v , w] or incomplete like [u, u, *]. These triples are organized in a stack structure, which is simply a sequence of triples where the last one, the top triple, determines the action to be executed. Hence the stack at any given time in the computation has the form TI, . . . ,Tt, and Tt is the top triple. The algorithm is arranged in such a way that Tt is always a complete triple which we denote as [ut, v t , wt]. If t > 1, then we will denote the preceding triple [ u t - ~ v, t - 1 , wt-l] if it is complete and [ u t - l , u t - l , *] if it is incomplete. In order to compute p ( z ; a ) the initial stack is [0, z,*][l,z,z]. If at some point in the computation ut = 0 this means halting and the output is wt. The algorithm consists of 11 rules, one for each possible value of u l , from 1 to 11.

=

Rule 1: If ut = 1, replace top triple by [2,vt,w'] where g(wt;u) w'. If g ( w t ; a ) is undefined the computation is blocked and no output is produced. Rule 2: If ut = 2 and wt = 0 replace top triple with [3,vt,vr]. If wt top triple with [8, v t , vt]. Rule 3: If ut = 3 replace top triple with [4,vt,w'] where a ( w t ) undefined computation is blocked.

= w'.

#0

replace

If a ( w ) is

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Rule 4: If ut = 4 and w t = 0 replace top triple with [ 5 , v t ,01. If wt # 0 replace top triple with [ 6 , v t , 11. Rule 5: If

ut

= 5 replace top triple with [7, vt, wt].

Rule 6: If ut = 6 replace top triple with [7, vt, wt]. Rule 7: If

tit

= 7 replace top triple with [ll,v t , wt].

Rule 8: If ut = 8 replace top triple with [Q, vt, wt

+ 11.

Rule 9: If ut = 9 replace top triple with the following triples: [lo, v t , *][1, wt,.it]. Rule 10: If

ut

= 10 replace top triple with [ l l ,v t , w].

Rule 11: If ut = 11 and the triple below the top triple is both triples with [ut-l, v t - l , wt].

[ut-l, v t - 1 ,

*] replace

Note that the algorithm described above is simply a set of formal rules to manipulate the stack. The notions of recursive equation, solution, etc., play no role in the execution of the algorithm. Note also that the algorithm works even if the given functionals are not monotonic. This is consistent with our discussion above where we made clear that the definition of basic recursion does not require monotonicity. We shall see later that in dealing with functional recursion the algorithms can be executed only if monotonicity is assumed. We use the algorithm above to prove that the functional p is p-recursive in g. We knew already that this is the case, since in the analysis of Example 5.2 we identified the minimal solution using unbounded minimalization. The argument here depends only on the algorithm that computes p and applies, in fact, to any recursive equation. The basic idea is to encode the whole algorithm until it reduces to some operations applied to numbers. The crucial operations are primitive recursion and unbounded minimalization, so the function is obtained by p-recursive operations. We start by encoding the triples, using the encoding explained in the preceding chapter. A complete triple [ u , v , w ]is encoded in the number < u , v , w > . An incomplete triple [u, v , *] is encoded in the number < u , v>. Finally, a stack [ul, v1, wl]. . . [ u t , vt,wt] is encoded in < < u l , v l , w l > , . . . , < u t , vt, wt>>. In order to operate on these encodings we need several primitive recursive functions. Some of them have already been introduced. For example, if t =

Chapter 5. Basic Recursion

81

< X I , .. . , x k > , then [x]i = x i , i = 1,.. .,k. We introduce a few more and leave to the reader to write the primitive recursive specifications. The binary primitive recursive function c c is such that whenever cc(x, y) = x' and x = < X I , . . . , xk>, then x' = < X I , . . . , xk,y>. The unary functions rd and dr are such that whenever x = < X I , . . . ,X k , xk+1>, then r d ( x ) = < X I , . . . , X k > and d r ( x ) = X k + l . We proceed now to define a ( 1 , l ) ary functional trp such that whenever y encodes a stack S and trp(y; a) N y', then y' encodes the stack S' obtained by applying to S the corresponding rule among those which define the algorithm. It is easy to see that the corresponding rule is determined by the value [dr(y)]l. This means we must use a specification by general definition by cases, which involves exactly thirteen cases. We write only a few of the cases, and leave to the reader to complete the specification.

ifeq([dr(y)l1,11) 34 0.

-0

N

It is clear that the functional tr, is primitive recursive in g. Now we introduce a new (2,l)-ary functional ev, which describes the computation of p ( x ; a ) and generates one by one the stacks derived by the computation rules. evp(O,x;a) evp(y

+ 1,x;a)

N

.

N

trp(evp(y,x; a);a).

Now we must determine at which step the computation halts, i.e., which is the value y such that evp(y, x ; a)is a halting stack. This determination can be obtained

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82

by using unbounded minimalization. Hence we set h such that:

It follows that we can express the functional p in the form

and this shows that p is p-recursive in g. Although we have proved that the functional computed by the stack algorithm is p-recursive in g, we have not yet proved that such functional is the minimal solution of the recursive equation. Clearly, it is sufficient to prove that the functional p computed by the algorithm satisfies conditions RC 1 and RC 2 above. To prove condition RC 1, assume U 2~ v where p denotes now the functional computed by the algorithm. This means we can use the non-recursive stack algorithm to evaluate U and the output is v . If we change from the non-recursive to a recursive evaluation, by computing values of p rather than assuming they are given, we shall get exactly the same output. This means that p ( z ;a) N v . To prove condition RC 2, assume p’ is another functional such that whenever U’ N v , then p‘(e;a)N v holds for all values of c and a,and U’ is the result of replacing p with p‘ in U . To show that p’ is an extension of p we prove by induction on the number of recursive calls that whenever p ( z ; a )N v , then p ’ ( c ; a )N v. If there is no recursive call the evaluation of U is independent of p , hence U’ N v and p ’ ( c ; a)N v . If the evaluation of p ( c ;a) involves recursive calls we replace the calls by values of p’, which by the induction hypothesis are defined and consistent with the values of p . By doing this we get a non-recursive evaluation of U’, hence U’2~ v and p ’ ( e ;a) N v .

Theorem 5.2 A Let T be a class of funclionals. The following conditions are equivalent:

(i) 3 is closed under p-recursive operations.

(ii) T is closed under basic recursion and contains the numerical functions s and Pd. If (i) holds and p ( z ; a)N U is a recursive equation where U is a recursive basic 3-term we can apply the method described in the example above and PROOF.

Chapter 5. Basic Recursion

83

show that the minimal solution of the equation is ,+recursive in F. Conversely, if (ii) holds, then it follows from Theorem 5.1 that F is closed under F-recursive 0 operations. Corollary 5.2.1 Let F be an arbitrary class of functionals. Then MR(F) is the unique class that satisfies the following two conditions:

( i ) MR(F) is an extension of 7 that is closed under basic recursion and contains the numerical functions s and p d ,

(ii) If F' is an extension of 7 that i s closed under basic recursion and contains the numerical functions s and p d , then F' is an extension of MR(7). 5.2 it follows that MR(F) satisfies (i). If F' satisfies the conditions of (ii) it follows again from Theorem 5.2 that F' is closed under precursive operations, hence 3' is an extension of MR(F). The uniqueness of MR(F) is trivial. 0 PROOF. From Theorem

EXERCISES

5.1 An (r,s)-ary numerical mapping is a mapping from N'ig to N , hence it is an element of MAP(N'ig,N) not necessarily monotonic. Assume F is a class of numerical mappings and define recursive basic F-terms in p;z;a in the usual way. If U is a recursive basic 3-term in p;z;a,where p is a of arity (k,m ) , then U induces a mapping T from MAP(Nkim,N) to MAP(Nkpm,N), where T p = pl and p l ( z ; a)1:U . Prove that T is monotonic, hence T is a monotonic transformation on MAP(N ' v m , N). 5.2 Let T be a class of functionals. Define formally the class MR(F) and extend to this definition the result of Theorem 4.3. 5.3 Write the complete definition by cases of the functional trp introduced in the text.

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84

5.4 Prove that during the execution of the stack algorithm described in the text the top of the stack is always a complete tuple. 5.5 Using as a model the programs in Example 4.2, define a programming language

AL[F] (for assignment language) where all functionals which are p-recursive in F are computable. 5.6 Using the programming language A L [ q of Exercise 5.5, prove that if

+

f is

F ,there is a (k 1,m)-ary functional h , which is primitive recursive in 3,and a primitive recursive unary function p such that f(z;a)N p ( h ( g ( z ;a ) z ;a))where g ( z ; a)N (py)h(y 1,z; a). a (k,m)-ary functional which is p-recursive in

+

Notes Basic recursion is a natural extension of what we called recursion in Sanchis [27]. It is important because it is sufficient to support a weak version of Church’s thesis, which is discussed in the next chapter. We find again that this type of recursion can be eliminated in terms of explicit algorithms, but this is only a local phenomenon which cannot be extended. In the general case where we have functional recursion we must take recursion as a primitive. The reduction of basic recursion to explicit computation is done by a stack algorithm, exactly as w a s done for numerical functions in Sanchis [27]. The crucial point here is that function arguments behave as fixed parameters and do not interfere with the stack evaluation. We have already mentioned that up to this point we do not need the restriction to monotonic mappings (= functionals). In fact, basic recursion is well defined even when non-monotonic mappings are involved. This is essentially due to the restricted role of the function arguments.

Chapter 6

Church’s Thesis In this chapter we extend the well known Church’s thesis to functionals. Essentially, we give a mathematical characterization of the functionals which are computable relative to a given class 3 of functionals. At this stage we want to preserve the basic form of Church’s thesis and impose a number of important restrictions. This means that the thesis as given here covers only a proper subset of what can be considered legitimate functional computations. A more inclusive thesis is proposed later. The restrictions are intended to exclude algorithms containing recursive calls. We also require that functionals from the class 3 must be called in a particular simple way, which essentially excludes functional substitution. By excluding algorithms containing recursive calls we impose a restriction where all algorithms must be explicit. We cannot give a formal definition of this notion since, in fact, we cannot define what an algorithm is. Still, we think the idea is clear enough to support a meaningful discussion. First-order computability has been organized on the basis of explicit algorithms and this was right, for recursion can be eliminated at that level. We shall show that the same situation is valid in the theory of functionals, but only up to some point. The exclusion of functional substitution is in the same vein, for the point where recursion cannot be eliminated and must be taken as a primitive is determined by the possibility of meaningful functional substitutions. So, only by combining both constructions can we introduce procedures that are beyond the standard Church’s thesis. When the thesis given here is restricted to numerical functions we obtain the original Church’s thesis.

85

86

L.E. Sanchis

We start with a discussion, necessarily vague and informal, in which we try to characterize the algorithms we want to consider at this stage. We assume that an algorithm operates on a well defined formal structure (vectors, matrices, trees, etc.) and that such structures may contain numerical values and symbols. The algorithm determines a computation which depends on the input, and a computation is a sequence, finite or infinite, of configurations. A configuration is simply the formal structure with a number of numerical values and symbols. Essentially, the algorithm determines how a configuration occurring in the computation changes to the next configuration. More precisely, the algorithm consists of a finite number of computational rules which prescribe in a deterministic way which is the next configuration after any given configuration. Furthermore, the algorithm determines in which conditions the computation halts, and which is the output of the computation. We shall introduce later several strong assumptions concerning the nature of the computation rules. At this stage we say only that each rule in the hlgorithm must be self-contained and independent of the whole algorithm of which it is a part. Essentially, this means that we require the algorithm to be erplicit and that it exclude recursive instructions where the total algorithm itself is called upon in the execution of a particular instruction. This does not mean we object to recursive instructions as legitimate computational operations. They must be excluded in the formulation of the traditional Church’s thesis, but should be included in a more comprehensive assertion. On the other hand, it is important to note that some forms of recursion can be reduced to explicit algorithms, and fall under the scope of Church’s thesis. Since an algorithm computes a particular (k,m)-ary functional f, the rules to compute f(z;a )will depend on the input (2; a ) .We assume that any rule involving a function input a will be of the form: I= a ( y ) where y is a value generated in the computation, i.e., the value y is a part of the configuration at the time the instruction is executed. To execute such an instruction we assume that an external agent or oracle will provide the value a ( y ) if it is defined and will remain silent otherwise, a situation that we describe by saying that the computation has been blocked and no output produced. Furthermore, the algorithm that computes the functional f is relative to the given class 7 , and some instructions will refer to functionals in 7 . We assume here that a call to a functional g in 7 ,say of arity (s,r ) , will be of the form z =

Chapter 6. Church’s Thesis

87

g(y1,. . . , ys; a i l , .. . , a i r ) where the input of the computation is a =

a 1 , . . .,a,,,,

and a i l , .. . ,ai, are variables in a.Again, the values y l , . . . , ys are generated in the computation, and the value of g is provided by an external oracle, if it is defined; otherwise the computation is blocked. An algorithm that satisfies the above conditions is called an explicit algorithm. Although this is not a formal definition, it is clear that the conditions are satisfied by many algorithms found in applications. The stack algorithm described in the preceding chapter is an explicit algorithm. Any machine language program for a standard computer is an explicit algorithm, but a program written in a higher order language is not necessarily an explicit algorithm. Church’s thesis for numerical functions implies that any computable numerical function is computed by an explicit algorithm (for example, in a Turing machine). The class of all functionals that are computed by explicit algorithms relative to a class 3 of functionals is denoted by E A ( 3 ) . If 7 = {g} we write EA(g), and if F = 0 we write EA. EXAMPLE 6.1 The algorithm to compute f u ( c 1 , t 2 ; alra2)in Example 4.2 provides a simple example of an explicit algorithm. Here the formal structure is a vector (oo,v1, 212, 213) with four numerical components. The value vo denotes the instruction to be executed, v1 is the contents of X I , v2 the contents of X 2 , and 213 the contents of X 3 . Typical rules in this algorithm are as follows: From configuration (1,211,212,213), move to configuration (2, e l , v 2 , 213); from configuration (4, v1, v 2 , O ) , move to ( 5 , v l , v 2 , 0 ) ; from configuration ( 4 , v l , v 2 , v 3 ) , where v3 # 0, move to (7,211, v2, 213); from configuration (5,211,212, Q), move to configuration (6,211, ui, 213) where a l ( v 1 ) tr vb (here the value v $ is provided by the oracle); from configuration (8,211, 212, 213), move to configuration (0, v;,212,713) where g ( v l , 212; a1) N v; (value v; provided by the oracle); etc. A configuration of the form ( 0 , v l , 712,213)is halting with output v1. The initial configuration is (1,0, 0,O).

6.2 Consider the following specification by basic primitive recursion where f is a (2,l)-ary functional, and h is a given (3,l)-ary functional: EXAMPLE

f(0,z;a)

N

f ( Y + 1, I ;a )

4.1

2

h(f(Y, c;a ) ,Y, 2;a ) .

The following program computes f ( y , c ; a ) assuming that locations X I and X2 contain the values y and c, respectively:

L.E. Sanchis

88

1: 2: 3: 4: 5: 6: 7:

X3+a(Xz) If X1 = 0 goto 7 else goto 3

x, + h(X3,x4,xz;a) xltX1-l XqtX4+1 Goto 2 Halt

This program is actually an explicit algorithm operating on vectors of the form ( V O , ~ 1 ~ ~ 3~, 2 21 4 ),.The rules of the algorithm can be derived from the instructions in the program, as in Example 6.1. To compute f(y, 2;a) the initial configuration , and the is (1,y, z,O, 0). The halting configurations are of the form (7,211, ~ 2 , 2 1 3 vq), output is 213. We conclude that f is in the class EA({h}). Using the preceding notation we can formulate Church’s thesis in the following compact form: If F is an arbitrary class of functionals, then MR(F) = E A ( 7 ) . We want to discuss the evidence for this assertion. We shall not give a full discussion of the inclusion MR(F) E A ( T ) . Still, there are some subtle points and we give some indications about the general approach. Essentially, what is required here is to define a formal system, programming language, machine, etc., in which the notion of computable functional relative to the class 7 can be formally defined. Once this is done, it is sufficient to show that the class of all formally computable functionals contains F and is closed under p-recursive operations. There are several choices for a convenient formalization as proposed above. The general system we introduce in Chapters 9 and 10 can be used, but it is unnecessarily complicated for the present purpose. The best approach is to define a formal language along the lines of Example 6.2, which is very convenient for handling closure under primitive recursion and unbounded minimalization. The converse relation, i.e., E A ( F ) E MR(T) is the heart of Church’s thesis and cannot be proved in any satisfactory way since the definition of explicit algorithms is essentially vague and informal. Still, there is a kind of canonical procedure which can be used to show that if an explicit algorithm A is given that computes a (k,m)-ary functional f from class T , then the procedure describes how f can be obtained by p-recursive operations from

Chapter 6. Church’s Thesis

89

+

class F.The procedure shows that there is a (k 1,n)-ary function e v A primitive recursive in F ,and a unary primitive recursive function out such that

+

where g(z;a)N (py)evA(y 1,z; a). The existence of a canonical procedure to reduce computations to p-recursive operations was discussed in Sanchis [27]. It is of some importance because it makes explicit some conditions that are only implicit in Church’s thesis. First, the algorithm under consideration must be explicit, for the procedure does not work with recursive algorithms. Second, the transformation rules in the algorithms must be primitive recursive, or at least must belong to a class that can be formalized in terms of p-recursive operations. The procedure is quite simple. First, we assume that the configurations in the computation induced by the algorithm can be encoded in numbers so they can be identified with numbers. Such encoding can be done in a number of ways, and we require only that the functions involved be primitive recursive. We have already seen that a vector (21,.. . , zk) can be encoded in the number < r l , . . . , t k > . Matrices can be encoded as sequences of vectors and, in general, there is no doubt that the encoding can be performed on any well defined formal structure. Once the configurations have been encoded it follows that each of the rules in the algorithm becomes a numerical function. We assume now that each of these functions are primitive recursive. By combining all the functions and using definition by cases we obtain a (k 1,m)-ary primitive recursive functional t r A such that if during the computation of f(z; a)a configuration with encoding y changes to a configuration with encoding y‘, then t r A ( y , Z ; a)N y’. We assume y’ = 0 when y is a halting configuration. We conclude that the functional t r A is primitive recursive in F.

+

The specification of the (k primitive recursion: e v A ( 0 , z ; a) eva(y 1 , z;a)

+

+ l,rn)-ary function e v A

is straightforward using

N

the encoding of the initial configuration

N

t r A ( e v A ( Y , z;a),?a).

It appears completely natural to assume a unary primitive recursive function out such that out(y) = output of the computation when y is a halting configuration.

L.E. Sanchis

90

+

Finally, the (k,m)-ary function g such that g ( z ; a) N (py)evA(y 1,z; a) is precursive in F. Now we can write the functional f computed by the algorithm A in the form

f(z;a) out(evA(g(z;

z;a)),

and it follows that f is p-recursive in F. The crucial application of primitive recursion takes place in the definition of the functional e v A which describes the computation as a deterministic process where each step is determined by the preceding state. On the other hand, unbounded minimalization has only one critical role, namely, to determine whether the computation halts, and a t which step. If the computation does not halt, the operation is undefined. Both operations are crucial and necessary. Church’s thesis means essentially that the computations induced by a given explicit algorithm can be described in terms of primitive recursion and unbounded minimalization. As an example, we discuss the algorithm of Example 6.2 where a (2,l)-ary functional f is computed from a given functional h, which we assume is in a class F.Of course, in this example we know already that f is obtained by basic primitive recursion in F ,so the construction of Church’s thesis is not necessary. Our purpose is to show that it is enough to have the algorithm to show that f is p-recursive in F. w e consider first the specification of the functional t r A . The algorithm in Example 6 . 2 operates on vectors of the form (vo,v 1 , v z , v 3 ,v4) which we encode in the number < vo, v 1 , ~2,713,v4 >. The specification of t r A is by cases as follows:

91

Chapter 6. Church’s Thesis

To write the specification of e v A we note that the initial configuration to compute f ( z l , 2 2 ; a ) is (1,21,22,0,0). Hence

evA(0,21,22;a) eVA(y

+ 1,21,22;a)

N

trA(evA(Y, 2 1 , 2 2 ; a),21,22; a ) ,

Finally, the output functions is o u t ( y ) N [y]4. It follows that

where g(z1,xz;a) ( w ) e v A ( y + l , z l , z z ; a ) . Note that the value of trA(Y, 21,22;a ) is actually independent of 21 and 22, which only appear in the specification of evA(O,zl,zz;a). On the other hand, t r A ( Y , 2 1 , 2 2 ; a) depends on a. In general, the value of trA(Y,Z; a)depends on all the input values. The preceding analysis shows that whenever a functional f is computed by an explicit algorithm a normal form can be obtained where the functional f is expressed in terms of primitive recursion and unbounded minirnalization. I t is easy to check that the analysis fails if the algorithm contains recursive calls or some form of

92

L.E. Sanchis

functional substitution. On the other hand, it is possible to argue that in a number of situations recursive calls can be eliminated. This is, in fact, the case, as we shall show presently, and Church’s thesis applies to functionals computed by some forms of recursive algorithm. We describe below a general procedure where a recursive algorithm can be transformed into an equivalent explicit algorithm. An application of the procedure is given in Example 6.3. This is a significant result, for we assume that recursion is a computable structure and must be covered by Church’s thesis. The procedure applies to algorithms that are explicit in the sense explained above, and that may contain recursive calls (as in instruction 3 of the algorithm in Example 6.3). Applications of functional substitution are not allowed. Consider a (k,m)-ary functional f which is computed by a n algorithm A which contains recursive calls and otherwise satisfies the conditions of an explicit algorithm. We assume that such recursive calls are always of the form z = f(y1, . . . , yk; a),where y1,. . . , gk are numerical values generated in the computation, and a are the input functions of the computation. We denote with the letter c the configurations in the algorithm A . We want to change A to an explicit algorithm A’ which also computes the functional f. The configurations in A’ are of the form (c1, . . . , c t ) , where c l , . . . , ct are configurations in A . The length t of this structure will change during the computation induced by A‘. The initial configuration is (c), where c is the initial configuration in A . We apply the rules of A until a recursive call appears, say of the form 2 = f(y1,. . . , yk; a ) , at configuration(c’). Here we extend (c’) to (c’, CI), where c1 is the initial configuration in A , to compute f(y1,. . . , yk;a). We continue now with the rules of A applied to c1. If a new recursive call takes place, say a t configuration ( 8 , ci), we extend the configuration to (c’, c i , cp). Assume the computation started with c2 halts. In this case, we return the output value to c; and continue the computation from (c’, c;). If the computation with c’, ever halts, we return the value to c’ and continue the computation of (c’). E X A M P L E 6 . 3 Consider the following recursive algorithm

ary functional f :

A which computes a (3,2)-

93

Chapter 6. Church’s Thesis

1 : If X1 = 0 goto 6 else goto 2 2 : x1 + x1- 1 3 : x1 f(X1, X3, xz;Q 1 , .z) 4 : x1 XI 1 Halt 5:

-

+

+

6 : xz Ql(X2) 7 : x3 aZ(x3) 8 : XI + x1 +xz +x3 9 : Halt +

+

We assume that in order to compute f ( z l , q ,2 3 ;01, 02) the initial configuration is given by 11 in XI,by 22 in X z , and by 23 in X3. The initial instruction is the first in the list. The computation halts if and only if instructions 5 or 9 are reached. The output is in variable X I . More formally, the algorithm deals with quadruples of the form (vo,v1, vz,v3) where vo is a number between 1 and 9, and vl, v2,v3 are the contents of variables X I , Xz, and X3, respectively. The initial configuration is (1, 21,22,23). A halting configuration is of the form (vo,v1, ~ 2 ~ 0 with 3 ) vo = 5 or 9. The output is v1. To eliminate the recursive call we must write another algorithm A’ where the configurations are of the form (Vl, . . . , &) where V1,. . . & are configurations in A . There is no need here to write explicitly the computation rules for A’. It is sufficient to give as an example the computation of f ( 2 , 1 , 2 ; a],cr~), where we assume crl(1) N 2, and 4 2 ) N 3.

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lation to an explicit algorithm is essential, for only in this case can we encode the algorithm in a primitive recursive functional tra. On the other hand, the recursive call discussed in Example 6.3 is of a very special form, for we exclude recursion via functional substitution. Still, in dealing with numerical functions rather than with functionals, functional substitution is not available, and the recursion in Example 5.3 is actually universal. For this reason Church's thesis is a universal statement when applied to functions, and it is only relative when applied to functionals. Theorem 6.1 Let F and F' be classes of functionals such that 3' C_ MR(F) and 3 is closed under functional substitution with functionals f r o m 3'. Then MR(3) is closed under functional substitution with functionals from 3'.

The same argument as in Theorem 4.8. Corollary 6.1.2. MR is closed under functional substitution.

PROOF.

PROOF. Immediate from Theorem 6.2 with

F = 0.

0

0

95

Chapter 6. Church’s Thesis EXERCISES

6.1 Let F be a class that contains only quasi-total functionals, and f a (k,m)-ary functional which is computed by an explicit algorithm relative to 3. Refine the analysis of Church’s thesis and prove there is a unary primitive recursive function g and a ( 6 + 1,m)-ary functional h primitive recursive in F ,such that the relation f(z;a )

2

g((lly)h(y, z ; a ) )

holds whenever a is a list of total functions. 6.2 Use the programming language outlined in Exercise 5.5 to show that the rela-

tion MR(F)

E A ( F ) holds.

6.3 Write the complete set of rules of the explicit algorithm that computes the

functional in Example 6.3, and use the analysis of Church’s thesis to show that the functional is p-recursive.

Notes We approach Church’s thesis here as in [27], moving from explicit algorithms to p-recursive operations. The underlying hypothesis, which is seldom discussed, is that recursive algorithms can be reduced to explicit algorithms. We think this assumption is valid only when recursion is taken in the sense of basic recursion, where no substitution is possible in function arguments. The general form of the reduction is discussed in some detail in Example 6.3, where a recursive algorithm is given and transformed into an explicit algorithm. When we move beyond basic recursion to functional recursion this reduction is not possible anymore, and we must take recursion as a primitive. This situation is reflected in the extended Church’s thesis. Up to this point we have been dealing essentially with first-order computations. If we consider only p-recursive numerical functions, even relativized to arbitrary numerical functions, we get the theory of [27]. If we relativize to total functions we get the classical theory of [14].

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Chapter 7

f i n c t ional Recursion As we enter into one of the crucial constructions in the theory of computable functionals, we again note that the preceding work, up to p-recursive operations, is actually independent of the assumption that functionals are monotonic mappings. In fact, all the operations involved-substitution, distribution, primitive recursion, and unbounded minimalization-are well defined when applied to mappings not necessarily monotonic. The same is true for basic recursion, for recursive basic terms are monotonic even if the given functionals are not monotonic. Moreover, the notion of explicit algorithm does not involve monotonicity. We must conclude that Church’s thesis, as defined in the preceding chapter, applies in general to classes of mappings which there is no need to assume are monotonic. The type of recursion we introduce now involves functional substitution, and it will be clear that such a recursion is well defined only when the functionals are monotonic. Similarly, the associated notion of computability to be discussed later requires the assumption of monotonicity. On the other hand, functional substitution is well defined with non-monotonic mappings, and we have already proved some closure properties in Chapter 4 that are independent of monotonicity assumptions. But functional recursion, which is an extension of basic recursion allowing for functional substitution, is well defined only when monotonicity is assumed. Furthermore, we shall define in the next chapter recursive algorithms where the computation again requires montonicity. Both constructions-functional recursion and recursive algorithms-are related by the extended Church’s thesis.

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98

The initial step in our construction is to extend the notion of basic F-term. We now define functional F-terms in variables 2; a by six rules, FT 1 to FT 6, where FT 1, FT 3, and FT 4 are obtained by changing “basic F-term” to “functional Fterm” in rules BT 1, BT 3, and BT 4, respectively. Rule FT 2 takes the following form:

FT 2: If g is an (s,r)-ary functional in the class F , s, r 2 0 , U 1 , . . . , Us are functional F-terms in z;a,and Vl,. . . , V, are functional F-terms in the variables y,z;a , then g ( U 1 , . . . , U , ; (Xy)Vl,. . . , (Ay)V,) is a functional F-term in z;a. The new rules FT 5 and FT 6 are as follows:

FT 5: If V is a functional F-term in the variables z;a,and U is a functional Fterm in the variables y,z;a,then ( C y < V)U is a functional F-term in the variables z;a. FT 6: If V is a functional F-term in the variables 2; a,and U is a functional Fterm in the variables y,z;a,then (IIy < V ) U is a functional F-term in the variables z;a. If U is a functional F-term in 2; a,then U contains free variables, which must be in the list z;a,and possibly bound variables if terms are introduced by rules FT 2, FT 5, or FT 6. Such bound variables are not in the list a;a.As given, rule FT 2 requires that all A-terms be written with the same prefix (Xy). In practice we may allow that different prefixes be used with different terms. Functional terms are defined with a view to the extension by functional recursion, and the operations included in the definitions are determined by the applications of functional recursion, particularly the formulation of the extended Church’s thesis. Still, the crucial test for the scope of the definition is the specification of the universal interpreter in Chapter 12. We shall see there that we do not need both bounded sum and bounded product in the definition of functional terms, for one of them is sufficient. At this stage we do not exclude the possibility of eliminating both constructions in the definition. The manipulation of variables in the construction of functional terms is slightly more complicated than in the definition of basic terms. This is due to the presence of abstraction and the associated binding process. This affects only the numerical variables. The function variables, denoted by a in the rules, are fixed throughout the whole construction of a functional term.

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99

Note that if a is a unary function, then a = ( X y ) a ( y ) ,hence we can use distribution in writing functional terms since each variable can be expressed using rules FT 4 and FT 2.

7.1 Assume the class 3 contains a (1, 1)-ary functional g and the unary function s . The following is a functional F-term in t;a: EXAMPLE

Let U be a functional F-term in x;a. Then U induces a functional f u such that f u ( z ; a ) N U . This is self-explanatory since all operations in U ,including A-abstraction, already have been defined. Still, there is some interest in making this specification explicitly by induction on the construction of U. There are six rules in the specification of f u , each corresponding to a construction rule. Rules DFT 1, DFT 3, and DFT 4 are similar to rules DBT 1, DBT 3, and DBT 4. Rule DFT 2 takes the following form:

DFT 2: Let U be of the form g ( U 1 , . . . , Ua;( X Y ) ~ ., . . , (Xy)V,.) by rule FT 2. We assume functionals f i ( z ;a ) 1~ f u , ( x ;a ) , i = 1,.. ., s, g j ( y , x ;a ) N f v j ( y , z ;a ) , j = 1,.. . , r are given by the induction. Using basic operations and functional substitution we get g'(y, 2; a)? g ( y ; ( X y ) g l ( y , za ; ) ,. . ., ( X y ) g , ( y , z ; a ) ) ,and using basic operations we get f u ( x ;a ) 1: g'( f r ( z ;a ) , . .., f a ( Z ; a ) , z ;a ) . Rules DFT 5 and DFT 6 correspond to the operations of bounded sum and bounded product. Here we refer the reader to the discussion in Chapter 4 in terms of primitive recursive operations. Theorem 7.1 Let F be a class of functionals closed under primitive recursive operations and functional substitution. If U is a functional F - t e r m , then fu i s F computable. PROOF.

We use induction on the construction of U by rules FT 1 to FT 6. All

cases are straightforward and are left to the reader.

0

In dealing with basic terms in Chapter 4 we took the position that a functional fu induced by a basic F-term U is computable relative to the functionals in the class T (see Example 4.2). We claim a similar position here, although the correct form required to substantiate this position is not discussed here. This matter is considered in Chapter 8. Still, we can advance some ideas.

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For example, if U is the functional term of Example 7.1, we can compute fu(+;a) relative to the functional g, and this requires, of course, an oracle to provide values of g when required. In fact, the oracle must be used in two different ways, as there are two occurrences of g in U . We note that in order to compute fu(+;a ) we must ask the oracle for the value of g(z; p), where p = (Xy)g(y+l; a ) . To provide this information (if defined) the oracle needs the values of the function P, i.e., the values P(O), P(l),p(2), . . . whenever these values are defined. Hence we assume that for each y a computation is started to determine p(y), so an infinite number of computations are simultaneously executed, and in each of them the oracle for g is invoked. Note that we do not assume that ,8 is a total function, so it is possible that p(y) is undefined for some y, in which case the computation of p(y) is undefined. Even if this is the case we cannot assume that the oracle knows that P(y) is undefined. We have reached a critical point where we must clarify the role of the oracle within the general setting we assume in this work. The only way we can force the oracle to make a determination about the function p is by imposing the condition that g(+; is undefined when /3 is not total. In this case, the oracle must obtain all values p(y) for y = 0,1, . . ., otherwise the oracle remains silent and the computation is blocked without output. Within the frame we have assumed in this work we have no right to impose such a condition in the general case. This means we must organize the role of the oracle in such a way that it is irrelevant whether the totality of values of p is available or not.

a)

This situation is extremely delicate and forces us to refine the analysis. What we say is that the oracle expects values of p(y) for all y, and whenever sufficient values of @ are available, then the value of g(+;P) is reported. There is no doubt that in this particular example it would be correct to require that the totality of defined values of p be available to the oracle. Still, there are more involved situations where to impose such a requirement would result in a completely distorted computation (see discussion after Example 7.3). Finally, note that the setting we propose is consistent in the sense that the value of f u ( z ; a )so obtained is correct, for the functional g is monotonic, and any set of partial values of /3 that are sufficient to determine g(z; p) will provide the right value. The preceding analysis shows to what extent monotonicity is a necessary condition in order to compute algorithms relative to functionals that are defined with arguments not necessarily total. Functionals for which this condition fails and which

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101

are defined only for total arguments were called singular in Chapter 2. They are not excluded from our theory-on the contrary, they will play a crucial role in several important results (see Chapters 15, 16, and 17). A singular functional that is quasi-total is very similar to a total function in first-order computability, and it is well known that relativization to total functions implies many results that fail in the general case. We will find a similar situation in the theory of functionals. 7.2 We can write algorithms to execute the computations described in the preceding paragraphs. In the case of jr-r, where U is the term of Example 7.1, we need a main algorithm A1 with a subroutine A2. Both are written in the familiar language of locations (see Example 4.2). Here we assume that the numerical input of each algorithm is in location X1 and the output when the computation halts is in location XO. The algorithm A1 consisists of just one instruction: EXAMPLE

The subroutine

A2

consists of two instructions: x 1

xo

+

+

x1+ 1

s(X1;0 ) .

We proceed now to define recursive functional 3 - t e r m s in p ; x ;a , where p is a functional symbol of a fixed arity, say (s, m),and the arity of a is also rn. On the other hand, it is not assumed that the arity of z is s. The definition involves seven rules RFT 1 to RFT 7, where rules RFT 1 to RFT 6 are obtained from rules FT 1 to FT 6 by changing “functional F-term in x ;a (or in y,x;a)”to “recursive functional F-term in p ; z ;a (or in p ;y,x;a).’’ Rule RFT 7 is as follows:

RFT 7: If U 1 , . , . , U8 are recursive functional 3-terms in p ; z ;a,then p ( U 1 , . . . , U,; a)is a recursive functional F-term in p;x;a. This rule is similar to rule RBT 5 in the definition of basic recursion where we assumed, of course, that the arity of z was s. We cannot do the same here, for the variables of the construction change due to the applications of rules RFT 5 and RFT 6. Still, we are interested in terms where this condition is satisfied at the end of the construction. We say a recursive functional F-term in p;x;a is regular if the arity of p is ( k , m ) and the arity of z ; a is also ( k , m ) . A regular recursive

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102

functional F-term U in p ; z ;a , where p is (k,m)-ary, induces a transformation T on Fk,, where T p = p' and p ' ( z ; a ) N U . This transformation is monotonic, so if p l is an extension of p , and T p l = p i , then p; is an extension of p i . The proof of this relation involves a simple induction on the construction of U , and it is left to the reader. Note that it is critical to assume that the functional g (from the class F)in rule RFT 2 is monotonic. The monotonicity of the transformation T induced by a regular recursive functional 3-term U determines that T has a minimal fixed point, which we shall also denote by the symbol p . The functional p is the minimal solution of the recursive equation p ( z ; a ) N u. Note that in this context monotonicity plays a strictly mathematical role, as shown in the proof of Theorem 1.2. On the other hand, the discussion above was intended as an operational analysis to show that monotonicity is essential for the computation of functionals. In both cases functional substitution is involved. The operational role of monotonicity will be still more explicit in Chapter 8 when we discuss the computability of recursive algorithms. In both cases, mathematical or operational, the problem disappears if we assume the given functionals are singular. F'unctional Recursion. Let U be a regular recursive functional F-term in the variables p ; z ;a. We introduce a new ( k , m)-ary functional p as the minimal solution of the recursive equation p ( z ; a )N

u.

We say that p is obtained by functional F-recursion. A class 3 of functionals is closed under functional recursion if, whenever a functional p is obtained by functional 3-recursion, then p is F-computable.

Theorem 7.2 Let 3 be Then,

a

class of functionals closed under functional recursion.

(i) 3 is closed under basic recursion, basic operations, and functional substitution.

(ii) If F contains the numerical functions s and p d , then F is closed under precursive operations. PROOF. Case (i) is clear since a recursive basic F-term is also a recursive functional 0 F-term. Case (ii) follows from Theorem 5.2.

Chapter 7. Functional Recursion

103

7.3 Assume the class F contains a ( l , l ) - a r y functional g. Then the following is a regular recursive functional F-term in p ; t: EXAMPLE

P(T

1 = d";(xY)P(Y;)).

The function p is completely determined by the functional g. For example, if g ( z ; a ) 21 [z + O , a ( p d ( t ) ) ] then , the equation has only one solution, namely p ( z ) N o for all t,so p = c o . We propose the following algorithm A to compute the function p in Example 7.3 (compare with Example 7.2):

Note that this is a recursive algorithm, for it contains a call to itself. The execution of A with input 3: (in location XI) requires only one step (if defined), and such step consists of asking the oracle the value of g(z;/3), where P(y) is the result of executing the same algorithm A with input y. As explained above, the oracle needs enough values of /3 to determine g(+;/3), not necessarily all values of /3. In fact, it is clear that if we required that in order to report g ( r ; P ) all values of /3 must be available, then the algorithm is undefined for every t. At first sight it appears that the recursive equation and the algorithms describe the same construction. We do not think this is the case. The algorithm describes a process and belongs to operational mathematics. The recursive equation is a set theoretical construction. From this point of view it is not obvious that both constructions define the same object (i.e., the same function p ) . A proof of this relation is given in Example 7.4. Now assume the functional g in Example 7.3 is singular, so g(z; a) is undefined when a is not a total function. In this case the algorithm A above is always undefined, so it computes the totally undefined functional UD. If we consider the recursive equation in Example 7.3 under the same assumption, we find that it induces a transformation T where TUD = UD, so UD is the minimal fixed point. We can see that the assumption that functionals are singular simplifies the situation, but, a t the same time, trivializes functional recursion. Note that singularity is a common feature in most presententations of the theory of computable functions, and we have decided to take a more general approach. In order to prove properties of functionals that are determined by recursive specifications or computing algorithms we need mathematical principles, which in

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general are inductive principles. For recursive specifications we can use conditions RC 1 and RC 2 to prove that a given functional is the minimal solution of a recursive equation (see Example 5.2). In dealing with algorithms we can use induction on the computations, where it is shown that some property is preserved when a computation is derived from other computations, typically via a call to another algorithm. From this relation we can infer that all computations satisfy the given property. 7.4 Let p o be the function computed by algorithm A in Example 7.3. We want to show that po is the minimal solution of the recursive equation, and for this it is sufficient to prove that po satisfies conditions RC 1 and RC 2. Concerning RC 1, it is clear that if g(z;(Xy)po(y)) u v , then algorithm A with input z will return the value v , hence p o ( z ) N v . For condition RC 2 we assume p’ is another function such that whenever g(z;(Xy)p‘(y)) N v , then p ’ ( z ) 2: v . We must show that p‘ is an extension of po and prove by computation induction that whenever algorithm A with input z returns the value v , then g(z;(Xy)p‘(y)) 1: v and p‘(z) 1: v. In fact, the value v is returned when previous values of po are sufficient to report EXAMPLE

g(z;(Xy)po(y)) 2: v . For the values involved in this computation we can use the induction hypothesis and conclude p’ is an extension of po as far as those values, hence g(z; (Xy)p’(y)) N v and p ’ ( z ) 2: v .

Note that, although conditions RC 1 and RC 2 were proved for basic recursion, they can be applied for functional recursion and, in fact, for any type of recursion, for they depend only on the general principles available from Theorem 1.2. In applications we need a more general form of recursion where several functionals are specified simultaneously by recursive equations, one for each functional, and where each recursive equation involves all functionals introduced by the recursion. To formalize this notion we must define recursive functional F - t e r m in p 1 , . . ., p t ; z ;a,t 2 1, where p 1 , . . . ,pt are functional symbols of the same arity (s,n)and it is not assumed the arity of z is s. This is done by a straightforward extension of rules RFT 1 to RFT 7, where RFT 7 now applies to any of the symbols p l , . . . ,pt. The details are left to the reader. Let U be a recursive functional F-term in p 1 , . . . , p t ; z ;a. We say that U is regular if the arity of the symbols p 1 , . . ., p t is (k,m), and the arity of z;a is also

(k,m). Assume now that U1,. . . , Ut are regular recursive functional F-terms in

pl

, . . .,

105

Chapter 7. Functional Recursion pt;s; a . We can write a system of recursive equations of the form

P l ( s ; a ) N u1 . . .. . .. . . . . . Pt(z;a)

= ut.

To prove that this system has a minimal solution ( p i , . . ., p t ) is an easy application of Theorem 1.2. But we do not need to do this, for we can find the minimal solution using functional F-recursion. In fact, the application of Theorem 1.2 is quite simple in this situation. We need only to consider the domain F k , m x . . ‘ x F k , m , consisting of 2 copies of 3kFk,m,and consider the monotonic transformation induced by the system of recursive equations. The approach we use below is more operational and can be extended to cover situations that do not exactly fit the pattern assumed in the system, namely, that all the recursion symbols are of the same arity. First, we perform a transformation where each occurrence of p i ( - ; a ) in one of the terms U 1 , . . ., Ut is replaced by p ( i , - ; a ) ,i = 1,.. . , t . We denote the terms obtained in this way by U ; , . . . , Ui. They are considered terms in p ; y,x; a . With these terms we construct a new recursive functional F-term V as follows:

and this term can be used to write the recursive equation

We denote by p the minimal solution of this equation and set p i ( = ; a ) N p ( i , z ; a ) . We claim that the functionals p i , . . . ,p t introduced in this way are the minimal solution of the original system of recursive equations. In fact, they satisfy the crucial conditions R C 1 and R C 2 which are reformulated for simultaneous recursion as follows: R C 1: If U; N

ZI,

then p i ( z ; a ) N v , i = 1,.. .,t .

RC 2: Let p i , . . . , p i be given functionals and denote by U y the result of replacing p l , . . . , p t with p i , . . . , p i , respectively, in Ui, i = 1 , . . . , t . Assume also that whenever U;’ N v , then p:(a; a ) N ZI holds for all values of the variables. It follows that p: is an extension of p i .

106

L . E . Sanchis The proof of RC 1 is straightforward and, in fact, we immediately get equality,

i.e., p i ( = ; a ) N U i . For, P i ( z ; a) N N

a)

~ ( i l z ;

U,!(since p is the solution of p ( y , z ;a ) N V )

- U i (sincepj( z ; a ) ~ p ( j , z ; a ) , jl ,=. . . , t ) . To prove RC 2 we introduce a functional p’ such that

and denote by V’ the result of replacing p with p’ in V . Clearly, if V‘ N v, then p ’ ( y , z ; a ) E v, so p’ is an extension of p . Since p : ( z ; a ) N p ’ ( i , z ; a ) it follows that pi is an extension of p i . If functionals P I , . . . , pl are introduced as the minimal solution o f t simultaneous recursive equations with recursive functional F-terms, we say that they are obtained by simultaneous functional F-recursion. Clearly, standard functional recursion in F is a special case of simultaneous functional recursion. We say that F is closed under simultaneous functional recursion if whenever functionals pl , . . .,pt are obtained by simultaneous functional F-recursion, then they are F-computable.

Theorem 7.3 Let F be a class of functionals closed under functional recursion. If 3 contains the numerical functions s and pd, then F is closed under simultaneous functional recursion. PROOF. From the assumptions it follows that 3 is closed under p-recursive operations, hence the function eq is F-computable. If p is a functional obtained by simultaneous functional F-recursion, then the construction described above shows that p is obtained by numerical substitution in a functional p‘ obtained by functional 0 3-recursion. It follows that p’ is in the class 7 ,hence p is also in F.

7.5 Consider terms U1 and U2, where U1 is a recursive functional T-term i n p 1 , p z ; x 1 , x 2 ; c r land U2 isarecursive functionalF-terminpl,pz;xl;cr, and p 1 , p z are functional symbols of arity ( 2 , l ) and ( 1 , l ) , respectively. These constructions go beyond our rules, since p l and p2 are of different arity. Still, they are meaningful EXAMPLE

107

Chapter ‘7, Functional Recursion and we can write a system of recursive equations where P1(zlrzZ;a)

N

u1

pz(z1;a)

2:

uz

To find the minimal solutions of this system we replace the symbol pa by the (2,l)ary symbol p ; , with the intention of setting p ; ( z l , z z ; a ) N pz(t1;a). But this means that p z ( z I ; a )N p L ( q , O ; a ) , so we replace p z by p; in U l , U z , and obtain terms U i , U;. These are properly constructed recursive functional F-terms in p l , p ’ , ;2

1 , ~ a ,~so; we

can solve the standard system

= u;

Pl(z1,zZ;a) P;(zlzz;a) N

u;

Now from p i we get pz by the equation p ~ ( z 1 ; aN) p;(zl,O;a).This means that the solution of the initial system is obtained by constant substitution in functionals obtained by simultaneous functional 3-recursion. If F contains the function eq we can replace simultaneous functional 3-recursion with functional 3-recursion.

EXERCISES

7.1 Let g be a (0, 1)-ary functional such that g(; a ) is undefined if a is not a total function, g ( ; a ) N 0 if a ( y ) # 0 for some y, and g ( ; a )N 1 if a ( y ) N 0 for all y. Determine the minimal solution of the following functional recursive equation:

7.2 Let F be a class of functionals closed under p-recursive operations. Prove that 3 is closed under simultaneous basic recursion.

7.3 Let f be a (1,2)-ary functional. Prove there is a (1, 1)-ary functional p that satisfies the following conditions: (a) f ( z ;(Xy)p(y; a ) ,a ) N p(t; a ) for all z and a.

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108

(b) If (YO and PO are functions such that POis an extension of (Xy)f(y; PO,'YO), then PO is an extension of (Xz)p(z;(YO). 7.4 Define recursive functional F-terms in p l , . . . ,p t ; z ;a without assuming the symbols p l , . . . ,pt are of the same arity, say pi is of arity ( k i , m), and a is

of arity m. Consider a system of recursive equations where in the equation Ui the term Ui is in p 1 , . . . ,p t ; 21,.. ., x k , ; a . Reduce the minimal solution of this system to standard functional F-recursion.

p i ( z 1 , . . . , z k , ; a )2~

7.5 Extend the definition of functional recursion by changing rule RFT 7 and

allowing for functional substitution in the symbol p . Determine the minimal solution of the following recursive equation, where p is a (1, 1)-ary symbol:

The form of functional recursion considered in this chapter is minimal, for it allows functional substitution only in the given functionals. Hence, if there are no given functionals, or if they are simply functions, it follows that the recursion reduces to basic recursion. For a more general form of functional recursion, see the extension proposed in Exercise 7.5. Later we shall show that this type of recursion can be reduced to the restricted form considered here. We use functional recursion to give a formulation of the extended Church's thesis. The main application is the definition of the universal interpreter in Chapter 12. The central role that recursion plays in the theory of computable functionals was made explicit in Platek [23]. The original theory of Kleene was formulated in terms of induction rather than recursion. A different version of functional recursion is given in Kechris-Moschovakis [13]. This is a type of recursion intermediate between basic recursion and functional recursion as defined in this chapter.

Chapter 8

Recursive Algorithms In this work we take the position that recursion is a primitive form of computation that must be accounted for in any satisfactory version of Church’s thesis. Undoubtedly, recursion may be eliminated in some situations in terms of explicit computations. This appears to be the case in first-order computability that includes the theory of computable numerical functions, relativized also to numerical functions. A similar situation occurs with basic recursion which, in fact, is an extension of the first-order phenomenon. In this chapter we take up the general problem of computing functionals, relative to given functionals, and again we think that recursion should be a fundamental part of the computational process. The problem here is not the determination of the minimal fixed point of some type of transformation. We consider that this problem has been settled by Theorem 1.2. Our intention is to extend the idea of explicit algorithms, which we used to formulate Church’s thesis, and to advance an informal model of recursive computations. Later we shall use this model to formulate the extended Church’s thesis. We think that the model proposed here is sufficiently general. Simply, it consists of the previous model of explicit algorithms and the adjoining of new structures

that make recursive calls possible. Such structures are calls from one algorithm to another, not necessarily the same. An algorithm now is rather a finite set of algorithms related by recursive calls. We start with the definition of recursive operations which extend p-recursive 109

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operations and eventually are used to formulate an extension of Church’s thesis. A class 3 of functionals is closed under recursive operations if it contains the unary numerical functions s and pd and is closed under functional recursion. From this definition it follows that closure under recursive operations implies closure under p-recursive operations, basic recursion, primitive recursive operations, and basic operations. Furthermore, from Theorem 7.3 it follows that it also implies closure under simultaneous functional recursion. We insist that recursion in these definitions is a purely mathematical notion, without operational content, controlled by Theorem 1.2. On the other hand, operational recursion is a primitive computational concept that describes how recursive algorithms are computed. We do not have a formal definition of operational recursion, as we do not have a definition of computing or of algorithm. We do have a formal definition of mathematical recursion, at least in the context of set theory. If F is a class of functionals we denote by RC (F)the smallest class that contains F and is closed under recursive operations. The construction of RC (F)follows the usual pattern, and it is left to the reader. Functionals in RC (F)are said to be recursive in F. If F = (9) we write RC (9). The functionals in RC (9) are said to be recursive in g . Finally, we set RC = RC(0). The functionals in RC are recursive 0 functionals.

A system of recursive algorithms is a finite sequence of algorithms, say A l , described below. The system is relative to a class F of functionals, and when we want to stress this fact we shall denote it as a system of recursive F-algorithms. We denote by A = A l , . . . , A , a system of recursive 7-algorithms. Each algorithm in a system A is designed like the explicit algorithms of Chapter 6 but may contain extra rules that refer to the algorithms in the system. Each algorithm operates on a formal structure that is not necessarily the same for distinct algorithms in the system. The formal structure is finite and may contain numerical values or symbols, in which case it is called a configuration. Given an algorithm A; in the system and a legal input consisting of numerical values and function values, the algorithm determines a computation that is a sequence of configurations, each obtained from the preceding by one of the rules in the algorithm. Hence each algorithm in the system determines different computations that may continue indefinitely or halt with a well defined output. We allow also the possibility of a

. . . , A , , s 2 1, that satisfies conditions

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computation being blocked without output. Up to this point we have a situation completely similar to the one we considered for explicit algorithms. The difference is that the algorithms in a system have rules where one algorithm invokes or calls .another algorithm in the system. So it is possible for algorithm Ai to call algorithm A ; , or to call another algorithm A j , which may have a call to Ai, etc. These systems need not be strictly recursive in the sense that by a sequence of calls an algorithm may call itself, but we shall use the same notation even when this is not the case. We keep the discussion of recursive algorithms at a very general level and, as a consequence, the presentation may appear vague. This is a deliberate choice that we think is appropriate. Any particular formalization will undoubtedly provide more precision, but at the same time it will deprive these ideas of the intended general extension. Still, we note that formalizations are feasible and actually easy to construct. It is sufficient to start with the language of locations suggested in Chapter 6 to compute the p-recursive operations (see Example 6.2) and adjoin instructions where different algorithms are called, following the model described below. A formal model constructed along these lines is sufficient to compute the universal interpreter of Chapter 12, hence the recursive operations in general. The examples below are written in this particular language. We shall discuss in some detail the possible forms in which a call takes place in a system of recursive F-algorithms A = A1, . . ., A , . First, we note that it is assumed each algorithm Ai has a fixed arity (ki,rn) where ki depends on the algorithm Ai but m is fixed and equal for all the algorithms in the system. A call is a rule or instruction in a given algorithm that invokes other algorithms. The former is said to be the calling algorithm and the latter are the called algorithms. We discuss the manner in which calling algorithms are written and executed in the well-known language of locations involving locations X O X , I ,. . ., and each location contains numerical values that change during the computation. The execution of a recursive algorithmobviously depends on the manner in which the instructions are executed. While the explicit instructions are independent of the algorithm in which they appear, in the sense that they can be executed without reference to any algorithm in the system, a call instruction involves one or more algorithms. This is basically a recursive relation that we must take as a primitive. We describe the execution of recursive F-algorithms in a system A in terms of two ideal entities: one that we call the computing A - m a c h i n e , and another that we call the 3 - o r a c l e .

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The computing machine is a recursive machine and it has two different but related types of actions. First, the machine executes instructions during a computation involving a particular algorithm. In general, these instructions are explicit or calls involving algorithms in the system. Second, the computing machine computes algorithms with a given input and returns a value whenever the computation is defined. The machine is recursive in the sense that the execution of algorithms requires the execution of instructions, and the execution of instructions requires the execution of algorithms. We propose this as a primitive computational concept in order to formulate the extended Church’s thesis. On the other hand, if we work in the context of a given formalization, given by a formal machine or formal programming language, the recursive computation described above can be formalized, usually via some definition by induction. This is a well known phenomenon where, for example, the computation of a Turing machine can be formalized, although the notion of machine computation in general cannot be formalized. In the same way, we can define systems of algorithms for which the recursive computational behavior can be fully formalized, although this is not possible for the general notion of recursive computing machine we assume in our discussion. The use of the word “machine” here does not imply a reference to a mechanical artifact or computer. Rather, it refers to the assumption that the machine executes instructions in a purely formal manner following the rules of the algorithms in the system. The role of the F-oracle is standard in this type of discussion. In relation to an input function a,the oracle provides values of a(.) when required in a computation if such value is defined. In relation to a given functional g, the action is more involved and is discussed below. The oracle proposed here is a complex entity that provides information necessary for the computation and also uses information derived from the computation. We assume that in order to obtain this information the oracle is able to activate the machine, and even to start an infinite number of independent computations. At any rate, the information the oracle receives is only about the output of such computations, and nothing else. If a particular computation does not halt, this information is not available to the oracle.

To understand properly the mechanics of calling instructions we need some conventions regarding the manner in which inputs are handled when control is trans-

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ferred from one algorithm to another. The called algorithm must be provided with adequate inputs that are determined by the calling algorithm at the time of the call. The numerical inputs are obtained from fixed locations described in the instruction. The function inputs are always identical to the function inputs of the calling algorithm (recall that they have the same functional arity). We denote that constant function input by a . Since this input is transferred from one algorithm to another in each call, it is constant in the whole computation. Thus when the algorithm Aj appears as a called algorithm in some instruction, we describe the inputs of the call in the form A j [ - ; a ] , where the dash refers to a number of locations in the calling algorithm, and a is the function input of the calling algorithm. A more explicit description would be Aj [ X j , ,. . . ,X j , ; a ] ,which means that the arity of Aj is ( E , m ) . The remark above, in the sense that a computation is always a finite sequence of steps, refers, of course, t o halting computations. As usual, we may have non-halting computations that induce an infinite number of steps. So, a halting computation is finite, and at every step the amount of information available is also finite. This means the structure of a computation is exactly the same as in the case of explicit algorithms. The difference is in the nature of the steps, and here we allow for recursive calls where whole computations controlled by different algorithms are invoked. We allow two types of calls in a recursive algorithm, depending on whether the support of the call is one of the called algorithms or a functional in the class F. By support we mean the process that determines the output of the call whenever it is defined. This output is the only element that enters in the computation of the calling algorithm, and if the output is undefined the computation is blocked. More precisely, in the calling algorithm the call is a request for a value-it is just one step that is executed when the value is returned. A direct call is a call where the support is provided by the called algorithm. Such a call can be written in the following form

Xu

+

A j [ X j , , . . ., X j , ; a].

Here there is only one called algorithm, namely Aj of arity ( k , m ) . Note that Aj is also the support of the call. The symbol a denotes the function input in the calling algorithm. The execution of this call is straightforward and requires only the transfer of the contents of X j , , . . .,X j , in the calling algorithm to the proper input locations in algorithm Aj and the execution of A , until an output value i s

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obtained and reported to the calling algorit.hni. In the calling computation this step appears simply as the insertion of a numerical value into location Xu. The process by which the value is obtained, namely the computation of algorithm A j , is not a part of the calling computation. It should be clear that a direct call is essentially a recursive call of the type we have discussed in terms of explicit algorithms. We have shown in Chapter 6 that such calls can be eliminated (see Example 6.3), and the same process actually is available here as long as we have algorithms that involve only direct calls. The elimination fails if we have algorithms where indirect calls are involved. An indirect call involves a ( 1 , w)-ary functional g in the class F ,the support of the call, and v called algorithms of arity (kl,m),. . . , (k",m).The call is written as follows:

Xu

+

g ( - ; ( X y ) A j , [ ~ , - ; a .].,, .( X y ) A j , [ y , - ; a l ) ,

where the dashes represent sequences of locations of the proper arity in the calling algorithm. The execution of this instruction involves the oracle in an essential way since the functional g is not computed and it is assumed given in the class F. The numerical inputs for g are immediately given to the oracle and present no problem. The function inputs must be computed through the called algorithms Aj,, . . . , Aj,. This means the machine must compute Aj,[y, - ; a ]with y = 0 , 1 , 2 , etc., and similarly for the other algorithms. We assume that the machine reports to the oracle the output values of the computations and that the oracle reports the value of g as soon as sufficient values are available for g to be defined. This creates some ambiguity, for different sets of values can be available and sufficient for g to be defined. In fact, this ambiguity is irrelevant, for the functional g is monotonic. Here we refer the reader to Example 7.1, Example 7.2, and the discussion following the proof of Theorem 7.1. The crucial factor in an indirect call of the type described above is that the functional g is just given, and not computed by some algorithm. As a consequence, there is no way to eliminate the functional g. On the other hand, for the oracle to be able to report the value of the call, some number of extra computations is necessary via the called algorithms. These computations are isolated form the calling computation. Their role is to provide the information necessary for the oracle to complete the execution of the call. Both direct and indirect calls are essentially recursive in the sense that the output of whole computations is invoked. The execution of a direct call does not

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involve the oracle, only the machine. To execute an indirect call, action by the oracle and the machine is required. The algorithms A1 and A2 in Example 7.2 form a system of recursive algorithms, although in this case there is no proper recursion. The algorithm A proposed in Chapter 7 to compute the functional of Example 7.3 is by itself a system of recursive algorithms. Here the recursion is real, for the algorithm A calls itself.

8.1 We describe a system of F-algorithms consisting of two algorithms: Al which has arity (2,2), and A2 which has arity (1,2). They are described in terms of locations X o , X I , X z , . . . where X I ,X z , . , . contain the numerical input in a given computation, and X O contains the output at the time the computation halts. The function inputs, which are shared by the two algorithms, are denoted by a l , 0 2 . These inputs are not stored, for in general they are infinite objects. We assume they are under the control of the oracle which will provide values whenever they are required by the computing machine. The functional g from the class F is (1,l)-ary. The algorithm Al consists of the following instructions: EXAMPLE

+1 2: xz az(X2)+ 1 1:

x1

+

Cul(X1)

+

3:

x3

+

4: If X3

5:

Az[X3;a i ra21

= 0 goto 1 else goto 5

xo + x3 - 1

6 : Halt.

The algorithm A2 consists of the following instructions:

Algorithm A1 has one direct call in instruction 3. The execution of this call is straightforward: transfer the value of X3 in Al to X1 in A2 and execute A2 until computation halts, then transfer the value of X O in A2 to X s in A l . The function inputs a1, a2 are not affected and have the same values in both computations.

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Algorithm A2 has an indirect call in instruction 2. The execution of this instruction requires action from machine and oracle. The value of X1 in algorithm A2 is transferred to X2 in algorithm A1. The machine executes simultaneously A1 [y, X I ;01, 021 with y = 0 , 1 , 2 , .. . and reports to the oracle the output values of the computations. If sufficient values are produced for g to be defined, the oracle reports the value that is stored in X I . This process is considered to be one step in which a value is requested and eventually reported by the oracle. The internal process by which the value is obtained is not a part of the calling computation. At this stage the reader may be inclined to think that by introducing such a powerful recursive oracle we are pushing the difficulties under the rug of a purely verbal construction. To answer this we note first that we are not doing anything that goes beyond ordinary mathematical practice. The computations induced by A1 and A2 in Example 8.1 can be defined by induction in the frame of set theory. What we try to do is to give some operational content to a purely set-theoretical argument. We assume in our discussion that there is only one oracle for a given system of recursive algorithms. An alternative approach could be to assume that a unique oracle is created at the time a call is executed and that this oracle appoints new oracles to control sub-computations. We do not think these digressions help to clarify the basic situation. The oracle is an ideal element, and we cannot overcome this elemental fact. Efforts to infuse some operation content are useless. Note that the only extra information available to the oracle is determined by the computations induced by the call. In particular, if one computation is undefined, such information does not reach the oracle. We have already mentioned that this situation determines to a great extent the nature of the computational process. These considerations are relevant only if the functional g invoked in an indirect call has at least one function argument. If g is a numerical function, the role of the oracle takes the standard form and reduces to the valuation of g applied to given numerical arguments. Clearly, this means we are again in a context where recursion can be eliminated.

A crucial point in this discussion is the assumption that computations are finite objects. This is the case, not only in the sense that a computation consists of a finite number of steps, but also because at every step the configuration in the machine is a finite object.

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We have mentioned above that the whole construction can be resolved in the frame of set theory by using transfinite induction to define the computations induced by the algorithms in a given system. This is actually the standard approach to handle a plurality (finite or infinite) of computations. Later we shall find it useful to follow this approach, particularly because from a clear induction we can get more transparent proofs by induction. On the other hand, we must note that proofs by induction are also available through the operational approach. We call this type of proof computation induction, and it was used in Example 7.4. We do not attempt to formalize computation induction in this work. Examples of algorithms in this book are written using the familiar language of locations, or assignment language, as in Example 8.1. In general, we shall assume that primitive recursive functions can be used to write assignments. So, if g is a k-ary primitive recursive function and X i , X j , , . . . ,X j , are locations, then X i + g ( X i l , .. . , X i , ) is a valid assignment. In the same vein we must note that some of the requirements we have proposed for recursive algorithms may be relaxed in different languages. In particular, we may allow the arity of a given algorithm to be somewhat ambiguous, to be determined at the time of the call or execution. Let A be a system of recursive F-algorithms. If Ai is an element with arity ( k , n ) , then Ai computes a (k,n)-ary functional f. Essentially, this means that f(z;a)N the output of the computation of Ai with input (2; a).This computation may involve other algorithms in the system via direct or indirect calls. At any rate, the computation must be finite, performed in a finite number of steps. If f is computed in this way we say that f is recursively F-computable by Ai in A. The class of all functionals that are recursively 3-computable is denoted by RA(F). If F = 0 we write RA = RA(0). This is, of course, an informal definition, and we are looking for an extension of Church’s thesis which includes all functionals that are explicitly and recursively F-computable. In practice, the situation is not as general as discussed above. Given a system A of recursive F-algorithms, there is a functional f that we say is recursively F computable in A. This means there is some algorithm Ai in A that computes the functional f . From this point of view we may think of A , as being the main algorithm in A. In the general situation, each algorithm Ai in A computes a functional fi, and all these functionals are equivalent as far as the property of

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being computed in A is concerned. The notation R A ( F ) must be understood in this general sense, which means that no main algorithm is identified in A, and no main functional is distinguished among those computable in A. Our version of the extended Church’s thesis is as follows: If F is an arbitrary class of functionals, then R C ( F ) = RA(7). This equality is equivalent to two inclusions, and both are important for the RA(F) proper discussion of the extended Church’s thesis. The inclusion R C ( F ) will be proved later, using the universal interpreter. We shall show there that the language of locations, enlarged with direct and indirect calls, is sufficient to compute the functionals in the class RC(F7). The reader is advised to attempt some experiments at this stage to determine the difficulties of proving this result with the machinery presently available. The second inclusion here will be discussed in a manner that extends the method used for explicit algorithms in Chapter 6. Assume that A = A 1 , . . . , A, is a system of recursive F-algorithms and f l , . . . , f, are the functionals that are recursively 3computable by A 1 , . . . , A , in A, respectively. We assume the arity of fi is (ki, m), i = 1, . . . , s. Our assumptions about the algorithms in the system A are essentially the same as in the case of explicit algorithms considered in Chapter 6. In particular, it is assumed that the computational rules in the algorithms can be expressed by primitive recursive functionals via a proper encoding of the computational structures. This assumption does not apply to the calls in the algorithms, for the execution of such instructions requires values that must be determined outside the calling computation. These steps are formalized via functional recursion. We start by introducing a transition functional tri of arity (ki 1,m),i = 1 , .. . , s, that describes the rules of the algorithm via some encoding of the configurations peculiar to each algorithm. As long as these rules are explicit, only primitive recursive operations are required. Whenever we have a call-direct or indirect-where some algorithms are invoked, we use the functionals f i , . . . , f,. As a consequence, we get s evaluations of the form

+

where U1, . . . , Us are recursive functional P - t e r m s in f l ,

. . . , f,;y , z ; a and F’is an

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extension of T containing extra primitive recursive functionals that may be necessary for the description of the algorithms. We assume y is a halting configuration in the algorithm Ai exactly when t r i ( y , z i ; a ) 2: 0. Note that the functionals trl, . . . , tr, are determined in terms of the functionals f i , . . . , f a , which at this stage are known only via the computations determined by the algorithms. Later we determine the functionals fi , . . . f, in terms of trl, . . ., tr, and other functionals. The final result will be a system of simultaneous recursive equations. Next, we introduce the functionals evi, also of arity (bi 1,m), that describe, step-by-step, the computation of algorithm A, with a given input. In Chapter 6 these functionals were described by primitive recursion. Here we use the standard reduction of primitive recursion to basic recursion. For each i the recursive equation for evj is

+

evi(y, zi;a ) N [Y

+

K , tri(evi(pd(y), zi; a ) ,zi; a ) ]

where V, describes the initial configuration for algorithm Ai with input

zj;

a.

Note that in the same way that the functionals fi, . . . , fs determine the functionals trl , . . . , tr,, now these functionals determine the functionals evl, . . . ,ev,. This means that the intended functionals fi, . . . fs, trl, . . . tr,, evl, . . . ,ev, are solutions of the equations that are written above. We now proceed to the evaluation of the least y such that evi(y,zi; a )is a halting configuration. For this we need unbounded minimalization that is reduced to basic recursion in the usual way. We introduce a functional gj of arity ( k j + l , m)such that evj(gi(0,zi; a ) ,z j ; a ) ,if defined, is the halting configuration in the computation:

The treatment of primitive recursion and unbounded minimalization via recursion is derived from the proof of Theorem 5.1. Finally, we assume that for every i = 1 , . . ., s there is a primitive recursive function outi such that whenever y is a halting configuration, then outi(y) is the output of the computation. Now, using these functionals, we can write recursive equations for the functionals 1 1 , . . . , f a computed by the algorithms in the system A. We set for each i fi(zj; a ) N outj(evi(gi(0, z j ; a ) ,zi; a ) ) .

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In this way we obtain a system of 4 x s simultaneous recursive equations, with functionals f1 , . . . , f, , trl , . . . , tr, , evl , , . .,ev, , gi, . . . ,g,. These functionals are not of the same arity, but this difficulty can be handled as in Example 7.5. Furthermore, the given functionals in the recursion are functionals in 3 plus primitive recursive functionals. It follows that the functionals f1, . . . , fa are recursive in 3. We do not claim that the preceding argument is a proof of the relation RA(F) 5 RC(3). In fact, such a proof is impossible since we do not have a formal definition of the class RA(F). Still, it provides a canonical method to derive the recursiveness of a functional f which is computed by a system of recursive 3-algorithms. In practice, the possibility of the reduction can be verified by simple inspection, and the invoking of the extended Church's thesis in this direction amounts to a shortcut to describe the above transformation. Similar considerations, of course, apply to the original standard Church's thesis discussed in Chapter 6. As an example we write the specifications for the functionals involved in the computation of algorithms A1 and A2 in Example 8.1. It is sufficient to give the specifications for the (3,2)-ary functionals trl and the (2,2)-ary functional t r z .

Algorithm A1 involves four locations plus the label of the instruction to be executed, so we must consider quintuples (211, v ~v3, , v4, v5) where v1 denotes the , the contents of locations X o , XI, X z , X3, respectively. instruction and v z , ~ 3 , 2 1 4v5 The quintuple is encoded in the number < v l , vz, v3, v4, v5>. The specification of trl is by cases as follows:

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The (1,2)-ary functional denoted by fz is, of course, the functional computed by the algorithm Az. The (2,2)-ary functional t r 2 describes the algorithm A2. Here there are two locations, so we consider triples (v1, v ~ v3) , encoded in the usual way. The specification of trz is by cases as follows:

where U1 and Uz denote the terms described above using definition by cases, V1 = and V2 = . We set o u t l ( y ) = outz(y) = [ y ] ~ . Note that the term Uz involves functional substitution, so we are dealing with functional recursion rather than basic recursion. In this way we have resolved the system of recursive algorithms in Example 8.1 into eight simultaneous recursive equations, and we claim that the functionals computed by the algorithm are actually the minimal solutions of the recursive equations. For this purpose we take f1 and f~ to be the functionals computed by algorithms A1, Az, and t r l , t r 2 , e v l , e v z , 91, 92, the functionals that describe

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the computational process in the manner we have explained above. We want to show that these functionals satisfy conditions RC 1 and RC 2. Now condition RC 1 is trivial because we have written the equations in a manner that faithfully describes the computational relations, so we may say simply that the functionals are actually fixed points of the equations. Condition RC 2 is a minimality condition, and to prove that it is satisfied we need some form of induction. We shall use computation induction (already used in Example 7.4). First, we assume that t r ; , tra, e v i , eva, gi, ga, fl, f; are functionals that satisfy the assumptions of condition RC 2; i.e., these are the functionals denoted there by p i , . . . , p i . Using the induction hypothesis on the algorithms that compute f1 and f2, it follows easily that tri is an extension of trl and 2r; is an extension of trz. From this it follows that evi is an extension of evl and eva is an extension of evz. In a similar way this implies that gi is an extension of g1 and gi is an extension of g2. Finally, this implies that f; is an extension of f1 and f; is an extension of f2. This completes the proof that RC 2 is satisfied, hence the functionals derived from the computations are the minimal solution of the recursive equations. The conditions we impose in our definition of recursive algorithms are undoubtedly quite general, and fairly informal. Still, some of them involve severe restrictions that are imposed in order to keep the whole construction inside the frame of what we understand by computable. In particular, the restriction in the format of a direct call, where we do not allow for functional substitution, is substantial. EXERCISES

8.1 Write a recursive algorithm that computes the functional p of Exercise 7.1 8.2 Let 3 be a class of functionals. Assume g is a (1,l)-ary functional, h is a (2,l)-ary functional, and both are recursively 3-computable. Prove that the

(1, 1)-ary functional f such that

is also recursively F-computable. 8.3 Let

F be an arbitrary class of functional. Prove that R A ( 3 ) is closed under

p-recursive operations.

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Notes In this chapter we take operational recursion as a computational primitive and define recursive algorithms where recursive calls are executed via recursive computations. Clearly, the whole situation can be explained in the frame of set theory, and we are operating in the context of traditional mathematics. The point is that recursion has an essential operational content, and in computability theory such a content must be made explicit, rather than reducing it to a set theoretical construction. The distinction between operational recursion and mathematical recursion, defined by Theorem 1.2, may appear irrelevant to some readers, who may consider that both are aspects of the same phenomenon. This is a valid position, which we do not intend to contradict. Our point is that in the frame of computability theory the operational approach should be given primacy. Note that the inclusion R C ( F ) C R A ( F ) is by no means obvious. A final validation of the thesis is given in Chapter 12 via the universal interpreter. Our approach to recursive functionals, via the rule of functional recursion, departs from the treatment in Kleene [16] and is closer to the second approach in Kleene [MI. It is also related to the general theory of Platek [23], known to us via Moldestad [20]. Note that we consider functionals with partially defined arguments, as in Kleene [14] and Tourlakis [28]. Still, the traditional line with totally defined functions arguments predominates in the literature. For example, see Gandy [5] and [6], and Kechris and Moschovakis [13]. A recursive algorithm is essentially an algorithm written in a higher order programming language. This suggests that the functionals defined by such algorithms may play a role in the semantics of such languages.

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Chapter 9

Formalization: Structural Semantics The material in the preceding chapters has been essentially introductory, consisting of definitions and results that follow immediately from them. We start now a deeper analysis that will provide a powerful tool to study the theory of computable functionals. The main idea is to enlarge the type of functional recursion by allowing a more general form of functional substitution. In Chapter 10 we show that such a recursion can be computed by a reduction procedure, and later in Chapter 12 this procedure is formalized via functional recursion of Chapter 7. The main idea is the introduction of a general formalism, derived from the lambda calculus, that allows for recursive terms. This formalism contains the original system of functional terms and functional recursion and goes beyond by allowing new constructions. Furthermore, the original restrictions on the use of function variables are eliminated and functional substitution is admissible in full generality. The formalism can be used to define functionals, and this process is called structural semantics. The problem is that from the formal construction it is not clear that such functionals are computable by the recursive algorithms of Chapter 8. The problem is solved by introducing a reduction process that is shown to induce a valid computational procedure. This reduction takes the form of a reductional semantics, discussed in Chapter 10. We assume a fixed class T of functionals. Most definitions in this chapter are

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relative to the class F ,but we shall omit making this relation explicit in many cases. Here we use letters F , GI H to denote functionals in F. We continue the notation of the preceding chapters, so letters I , y, z , etc., denote elements of N, and letters f, g, h , etc., denote (k,m)-ary functionals k , m 2 0. We call these symbols informal variables. Recall that we cannot identify N with F o , ~ ,for the latter contains a bottom element that is not a number. The formal language we are about to introduce involves formal variables that are always of a given type. For every type we assume an infinite set of formal variables of that type. First there is a type-0, or numerical type. The formal variables of this type are given by the infinite list: v l , vz, . . . , v,,, . . . . Furthermore, for every pair of numbers k,rn 2 0 there is a type-(k, m),or functional type. In particular, we set 1 = (1,O) or function type. The formal variables of type-1 are given by the infinite list: q 1 , q 2 , . . . , qn, . . . . We do not make explicit which are the formal variables of types different from 0 or 1. Instead, we shall use symbols K , K ’ , K ” , ~ 1 K Z, , . . . to denote formal variables of arbitrary type, including type-0 and type-1. We shall use symbols v , v’, u”, vi, vil , via, . . . , v;, vi, . . . , to denote type-0 variables, and symbols q , q’, q”, q j , q j l , q j 2 , . . . , q;, 41. . . to denote type-1 variables. Note that v 1 , vz, . . . are the real type-0 variables in the system, and q 1 , qz, . . . are the real type-1 variables. Symbols v , , v’, . . . are metavariables that denote type-0 variables, and symbols q i , q’ . , . are metavariables that denote type-I variables. The symbols K , K ‘ , . . . are metavariables that denote variables of arbitrary type, including type-0 and type-I. We proceed now to define formal terms, constructed by means of formal variables, numerals, symbols denoting functionals in F , and a number of auxiliary symbols. The definition is by induction with the following inductive rules:

FM 1: A formal variable of type-0 (or type-(k, m))is a formal term of type-0 (or tYPe-(k, m>).

FM 2: A numeral is a formal term of type-0.

+

FM 3: If U is a formal term of type-0, then (U 1) and (U - 1) are formal terms of type-0. FM 4: If U is a formal term of type-(k, m),A, rn 2 0, U1,. . . u k are formal terms of type-0, and &, . . . , V, are formal terms of type-1, then { u } ( U , , . . . , uk;v 1 , . . . , Vm) is a formal term of type-0.

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127

FM 5: If U is a formal term of type-0, o i l , . . . , vik are different formal variables of type-0, and q j , , . . . , q,,,, are different formal variables of type-1, k, m 2 0, then (Av;, , . . . , vir;q,,, . . . , qj,)U is a formal term of type-(k, m). FM 6: If U is a formal term of type-(k, m ) ,k,m 2 0, and K is a formal variable of type-(k, m ) , then ( A K ) U is a formal term of type-(k, m). FM 7: If F is a (k,m)-ary functional in the class F ,k , m 2 0, U 1 , . . ., Ub are formal terms of type-0, and Vl,. . . , V, are formal terms of type-1, then F ( U 1 , . . . , uk;Vl, . . . , Vm) is a formal term of type-0.

FM 8: If U1, Uz, and term of type-0.

U3

are formal terms of type-0, then [Ul+ Uz,U3]is a formal

We intend to define two semantics for the above formalism. The first semantics is structural, where the value of a term is completely determined by the values of its sub-terms. The second is reductional, where the value of a term is given by a reduction. We shall prove both semantics are equivalent and assign the same value to every term. The preceding formalism follows the pattern of an applied lambda calculus extended with recursive terms of the form ( h n ) U . Still, we avoid some standard lambda calculus constructions where the lambda abstraction can be applied to terms of arbitrary type. In this way we avoid construction of terms where the semantics is outside the objects we are concerned with in this work. A construction of the form (An)U where U is of type K' introduces a higher type functional of type K + K' that we want to avoid here. Similarly, we restrict the standard application operation in such a way that the result is always a type-0 term. A construction by rule FM 5 introduces a formal term of the form (A-; - ) U , where the first dash represents a sequence of different type-0 variables, and the second a sequence of type-1 variables. We call (A-; -) a A-prefix. We shall say that the prefix (A-; -) binds all occurrences in U of the variables in the prefix. Such occurrences are said to be bound by the prefix, and the variables in the prefix are said to be bound in the construction. Note that only variables of type-0 or type-1 can be bound by a A-prefix. On the other hand, a construction by rule FM 6 of the form ( A K ) U has a A-prefix (Ax) that binds all the occurrences of K in U . Such occurrences are said to be bound by the prefix, and the variable IE is bound in the

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128

construction. Only functional variables of type-(E, rn) can be bound by a A-prefix. A type-1 variable qi can be bound by a A-prefix or by a A-prefix. An occurrence of a variable in a term U that is not bound by any prefix is said to be a free occurrence. A variable occurs free in U if it has at least one free occurrence in U . A variable is bound in U if it is bound by at least one prefix in the construction of U . It is possible that a variable occurs free in U and is also bound in U . Note that a free occurrence of a variable must be real, so the variable must occur outside the scope of any binding prefix. On the other hand, if a variable is bound in a term this only means an occurrence in a prefix. For example, the variable 01 is bound in (Av1;)U even if v1 does not occur at all in U . A term U is said to be closed if there is no variable that occurs free in U . The term U is linear if only variables of type-0 or type-1 have free occurrences in U . The term U is regular if no variable that occurs free in U is bound in the construction of u. 9.1 We use the variable v1,vz of type-0, q1,qz of type-1 to construct several formal terms: We set U1 = ( X V Z , V ~ ; ~ ~ ) { Q ~ which } ( V first ~ ; ) ,involves rule FM 4 and then rule FM 5. This term is closed, hence it is linear and regular. The variables bound in the construction are vz,v1, and q1. We set U2 = {Ul}(vl, v z ; q1) of type-0 by rule FM 4. Here the variables v l , v 2 , and q1 occur free and are also bound in the construction. This term is linear, but it is neither closed nor regular. We set U3 = { q ~ } ( U 2 ;of) type-0 by rule FM 4. The variables v l , v z , q l , and q z occur free, and the variables v l , v z , and q1 are bound in the construction. This term is linear, but it is neither closed nor regular. We assume K is a variable of type-(2,l) and we construct the formal term U4 = {K}(U~ 0 ;,q z ) of type-0 by rule FM 4. The variables v 1 , v z , q1, q z , and K occur free, and the variables v l , u ~ , q 1are bound in the construction. This term is neither linear, nor closed, nor regular. Finally, we set Us = (AK)(Av~, v z ; q2)U4 of type-(2,l) by rules FM 5 and FM 6. The only variable that occurs free is q1. The variables v l , v z , q l , qz, and n: are bound in the construction. This term is linear, but it is neither closed nor regular. EXAMPLE

Both semantics require that values be assigned to the free variables of a term

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129

to be evaluated. It is more convenient to assume an assignment mapping T that assigns a value to every variable in the formal system. This mapping is total and satisfies the condition that if vi is a type-0 variable, then T(Q) is a number, and if K is a type-(k, rn) variable, then T(K) is a (k,rn)-ary functional, i.e., an element of F k , m . Recall that we identify F1,o with F1 = the class of unary numerical functions. Given a formal term U and an assignment mapping T, then T determines the meaning or denotation of U in the manner we describe later. In general, this denotation is straightforward and can be derived from the usual meaning of the symbols occurring in U . We refer to this denotation as the structural semantics of the term U because it is actually determined by the structure of U . In other words, the denotations of the subterms of U determine the denotation of U . In the next chapter we shall consider a reductional semantics where the denotation of U is derived by a reduction process. In both semantics, the denotation of U depends on the particular assignment mapping T. Since an assignment mapping 7 is total, there is no danger in using the standard equality to denote the value of an application. We therefore write ~ ( v i )= z to indicate that the number z is assigned to the type-0 variable vi. We also write ~ ( q i= ) a to indicate that the unary function a is assigned to the type-1 variable qi. In general, if K is a formal variable of type-(k, rn) we write T(K)= f to indicate that the (k,m)-ary functional f is assigned to the variable K . Recall that we may use the symbol K to denote a type-0 formal variable. If this is the case, then the notation T(K) denotes a number, and we may also write T(K) = f with the understanding that f in this context denotes a number. In some situations we may want to change an assignment T to another assignment 7‘. If vi is a type-0 formal variable and z is a number, then 7’ = [ z / v i ] ~means that T’(K)= T(K) whenever K is a formal variable different from vi, and ~ ‘ ( v i )= 2. Similarly, if qi is a type-1 formal variable and a is a unary function, then T’ = [ Q / Q ~ ] T means that T‘(K) = T(K)for every formal variable K different from q i , and ~ ’ ( q i )= a. In general, if K is a formal variable of type-(k, rn) and f is a (k,rn)-ary functional, then 7‘ = [ ~ / K ] T means that T‘(K’) = T(K‘) for any formal variable K’ different from K , and #(K) = f. In general we may want to change an assignment r not only in one variable, but in several variables at the same time. In this case we extend the notation and write 7’ = [fi,. . .,f , / ~ l , .. . , K,]T where ~ 1 ,. .. , K, are different formal variables, K{ and fi are of the same arity (i.e., if ~1 is of type-0, then fi is a number, and if

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is of type-(k, m), then fi is a (k,m)-ary functional), i = 1 , .. ., s. This means that T ’ ( K ) = T ( K ) whenever K is a formal variable different from ~ 1 , .. . , K ~ and , ~i

= fi for i = 1,.. . , s. If U is a formal term, and r,r‘ are assignments, then T c u r’ means that whenever K is a formal variable that occurs free in U , then T ( K ) = T ’ ( K ) in case K is of type-0, and T ( K ) Ck,,,, T ‘ ( K ) in case K is of type-(k, m). If U is a formal term, and 7,r’ are assignments, then r =(I 7’ means that T ( K ) = T ’ ( K ) whenever K is a formal variable that occurs free in U . Clearly, the relation r =u 7’ is equivalent to the conjunction of T C_u r’ and r’ S,r 7. Note that if u is a closed term, then r Cu 7’and r =u r’ holds for all assignments T and 7’. T’(K~)

EXAMPLE

9.2 Assume ~(211)

=0

T’(v~= )

1

T

and r‘ are assignments such that: T(Q)

=1 0

r(q1) = UDi

~ ’ ( t ~ 2=)

r’(q1)= pd

The following relations are easily verified, where U1, defined in Example 9.1: T

=u, 7‘

?-

#u,

7’

T

#a,

7’

7

&,T I .

T(q2)

r’(q2)

U2, U 3 ,

=s =s

and Us are the terms

The structural semantics for the formal terms is given by a function U with two arguments, a formal term U , and an assignment r. The application of V is written U ( U ) ( r ) . This expression may be undefined if U is of type-0. If U is of type-(k, m), then U ( U ) ( r ) is always defined, and the value is given by the rules below. The determination of the value U ( U ) ( r )depends on the values of the subterms of U . So, even if U is of type-0, we may need to determine the values of subterms of functional terms to be able t o determine the numerical value of U . On the other hand, the determination of the value of a functional will require the values of type0 terms. Still, there is no circularity in the definition because in every reduction the terms are shorter. The crucial element of the definition is that a t every stage the value V ( U ) ( r ) is defined for all possible assignments T . For the definition to work properly we have to make some assumptions, which are explained below. As

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131

we define the function V we must check at every stage that these assumptions are being satisfied. We define the value of V ( U ) ( r )by induction on the construction of U , as given by rules FM 1 to FM 8. At every stage of the induction the value is defined for all assignments r , and in such a way that the following two conditions are satisfied: C 1: If U is of type-0 and V ( U ) ( r ) is defined, then V ( U ) ( r ) N z where number. Furthermore, if r T’,then V ( U ) ( r ‘ )N z.

cr~

I

is a

C 2: If U is of type-(k,m),k,m 2 0, then V ( U ) ( r )= f where f is a (k,m)-ary functional. Furthermore, if r C _ ~ Jr‘, then V ( U ) ( r ‘ )= f’ where f &k,m f‘.

We proceed now to define V ( U ) ( r )for each of the cases in the construction of the term U , as given by rules FM 1 to FM 8: Case FM 1. Here U is a variable K . We put V ( U ) ( r )N T ( K ) . Clearly, conditions C 1 and C 2 are valid. Case FM 2. Here U is a numeral with value I. We put V ( U ) ( r )1: I . Clearly, condition C 1 is valid. Case FM 3. Here U is either U’+ 1 or U‘- 1. We put V ( U ) ( r )N V( U ’ ) ( r ) 1 in the first case, and V ( U ) ( r )N V ( U ’ ) ( r )- 1 in the second case. Condition C 1 follows from the induction hypothesis on U’. Case FM 4. Here U is of the form { U ‘ } ( U l , . . . , uk;v 1 , .. . , v,), k,m 1 0. If one of the expressions V ( U i ) ( r )is undefined, i = 1 , . . ., k, then V ( U ) ( r )is undefined. Otherwise, let V ( U ’ ) ( r ) N f, V ( U i ) ( r )N zi, i = 1,. . . , k, and V ( V j ) ( r )= a , , j = 1 ,... , m . We set V ( u ) ( r )N ~ ( E ~ , . . . , c ~..., ; c om). Y ~ , Now the first part of condition C 1 is trivial. To prove the second part, assume V ( U ) ( r ) N

+

. . , x k ;( ~ 1 , .. . , a,)

the induction hypothesis it follows that V(U‘)(r’) F k , m , V(Ui)(r’)N zi, i = 1 , . . . , k,V ( K ) ( r ’ )= a; where a; is an extension of a j , j = 1 , . . . , m ,hence V ( U ) ( r ’ ) N f’(t1,. . . , zk;a:, . . . , a&) N I ,since f is monotonic, and f’ is an extension of f . Case FM 5. Here U is of the form (Xvi,, . . . , v i k ; q j l l . ..,q,,,,)U’. We set V ( U ) ( r )= f, where f(z1,.. . , t k ;a l l . .. , a m )N V(U’)(r’) and T’ = [ X I , . . . , X k ,a1, f(11,.

N I. By

= f’ where f’ is an extension o f f in

. . , , a m / v i l l . . ., v i k r q j , , . . ., qjrn]r. From condition C 1 on the given term U‘ it follows that f is monotonic, hence the first part of C 2 holds for U . Using again C 1 on U’ it follows that the second part of C 2 also holds for U . Case FM 6. Here U is of the form ( A K ) U ’ . To define V ( U ) ( r )we note that if we set Tf = V(U’)([f/.]r) = f’, then by C 2 applied to U‘ it follows that T is

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L.E. Sanchis

a monotonic transformation on F k , m . We set V ( U ) ( r )= f where f is the minimal fixed point of T. Since f is an element of F k , , it follows that the first part of condition C 2 holds for the term U . Next we assume T Cu T’ and introduce Tf = V ( U ’ ) ( [ f / ~ l r ) and T‘f’ = V(U’)([f’/~lr’). From condition C 2 applied to U’ it follows that whenever f s k , m f’, then Tf C k , m T’f’. From this relation it follows that the minimal fixed point of T’ is an extension of the minimal fixed point of T. This means that the second part of condition C 2 holds for U. Case FM 7. Here U is of the form F(U1, .. ., u k ; V1,. . ., Vm). If one of the expressions V ( U , ) ( T )is undefined, i = 1,.. . , 6 , then V ( U ) ( r )is undefined. Otherwise, let V ( U ~ ) ( TN) z,,i= 1,..., 6 , and V ( Q ) ( r )= aj,j = l , . .. , m . We set V ( U ) ( r )N F ( z 1 , . . . , x k ; a l , . . .,am). The first part of condition C 1 is trivial. The second part follows using the same argument as in case FM 4 and noting that F is monotonic. Case FM 8. Here U is of the form [Ul + U2,U3] and we set V ( U ) ( r ) N [V(U~)(T) + V(Vz)(r), V ( U 3 ) ( 7 ) ] .Condition C 1 follows from induction hypothesis. This completes the definition of the function V . In general, the different cases are handled in a manner clearly consistent with the intended meaning of the symbols involved. Only in case FM 6 is a new construction introduced, which is resolved via the minimal fixed point theorem of Chapter 1. By using this construction we go beyond the formalism of functional terms and functional recursion. Mathematically, the definition is satisfactory, a t least in the context of set theory. Operationally, it provides little information about the actual evaluation of type-0 terms. For this reason we must introduce another reductional semantics in the next chapter. We must also prove that the reductional semantics is equivalent to the structural semantics defined here. For this purpose we need several properties that are easily derived from the definition. Theorem 9.1 Let U be a formal term, with r and r’ assignments such that r =u T ’ . Then V ( U ) ( r )N V(U)(r’). PROOF.

function.

Immediate from conditions C 1 and C 2 in the definition of the semantics 0

Theorem 9.2 Let U be a formal t e r m of type-0 of the form { U ’ } ( U l , . . ., u k ; &, . . . , V,,,) under rule FM 4 , where U’ is a formal t e r m of type-(k, m ) of the form

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133

. . . , v i k ; q j , , . . . , qj,,,)U“

under rule FM 5. If r is an assignment such that i = 1,..., k, and V ( ~ ) ( T=) aj,j = 1,..., m, then V ( U ) ( r ) N v(u”)( [ ~ l , . . . , t k , a l , . . . , a m / v i r , . . . , u i k , q j l r . . . r q j m l T ) . (XVi,,

V(Ui)(T) N Zi,

PROOF. Immediate from combining cases FM 4 and FM 5 in the definition of the 0 semantics function.

Theorem 9.3 Let U be a formal term of type-(k,m) of the form (AK)U’ where U’ is also of type-(k,m). Assume V ( U ) ( T )= f , where f is a ( k , m ) - a r y functional. Then

Immediate since f is the minimal fixed point of the monotonic transformation T f = V ( u ‘ ) ( [ f / ~ ] r ) . 0

PROOF.

Theorem 9.4 Let U be a formal term of type-(k, m ) ,k , m 2 0 , and let v i , , . . . , v i k be different type-0 variables that do not occur free i n U , and q j 1 , . . . ,q j , be different type-1 variables that do not occur free in U . Then V ( U ) ( r ) = V((Xvj,, . . . , v i k ; q j 1 , . ’ . , ~ j , ) { U } ( v i l , . . . r ~ i k q; j . j , t . * . , q j m ) ) ( . ) .

By case FM 5 f’ is given by the expression f’(z1,.

. ., zk; air.. . , a m ) 21 V({U}(vi,,

*.

. ,v i k ; q j l , .

*

,qj,,,))(r’),

where 7’

= [ X I , . . ., z g , ~. .~. , ,a m / v i , j . . . , v i r , q j 1 , . . ., qj,,,]~.

Since the variables v i , , . . . ,vik,q j , , . . .,q j , do not occur free in U ,it follows that V ( U ) ( r ’ ) = V ( U ) ( r )= f . Hence by case FM 4 it follows that

hence f = f’

0

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L.E. Sanchis

The preceding results show that the structural semantics obeys the obvious rules determined by the meaning of the formal operations. To obtain more properties we must define formally the notion of substitution. Let U be a formal term of type-(k, m),~ 1 , ... , K , , a finite sequence of distinct variables, and W1,.. . , W, a finite sequence of formal terms, s 2 0. We , 1, . . . , s. We denote by assume Wi is of the same type as the variable ~ ii =

[Wl,.. . W,/Kl, . . . , K ~ ] Uthe result of simultaneously substituting the terms W1, . . . , W, for the free occurrences in U of the variables ~ 1 , ... , K , , respectively. More precisely, all free occurrences of the variable ~1 are replaced with the term W1, all free occurrences of ~2 are replaced with Wz, etc., but the replacements are executed simultaneously, not one after the other. It is easy to show by induction on the construction of U that the substitution [Wl,.. . , W s / ~ 1 , ...,K,]U produces a formal term of the same type as U . We say that a substitution [ W l } . . , W * / K.~ . , , K ~ ] isUfree if there is no variable that occurs free in one of the terms W1,.. . , W , and bound in the term U . Note that substitution by itself plays no role in the definition of the structural semantics given above. We shall see later that it is the crucial element in the reductional semantics of Chapter 10. At this stage we want to make sure that substitution preserves the meaning of terms, as given by the structural semantics. This is not necessarily the case if the substitution creates a collision of variables, where a variable that is free in a term becomes bound after the term is substituted in another term. To make sure this condition is satisfied we require that the substitution be free. Our definition is a bit too restrictive, but it is sufficient for our purposes. Theorem 9.5 Consider a free substitution U’ = [Wl,.. ., Ws/tcl,.. . , K,]U and assume V(Wi)(r)N f i , i = 1,. . ., s. Then V(U’)(T)% V ( U ) ( [ f l ,..., f , / K 1 , . . . ~$1.). }

We use induction on the construction of U by rules FM 1 to FM 8. The only cases that require discussion are FM 5 and FM 6, where the term U contains a binding prefix. The following remark is useful to deal with these cases. Assume the variable I G ~does not occur free in U . Then we can delete the element . . . , Wi,. . . / . . , ~ i .,. .from the substitution without changing the term U’, and we can also delete the element . . . , f i , . . . / . . . , ~ i . . . from the assignment without affecting the value of U . Hence it is sufficient to prove the equation under the assumption that all variables 6 1 , . . . , K , occur free in U . PROOF.

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135

With this understanding we consider case FM 6 where U is of the form (An)V. From the assumption above it follows that all variables ~ 1 , . .. , K , are different from K , and U’ is of the form (An)V’ where V‘ = [ W l , . . . , W , / n l , . . . , K , ] V . Since the substitution is free it follows that K does not occur free in any of the terms W1,. . . , W,. Now V(U‘)(T)= f‘ where f’ is the minimal fixed point of the transformation T such that

Tf

= W’)([f/+) = V ( V ) ( [ f l , . . ’, f s / K l , . . Ks”l.1.) = V(V)([f/.l[fl, ’ . * I f s / K l , . . . , K s 1 7 ) . 1

(ind. hYP.1

and this means U(U)([fl,. . . , f , / n l , . . . , K , ] T ) = f’. Note that the induction hypothesis is applied with the assignment [ f / ~ ] r for every functional f , rather than with the original assignment T. Furthermore, V ( w ; ) ( [ f / ~ ] r ) = f;, since K does not occur free in W;, i = 1 , . . . , s. The case corresponding to rule FM 5 is similar, and it is left to the reader. 0 Note that the restriction that the substitution is free is essential in the proof of Theorem 9.5. As a consequence, in dealing with substitutions, and particularly in the reductional semantics of Chapter 10, we shall need some general conditions to insure that substitutions performed in a given context are free. The best approach is to arrange the terms in such a way that there is no variable that occurs free in one term and is bound in the construction of another. In particular, the condition is satisfied when the given terms are subterms of a regular term.

. . . , v ; ~ q; j i , . . ., qj,,,)uo}(vl,.. ., uk;v1,.. . , vm),the substitution u‘ = [ U i , . . . , Uk,v1,.. . , Vm/Vil, . . . , v i k rq j l , . . ., qj,,,]Vo i s free, and there are 2 1 , . . .,zk such that V ( U ~ ) ( TN) i;, i = 1,. . ., k. Then V ( U ) ( T )N V ( U I ) ( T ) . Corollary 9.5.1 A s s u m e U i s a f o r m a l t e r m of the form {(Xvi,,

PROOF.

We set U ( v j ) ( ~=) a j , j = 1,.. . , rn, and

. . . , v i k ,q j , , . . . ,qjJr.

TI

= [q,. . . , X k , c q ,. . . , a , , , / ~ ; ~ ,

It follows that

V(U)(T)

N

V(VO)(T’)

N

V(U’)(r)

(Theorem 9.2) (Theorem 9.5) 0

Theorem 9.6 Let U be a regular formal t e r m of the f o r m ( A K ) ~ ’and , r some assignment. Then V([U/K]V’)(T) = V ( U ) ( r ) .

L.E. Sanchis

136 PROOF.

We set V ( U ) ( r )= f . It follows that

V ( [ U / n ] U ’ ) ( r ) = V ( C r ’ ) ( [ f / ~ : ] r )(Theorem 9.5) = f (Theorem 9.3). 0

Corollary 9.6.1 Let U be a regular formal term of the form {(h~:)u’}(u~, ...,u k ; Vl,. . . , Vm), and some assignment r . Then,

Immediate from Theorem 9.6 and case F M 4 in the definition of the semantics function. 0

PROOF.

If U is a formal term relative to a class F of functionals, and we want to make explicit this relation, we shall say that U is a formal F-term. If U is of type-(k, m), and V ( U ) ( T )= f for some assignment r , we say that the (k,rn)-ary functional f is f o r m a l l y r-definable from F. If U is closed this means that f is independent of r , we write V ( U ) = f and say that f is formally definable from F. We denote by FD(F) the class of all functionals that are formally definable from F. If F is empty we write simply FD . The elements of FD are said to be formally definable. The terminology in the preceding paragraph, where we refer to functionals as being definable, is intended to stress the fact that the structural semantics provides a minimum of computation information, so we cannot infer that a functional derived in this way is computable. Later, we shall prove that the functionals defined from a class F are also recursively F-computable. To obtain this result we must define the reductional semantics, prove the equivalence to the structural semantics and, finally, use functional recursion to define the universal interpreter in Chapter 12. 9.3 The binary function add is formally definable. A type-(2,O) term that formally defines add is the following term AD:

EXAMPLE

This term formalizes the standard definition of addition by primitive recursion. The bound variables v1 v2 can be taken arbitrarily. The symbol A: denotes some fixed type-(2,O) variable.

Chapter 9. Formalization: Structural Semantics

137

9 . 4 The binary function pr is formally definable by the following type(2,O) term PR:

EXAMPLE

9.5 Consider the (1, 1)-ary functional bs, such that bs(z;a) N (Cy < z)a(y). This functional is formally definable by the formal term BS such that:

EXAMPLE

9 . 6 Consider the (1,l)-ary functional bp such that bp(z;a) N (IIy z)a(y). This functional is formally definable by the formal term BP such that:

EXAMPLE


)

N

@k,'m(Z, 2; a ) .

0

If F is a reflexive structure with internal interpreter functional, and z is a number that satisfies the relation

((a,

S), f is a (k,m)-ary

f(2;a N @ k y z , 2; a),

we say that z is a (0, S)-index for f, or simply an index for f. So from Theorem 11.1it follows that a F consists exactly of those functionals that are indexed by the interpreter ((a, S). If z is an index for the functional f l then z represents f in the reflexive structure 7 . In practice, this means we can operate on the functional f via the index z to produce values that are also indexes of functionals. By properly choosing these operations we can obtain new functionals related to f and satisfying special relations. EXAMPLE 11.1 Note that Theorem 11.1 also applies to the functionals

which

are F-computable. For example, there is a number z such that (alJ(ylv; a)N ( a y z ,y, v ; a). Theorem 11.2 Let 7 be a reflexive structure with internal interpreter ((a, S), and f a (1, 1)-ary functional such that for some unary function (YO, ( A z ) f ( z ; a o )is a total extension of (Xz)(a(z; ao). Then f is not 7-computable.

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L.E. Sanchis

To get a contradiction, assume that f is F-computable and let f’ be the ( l , l ) - a r y functional such that

PROOF.

f ‘ ( x ; a ) 21 f ( S ( x ,2 ) ;a )

+ 1.

It follows that f’ is F-computable, hence there is a number

f ’ ( x ;a ) N

L

such that

CP(S(2, x ) ;a ) .

Furthermore, f’(z; ao) is defined, say, f ’ ( z ;ao) N v. From the above equations it 0 follows that v = v 1.

+

The preceding theorem, and the corollaries that follow, show the limitations implicit in the definition of reflexive structures. In general, they are forms of the halting problem in computability theory and are proved by diagonalization. Note that a reflexive structure is countable, so we can get functions outside the class by simple Cantor diagonalization. The examples given below are more specific and relate to the interpreter functional. Essentially, what is proved is that such a functional has no quasi-total extension.

Corollary 11.2.1 Let F be a reflecive structure with internal interpreter (CP, S ) , and f a ( 1 , l ) - a r y 7-computable functional that is an extension of CP. If a is an arbitrary unary function, then (Ax)f ( x ;a ) is not a total function. PROOF.

Immediate from Theorem 11.2.

0

Corollary 11.2.2 Let 7 be a reflexive structure with internal interpreter (CP,S). There is no ( 1 , l ) - a r y predicate P d-computable in F, and total function a0 such that P ( x ;ao) holds if and only if @(z;(YO) is defined. To get a contradiction, assume there is such a predicate P and total unary function ao. If x p is some F-computable dual characteristic functional of P we introduce a ( 1 , 1)-ary functional f such that

PROOF.

It follows that f is F-computable and ( A x ) f ( r ;ao) is a total extension of (Ax)CP(x; ao),contradicting Theorem 11.2.

0

Chapter 11. Interpreters

161

Corollary 11.2.3 Let F be a reflezive Structure with internal interpreter ((a, S ) . Then the predicate De is p-computable in F,but it is not d-computable in F.

It is p-computable in F ,for a partial characteristic functional is given by $*(z;a) N [(a(z; a)+ 0, 01. PROOF.

That it is not d-computable in

F follows from Corollary 11.2.2.

$0

of Da

0

11.2 If F is a reflexive structure with internal interpreter ((a, S), and f is a (k,m)-ary F-computable functional, then D j is p-computable in the class F. For a partial characteristic functional for D j is given by EXAMPLE

$r(z;a)N

[f(.;

a)

+

0,OI.

Let F be a recursive structure with internal interpreter ((a, S). From the general properties of the functions Sk discussed above, it follows that (ar+k,m(%,y, 2; a) N ( a ( S ' + k ( % , y,z); )

y),2); ) (aE,m(Sr(z,y),z; a).

N (a(Sk(S.(%, N

Theorem 11.3 Let T be a reflezive structure with internal interpreter (a,s), and U a functional F - t e r m in y, 2; a . There is a total primitive recursive function g such that (akv"(g(y), PROOF.

2;a)N

u.

Let f be the (k,m)-ary functional such that f ( Y , c ;a)N

u.

Since F is closed under basic operations and functional substitution it follows that f is F-computable, hence there is a number z such that f(y, 2;a)

N (ar+k,m(z,

y, z;a)

N (ak-m(Sr(2, y), 2;a).

0 Hence we take g(y) = S'(z,y). Applications of Theorem 11.3 will be referred to as the abstraction property for the reflexive structure F. In most cases the term U will be determined by the context.

162

L.E. Sanchis

The abstraction property is the standard way to exploit the indexing structure in a reflexive structure. The application of the property induces a total primitive recursive function that returns as values indexes of F-computable functionals. The totality of the induced function is crucial and holds even if the values are indexes of partially undefined functionals. This possibility is explicitly identified in Example 11.4 below. EXAMPLE 11.3 Let 7 be a reflexive structure with internal interpreter (a,S). By the abstraction property there are primitive recursive functions 91, gz, and 93 such that

11.4 Let F be a reflexive structure with internal interpreter (a,S) and f a 2-ary F-Computable function (not necessarily total). For given k and rn there is a 2-ary primitive recursive function g such that

EXAMPLE

Note that if f(y1, yz) is undefined the right side is undefined, hence the left side is also undefined, although g(y1, yz) is defined, since g is total. The abstraction property implies the existence of a primitive recursive function g ( y ) = S k ( z , v )with a fixed number z. Since the function g is obtained from S by substitution, some properties of the latter will extend to the former. For example, if S is 1-1, then g is also 1-1. Furthermore, if S satisfies the condition z < S ( z , z ) and < S(z, z),the same increasing properties extend to the function g. Since in applications the numerical function S is an encoding, it follows that the preceding conditions are satisfied in most cases. In particular, they are satisfied by the universal interpreter of Chapter 12. These conditions apply immediately to the abstraction property, where a function g is introduced by substitution in the function S. In this case, it follows that y < g(y) < g(g(y)), etc. Several applications of this situation are discussed in Sanchis [27]. 'In most cases it is perfectly possible to omit the superscripts in the notation @'J"'for ' the arity can be recaptured from the arguments to which the functional

Chapter 11. Interpreters

163

is applied. So we may write simply ib(r,z;a),assuming 6 is the arity of z and rn is the arity of a. Still, there is some danger of confusion, for ib in the interpreter (@,S) is (1, 1)-ary and is also (1, 1)-ary. By omitting the superscripts in the latter we identify both functionals, which is not consistent with the definitions. We must be careful to avoid this type of ambiguity. @'I'

A better solution is to introduce a new type of notation, where instead of ibk~m(t,z; a)we write { ~ } ~ v ~a). ( zNow ; we can omit the superscripts and write { z } ( z ; a)without any danger of ambiguity. This has the extra advantage that in case f is a (k,m)-ary F-computable functional, and z is a number such that f ( z ;a) N @ k i m ( ~ ,a) z ; we can now write

and identify f with

{z}~!" and

{z}.

We can now write f = { z } ~ ~ = " {z}.

The notations { z } ~ ~and " {z} can be extended to more general situations where, instead of a fixed number z, we have an expression U that may be undefined for some values of the variables in U . In this case { U } k J " and {U} denote the (k,m)-ary functional such that { U } k ' " ( z ; a)N @""(U, a;a). The functional {U}ksm depends on the values of the variables in U (which must be different from the variables 2 ; a).If U is undefined, then {U}k,m = UDk,,. If the variables in the term U are y;p, we can introduce a functional g(y; p) N U and { U } = {g(y;p)}. But this transformation is trivial since g is still undefined when U is undefined. If U contains only numerical variables, then we can use the abstraction property and introduce a total primitive recursive function g such that {U}= {g(y)} (see Example 11.5). Note that in this case the function g is primitive recursive even if the term U contains functions that are not primitive recursive. This use of {U} must not be confused with the similar notation used in the definition of formal terms in Chapter 9. Here the expression U is numerical, and in a formal term {U}is used only when U is a type-(b, rn) term.

EXAMPLE 11.5 Let U be a functional F-term in the numerical variables y, where F is a reflexive structure with interpreter (a,S). By the abstraction property there

L.E. Sanchis

164

is a primitive recursive function g such that

hence { g ( y ) ) k ~ m = {U)kim and {g(y)} = { U ) . Note that in case U is undefined, g(y) is still defined and {g(y)}

= UD.

Theorem 11.4 Let 3 be a reflexive structure with internal interpreter (a,S ) , and f a (k 1,m)-ary F-computable functional. Then,

+

(i) If rn = 1 there is a number z such that f ( z , z ; a ) z @ ( S k ( z , x )a; ) . (ii) If m 2 0 there is a number z such that f(z,z; a )N PROOF.

{ Z } ~ ~ ~ a () X.;

To prove part (i), assume z' is a number such that

E+1 I @(S ( 2 Y, 2); a) 2 1

Take z = S(z',

2').

f W Y , Y), 2 ; a).

It follows that f ( z , 2; a)

=

f(S(z', z'), 2; a)

2:

@(Sk++',

z',

z

@(Sk(S(Z',

ZI), 2); a)

N

@ ( S k ( z , 2); a).

Part (ii) follows by applying part (i) to the (k f ( y , z ; [ a ] l...,[aim). ,

2); a)

+ 1,1)-ary functional f ' ( y , z ; a )

N

0

Corollary 11.4.1 Let 3 be a reflexive structure with internal interpreter (O,S), and g a total unary F-computable function. Assume k,m 2 0. There is a number z such that { z ) ~ ~ "= ' {g(z))k~".

+

Introduce the (k 1, m)-ary functional f(y,z; a ) N { g ( y ) ) k ~ m ( a), z ; and 0 take z as given by Theorem 11.4 (ii). PROOF.

Theorem 11.5 Let 7 be a reflexive structure with internal interpreter (a,S ) . Assume p ( ~a) ; N u is a recursive equation where u zs a recursive functional F-term in p ; x ;a . There i s a solution of the equation which is 3-computable.

Chapter 11. Interpreters

165

Replace all occurrences of p ( - ; -) in U by @kim(z,-; -) where z is a new variable. In this way we get a non-recursive specification of the form p ' ( z , z ; a)N U', and p' is T-computable. By Theorem 11.4 there is a number z such that p ' ( z , z ;a)21 @kJ'"''~, z;a).If we put p ( z ; a)N p'(z,z;a), it follows that p is T-computable and 0 also a solution of the recursive equation.

PROOF.

Corollary 11.5.1 If F is a reflexive structure, then 3 is closed under primitive recursion, unbounded minimalization, and basic recursion. In the proof of Theorem 5.1 it is shown how primitive recursion and unbounded minimalization can be obtained using basic recursion. In fact, in both cases the recursive equations have a unique solution, so from Theorem 11.5 it follows that the solutions are T-computable. From Theorem 5.2 it follows that F is closed under basic recursion. 0 PROOF.

Let T be a reflexive structure with internal interpreter (a,S), and PO a ( 2 , l ) ary T-computable functional. We say that PO is a (a,S)-preorder in T if whenever a is a total function, then the following three conditions are satisfied: P O 1: If po(z, z'; a ) is undefined, then both @ ( z ; a ) and @(z'; a ) are undefined. P O 2: If po(z, 2;a ) 11 0, then either O(z; a) is defined or

P O 3: If po(z, z'; a ) $ 0 , then

@(z'; a ) is

a(%'; a)is undefined.

defined.

A preorder functional PO is not necessarily quasi-total, but po(z, z'; a ) is defined (when a is total) in case either @ ( z ;a ) or @(z'; a ) is defined. Furthermore, from the value of po(z, z'; a)we can derive information about O ( z ;a ) and a("';a ) . This is clear from condition P O 3. Note that we can always arrange the functional in a ) is defined, and this implies that such a way that po(z, 2'; a ) N 0 only when a(%'; @(z; a ) is defined. We cannot expect that a functional of this type is available in any reflexive structure, even if we restrict the conditions to applications with total function arguments. In Chapter 15 we give an application where a preorder functional is constructed for classes of the form R C ( T ) where F is a finite normal class. Theorem 11.6 Let T be a reflexive structure with internal interpreter (a,S ) , and 2 0 there is a (k 1, 1)-ary T-computable func-

PO a (@,S)-preorder in T . If k

+

L.E. Sanchis

166

, a) is defined, then tional h such that i f a is a total function, and @ ( S k + ' ( z , zy); cP(Sk+'(z,z,h ( z , z ;a)); a) is also defined. PROOF.

We introduce several auxiliary functions. The number

z1

is taken such

that 2, y);

@(Sk+2(2',Z,

The functions

f1, f2,

and

f3

f4(w,

cp(Sk+'(Z, 5,y

+ 1); a).

are primitive recursive defined by substitution:

fz(2,z)

= =

f3(w, z , z )

=

fl(Z)

We introduce also a (k

a)21

S(Zl,Z)

Sk+'(z, 2 , O ) S"+'(W,

fl(z), 2).

+ 2,l)-ary functional f4: z , 2; a) N Sk+'(%,

2 , cp(f3(w,

z , z);a)

+ 1).

Finally, we introduce a (k + 2,l)-ary functional h' using definition by cases: h'(w, z , z;a)

N

0 if p o ( f ~ ( za), , f3(w, z , 2); a)N 0 0 if po(fz(z, a),f4(w, z , z;a);a)N 0

N

W + ' (fi(z), W,

N

2);

a)

+ 1 if po(fz(z, z),f4(w,

0,s; a);a)

90

It follows that h' is F-computable, hence using Theorem 11.4 (i) we fix a number w such that h'(w, z , z ; a) N cp(Sk+'(W, 2, z);a) Let PI and P2 be (k

+ 2,l)-ary predicates such that:

PI(%,2, y; a) @(Sk+'(z,

2, y);

a) is defined.

P z ( z , x , v ; a )h~' ( w , z , 2 ; a ) ~ v A c p ( S ~ + l ( z , 2 , wis) ;defined. a)

Assume PO is a (a, S)-preorder and a is a total function. We prove by induction on y that for all z , z , if Pl(z,z,y; a),then there is v such that Pz(z,z, w ; a). First, , a) is defined. From P O 1 it follows that we assume y = 0 and that @ ( S k + ' ( z , z0); PO(fZ(Z,

21, f3(w, 2, z);a)N t .

If t = 0, then h ' ( w , z , z ; a ) N 0 and Pz(z,z,O;a)holds. If t # 0, then from PO 3 it follows that @(f3(w, z,z); a) N @(Sk+'(w, fl(z),z); a) N u . Furthermore, f*(w, z , z ; a )N S'+'(z,z, u 1). We conclude that

+

PO(fZ(Z,

z),f4(w, 2 3 2 ; a);a)2 t'.

167

Chapter 11. Interpreters

If t’ = 0 we have again h’(w, z , z ; a) N 0 and Pz(z,z, 0; a)holds. If 2’ # 0, then from P O 3 it follows that @ ( f 4 ( w z, , z ; @);a)is defined, hence @ ( S k + ’ ( z , au, 1);a) is defined. Furthermore,

+

h’(w, z , 2 ; a )

N

@ ( S k + l ( W , !I(%),

N

u+l.

z);a)

+1

+

It follows that P 2 ( z I 2 , v;a) holds with v = u 1. Now we consider the case y 1, assuming the relation holds for y’ 5 y. We assume @(Sk+l(z,z,y 1);a) is defined, and by the induction hypothesis we can is undefined. It follows that assume that in case y’ 5 y, then @(SE+’(z,z,y’);a) @(Sk+l(fl(z),z,y);a) is defined, and by the induction hypothesis there is v’ such v’); a) is defined. This means that that h‘(w, fl(z),z; a) N v’ and @(Sk+’(f1(z),z, @ ( S k + ’ ( z , z , v ’ + l ) ; a )isdefined, so weneed only toshow that h ’ ( w , z , z ; a )N v ’ + l is defined, so Pz(z,z, v;a) is satisfied with v = v’ 1. Note that from our assumption above O(Sk+’(z,z,O); a ) is undefined, hence @ ( f 2 ( z , z ) ; a) is undefined. Furthermore, @(Sk+’ ( w l f l ( z ) , z )a) ; N h’(w, fl(z), z ; a )N v’, hence @(fs ( w , z , z ) ; a) is defined, hence from P O 2 it follows that

+

+

+

P O ( f i ( Z , 21,f 3 ( w ,z , z);

We know that @(Sk+’(t,z, v’ defined. I t follows that

$ 0.

+ 1);a) is defined,

hence

@ ( f 4 ( w ,z , z ; a);a) is

We conclude that

+

h’(w, z,v’ l),

N

@(Sk+’(W,

=

h’(w,

N

v’+ 1.

fl(Z),

f l ( % )2); , a)

+

+1

z;a) 1

If we take h ( z , z ;a) N h’(w, z,z; a)and assume PO is a (a,S)-preorder, it follows , a) is defined, then @ ( S k ( z , zh, ( z , z ;a ) ;a) is defined. that whenever @ ( S k ( z , zy); 0

Note that the proof of the preceding theorem does not involve functional substitution; in fact, it is identical to the proof given in Sanchis [27] for numerical

L.E. Sanchis

168

structures. Essentially, the proof involves a definition by cases and one application of the recursion theorem (Theorem 11.4 (i)). In applications the real challenge is the construction of the preorder functional, particularly the proof that it is an element of the reflexive structure. In the construction of Chapter 15 the construction is relatively simple because we have available functional recursion, which is not always the case in other presentations. Still, the construction involves many cases derived from the universal interpreter of Chapter 12. If F is a reflexive structure with internal interpreter (a,S), and PO is a (@,S)preorder in F ,we say that F is preordered. Let P be a (k 1,m)-ary predicate. A selector for P is a (k,m)-ary functional f that satisfies the following two conditions:

+

SL 1: If there is y such that P(y,z; a)holds and the functions in a are total, then f(z;a)is defined.

SL 2: If f(s;a)N y, then P ( y , z ; a)holds. We say that a class F of functionals has the p-selector property if whenever P is a (k 1,m)-ary predicate that is p-computable in F ,then there is a selector for P that is F-computable. 3 has the d-selector property if whenever P is a (k l,rn)-ary predicate that is d-computable in F ,then there is a selector for P that is F-computable.

+

+

11.6 Let P be a (L+l, m)-ary predicate, and x p some dual characteristic functional of P. Then the functional f such that EXAMPLE

is a selector functional for P. I t follows that if a class 3 of functionals is closed under unbounded minimalization, then F has the d-selector property. Theorem 11.7 Let

F be a reflexive structure. Then,

(i) F has the d-selector property.

(ii) If F is preordered, then F has the p-selector property. Part (i) follows from Corollary 11.5.1 and the argument in Example 11.6. To prove (ii) we aisume F is preordered, so there is an internal interpreter (a,S) PROOF.

169

Chapter 11. Interpreters

for F and a (2,l)-ary F-computable functional PO that is a ((a, S)-preorder in F. Assume P is a (k + 1,m)-ary predicate that is p-computable in F,hence there is a partial characteristic functional $p that is F-computable. Let z be an index such that 2, Y);

@(S"+'(Z,

a)2 M Y , 2; [all,.. . , [a],),

and set f(2;a)N [ $ P ( h ( Z , 2; ), 2 ; a)+ h ( 2 , z ; ), h ( z , 2 ;)]

where h is the (k + 1, 1)-ary functional given by Theorem 11.6. To prove that f is a selector functional for P we first check condition SL 1 and assume there is a y such that P(y,z;a) holds, where the functions in a are total. From condition P C F 1 it follows that (a(Sk++'(z,z, y; ))is defined, and from Theorem 11.6 it follows that f(z;a)is defined. To check condition SL 2, assume f(z;a)N y. From the specification of f it follows that $p(y,z; a) is defined, hence P(y,z; a)holds by condition P C F 2. 0 EXERCISES

11.1 Let F be a reflexive structure with internal interpreter ((a, S), and f a (k,m)ary F-computable functional. Prove there is a k-ary primitive recursive function g such that {g(z))(y;

11.2 Let

F

= {f(z;a))(y;

a)

a).

be a reflexive structure with internal interpreter

(a,S).

Prove:

(a) There is a unary primitive recursive function g such that

(b) There is a number

such that {z)(z;)N {z)(z;) holds for all z.

11.3 Let F be a reflexive structure with internal interpreter (a,S). Assume k,m 2 0. Prove there is a unary primitive recursive function g such that

holds for all y.

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170

11.4 Let F be a reflexive structure with internal interpreter ((a,S). Consider the (2,l)-ary functional PO such that PO(%,2’; a ) is undefined in case a is not a total function, and in case a is a total function, then PO(%,2’; a )

N

E

Prove that

PO

0 if (a(z;a ) is defined or 1 otherwise.

(a(%’; a ) is

undefined.

is not F-computable.

11.5 Let F be a class of functionals closed under basic operations and assume has the p-selector property. Prove that F p is closed under disjunction.

F

11.6 Let F be a class of functionals closed under basic operations and assume F has the p-selector property. Let D be a (1, 1)-ary predicate which is p-computable in F. Prove there is a (2,l)-ary functional POD which is F-computable and satisfies the following conditions whenever a is a total function: (a) If poD(t,y; a ) is undefined, then neither

D(+; a ) nor D(y; a ) holds.

(b) If PO,(+, y; a ) N 0, then D(+;a) holds. ( c ) If

POD(+, y; a) $

0, then D(y; a) holds.

Notes Interpreters are a classical tool in computability theory. They play a relatively minor role in our presentation, for we have available a general form of functional recursion. In fact, the universal interpreter of Chapter 12 is defined using functional recursion. The most important application in this work is Theorem 11.6, which is taken from Hinman [lo]. A slightly different proof is given in Fenstad [4]. More information about interpreters and reflexive structures is given in Sanchis ~71.

Chapter 12

A Universal Interpreter In this chapter we show how to construct reflexive structures, using a form of recursion that simulates the inductive relation R of Chapter 10. In fact, we give a general procedure to construct interpreters for classes of the form RC(T0) where 30is a finite class of functionals. Since the procedure is uniform we refer to the construction as a universal interpreter. We shall consider a class 3’0 = { G I , .. . ,Gt}, t 2 0, where each Gi is a (1, 1)-ary functional, i = 1,.. ., t . It will be clear that the procedure applies to functionals of any arity, and the restriction to arity (1,l) is adopted for reasons of simplicity. The construction of an interpreter for a class of the form RC(T0) proves that such classes are reflexive structures and all the results derived in Chapter 11 are valid in them. In particular, the selector theorem applies for classes where a preorder functional is available. Examples of such classes are given in Chapter 15. The interpreter for the class RC(F0) will be denoted in the form (@,S), where CP depends on FO but S is independent of TO.The construction is slightly more general, for we specify a (1,rn)-ary functional 0, for each rn 2 0, and each one depending on TO.Finally, we take @ = @ I . This requires an extra effort but will provide some extra applications. The specification of amis essentially a translation of the inductive relation R with FO as the given class of functionals. A delicate point in the translation of R ( U , r , p , y) is the representation of the assignment T. First, we note that in the translation the term U is assumed to be regular and linear, so we are interested 171

172

L.E. Sanchis

only in the values for type-0 and type-1 variables. Second, it is clear that we need to represent only the values for those variables which occur free in U . More precisely, although the relation R is defined by a non-finitary induction, in the derivation of R ( U , ~ , py), only the values of 7 for variables that occur free in U are necessary. Furthermore, since we assume that the term U is linear, it follows that the only free variables in U are type-0 and type-1. Note that U may contain occurrences of type (k,m)-variables, but these occurrences must be bound. Let z and p be numbers, and a = a1,. . . , a, a sequence of unary functions. We say that the assignment 7 is consistent with the triple ( z , p , a )if whenever 1 5 i < p , then ~ ( v ; = ) [I] , u represents U in a sense which is explained below, the assignment 7 is consistent with ( z , p , a )and the regular linear term U satisfies the conditions described above. We start by defining the manner in which a number u represents a formal term U of arbitrary type. Note that in applications the term U is linear, and the only free variables in U are t y p e 0 or type-1. Still, the term U may contain bound variables of arbitrary type, and these variables must be taken into account in the definition of the representation. We define the numerical representation by an induction that parallels the construction of terms by rules F M 1 to FM 8. The induction involves rules RP 0 to RP 9 as follows:

RP 0: If u = , i 2 1, then u represents the type-0 variable vi. RP 1: If u = , j 2 l , s , r 1 0, then u represents a type-(s,r) variable. In particular, represents the type-1 variable q j .

Chapter 12. A Universal Interpreter

RP 2: If u = ,z 2 0, then u represents a numeral with value

173 2.

RP 3 : If u = < 3 , u l > , where u1 represents a type-0 term U1, then u represents the type-0 term (Ul 1).

+

RP 4: If u = , where u1 represents the type-0 term U1, then u represents the type-0 term (Ul - 1). RP 5: If u = , < w 1 , . . .,w,>>,s,r 2 0, where uo represents a type-(s, r ) term UO,u; represents a type-0 term U;, i = 1,. . . , s , and wj represents a type-1 term W j , j = 1 , .. ., r , then u represents the type-0 term ( U o } ( U l , .. . , Ua; W I , .. ., wr). RP 6: If u = > , s , r 2 0, where uo represents a type-0 term UO, 2 1 1 , . . . , u, represent different type-0 variables v;,, . . . , v i , , and w 1 , .. . , wr represent different type-1 variables qj, , . . . , q j ? , then U represents the type-(s, r ) term ( A V ; ~ , . . . , v ; , ; q j , , .. . ,qjT)Uo. RP 7 : If u = < 7 , u l , u 2 > , where u1 represents a type-(s,r) variable ~ , s , 2r 0, and uz represents a type-(s, r ) term U z , then u represents the type-(s, r ) term (A.)Uz. RP 8 : If u = < 8 , i , u 1 , ~ 2 >where , 1 5 i 5 t , u1 represents a t y p e 0 term U1, and u2 represents a type-1 term U2,then u represents the term G;(Ul;Uz). RP 9: If -u = < ~ , u ~ , u z , uwhere ~ > , u l , u ~ , u 3represent t y p e 0 terms U1,Uz,U3, then u represents the type-0 term [U, -+ Uz,u s ] . 12.1 The term U1 = { q z } ( v z ; )is represented by the number ul = , , O>. The term Uz = (A; q2)Ul is represented by the number u2 = . The term U3 = ( A K ) U can ~ be represented by the number u3 = , if represents K .

EXAMPLE

12.2 The term U = (AQ; qz){qz}(vz;)is represented by the number u = , 0 > , < < 0 , 2 > , < l , 2 , 1 , 0 > > > .

EXAMPLE

The above conventions define for every formal term U a number u that represents U . It should be clear that from the given number u we can derive the complete structure of the term U . More precisely, any structural information about the term U can be obtained from the number u by a primitive recursive function.

L.E. Sanchis

174

Note that the indexes for a given functional in the universal interpreters are triples z = , where u represents the term U that computes the functional, and p is a number that covers U . In order to write the recursive specification of the functional amwe need a number of primitive recursive functions. As we assume that the basic techniques of primitive recursive definitions are known to the reader, we shall omit writing formal specifications for these functions. Instead, we shall give explanations that describe which kinds of transformations are performed when the functions are applied under given constraints. The action outside these constraints is irrelevant. The reader is encouraged to work out the specifications of the primitive recursive functions that are involved in the recursion. The most important are the functions ex, rpm,and st, which are explained below. Some of them may require a substantial amount of work. At any rate, note that it is sufficient to show that the functions are recursive in order to determine that the interpreter a, is recursive in &. First, we need some functions that are related to the encoding of tuples. We already know the binary function which, in case x = < q ,... , x k > , then [z]i = q , i = 1, ...,6. Here we shall write dl(z,y) = [z],,dz(r,yl,yz) = dl(dl(z,yl),yz), d3(2, y1,yz,y3) = dl(dz(z, y1,yz), y3), etc. In practice we shall omit the subscripts in dl, dz, d3, etc., since they can be derived from the number of arguments in ap. , t k > , then d ( t , 0) = k, and if k < y, plications. Note that whenever z = < q ,.. then d(z, y) = 0. We need also a concatenation binary function cc, such that whenever t = < q , . . . , z d >and y = < y 1 , . . . , y r > , then c c ( t , y ) = < q , ..., z,,y1,.. .,yr>. Note that cc(,) = < X I , .. . , x k , y>. Concerning the representation of an assignment T by a triple ( z , p , a )we need a function that changes T via z and p . We introduce a 3-ary function ex such that e x ( z , p , y ) = x’ where d(z’,v) = d ( z , v ) if v # p , and d(t’,v) = y if v = p. This amounts to defining T’ = [y/vP]r.

yI.[

The representation of terms by numbers induces several important numerical functions. The unary function rp, is such that rpm(z) = 0 if and only if z = < u , t , p > where u represents a regular linear type-0 term U , p covers U , and whenever a type-1 variable q, occurs free in U ,then 1 5 j 5 m. The 3-ary function s t describes substitution. If st(u, w,t ) = u’, where u represents a formal term U ,w = <w1,. . . , w , > , t = < t l , . . . ,t,>, s 2 0, ‘1ui represents a variable l c i , and ti represents a term Wi of the same type as I E ~ i, = 1,. . . , s, then

175

Chapter 12. A Universal Interpreter u’ represents the term U‘ = [Wl,. . . , W s / K l l . .

. , K,]U

1 2 . 3 From the explanation above we can see that st(u1, < < 1 , 2 , 1 , 0 > , < 0 , 2 > > , < < 1 , 1 , 1 , 0 > , < 0 , 1 > > ) = u’ where, in case u1 is the number of Example 12.1, then u’ represents { q l ) ( q ; ) .

EXAMPLE

We need several functions t o control the cases in the specification of @,(z;a). First, we need rp,(z) = 0, hence z = < u , z , p > . We need also t h e value d(u, 1) = d ( z , 1, l ) , so we set the binary function rpA such t h a t

Furthermore, when rpA(z, 5) = 0 we have to consider three subcases depending o n the value d(u, 2 , l ) = d(z, 1 , 2 , 1 ) = 1, 6, or 7. So we introduce the binary function eq, such that eqm(r,i) = rp&(z, 5)

+ eq(d(z, 1 , 2 , 1 ) ,4.

Furthermore, when eq,(z, 1) = 0 we need the value d ( u , 2 , 2 ) = d ( r , 1 , 2 , 2 ) = j to determine which is t h e input function aj which is applied in this case, so we set t h e binary function e q k such that e q A ( r , j ) = eqm(z, 1)

+ eq(d(z, 1 , 2 , 2 ) , j ) .

Finally, when r p k ( z , 8) = 0 we need the value d ( u , 2) = d ( z , 1 , 2 ) = i to determine the given functional Gi,so we introduce the binary function rg, such t h a t

We call the primitive recursive functions rpA, e q m , e q h , and rg, functions. We summarize their properties as follows: rpA(z,y) = 0 eqm(r,i) = 0 eqA(z, j ) = 0

rg,(z, i ) = 0

iff iff iff iff

the case

rp,(z) = 0 A d ( r , 1 , 1 ) = y rpA(z,5) = 0 A d ( r , 1 , 2 , 1 ) = i eq,(r, 1) = 0 A d ( z , 1 , 2 , 2 ) = j rp&(z,8) = 0 A d ( r , 1 , 2 ) = i

T h e above functions describe essentially the basic manipulations t h a t are required t o evaluate @ m ( z ; a )Still, . when they are applied in the recursive equations we get fairly complicated expressions, and it is convenient to introduce a number of abbreviations. These are given in the following eight functions:

L.E. Sanchis

176

The functions 91,.. . ,ga are given by substitution with primitive recursive functions, so they are clearly primitive recursive. They describe how an index of the form z = is manipulated during the reduction process in the definition of the relation R. The reader is encouraged to carry out the intended transformations, for these are necessary to understand the different cases in the specification of the functional ambelow. ,,,, which Now we proceed to write the recursive specification of the functional @ depends on the number m and also on the class 70= {GI,. . . ,Gt}.The specification is derived from rules RD 1 to RD 9 which define the relation R.To make explicit this relation we write to the left of each equation a reference to the original rule. Note that RD 5 breaks into m equations, depending on K = q1, K = q 2 , . . . , K = qm. Recall that the intended assignment of qj is a,. On the other hand, rule RD 8 breaks into t equations, depending on F = GI, F = G2,.. . , F = Gt.

RD RD RD RD RD

1 2 3 4 5.1

177

Chapter 12. A Universal Interpreter

.............................. RD 8.t

N

RD 9

2:

Gt(@rn(gl(z,d(z,1,3));a); if rg,(z,t) 2: 0 (Ay)@VTl(g7(z,y); a)) [@m(g8(%,2); a) @&8(Z, 3); 4, if rph(z,9) N 0 @m(gs(z,4);all if r p & ( z , 9 ) $ 0 UDl,m(z;a)

-

N

12.4 In order to evaluate @z(z;pd,s), where z = < u l , < 1 , 2 > , 3 > and u1 is the value from Example 12.1, we note 1 that eqa(z, 2) = 0, hence equation RD 5.2 applies: @Z(z;pd,s) N s(@z( , 3 > ; p d , s ) ) . EXAMPLE

Now we note that rp’,(, 0) = 0, hence 1 equation RD 1 applies: @z( , 3>; p d , ~ N ) 2.

We conclude that

@z(z; pd,s)

N

~ ( 2N ) 3.

Theorem 12.1 Assume R ( U , r , p , y ) , z = < u , z , p > , and rprn(z) = 0, where u represents the regular linear t e r m U . If a= a1,.. . , a r nand r is an assignment ) y. consistent with the triple ( z , p , a ) , then a m ( r ; aN PROOF. We use induction on the definition of R(U,r,p , y). All cases are straightforward and follow easily from the rules and the induction hypothesis. We discuss only case RD 8.i, where U = Gj(U1; V1) is represented by u = . From the rule we know that G ; ( x ’ ; p )N y, where ‘R(V1, r , p , z ’ ) holds, and the unary function /? is (r,p)-computed by Vl. By the induction hypothesis, noting that g l ( z , d ( z , l , 3 ) ) = < u l , x , p > , we get @,,,(gI(z,d(z,1,3));a) N 3’. We claim also

L.E. Sanchis

178

that the function (Xy)@,(g,(z, y ) ; a ) is an extension of p. Assume p ( z ) 2: z”. It follows that R ( { V 1 } ( u p ;[)z,/ v p p ] r , p +1,x”) holds, and by the induction hypothesis @,( ; a ) N d’, where u’ represents { V l } ( v P ; )Since . g7(2, y) = < u ‘ , e x ( z , p , y ) , p + 1>, we have @,(gT(z,y);a) N z”. Finally, we know that Gi

is monotonic, hence @ , ( z ; a ) N

N

Gi(@,(gl(z,d(z, 1 , 3 ) ) ; a ) ; (XY)@,(g7(Z, y ) ; a ) )

G i ( d;p) N y.

0

To prove the converse of the preceding theorem we introduce a new (1, m)-ary functional @A as follows: @&(%;a)N V ( U ) ( T )

1:

UDl,,(z; a)

Theorem 12.2 The functional

if rpm(z) = 0, and z = < u , z , p > where u represents the regular linear term U ,and the assignment r is consistent with (z,p , a); otherwise.

@A is an extension

of

am.

PROOF. We recall that 0, is the minimal fixed point of a monotonic transformation Tf = f’, where f and f’ are (1, m)-ary functionals. The transformation T is defined by the recursive equations for ,@, so f’ is obtained from f by the same equations, but writing f instead of .@, We are interested in the result of applying T to @A,so we write T@A = @$.To prove that @A is an extension of amit is .: sufficient to show that @$is closed under T ,i.e., that @A is an extension of a Hence we must show that whenever @ L ( z ; a N ) y, then @ $ ( z ; a )N y. The proof consists of analyzing each case of the specification of @$from @A, assuming @%(%;a) N y and proving @;(%;a) N y. For example, assume eq,(z, 7) = 0, hence @;(%;a) N @‘,(gs(z);a) N y. It follows immediately that z = < u , z , p > where u represents a term U = { ( A K ) W } ( U.~. .,Us; , V1,.. . , V?)and gS(z) = where 21’ represents a term U’ = {[(hc)w/~]w}(Ul,. . . ,Us; V l , .. ., Vr). The condition @ L ( g s ( z ) ; a N ) y means V ( U ’ ) ( T )N y, hence we have @ L ( z ; a )N V ( U ) ( r ) N V ( U ’ ) ( r )N y by Corollary 9.6.1. The case where eq,(z, 6) = 0 can be handled in a similar way using Corollary 9.5.1. We consider also the case rg,(z, i) = 0, so z = < u , x , p > and u = represents the term U = Gi(U1;U z ) . Here @:(z;a) 21 y means

G ~ ( @ ~ ( < ~ ~ , X , P > ; ~ ) ; ( X Y ) @ ~ 1>;a)) ( < ~ ’N~Y, ~ X ( ~ , P , Y )

Chapter 12. A Universal Inlerpreter

179

Theorem 12.3 Assume U is a regular linear type-0 t e r m and rpm(z) = 0, where I = < u , x , p > , u represents U , and r is consistent with ( x , p , a ) . The following conditions are equivalent:

(ii) @,(z;a)

N

y.

(iii) V ( U ) ( T )N y. The implication from (i) to (ii) follows from Theorem 12.1. The implication from (ii) to (iii) follows from Theorem 12.2. Finally, the implication from (iii) 0 to (i) follows from Theorem 10.8 (i).

PROOF.

The proof of Theorem 12.3 is, essentially a translation of the proof of the equivalence between structural and reductional semantics in Chapter 10, and we take advantage of this situation to simplify the argument. For example, the proof of Theorem 12.1 shows that the definition of the relation R translates into the definition of the interpreter am.Instead of proving the converse relation we use Theorem 12.2, and so we avoid the argument in Theorem 10.2. This is more convenient because the evaluation V is more algebraic than the relation R. The crucial implication from (iii) to (i) is the main theorem in Chapter 10. We are now in a position to complete the definition of the interpreter for the . . , Gt}is a finite set of given (1, 1)-ary functionals. class RC(Fo), where Fo = {GI,. The primitive recursive function S is taken as follows:

180

L.E. Sanchis

S ( z , v ) = . It follows that S k ( < u , O , p > ; z ) = < u , < z > , p > . We shall show that ( Q 1 , S ) is an interpreter for RC(F0). Theorem 12.4 Let f be a ( k , m ) - a r yfunctional, k , m tions are equivalent:

2 0. The following condi-

(i) f is recursive in Fo (ii) f is formally definable f r o m Fo. (iii) There is a number z such that f(z;a) N O m ( S k ( ~ , a). ~); PROOF. The implication from (i) to (ii) follows from Theorem 9.7. To prove the implication from (ii) to (iii), assume V ( V ) = f , where V is a closed type( k , m ) term. We assume V has been constructed in such a way that the type-0 variables 211,. . . , v k and the type-1 variables q l , . . .,q, do not occur in V . Take U = { V } ( v l , . . . , v k ;q l , . . . , q,), and let z = , where u represents U and p covers U . From Theorem 12.3 (ii) and (iii) it follows that f(z;a)N am(Sk(z,z);a).

The implication from (iii) to (i) is clear, for the functional amis recursive in TOand f is obtained from 0, by a numerical substitution involving a primitive recursive function. 0 The proof of Theorem 12.4 shows that whenever f is a functional recursive in Fo,then an index for f relative to the universal interpreter is given by < u , 0, p > , where the number p can be taken almost arbitrarily. We have already explained that, notwithstanding this situation, the parameter p plays an important role in determining the new variables that must be added during the reduction. This determination is not essential for the induction which defines the relation R,but it is essential for the recursive definition of the functional am.

Corollary 12.4.1 The class R C ( F 0 ) is a reflexive structure with internal interpreter (a1,S). Clearly, RC(F0) satisfies the closure properties of a reflexive structure, 0 and (Q1,S) is an internal interpreter by Theorem 12.4 (i) and (iii).

PROOF.

Chapter 12. A Universal Interpreter

181

The preceding results have been obtained under the assumption that the finite class 3 0 contains only (l,l)-ary functionals. It is obvious that the procedure can be extended to functionals of arbitrary arity. The only change takes place in the equations corresponding to rule RD 8 in the recursive specification of a,,,. All the theorems and corollary proved above remain valid with this extension. In particular, R C ( F ) is a reflexive structure for any finite class 3, with interpreter (a1,S).

Theorem 12.5 Let F be an arbitrary class of functionals. Then R C ( 3 ) = FD(F). The inclusion R C ( F ) FD(F) follows from Corollary 9.7.1. To prove the converse, assume f is formally defined from F ,so there is a closed term V such that V ( V )= f . Since V contains only a finite number of functionals from the class 3,we conclude there is a finite class 30C F such that f is formally definable from Fo.From Theorem 12.4 (in the extended form discussed above) ,it follows that f is 0 hence recursive in 3. recursive in 30, PROOF.

Now we can satisfy the claim that the functionals recursive in a class 3 are recursively 3-computable. This assertion is a part of the extended Church’s thesis, which was left open in the discussion in Chapter 8. The problem can be solved now because the functionals a,, m 2 0 are recursively 3-computable. More precisely, if f is a (), m)-ary functional recursive in 3,then there is a finite class 30 3, and there is a number z such that f(z;a) N @ , , , ( S k ( z , z ) ; a ) where , amis specified in terms of TO.Given a recursive 3’0algorithm to compute a,,,, we can get an algorithm to compute f by initializing some fixed location with the value S k ( z , z ) .

For example, a system A of recursive algorithms that computes @ 1 relative to a class 30= {GI,. . . , G t } consists of two algorithms A1 and A2 written in the language of locations, where A1 is the main algorithm of arity (1, l), with input in X1 and output in X O , and A2 is an auxiliary algorithm of arity (2,1), with inputs in X I ,X2 and output in X O . To avoid complications we may allow to write assignments with arbitrary primitive recursive functions. The algorithm A1 computes @ l ( t ; a )assuming , the value of t is in location X I , by organizing the different cases in the recursive specification. The initial

L.E. Sanchis

182 instructions may be taken as follows: 1 : XZ +- rpi(X1,O) 2 : If XZ = 0 goto 3 else goto 5

3 : xo +d(X1,2,d(X1,1,2)) 4 : Halt 5 : XZ + rp:(X1,2) 6 : If XZ = 0 goto 7 else goto 9 7 : Xo+d(X1,1,2) 8 : Halt 9 : X2 rp;(X1,3) 10 : If X2 = 0 goto 11 else goto 15 +-

11: x3 + g l ( x l , d ( x l , L 2 ) ) 12 : X3 + A ~ [ X ~ ; C Y ] 13: X o + - X 3 + 1 14 : Halt

........................... Note that in instruction 12 the algorithm A1 contains a direct call to itself. In fact, most cases in the recursion are handled via this type of direct call. The execution of case RD 6 requires a short sub-routine to take care of the bounded product. We note here that, in fact, there is no need to evaluate the product, for the only purpose is to make sure it is defined. The whole case can be handled with the following instructions:

x3

i: +-d(Xi,1,3,0) i + l : If X3 = 0 goto i 2 else goto i i + 2 : x3 g3(x1) i + 3 : xo + A1 [X3;4 i + 4 : Halt i + 5 : x4 + 0

i+6:

x 5

i+7:

x 5

+

gz(X1,X4) A1 [X5;4 i + 8 : x4 = x4 + 1 i S 9 : If X3 = X4 goto i

+5

+

+

+ 2 else goto i + 6.

Chapter 12. A Universal Interpreter

183

The second algorithm, AZ, is required to handle case RD 8. This is an indirect call in A1 as follows:

The algorithm A2 computes 0 l ( g 7 ( z , y ) ; a ) with value y in X1 and z in Xz. The instructions are as follows:

Theorem 12.6 Let 3 be

a

class that contains only numerical functions. Then

M R ( F ) = RC(3).

We need only to prove the inclusion R C ( F ) C MR(3). Assume the functional f is recursive in 7 . Then f is recursive in a finite subset 30 3, and by Theorem 12.4 it follows that f(z;a)2: O , ( S k ( z , z ) ; a)for some number z . Since 30 contains only numerical functions the specification of 0, does not involve functional substitution and algorithm A2 above is not necessary. In this cme the recursion in the algorithm that computes 0, can be eliminated by the method of Example 6.3, so 0, is computed by an explicit algorithm. By the standard Church’s thesis this 0 means that 0, is p-recursive in 7 ,hence f is also p-recursive in 3. PROOF.

Corollary 12.6.1 MR = RC. PROOF.

Immediate from Theorem 12.6 with 3 = 8.

0

We can use the universal interpreter to define the jump operation on numerical functions. Actually, we need only the interpreter 0 0 , relative to a class FO that contains only numerical functions. Let p be a numerical function and aP= 00the universal interpreter relative to the class Fo = { p } . From Theorem 12.4 (i) and (iii) it follows that if h is a k-ary numerical function, then h is recursive in p if and only if there is a number z such that

h ( z ) N i P p ( S k ( r ,z ) ) .

L.E. Sanchis

184 The jump of p is the unary numerical function pj such that: Pj(Z)

N

0

N 1

if if

aP(t)is defined aP(z)is not

defined.

Clearly, pj is the unique dual characteristic function of the predicate Da,,. Note that p is not assumed to be a total function, but pj is always a total function. The theory of the jump is outside the material we cover in this work. It is essentially a first-order construction that applies to numerical functions. The function pj has some very special properties, and it is minimal with such properties. In'the first place, pj is a total function, even if p is not a total function. Second, the function p is p-recursive in pJ . Finally, pJ is not recursive in p. These conditions are proved in the next theorem. Note that the construction can be iterated, starting with a primitive recursive function p. In this way we generate p,p. p - , . . ., which is a sequence of functions J' U of increasing complexity. These results are applied later in the theory of continuous functionals in Chapter 14. Theorem 12.7 Let p be a numerical function and p the j u m p of p. Then, J

(i) If h is a numerical function recursive in p , then h is recursive an pj.

(ii) pj is not recursive in p. PROOF.

To prove (i) note that there is a number z such that

ap(Sk+'(z,Y,z ) ) = [eq(y, h ( z ) ) -, 0, UDI It follows that

h ( z ) N (py)pj(sk+'(z, Y, 2)). To prove (ii) assume p is recursive in p , and introduce the unary function f J such that

+

f(z) 2: [Pj(S(z,z))--* @p(S(ziz)) 1 i O I . It follows that f is a total function recursive in p, and there is a number z such that

f(.> = @p(S(Z, .I). Since f(z) is defined this means pj(S(z,z)) have a contradiction.

N

0, hence f ( z ) N (a,(S(z,z))

+ 1, so we 0

Chapter 12. A Universal Interpreter

185

Theorem 12.8 Let p be a numerical function and h a total k function recursive in p . Let g be the k-ary function such that g(z)

=

0

N

1

+ 1-ary numerical

if h(y, z)N 0 f o r all y otherwise.

Then g is recursive i n p . . J

PROOF.

If we set h‘(y,z) N [h(y,z)

+ 1,0] there

is a number z such that

The relation where a formal term U r-computes a functional f was defined in Chapter 10 relative to an assignment T. This is, of course, necessary, for in general the term U may contain free variables that must be accounted for. If U is a closed term without free variables the reference to T is irrelevant, so we shall write that U 70-computes the functional f , where TO denotes an empty assignment. Note that if U formally defines the functional f from F ,then the formal F-term U To-computes the functional f .

12.5 If the formal F-term U To-computes the functional f this does not mean that f is formally definable from F.Rather, it means that U formally defines from F a functional f’ which is an extension o f f . On the other hand, i f f is maximal so it has no proper extension, then we can say that f is formally defined from F,hence it is recursive in T . EXAMPLE

Theorem 12.9 Let f be a singular functional which is quasi-similar to a functional f ’ . If a total numerical function h i s recursive in f , then h is recursive in f’. If h is recursive in f there is a formal f-term U that defines h , hence U ro-computes h. By Corollary 10.9.1 the term U’ To-computes h , where U’ is the translation of U where f is replaced by f’. Since h is total, this means that U’ 0 formally defines h , so h is recursive in f ’ . PROOF.

L.E. Sanchis

186 EXERCISES

12.1 Evaluate the expression s t ( u 2 , < < 1 , 1 , 1 , 0 > , < 0 , 2 > > , < < 1 , 2 , 1 , 0 > , >) where u2 is the value given in Example 12.1. 12.2 Let F be a class of functionals closed under basic operations and functional substitution. Assume RC C F. Prove the following conditions are equivalent: (a) 3 is a reflexive structure.

(b) There is an F-computable (2,l)-ary functional ch’ such that whenever f is a (1, 1)-ary functional in 3 there is a number z such that f(2;a ) 2:

iP’(z,z;a ) .

( c ) There is an F-computable (1, 1)-ary functional ch” such that whenever f is a (k,1)-ary functional in F there is a b-ary primitive recursive function g such that f(z;a ) N ch.”(g(z); a ) .

12.3 Evaluate chz(;pd,s), where u’ is the value in Example 12.3. 12.4 Discuss case RD 5 in the proof of Theorem 12.1, where U = {tc)(Ul,.. .,Us;

Vl, . .

. I

K).

12.5 Discuss the case rp&(z, 9) = 0 in the proof of Theorem 12.2. 12.6 Assume the functional G1 is (2,2)-ary. Write a rule t o represent a term of I , Vwrite ~ ) , the equation to specify the value of the form G ~ ( U ~ , U ~ ; V and ch,(z;a) in case RD 8.1.

Notes T h e construction of the universal interpreter is by itself of great importance. There are also a number of substantial extra consequences that are applied later in this work. For example, the fact that definability implies recursiveness means that the powerful type of recursion available via the formal terms can be reduced to the relatively simple functional recursion. Furthermore, the recursion used to define the interpreter requires functional substitution only with the given functionals, which means that the function arguments enter the computation as simple parameters.

Chapter 12. A Universal Interpreter

187

Both the universal interpreter of Sanchis [27] and the one constructed here are defined by recursion, but they embody quite different types of constructions. The former involves a form of self-referential definition which induces the reflexive property in the interpreter. This idea originates with Kleene [16], and it is also used in Hinman [lo] and Tourlakis [28]. The construction we use here is more traditional, for the interpreter is derived by arithmetization of a universal computational formalism, in this case the reductional semantics for the formal terms. Corollary 12.5.1 comes from Tugu6 [29].

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Chapter 13

Enurneration In this chapter we are concerned primarily with classes of predicates, which can be generated in a number of ways. We assume such classes satisfy relevant closure properties and look at the functionals that are defined from them. If P is a class of predicates, then F g contains the functionals whose graph is in P. We shall see that Pg may satisfy important closure properties, depending, of course, on the closure properties of the class P. The relations between the classes 3 and 3 d and also between 3 and 3 p g , g’ were discussed in Sanchis [27] in terms of numerical functions. Here we consider functionals and predicates with non-total arguments, and this creates new problems. We propose new techniques which are useful in dealing with this situation. In dealing with classes of the form 3 d (or 3 ~ it )is important to take into account that the predicates are determined by their dual characteristic functionals (or by their partial characteristic functionals in the case of 3 p ) , and these functionals are not uniquely determined by the predicates. Still, the classes 3 d satisfy a number of closure properties, and these are sufficient for the applications in this chapter. Two new constructions which are used to define new predicates from given predicates are introduced in this chapter. Both are special forms of unbounded numerical quantification, existential or universal. Logically, they can be considered to be equivalent, since one can be defined from the other. In computability theory there is a natural hierarchy where existential quantification is more constructive than universal quantification. Here we are concerned primarily with the former. 189

L.E. Sanchis

190

The operation of unbounded quantification takes two forms, well known as existential and universal quantification. They apply to (k 1,m)-ary predicates and produce (k,m)-ary predicates, k, m 2 0. The formal definitions are as follows:

+

Universal U n b o u n d e d Quantification. Let Q be a (k We introduce a (k,m)-ary predicate P such that

P ( z ;a)

=

+ 1,m)-ary predicate.

(VY)Q(Y,2; Q) Q(y, z;a)holds for all y

We say that P is obtained from Q by universal unbounded quantification. Existential Unbounded Quantification. Let Q be a (k+1, m)-ary predicate. We introduce a ( k ,m)-ary predicate P such that

P ( z ;a)

3

( ~ Y ) Q ( Y2;, a) there is y such that Q(y,z; a)holds

We say that P is obtained from Q by existential unbounded quantification. Both quantifications, universal and existential, are non-finitary, for they determine a boolean value in terms of an infinite collection of values. So, they differ from the bounded quantifications of Chapter 4, which are finitary. For this reason we cannot expect that classes of the form 3 d or Fp are closed under unbounded quantification, even if 3 satisfies strong closure properties. Still, we shall see that this is possible in special cases. The determination of this type of closure is an important characterization of a given class 3 of functionals. In dealing with predicates in terms of functionals we must use characteristic functionals, and we know these are to some extent ambiguous, for in general a predicate has many partial characteristic functionals and many dual characteristic functionals. For this reason we must be very careful in translating definition by cases given by predicates into definition by cases given by functionals. On the other hand, this translation may be necessary if we want to insure monotonicity in the specification. For example, a specification of the form

h(z;a)

N

f i ( z ; a ) if

N

f2(z;a)

if

Q(z;a) -&(%;a),

is not necessarily monotonic and the mapping h is not necessarily a functional. A

191

Chapter 13. Enumeration monotonic approximation can be obtained by the specification h(z;a)

2:

f i ( z ; a ) if

N

fZ(z;a)

if

0 $ 0,

X Q ( Z ; ~ )N

XQ(z;a)

but this specification depends on the particular dual characteristic functional XQ. These problems do not appear in dealing with numerical functions, for a numerical predicate has a unique partial characteristic function and a unique dual characteristic function, and the translation above is faithful to the original specification. To handle this situation it is convenient to introduce the following notation. If f and g are (k,m)-ary functionals we say that f is a similar extension of g if f is an extension of g in the usual sense and f is also similar to g. The latter condition means that f(z;a ) N g(z; a ) whenever the a are total functions. We write g Es f to denote that f is a similar extension of g. This relation is clearly reflexive, transitive, and anti-symmetric.. When we say that a functional f is a similar extension of a functional g, this means that f is more defined than g, but only for non-total arguments. Since functionals are monotonic, and the values for non-total arguments are determined by the values for total arguments, the changes that f may introduce by being more defined than g are predictable from the common values of f and g for total arguments. So, we may say that f contains more explicit information, but such information was already implicit in g. Let F and F' be classes of functionals. We say that F' is a similar extension of F if 7 F',and whenever f is a functional in F' there is a functional g in F such that g Cs f. We write F c s 3' to denote that F' is a similar extension of F. This relation is clearly reflexive, transitive, and anti-symmetric. If F is a class of functionals, then 3; is the class that contains all functionals that are similar extensions of functionals in F. Clearly, F G s Fs. Furthermore, if F CS F',then F' CS Fs.

c

13.1 Let f be a (k,m)-ary functional and Gf the graph predicate o f f . I f f ' is a selector functional for G,, then f is a similar extension o f f ' . Hence, iff' is an element of the class F ,then f is an element of the class Fs.

EXAMPLE

13.2 If g is a k-ary numerical function, then the only similar extension of g is g itself. If 3 is a class of numerical functions, then F = 3;. EXAMPLE

L.E. Sanchis

192

c

The subscript “s” is clearly monotonic, so if F F’,then FS E F’s. Furthermore, it can be concatenated with other subscripts. For example, if F is a class of functionals, then Fsd and FSpare classes of predicates, FSdg and Fspg are classes of functionals. Let P be a predicate and F a class of functionals. If P is d-computable in the class PR(F) we say that P is primitive recursive in F. We denote by PRd(F) the class of all predicates that are primitive recursive in F. If 7 is empty we write simply PRd = PRd(0). w e say that the predicates in PRd are primitive recursive. If the predicate P is d-computable in the class MR(F) we say that P is precursive in 3. If P is p-computable in MR(F) we say that P is partially p-recursive in 3.We denote by MRd(F) the class of all predicates p-recursive in F ,and by MRp(7) the class of all predicates partially p-recursive in F. If F is empty we use the notations MRd and MRp. The predicates in MRd are p-recursive, and the predicates in M R p are partially p-recursive. If the predicate P is d-computable in the class RC(F) we say that P is recursive in 3. If P is p-computable in RC(F) we say that P is partially recursive in F.We denote by RCd(F) the class of all predicates recursive in F ,and by RCp(F) the class of all predicates partially recursive in F. If F is empty we use the notation RCd = the class of recursive predicates, and RCp = the class of partially recursive predicates. Note that in relation to the class PR(F) we introduce only the class PRd(F) of predicates that are primitive recursive in F. Of course, the class PRp(F) is also defined, but it is not used in our discussion. On the other hand, the classes MRp(F) and RCp(F) are important extensions of the classes MRd(F) and RCd, respectively. Finally, note that PR(F) C MR(3) C RC(F), hence PRd(F) MRd(F) & ( F ) and MRp(F) RCp(F).

c

c

Theorem 13.1 Let 3 be a class of functionals. Then,

(i) The class PRd(T) (MRd(F)) (RCd(F)) is closed under conjunction, disjunction, negation, universal bounded quantification, existential bounded quantification, distribution, and numerical substitution with quasi-total functionals primitive recursive (p-recursive) (recursive) in F.

(ii) The class MRp(7) (RCp(F)) is closed under conjunction, universal bounded quantification, distribution, and numerical substitution with functionals p recursive (recursive) in F.

Chapter 13. Enumeration

193

(iii) The class RCd(3) is closed under functional substitution with quasi-total functionals recursive in 3. Parts (i) and (ii) follow from Theorem 3.1, Theorem 3.3, and Theorem 4.6. Part (iii) is clear from the definitions, noting that the requirement that the functionals be quasi-total is necessary to make sure that conditions PCF 1 and DCF 1 are satisfied. 0 PROOF.

13.3 If 3 is a class of functionals closed under p-recursive operations, then the functional U D is 3-computable. If P is a predicate which is d-computable in F there is a dual characteristic functional x p which is 3-computable. From this we get a partial characteristic functional $ p , where EXAMPLE

This proves that 3 d C 3 p . It follows that MRd(3)

MRp(F) and RCd(3) C

RCP ( 3 ) . If P is a class of predicates and Q is a predicate obtained from a predicate in P by existential unbounded quantification we say that Q is P-enumerable. We denote by Pe the class of all predicates that are P-enumerable. If Q is a predicate obtained by universal unbounded quantification from a predicate in P we say that Q is Pcoenumerable. We denote by Pu the class of all predicates that are P-coenumerable. We denote by P c the class of all predicates that are obtained by the negation of a predicate in the class P. It follows that Pe = Pcuc, PU = Pcec, PCC = P, Pe, = PCU , and PUC = Pce. Theorem 13.2 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. Then,

(i) P e l P,, and P c are closed under numerical substitution with primitive recursive functions.

(ii) P

Pe and P

Pu.

(iii) Pe is closed under existential unbounded quantification, and PU i s closed under universal unbounded quantification.

(iv) If P is closed under distribution, then P, and PU are closed under distribution.

L.E. Sanchis

194

(v) If P is closed under conjunction, then Pe and

P, are closed under conjunction.

(vi) I f P is closed under disjunction, then Pe and PU are closed under disjunction. (vii) If P is closed under universal bounded quantification, then Pe and P, are closed under universal bounded quantafication. (viii) If P is closed under existential bounded quantification, then Pe and PU are closed under existential bounded quantification. The proof of part (i) is straightforward, noting that in the case of P, the conclusion follows because the primitive recursive functions are total. Part (ii) is also straightforward since by numerical substitution with projection functions we can add to a predicate in P a new dummy variable that can be eliminated by existential or universal quantification. To prove (iii) note that if Q is (k 2, m)-ary predicate in P , and P is a predicate obtained by two existential quantifications in the form: PROOF.

+

P ( z ;QL)

PY)PV)Q(Y, v , 2; a),

P ( z ;QL)

(3y)Q([yIi, [YIz,

then 2; a),

P is in Pe. A similar argument applies to P". Part (iv) is straightforward from the assumption. For parts (v) to (viii) we consider first the class Pe. If P = PI A Pz, where PI and Pz are in Pe, then there are predicates Q1 and Q z in P such that

so

P ( z ;a)

(3Y)Qi(y,z;

A ( ~ Y ) Q z ( Y2; , a) ( ~ Y ) ( Q ~ ( [ Y ] ~ , ~A; QQ Lz () [ Y ] z ; ~ ; Q L ) ) ,

so P is in Pe.If P = PIV Pz we use the same technique of exporting the existential

quantifiers, but here there is no need to contract them, for they can be identified. To prove part (vii), assume P is a predicate of the form

where Q is in the class P. It follows that

195

Chapter 13. Enumeration

so P is in the class Pe. Part (viii) for Pe is trivial, for it reduces to the interchange

of two existential quantifiers. To prove (v) to (viii) for P, we use duality. For example, assume P is closed under conjunction. Then Pc is closed under disjunction, hence Pce is also closed 0 under disjunction, and Pcec = P , is closed under conjunction.

Corollary 13.2.1. Assume

F is closed under primitive recursive operations. Then

(i) Fde is closed under existential unbounded quantification, distribution, conjunction, disjunction, universal bounded quantification, existential bounded quantification, and numerical substitution with quasi-total T-computable functionals.

(ii) Tpe is closed under existential unbounded quantification, distribution, conjunction, universal bounded quantification, and numerical substitution with F-computable functionals. (iii) If Pi is Fd-enumerable and P2 is Fp-enumerable, then P = PI v P2 is Fpenumerable. PROOF. Parts (i) and (ii) follow easily from Theorem 13.2 with Theorem 3.1, Theorem 3.3, and Theorem 4.6. To prove part (iii), assume Q1 is a predicate in Fd and Qz is a predicate in 3p such that Pi(2;Q) ( ~ Y ) Q ~ ( Y2;, a) Pz(2;Q) ( ~ Y ) Q z ( Y a; , a). It follows that

P ( z ;0) I (%)(Qi(y, a;Q) V Qz(Y, hence P is in

2; a)),

Fpe by Theorem 3.3 (iii).

0

Theorem 13.3 Let T be a class of functionals closed under p-recursive operations. Then 3 d = 3 d e fI Fdu.

The inclusiond' . E Tde n Tdu follows from Theorem 13.2 (ii). To prove the converse, assume P is a predicate such that PROOF.

P ( z ;a)

=

2; Q) (VY)QZ(Y, 2; a),

(3Y)Ql(Y,

L.E. Sanchis

196

where Q1 and Q2 are predicates d-computable in F. Assume x1 is a dual characteristic functional of Q1 which is F-computable, and xz is a dual characteristic functional of Q2 which is F-computable. We introduce a functional f such that

Clearly, f is 3-computable, so the functional

is also F-computable. We claim that x is a dual characteristic functional of P . To prove this we must check that conditions DCF 1 and DCF 2 are satisfied. To check DCF 1, assume the functions a are total. It follows that x l ( y l z ;a) and ~ 2 ( y , z ; a are ) defined for all y, hence f ( y , z ; a ) is defined for all y. We must show that there is some y such that f ( y , z ; a)N 0. If there is y such that u x l ( y , z ; a)N 0, then f ( y , z ; a)N 0. Otherwise, we have p(z;a), so there is y such that x2(y,z; a)N 0, hence f ( y , z ; a)N 0. This means x(z;a)is defined whenever the a are total functions. To prove DCF 2, note that if x(z;a)N v, then either v = 0 or v = 1. If v = 0 there is y such that x1(y,z; a)N 0, hence P ( z ;a)holds. If v = 1 there is y such 0 that x l ( y , z ; a)E 1, and f(y, z;a)N 0, so P ( z ;a)does not hold.

13.4 Let F be an arbitrary class of functionals. It follows from Theorem 13.3 that M R d ( F ) = M R d e ( F ) n M R d U ( F ) and R C d ( 7 ) = RCde(F) n RCdu(F).

EXAMPLE

Corollary 13.3.1 Let MRde

F be an arbitrary class of functionals. Then M R d ( F ) =

MRdec(F).

From Theorem 13.3we know that M R d ( F ) = M R d e ( F ) f l M R d U ( F ) . Since M R d ( F ) is closed under negation, we get

PROOF.

MRd(F)

*

= MRde(F) MRdcec(7) = MRde(F) n M R d e c ( F ) . 0

The result of Corollary 13.3.1can be expressed in the following form: a predicate P is p-recursive in the class F if and only if both P and the negation of P are M R d (F)-enumerable.

Chapter 13. Enumeration

197

Let P be a class of predicates. We recall that Pg is a class of functionals where f is in P g if and only if the graph predicate G, is in F. It is convenient to introduce another subscript "#," where P# = Peg. A functional in P# is said to be ?-enumerable.

13.5 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. It follows that P Pe, hence Pg C Peg = P#. If P is also closed under existential unbounded quantification we have Pe P , hence P# Pg,so Pg = P#. EXAMPLE

c

13.6 Assume the class T is closed under primitive recursive operations and f is a quasi-total functional in 3. It follows that a dual characteristic functional xf of G, is given by

EXAMPLE

X,(Y, a;a)2 eq(Y,f(z;a)), hence f is in the chss Tdg C Td#.

13.7 Assume the class 7 is closed under primitive recursive operations, contains UD , and f is a functional in 3. It follows that a a partial characteristic functional t+b, of G, is given by: EXAMPLE

hence 3 E 3 p g

Tp#.

P. Assume P is closed under numerical substitution with primitive recursive functions, distribution, conjunction, disjunction, universal bounded quantification, and existential

Theorem 13.4 Let P be a class of predicates such that PRd

unbounded quantification. Then Pg is closed under p-recursive operations. PROOF. From the assumption PRd C P it follows that Pg contains the basic functionals and the function s (see Example 13.6) and, furthermore, the equality predicate = is in P . We must prove that Pg is closed under numerical substitution, distribution, definition by cases, primitive recursion, and unbounded minimalization. The general approach is to prove that whenever a functional h is obtained by one of the rules, then G h can be expressed using the closure properties of P. If h is obtained by numerical substitution in the form

L.E.Sanchis

198

where the prefix (3yl . . .y a ) stands for ( 3 ~ ~ , . () 3 . ~ ~ ) . Closure under distribution follows immediately from the closure of P under distribution. If h is obtained using definition by cases in the form h ( z ; a ) N [ s ( z ; a )----* we can express

Gh

f 1 ( z ; a )f,2 ( z ; a ) ]

as follows:

Now assume the functional h is given by primitive recursion in the form

h(0,z;a)

h(v

+ 1,z;a)

We express the predicate

Gh

N

g(2;a)

N

f ( h ( v ,2; a ) ,v , z ; a).

as follows:

( 3 ~ ) G g ( [ ~z;]a) i, A (Vz < v ) G f ( [ ~ ] z [+~~] ,z + l t2; z , a) A [w]u+1= Y .

% ( Y , v , z ; a)

Finally, we consider an application of unbounded minimalization where

We express the predicate

Gh

in the form

0

The crucial assumption in the proof of Theorem 13.4 is that the class P is closed under existential unbounded quantification. The importance of existential quantification can be seen from the treatment of primitive recursion and unbounded minimalization. In both cases the quantifiers asserts the existence of a computation where the output is the value of the functional h.

Corollary 13.4.1. If T is a class of functionals closed under primitive recursive operations, then F,j# is closed under p-recursive operations.

Chapter 19. Enumeration

199

F ,hence w e apply Theorem 13.4 with P = Tde, noting that PR PRd C ’d. C F d e . The closure properties follow from Theorem 3.1, Theorem 3.3, 0 Theorem 4.6, and Theorem 13.2. PROOF.

Corollary 13.4.2. Let 7 be a class of functionals closed under p-recursive oper-

ations. Then

(i)

T p e is closed under disjunction and existential bounded quantification..

(ii) T p # is closed under p-recursive operations. To prove (i), assume P = PI V Pz where P1,Pz are in T p e , so there are predicates Q1, Qz in Fp and PROOF.

We introduce a predicate Q such that

To prove that Q is in Tpwe show there is a partial characteristic functional of Q in F.We set

where $1 is a partial characteristic functional of Q1 in T , and $2 is a partial characteristic functional of Q2 in T . Since Q is in Tp, and we can express P in the form

P ( z ;a)

=

(~Y)Q(Y a; , 4,

and it follows that P is in 3 p e . Closure under existential bounded quantification is trivial. To prove (ii), we apply Theorem 13.4 with P = T p e . From Example 13.3 we have PRd E T p .The closure properties follow from Theorem 3.1, Theorem 3.3, and 0 Theorem 4.6, with Theorem 13.2.

Theorem 13.5 Let T be a class of functionals closed under primitive recursive operations, and P a class of predicates closed under substitution with primitive

L.E. Sanchis

200

recursive functions. Assume every predicate in P has a selector in the class 3. I f f is a (k, m ) - a r y P-enumerable functional, then there is a (k, m ) - a r y functional h in f, where the class 3, and a unary primitive recursive function g, such that f’

ss

f’(z;a)= g ( h( z ; a)). PROOF.

Since f is in

P# there is a predicate Q in P

such that

We set Q’(y,z; a)3 Q([y]l, [y]2,z; a),take h a selector functional for Q’ in 3, and 0 a)N g(h(z ; a)). set g(y) = [yI2. It follows that f’ ss f, where f’(z;

Corollary 13.5.1 Let T be a class of functionals closed under primitive recursive operations. If 3 has the p-selector property, then T CS 3 p # . From Example 13.7, noting that UD is the only selector of a false predicate, we know that 3 3 p # . Hence from Theorem 13.5 it follows that 3 ssF P # .

PROOF.

s

0

Corollary 13.5.2 If T is a class that contains only quasi-total functionals, then MR(3)

GS

PRd#(F).

PROOF. From Example 13.6 we know that 3

s PR(3) c PRd#(F).

Since PRd# ( T ) is closed under p-recursive operations we have M R ( F ) E PRd#(F). From Theorem 13.5, with the construction in Example 11.6, we get M R ( F ) PRd#(F)

cs

0

These results apply to numerical functions, where CS is equivalent to =. Note that if P is a k-ary numerical predicate, then P has exactly one dual characteristic function x p , which is a total function. Furthermore, if C is a class of numerical functions, then MR(C) = RC(C) by Theorem 12.6. The result of Corollary 13.5.2 is essentially a generalization of the well known Kleene normal form for recursive numerical functions. The original version (see Kleene [14]) reduces the recursive functions by the application of existential unbounded quantification to a primitive recursive predicate. The same reduction holds for functions that are recursive in a total recursive function. The extension

20 1

Chapter 19. Enumeration

we present here applies to functionals that are p-recursive in given quasi-total funcwhich is a kind of weak tionals. The relation is expressed by the inclusion equivalence. If we restrict the reduction to numerical functions we get the original Kleene normal form. An equivalent formulation is given in the next theorem. Note that these results apply only to p-recursive operations and not to recursive operations in general. A formulation for recursive operations is given in Chapter 17.

s8,

Theorem 13.6 Let 3 be a class that contains only quasi-total functionals, and f a k-ary function. The following conditions are equivalent:

(i) f is p-recursive in 3 (ii) There is a k + 1-ary numerical predicate Q primitive recursive in 3, and a unary primitive recursive function g such that

To prove that (i) implies (ii), note that from Corollary 13.5.2 it follows that f is in PRd#(F), and the construction of Theorem 13.5 shows that there is a k-ary function f’ such that f ’ f and f ‘ ( z )= g ( h ( z ) ) , where g is a primitive recursive function, and h is a selector function for a predicate Q in PRd(3). Now since f and f ’ are numerical functions, this means f = f ‘ . Furthermore, the selector where Q is primitive recursive in 3 function h is of the form h ( z ) N (~Y)XQ(Y,Z), (see Example 11.6). 0 The implication from (ii) to (i) is trivial. PROOF.

cs

EXERCISES

13.1 Assume 3 and 3’are classes of functionals such that 3 CS 3’.Prove that 3 d = FA and 3 p = 3;. 13.2 If 3 is a class of functionals, then Ft denotes the class of all quasi-total functionals in 3. Assume 3 is closed under p-recursive operations and prove:

L.E. Sanchis

202 Fts = F d # t

(b) F d = F d # d . 13.3 Let C and C’ be classes of numerical functions. Prove that C if C = C’.

csC’ if and only

13.4 Assume F is a class of functionals closed under primitive recursive operations, P is a class of predicates closed under substitution with primitive recursive numerical functions, and every predicate in P has a selector functional in F . Prove that every predicate in Pe has a selector functional in 3. 13.5 Assume the class F is closed under basic operations and F p is closed under existential unbounded quantification. Prove that F p is closed under disjunction. 13.6 Assume a class 7 of functionals is closed under basic operations and has the p-selector property. Prove that F p is closed under existential unbounded quantification. 13.7 Let 7 be an arbitrary class of functionals. Prove Fpgs = Fpg. 13.8 Let 3 be closed under p-recursive operations and assume F has the p-selector property. Prove:

13.9 Let P be a class of predicates closed under substitution with numerical primitive recursive functions, negation, conjunction, and universal bounded quantification. Assume PI and Pz are P-enumerable (k,rn)-ary predicates. Prove there are ?-enumerable (k,rn)-ary predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of

Q1 and P2 is a boolean extension of

(b) Q1 V Qz is a boolean extension of PI V P2. N

(c)

Qz is a boolean extension of Q1.

Q2.

Chapter 13. Enumeration

203

13.10 Assume F is a class of functionals closed under primitive recursive operations, and F has the p-selector property. Let PI and P2 be Fp-enumerable ( k , m)-ary predicates. Prove there are 3p-enumerable ( k , m)-predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of

(b)

Q1 V Q 2

Q1

and Pz is a boolean extension of

Q2.

is a boolean extension of PI V P2.

N

( c ) Q2 is a boolean extension of

Q1.

Notes Enumeration plays a relatively minor role in the theory of recursive functionals. Still, it is important in dealing with p-recursive operations. For example, Corollary 13.5.2 characterizes the functionals that are p-recursive in a class F that contains only quasi-total functionals in terms of the functionals that are primitive recursive in F. It is easy to see that this result is a generalization of the traditional Kleene normal form for numerical functions recursive in total functions. To get similar results for the class R C F we must require the class F to be normal, and we must use hyperenumeration rather than enumeration. These results are proved in Chapter 17. Note that the relation 3 5, F' between classes of functionals is intended as a form of weak equality or equivalence. For example, it implies 3 d = 3 : and 3 p = 3.;.

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Chapter 14

Continuous Functionals In the first part of this work we have presented a theory of computable functionals, relative to a given class 7 of functionals, and we have asserted an extended Church’s thesis where the class of all such functionals is identical with the class R C ( F ) . Now we study the situation that arises when the given class 7 is restricted to a special type of functional. Such restrictions are important when the class R C ( 3 ) also satisfies them, which means that the particular properties of the functionals in the class 3 are preserved by recursive operations. We discuss here functionals where the evaluation involves only a finite number of values of the input functions. We call these functionals continuous, which is a

notation used in many places. The defining conditions are easy to formulate, for we consider functionals where the function arguments are partial functions. A similar notion of countable functionals has been studied in a different context where the function arguments are required to be total functions. In this case the definition must be expressed by means of some topological structure that makes explicit the relation of a total function to its finite subfunctions. The two theories can be related via the notion of singular functional, which is undefined for non-total arguments. The main purpose of this chapter is to prove a classical theorem from Grilliot [7] which characterizes those functionals that are effectively discontinuous. The original presentation w a s given using the topological definition with singular functionals. The singularity condition is still crucial in the main result, but some parts can be extended to non-singular functionals.

205

206

L.E. Sanchis

Let f be a (k,m)-ary functional. we say that f is continuous if whenever f(z;a) is defined, with a = a l , .. . ,am,then there is a’= a;, . . .,a;, where a: is a finite function and a: 21 airi = 1,.. . , rn, and f(z;a’)is also defined. Note that by the monotonicity of f we have f(z;a)N f(z; a’). 14.1 A k-ary numerical function, considered as a (k,0)-ary functional is always continuous. The (1, 1)-ary application functional ap(z;a ) N a(.) is also continuous, for if ap(z0;a ) is defined we can take a’(.) 1: a(.) if z = 2 0 , and &’(I) a , a’ is finite, and f(zo; a’) is also defined. undefined if z # 2 0 . Clearly, a’

EXAMPLE

EXAMPLE 14.2 Let f be a (1, 1)-ary functional such that f(z; a ) is undefined whenever a is a finite function, then f is continuous if and only if f = UD1,1. EXAMPLE

14.3 Let E be the (0, 1)-ary functional specified in the following form:

E(;a)

2:

0

N

1

N

if ( p ) a ( z ) is defined. if a(.) 9 0 for all I. UD(;a) otherwise.

Note that E(;a ) is undefined if and only if there is 2 such that a ( z ) is undefined, and for every I’ < z we have a(z’) 9 0. This functional is partially continuous, in the sense that whenever E(;a ) N 0, then there is a finite function a’ a such a that E(;a’) N 0. On the other hand, if E(; a ) N 1, then there is no finite a’ such that E ( ; a ’ )2: 1. So E is not continuous. The functional E in Example 14.3 may appear to be artificial, in the sense that E(;a) is undefined when a is a non-total function such that ( p ) a ( z ) is undefined. This makes it impossible for E to satisfy the condition of continuity when E(; a ) N 1. So it seems that E is not continuous, because we are arbitrarily restricting its domain of definition. This is a delicate point that requires some consideration. First, we note that it is impossible to extend E to obtain a continuous functional. So, E is not continuous in a very essential way. This suggests that our definition of continuity is too restrictive and that a functional should be considered continuous when some extension is available that satisfies the conditions in our definition. A more extensive definition of continuity is certainly possible and, in fact, it is used in the standard topological theory-for example, in Grilliot [7]. From a computational point of view we think this extension is undesirable. We do not want

Chapter 14. Continuous Functionals

207

a definition where the output values are determined by finite inputs but, rather, we

want the output values to be computed from finite inputs. Note that the results discussed in this chapter have been derived originally from the topological definition and can be formulated without difficulty in a more inclusive frame. We want to show that they also can be formulated in our theory, although at some points we must impose singularity restrictions which are closer to the topological approach. We denote by CT the class of all continuous functionals. We want to show that the class CT is closed under recursive operations. In proving that a given functional f is continuous we shall rely on informal arguments where the evaluation of f ( z ; a ) ,when defined, is shown to require only a finite number of values of the input functions a = c q , . . . , a,.

Theorem 14.1 The class CT is closed under p-recursive operations. Clearly, the basic functionals and the function s are continuous. Closure under numerical substitution, distribution, and definition by cases preserves the continuity of the given functionals. In primitive recursion we note that the evaluation of f ( y , z ;a ) is a finite process where we evaluate f ( 0 , z ;a ) ,. . . , f(y,z; a),and a t every stage of the process only a finite number of values of the input functions are required, so the whole process involves only a finite number of values. In unbounded 0 minimalization we have a similar situation.

PROOF.

Theorem 14.2 The class CT is closed under functional substitution. PROOF.

An application of functional substitution takes the form

Iv , where Pi(y) 11 g i ( y , z ; a ) , i = 1 , ...,s. If h ( z ; a ) N v , then f ( z ; P I , . ..,Pb)I Since f is continuous we need only a finite number of values of the functions P I , . :.’ , Ps. Since gi is continuous we need only finite number of values of the input functions a to evaluate P i ( y ) . We conclude that the whole process requires only a 0 finite number of values of the functions a.

Corollary 14.2.1 I f f is a ( k , m ) - a r y functional gauen by f ( z ; a ) N U ,where U is a functional C T - t e r m , then f is continuous.

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208

The functional f can be obtained by a sequence of applications of precursive operations and functional substitution. From Theorem 14.1 and Theorem 0 14.2 it follows that f is continuous. PROOF.

Theorem 14.3 If F 5

Fk,m

n CT and

is consistent, then LUBk,,(F)

is contin-

uous.

If LUBb,m(7) = g and g(z;a ) N v it follows from the proof of Theorem 2.1 that there is f in F such that f ( z ; a )r~ v . Since f is continuous there is finite a’such that a’C_l a and f ( z ;a’)N v . It follows that g ( z ; a’)N v , so g is 0 continuous. PROOF.

Theorem 14.4 Let T be a monotonic transformation on F k 0 msuch that whenever T g = g‘ and g i s continuous, then g’ is continuous. Then the minimal fixed point of T is continuous. PROOF. From the proof of Theorem 1.2 we know that g = g v , where the transfinite sequence {g,,} is obtained by applying the functional T and constructing least upper bounds. Since both operations produce continuous functionals, it follows that gv is 0 continuous.

Theorem 14.5 The class CT i s closed under recursive operations. Assume p is the minimal solution of a recursive equation p ( z ; a)N U , where U is a recursive functional CT-term in p ; z ;a. Then p is the minimal fixed point of monotonic transformation T p = p’ where p ’ ( z ; a) N U . From Corollary 14.2.1 it follows that whenever p is continuous, then p’ is also continuous. From Theorem 14.4 it follows that the minimal fixed point p is continuous. 0 PROOF.

Corollary 14.5.1 If F is a class that contains only continuous functionals, then R C ( F ) contains only continuous functionals. PROOF.

RC(F)

Since 3 CT.

CT, and CTis closed under recursive operations, it follows that 0

We study now the functionals which are not continuous at some function input. We have seen already an example of this situation, the functional E of Example 14.3,

209

Chapter 1.4. Continuous Functionals

where E ( ; a ) is not continuous at a whenever a is a function such that a(.) $ 0 for every x. It is convenient to restrict the discussion to (0,l)-ary functionals, which are sufficient to formulate the main results, and avoid notational complexities that appear when other arities are involved. If f is a ( 0 ,1)-ary functional and a is a unary function, we will write f(a)rather than f ( ; a ) . This applies, in particular, to the functional E where the application is written E ( a ) . I f f is (0, 1)-ary, f ( a ) is defined, and there is no finite unary functional a’such that a’ 2, a and f ( a ’ ) is defined, we shall say that f is discontinuous at a. I t follows that whenever a ( c ) 0 for all 2,then E is discontinuous at a. We have already mentioned the possibility that a functional f be discontinuous at a function a and, at the same time, some extension o f f be continuous at a. We know that this is not the case for the functional E, for if E is discontinuous at some a,then the same relation holds for any extension of E. Situations like this play an important role in the theory, and we introduce a type of structure that determines when a functional is essentially discontinuous at some point. This structure takes the form of a factorization of a total function p into two total functions d and r. At this stage we do not impose any restrictions on the functions d and r , and we discuss a number of examples. Later we shall be interested in factorizations satisfying special conditions. Let ( d , r ) be a pair of numerical functions, where d is a total binary function, and r is a total unary function. We say that the pair ( d , r ) is a factorization of a total unary function p if the relation p(y) 21 d ( r ( x ) ,y) is satisfied whenever y 5 x. Note that p is completely determined by the pair ( d , r ) ; in fact, p(y) -N d ( r ( y ) ,y ) holds for all y .

Theorem 14.6 Let d be a total binary function and r a total unary function. If p is a total unary function the following conditions are equivalent:

(i) ( d , r ) is a factorization of p. (ii) The relation d ( r ( z ) , y ) 21 d ( r ( z ’ ) , y ) holds whenever y 5 c 5

c’, and

p(y)

N

d ( r ( y ) ,y) holds f o r all y . PROOF.

and y

The implication form (i) to (ii) is clear from the definition. If (ii) holds 2: d ( r ( z ) ,y ) , hence ( d , r ) is a factorization of p.

5 y 5 x it follows that p(y)

0

210

L.E. Sanchis

14.4 Let d o ( z , y ) = [ z ] ~ +If~ ,f.3 is a total function we set r p ( c ) = . Then (do, rp) is a factorization of p, which we call the canonical factorization of p. Note that do is primitive recursive, and rp is primitive recursive in p. EXAMPLE

14.5 Let do be as in the preceding example. I f f is a total unary function 5 f ( + ) holds for all I , and p is a total unary function, we set r & ( z )= .Clearly, (do, rb) is a factorization of /3. Here r& is primitive recursive in p and f .

EXAMPLE

such that

2:

EXAMPLE

14.6 Let

p be

a total function. we set d p ( z , y )

N

p ( y ) if y

5

I,

and

d p ( z , y ) N 0 otherwise. If r ( z ) = z,then ( d p , r ) is a factorization of /3.

Let d be a total binary function, r and p total unary functions. I f f is a (0, 1)-ary functional we introduce the unary function f,j,,.(z) N f ( ( X y ) d ( r ( z ) , y ) ) .If f(P) is defined, ( d , r ) is a factorization of /3, and fd,,.(z) $ f ( P ) for all 2, we say that ( d , r ) is a deflector f o r f at p. 14.7 Let /3 be a total function such that p ( z ) $ 0 for all I . Then (do, r p ) is a deflector for the functional E at p. On the other hand, if p is a function such that ( p z ) p ( c ) is defined, then (do, rp) is not a deflector for E at p. In fact, in this case there is no possible deflector for E at p. EXAMPLE

Theorem 14.7 Let f be a ( 0 , 1)-ary functional and assume ( d , r ) is a deflector for f at p. Then

(i) i f f ‘ is an extension o f f , then f’ is discontinuous at /3; (ii) the functional E is recursive in {f,d , r};

(iii) i f f ’ is a (0, 1)-ary functional such that f(a)N f ’ ( a ) whenever f(a)is defined and a is a total function, then ( d , r ) is a deflector f o r f’ at p. To prove (i), assume there is a finite function p’ ,f3 and f’(P’) is defined. Then there is an I such that ( X y ) d ( r ( z ) y, ) is an extension of p. This means that f ( P ) N f ’ ( P ) N f;,,.(z) N f d , , . ( z ) , and this is a contradiction. To prove (ii) we introduce a (1, 1)-ary functional g such that PROOF.

Chapter 14. Continuous Functionals

so g is recursive in { d , r } . Since

E(a)

N

f(P)

211 is defined, let

f ( p ) N 2).

[eq(v, f((Xy)g(y;a)))

+

We claim that

1901.

To prove the claim we first m u m e ( p z ) a ( z )N 20.If y 5 20, then ( p z < y)a(z) N y, hence g(y; a) N d(r(y), y) N p(y) 1: d ( r ( z o ) ,y). On the other hand, if zo < y we have ( p< y)a(z) N 20, hence g(y; a) N d(r(zO),y). From this it follows that

=

e d v , f((Xy)g(y;a))) 1.

Assume now that a(.) g(y; a)N d(r(y),Y)

$ 0 for all z. In this case ( p z < y)a(z)

N

y, hence

= P(Y), and

Finally, assume E ( a ) is undefined so there is to such that a ( z 0 ) is undefined, and a(.) 0 when z < 20. In this case we have g(y;a) N d(r(y),y) N P(y) if y 5 20, and g(y;a) is undefined when y > 20, so (Ay)g(y;a) is a finite function, and /3 is an extension of (Xy)g(y;a). Since f is discontinuous at P it follows that f((Xy)g(y; a)) is undefined. This completes the proof of (ii). To prove (iii) we note that all functions involved in the definition of a deflector 0 are total functions, so f and f’ agree on those functions. EXAMPLE 14.8 Assume P is a total function and the factorization ( d p , r) of Example

14.6 is a deflector for f at 14.7 (ii) is given by

P. In this case the functional g in the proof

g(Y;a)

=

dP((PZ

of Theorem

< Y ) 4 P ) , Y).

The proof shows that whenever ( p z ) a ( z ) N 20, then g(y;a) N dp(z0,y) for all y, hence f((Xy)g(y;a)) $ f(@. On the other hand, if a is total but ( p z ) a ( z ) is undefined, then g(y;a) N p(y) for all y, so f((Xy)g(y; a))N I(@). Otherwise, E ( a ) is undefined and p i s an extension of the finite function (Xy)g(y; a),so f((Xy)g(y; a)) is undefined. Part (ii) of Theorem 14.7 was proved by Grilliot for singular functionals, which are undefined for non-total arguments, using a singular version of the functional E. In our version neither the functional E nor the functional f are singular. The result implies a kind of minimality condition for E in terms of recursiveness. We shall see later that in applications we may need to assume that the functional f is singular.

L.E. Sanchis

212

In the rest of the chapter we prove a converse to Theorem 14.7, where from the fact that the functional E is recursive in a functional f it follows that f has a deflector at some p recursives in f , This result is also due to Grilliot. Note that the proof of Theorem 14.7 (ii) uses only p-recursiion with functional substitution. Functional recursion is not necessary.

Corollary 14.7.1 Let f be a (0, 1)-ary functional. Assume there is a total function p recursive in f , such that f (p) is defined, and whenever f o r Q total function a there i s a number zo such that a(.) N 0 if z > 20, !hen f ( a )$ f ( p ) . Then E is recursive in f . PROOF. Consider the factorization ( d p , r ) of Example 14.6. Clearly, this is a deflector for f at p, hence E is recursive in { f,d p , r}, Since dp is primitive recursive in p, p is recursive in f , and r is primitive recursive, it follows that E is recursive 0 in f.

14.9 Let E’ be the (0, 1)-ary functional such that E’(a) is undefined if is non-total, and E’(a) N E(a) if a is a total function. This means that

EXAMPLE Q

E’(a)

=

[E((Ay)s(a(y)))

+

E(a), E(a)l,

hence E’ is recursive in E. On the other hand, E is recursive in E’, for we can apply Corollary 14.7.1 with p = cl. The functional E‘ is recursively equivalent to E and, furthermore, is singular. We shall find the latter condition to be very useful in some situations. The proof of the converse to Theorem 14.7 (ii) requires a number of strong assumptions about the functional f. We start by obtaining a result that relates the functional E and the jump operator of Chapter 12. Note again that only p-recursion with functional substitution is required. Functional recursion is not necessary.

Theorem 14.8 If p is a total function, then pj is recursive in {E,p}. Let aP be the unary interpreter for the numerical functions in RC(p). From Theorem 13.6 we know that there is a primitive recursive function g and a total binary function h primitive recursive in p , such that

PROOF.

213

Chapter 14. Continuous Functionals

Hence we can express pj in the form

0

A class F of functionals is said to be weakly normal if the functional E is recursive in F. A functional f is weakly normal if E is recursive in f . The functional E’ in Example 14.10 is weakly normal. A continuous functional is not weakly normal. Corollary 14.8.1 If F is a weakly normal class of functionals, and p i s a total numerical function recursive in F,then p is also recursive in F. J

PROOF.

Immediate from Theorem 14.8.

0

If ,B is a total unary function there is a canonical factorization of ,b of the form (do, PO) as explained in Example 14.5. The function do is primitive recursive, and

rp is primitive recursive in p. Note that there is a primitive recursive (1, 1)-ary ) r p ( x ) when /3 is a total function. In fact, the functional r such that ~ ( t , p N functional r is given by the following explicit specification: P(Z,

a)

N

( e x p ( 2 ,t

+ 1)

x (IIi 5 t ) e x p ( p n ( i

+ I), a ( i ) ) )

1.

Note that it is possible that r ( x ,a)is defined in cases where a is not a total function. Let f be a (0,1)-ary functional and p a total unary function. We say that f is effectively discontinuous at /3 if there is a deflector for f at p of the form (do, r ) where r is recursive in f. Note that in this case p is recursive in f . We say that f is eflectively discontinuous if there is a total unary function such that f is effectively discontinuous at p. EXAMPLE 14.10 The functional E is effectively discontinuous at any unary function

/3 which is recursive in E and satisfies the condition

p(x) 74 0 for all E , with deflector

( d 0 , r p ) . This is also true for the functional E’of Example 14.10. Note that if p is a total function such that p(z) N 0 for some 2 , then E’is discontinuous but not effectively discontinuous at ,B. In fact, there is no possible deflector for E’ at p.

In relation to the function do we introduce the primitive recursive binary function ext such that ext(v, ).

N

KEY < [ ~ l o ) e q ( d o ( Y), t , do(v, Y)).

L.E. Sanchis

2 14

+

Note that in case z = , then [z]~ = 2’ 1 , and whenever y I z’, then d o ( z , y ) 2: p(y). In general, if z < v , then ext(rp(w),rp(z)) N 0. T h e function ext is, of course, total. Now we fix a (0, 1)-ary functional f which, in general, will be assumed to be quasi-total, and we introduce a unary numerical function p such t h a t

I f f is quasi-total, then p is a total function. Theorem 14.9 Assume f is quasi-total and /? i s a total unary function recursive i n f . I f f is not effectively discontinuous at p, there is a number 20 such that whenever ezt(z’, r p ( z 0 ) ) N 0 , then f ( P ) N p ( z ’ ) . Assume there is no such number 2 0 . Then for every there is z’ such t h a t ext(z’, r p ( z ) ) 2: 0 and f ( p ) 34 p ( z ‘ ) . Let r‘ be the function such that

PROOF.

I t follows that r’ is a total function recursive in f , and ( d o , r’) is a deflector for f 0 a t ,B, so f is effectively discontinuous at 0. Corollary 14.9.1 Assume f is quasi-total, p is a total unary function recursive in f , f is not effectively discontinuous at /3, and there is a number x1 such that p ( z ’ ) N p ( r p ( z 1 ) ) whenever ezt(z’, r p ( z 1 ) ) N 0. Then j ( p ) N p ( r p ( z 1 ) ) .

14.1 1 We show now how Theorem 14.9 and Corollary 14.9.1 can be used to evaluate f (p) when f is quasi-total, p is recursive in f , and f is not effectively discontinuous a t p. Simply, we search for the least z such t h a t p ( z ’ ) N p ( r ( z , , B ) ) EXAMPLE

when ext(z’, r ( z , p ) ) N 0, and we set f ( P ) N p ( r ( z , P ) ) . To formalize the searching we introduce a binary function p’ and a unary function g. T h e function p‘ is primitive recursive in p :

Chapter 14. Continuous Functionals

215

The function g , which is not recursive in p , is given by the following specification: g(z)

2:

N

0 1

if p’(z’, z)N O for all otherwise.

2’

We can rewrite the expression above as

This relation holds, provided that f is quasi-total, p is a total function recursive in f , and f is not effectively discontinuous at p. Theorem 14.10 Let f be a quasi-total functional, and pj the j u m p of the unary function p above. I f f is not effectively discontinuous there is a functional f‘ which is recursive in p . , and f is quasi-similar to f ’ . J

PROOF.

We take the last expression in Example 14.12 and set

We know that r is primitive recursive, p is recursive in pj b y Theorem 12.7 (i), and f’ is recursive in p j . Since f is not effectively discontinuous it follows that whenever ,B is a total function recursive in f we have

g is recursive in pj by Theorem 12.8. It follows that

f ( P ) = f’(P), so f is quasi-similar to f’,

0

Corollary 14.10.1 Let f be singular and quasi-total. Assume f i s not effectively discontinuous. There is a function p recursive in f such that if h is a total function recursive in f , then h i s recursive an p . J

PROOF. We know that the functional f is singular and it is quasi-similar to f’ that is recursive in pj. By Theorem 12.9 it follows that h is recursive in f’, hence h is 0 recursive in p

J’

Theorem 14.11 Let f be a ( 0 , 1)-ary quasi-total, singular functional. The following conditions are equivalent:

( i ) f is weakly normal.

216

L.E. Sanchis

(ii) If p is a total function recursive in f , then pJ. is also recursive in f . (iii) f is effectively discontinuous. (iv) There i s a total function @, and a deflector ( d , r )f o r f at @, where both d and r are recursive in f.

The implication from (i) to (ii) follows from Theorem 14.8. The implication from (ii) to (iii) follows from Corollary 14.10.1. The implication from (iii) to (iv) is trivial. The implication from (iv) to (i) follows from Theorem 14.7 (ii). 0

PROOF.

EXERCISES

14.1 Give an example of a non-continuous (0, 1)-ary functional f such that whenever p is a total unary function there is finite @’ @ and f ( p ) is defined. 14.2 Give an example of a continuous quasi-total (0, 1)-ary functional f such that there is an extension o f f which is not continuous. 14.3 Assume f is a (0, 1)-ary quasi-total functional and @ is a total unary function. Prove the following conditions are equivalent: (a) I f f ’ is an extension o f f , and

@’

@ where

@’

is finite, then f ’ ( P ’ ) is

undefined.

(b) There is a deflector ( d , r ) for f at @ where r is a primitive recursive function. ( c ) There is a deflector for f at @.

(d) If f’ is similar to f , then there is a deflector for f’ at @. 14.4 Let f be a quasi-total (0, 1)-ary functional. Prove the following conditions are equivalent : (a) If @ is a total function there is no deflector for

f a t p.

(b) There is a functional f’ similar to f , and f’ is recursive in a total numerical function.

Chapter 14. Continuous Functionals

217

( c ) There is a continuous functional similar to f

14.5 Let f be a continuous quasi-total (0, 1)-ary functional. Prove that there is a continuous functional f’ similar to f , and f’ is recursive in a total numerical function.

14.6 Let Eo be the (0, 1)-ary functional such that Eo(Q)is undefined if Q is not a total function, and Eo(Q)cz 0 if Q is a total function. Prove that Eo is recursive in E but E is not recursive in Eo.

The theory of continuous functionals, in the context of recursive functionals, was initiated in Kleene [17]. A similar theory, oriented to the foundations of mathematics, is given in Kreisel [19]. The standard terminology is countable rather than continuous. The literature on countable functionals is rather extensive. For references, see Norman [22]. Theorem 14.11 is derived from Grilliot [7]. Degrees of continuous functionals are studied in Hinman [la].

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Chapter 15

A Selector Theorem In this chapter we prove that reflexive structures of the form RC(F), where F is a finite set of functionals, have the p-selector property, provided the functionals in 3 satisfy a number of restrictions. We apply here Theorem 11.7 (ii), so the result requires the definition of a preorder functional. The construction involves functional recursion with a fairly large number of cases. If (@,S) is an interpreter for the class RC(F), then a preorder functional PO satisfies a number of conditions. First, po(z,z’; a) is defined in case either @ ( z ;a) is defined or @ ( z ’ ; a )is defined. Second, if p o ( z , z ’ ; a )$ 0, then @ ( z ’ ; a )is defined. Finally, we assume that if po(z, z’;a ) $ 0, then @ ( z ;a) is defined. The last condition is stronger than the one imposed in the original definition. The preorder functional PO is specified by functional recursion derived from the universal interpreter of chapter 12, where @ = @ I is specified by functional recursion with equations RD 1 t o RD 9. We shall consider only the case where 3 = {G}, and G is a (1, 1)-ary functional. In order to specify PO we must assume that G is weakly normal, so E is recursive in G. To prove that PO is actually a preorder functional we must assume that G is quasi-total and singular. The preorder functional defined for the class RC(G) is a (2,l)-ary functional PO, specified by functional recursion. The cases of this recursion are derived from

the cases in the specification of @ in Chapter 12, and we use notation rpi, eqi, rg; , etc., with exactly the same meaning. As usual, the order of the equations in a 219

220

L.E. Sanchis

recursive system is essential, although in some disjoint cases the interchange of some equations will not change the specification. We introduce t h e different equations in the order they take in the specification. At this stage we make only one assumption about G, namely, that the functional E is recursive in G. This means that G is weakly normal in the sense of Chapter 14. T h e specification of PO involves two functionals t h a t are recursive in E, namely, E’ and E”. T h e functional E’ was defined in Chapter 14. T h e specification of E” is as follows: E”(a) is undefined if a is not a total function. If a(.) N 0 for all I, then E”(a) N 0. If a(.) 74 0 for some 2, then E”(a) N 1. T h e specification of PO follows a typical strategy, where the specification of p o ( z , z ’ ; a )depends on the specifications of @ ( % ; a and ) @ ( z ’ ; aby ) rules RD 1 t o RD 9. If @ ( z ; a )depends on @(y;a) (among other conditions), and @(“’;a) depends on @(y’; a ) among other conditions), then the specification of po(z,2’; a ) depends recursively on po(y,y‘; a ) . To organize these relations we assume that if po(y,y’; a ) N 0,then @(y; a ) is defined, and in case po(y,y’;a ) 34 0,then @(y’; a ) is defined. We assume also t h a t in case po(y,y’;a ) is undefined, then both @(y; a ) and @(y’; a ) are undefined. T h e preceding explanation should help the reader to understand the motivation for the different cases in the recursion. A complete explanation will follow from the proofs given a t the end of this chapter. Each case in the specification of po(z,z’;a)is denoted by a pair of numbers ( i , j ) ,which means that z is controlled by case RD i, and z’ by case RD j in the ’ ;) , respectively. In the first two equations where specification of @(P; a ) and @ ( t a i = 1 and i = 2, the parameter j is not relevant and we write ( 1 , O ) and ( 2 , O ) . Similarly, in equations 3 and 4 where j = 1 and j = 2, t h e parameter i is not relevant and we write ( 0 , l ) and (0,2). Case ( 1 , O ) : Here rp\(t,0) N 0 and z’ is arbitrary. po(z,z’;a ) N 0 Case ( 2 , O ) : Here rpi(z,2) 0 and z’ is arbitrary. po(2,2’;a ) N 0 Case ( 0 , l ) : Here z is arbitrary and rpi(z’,0) N 0. po(z,2’;a ) N 1 Case ( 0 , 2 ) : Here P is arbitrary and rp;(z’, 2) N 0. po(z,z’; a ) N 1

Chapter 15. A Selector Theorem

22 1

Note that in cases (1,O) and (2,O) po(z, z ; a) is defined even if @ ( z ’ a ; ) is undefined. Similarly, in cases ( 0 , l ) and (0,2) po(z, z’; a ) is defined even if @ ( z ;a ) is undefined.

po(z,z‘; a)N c y q z ; a ) ) Case ( 4 , j ) , j = 3 , 4 , 5 , 6 , 7 , 8 , 9 ,or 10. The equations are identical to the equations in case ( 3 , j ) .

L.E. Sanchis

222

Case ( 5 , j ) , j = 3 , 4 , 5 , 6 , 7 , 8 , 9 ,or 10. The recursive equation for case ( 5 , j ) is obtained from the equation in case ( 3 , j ) above, by the replacement of gl(z, d ( z , 1,2)) with gl(z, d ( z , 1,3,1)). Case (6,3): Here eql((z, 6) abbreviations:

N

0 and rp;(z’, 3)

2:

0. We introduce the following

Case ( 6 , j ) , j = 4,5, or 7. The equation in case ( 6 , j ) is similar to the equation in case (6,3), replacing gl(z‘, d ( z f , l ,2)) with gl(z‘, d(z‘, 1 , 3 , 1 ) ) if j = 5, and with gS(z’) if j = 7. No change is necessary when j = 4. Case (6,6): Here eql(z, 6) abbreviations:

N

0 and eql(z’, 6)

Case (6,8): Here eql(z,6) abbreviations:

2~

0 and rgl(z‘, 1) N 0. We introduce the following

Case (6,9): Here e q l ( 6 , l ) abbreviations:

N

0 and rp;(z‘, 9)

N

N

0. We introduce the following

0. We introduce the following

223

Chapter 15. A Selector Theorem

Case (6.10): Here eq1(6,1) 2: 0 and rp;(z’,9) $ 0. P O ( % ,t’;a) N cO(@(z;a)).

Case ( 7 , j ) , j = 3 , 4 , 5 , 6 , 7 , 8 , 9 ,or 10. The recursive equation for the case ( 7 , j ) is obtained from the recursive equation in case ( 3 , j ) above, by the replacement of g1(z, d ( t , 1 , 2 ) ) with g5(z). Case (8,3): Here rg,(z, 1) N 0 and rpi(t’, 3) ations:

11 0.

we set the following abbrevi-

A = PO(L?l(t., d(t,1,3)),!7l(Z’, 4 2 ’ 9 192)); a) B = E”((AY)P0(!77(Z,Y), g1(z’, 4 2 ‘ , L2)); a)) p o ( r , z‘; a)N [A B, 11 -+

Case ( 8 , j ) , j = 4,5,or 7. The equation in case ( 8 , j ) is similar to the equation in case (8,3), replacing gl(r’d(t’, 1,2)) with gl(z’, d(z’, 1,3,1)) if j = 5, and with gS(t’) if j = 7. No change is necessary if j = 4. Case (8,6): Here r g l ( t , 1) N 0 and eql(z’, 6) N 0. We set the following abbreviations:

A = E”((AY)(ni < d(z’, 1,3, o))PO(g~(z,Y), gdz‘, i ) ;a)) B = P 4 7 l ( Z , 4 t . j 1 , 3 ) ) , 9 3 ( 4 ;a) c = E”((~y)po(g7(z,Y), 93(2’); a)) D = (ni< d(z’, 1,3,O))po(gi(z, d(2,1,3)),9~(2’,i); a) PO(%,t‘; a)N [ A -+ [B -+ O,D], [C + B , 111. Case (8,8): Here r g l ( z , 1) c! 0 and rgl(t’,1) 2: 0. We introduce the following abbreviations:

L.E. Sanchis

224

Case (8,lO): Here r g l ( z , 1) N 0 and rp;(z‘, 9) $ 0. PO(%,2 ; a )N

c0(3(2;

Case (9,3): Here rp;(z, 9)

N

a)).

0 and rpi(z’,3) N 0. We set the following abbre-

viations:

Case (9, j ) , j = 4 , 5 , or 7. T h e recursive equation in case (9,j)is similar to the equation in case (9,3), replacing gl(z’, d(z‘, 1, 2)) with gl(z’, d(z’, 1 , 3 , 1 ) ) if j = 5, and with gs(z’) if j = 7. N o change is necessary if j = 4. Case (9,G):Here rp;(z, 9) viations:

N

0 and eql(z’, 6)

N

0. We set the following abbre-

Chapter 15. A Selector Theorem

225

L.E. Sanchis

226

A = po(gs(z, 2), ga(z’, 2); a) B = @(gS(Z,2); a) c = @(ga(z’,2); a) D = PO(g8(z, 3), !?8(z’,2); a) E = po(ga(z,4), ga(z‘, 2); a) D’ = Po(g8(Z, 217 !?8(2’, 3); a) E’ = po(ga(z, 2), ga(r’, 4); a) F = po(ga(r, 3), ga(Z’, 3); a) G = Po(gS(z, 3), ga(z’, 4); a) H = po(ga(z, 41, ga(z’, 3); 0) = Po(gS(z, 4 ) , !?8(z’, 4); a) M = [B + [ D --* 0, [C F, GI], [ E + 0, [C --* H , I~III N = [C --* [D’ + [B + F, HI, 11, [E’ --* [B --* G, K ] ,111 p o ( r ,z’; a)N [ A + M , N ] . +

Case (9,lO): Here r p i ( z , 9) N 0 and r p i ( z ’ , 9 ) $0. p o ( z , z ’ ; a) N

CO(@(%,

a)).

Case (10,O): Here rp;(z,9) $ 0 and z‘ is arbitrary. po(2,z’;a)N d(@(z‘; a)).

This completes the specification of the functional PO recursive in G under the assumption that G is weakly normal. To prove that PO satisfies conditions P O 1, P O 2, and P O 3 we will need extra assumptions about G. The process that determines the equations is, in fact, quite mechanical and fairly straightforward. To obtain the value of po(z, 2’; a) we consider the equations for @ ( z ; a and ) @(,’;a)which, in general, involve a number of subterms of the form @(yl;a),‘P(y2;a),. . . for @ ( z ; a) and @(A; a),a(&; a),. . . for @(r‘;a).The idea is to evaluate recursively all possible combinations po(y1, y;; a),po(y1, &; a), po(y2, y;; a),po(y2, &; a),. . . , and organize these values in a tree from which information about @ ( z ; a) and @(z’; a) can be determined. Finally, this information is used to determine p o ( z , z’; a). Each case in the definition of PO is given by an expression which describes an evaluation by cases. The reader is advised to expand these expressions in a tree form, from which the different sequences of cases can be derived as branches in the tree. For example, in case ( 9 , 9 ) the tree expansion takes the following form:

Chapter 15. A Selector Theorem

C

/ 0

D

/

227 A

/ \E

' '\ F

O

G

F

'\

G

Note that a branch to the left corresponds to a case = 0, and a branch to the right corresponds to a case # 0. The preceding tree describes seven possible branches, and each branch describes a possible evaluation. For example, the leftmost branch is A , C , D, 0 and corresponds to A N 0, C N 0, D N 0. The output of the evaluation is 0. Another possible branch is A , C , E , G, where A N 0, C N 1, and E N 1. The output of the evaluation is the value of G. Each case in the specification of PO has been written trying to satisfy two different goals. We refer to these goals as consistency and completeness. They are defined in the rest of this chapter. To satisfy one of the goals, we risk an injury to the other goal. The final proof shows that both goals are satisfied. Note that in general there are many ways to arrange the cases in such a way that both consistency and completeness holds, so there are many different preorder functionals that can be obtained by the method described above. For example, see Exercise 15.2. We say that an application of the form po(z, z'; a) is consistent where a is an arbitrary function, if p o ( z , z ' ; a ) N v and either v = 0 and @(%;a)is defined, or v = 1 and @("';a)is defined. Theorem 15.1 Let G be quasi-total, a total, and po(z, z ; a) N U where U is an expression determined by the corresponding recursive equation. Assume that U N v and that every application of po in the evaluation is consistent. Then, either v = 0 and @ ( z ; a) is defined, o r v = 1 and a(%'; a) is defined.

We must consider each possible form of the term U ,as determined by the recursive equations, and show that in every case the condition is satisfied. We discuss a few cases. PROOF.

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228

Consider the case (3,8), so po(z, z ; a ) N [ A + 0, B] N v .

There are two possible branches here, namely ( A , O ) where A N 0 and 2) = 0, or ( A , B ) where A N 1 and B N v . In the branch (A,O) the condition A TZI 0 and the consistency of PO implies that @(gl(z,d ( z , 1,2)); a ) is defined, hence @ ( z ;a ) is defined by RD 3, for a is a total function. In the branch ( A , B ) we consider v = 0 and v = 1. If B N v = 0, then there is at least one y such that p o ( g l ( z , d ( z , 1,2)),g7(z’,y);a) N 0, hence from the consistency of PO we get @(gl(z,d(z,1 , 2 ) ) ; a ) is defined, and @ ( z ; a )is defined by RD 3. If B N v = 1, this means that po(gl(z,d(z,1,2)),g7(z’,y);a) N 1 for every y, hence @(g7(z’,y);a) is defined for every y. On the other hand, from A N 1 we get that @(gl(z’,d(z’, 1,3)); a ) is defined. Since G is quasi-total it follows that @(z’;a ) is defined by RD 8. In case (6,8) the equation is of the form p o ( z , 2’; a ) N [ A 4 [B 4 0, D], [C 4 D , 1]],

and four branches must be considered. We discuss only the branch ( A , C , D ) , where A N 1, C N 0, and D N v. If v = 0 we get for at least one y that PO(gg(Z), g7(Z‘, y); a) 2: 0, hence @(g3(Z);a ) is defined. From C N 0 we get for at least one y that ( X i < d ( z , 1,3,O))po(gz(z, i ) , g7(z’, y); a ) N 0, hence @(gz(z,i ) ;a) is defined for i < d(z, 1 , 3 , 0 ) ) , and (IIi < d ( z , 1 , 3 , 0 ) ) @(ga(z,i ) ; a) is defined. We conclude that @ ( z ;a ) is defined by RD 6. If v = 1 we get that @(g7(z’,y); a ) is defined for every y, and from A N 1 we get that @(gl(z’,d(z’, 1, 3)); a) is defined. We conclude that @(z’;a)is defined by RD 8. In case (9,3) there are three branches and we consider the branch ( A , B , D ) , where A N 0, B $i 0, and D N v. If 2) = 0 we get @(gs(z,4); a) is defined and from B $i 0 we see that @ ( z ;a ) is defined by RD 9. If v = 1 we get @(gl(z’, 1,2)); a ) is defined. Since a is total we conclude that @(z; a ) is defined by RD 3. In caae (9,9) there are twelve branches and we consider the branch ( A ,B , E , C, H ) , where A N 0, B $i 0, E N 1, C N 0, and H N v . If v = 0 we get @(ga(z,4);a ) is defined, and from B $i 0 i t follows that @ ( % ; ais) defined. If v = 1 we get 0 @(ga(z’,3); a) is defined and from C = 0 it follows that @(z’;a) is defined. Corollary 15.1.1 Assume G is quasi-total. If p o ( z , z ; a) is defined, and a is a total function, then the application p o ( z , z’; a ) is consistent.

Chapter 15. A Selector Theorem PROOF.

229

We know PO is the minimal fixed point of a functional transformation

Tf = f’, where f and f‘ are (2,l)-ary functionals, so Tpo = PO, and whenever Tf C ~ , fJ, then f is an extension of PO. To apply this property we introduce a new (2,l)-ary functional PO’ such that: po’(z,2‘; a)

N

0

N

1

if whenever a’ is a total extension of a,then PO(%, z’; a‘) N 0 and @ ( z ; a‘) is defined. if whenever a’ is a total extension of a,then po(z, z’; a‘) 21 1 and O(z’; a’)is defined.

The functional PO’ is undefined outside these two cases. Note that when a is a total function, then po’(z,z‘; a) is defined if and only if the application po(z, z’; a) is consistent, and if this is the case, then po(z, z‘; a) N PO‘(%, z‘; a). Now we set Tpo’= PO” and prove that PO’is an extension of po”. If po”(z, z’; a) N v where a is an arbitrary function, then there is a case in the specification of PO where po(z, z’; a) N U , po”(z, 2’; a) N U1 N v, and lJ1 is obtained from U by replacing all occurrences of PO with PO’. To show that po’(z, z’; a)N v we take a total function a’ which is an extension of a and set U’ by replacing a with a’ in U , and U[ by replacing a with a’in U1. By monotonicity it follows that U; 21 v , and it is clear that the evaluation U; N v can be translated into an evaluation U’ I Iv . Furthermore, since the evaluation of U’ uses PO where the evaluation of V; uses PO’, it follows that the applications of PO are consistent. From Theorem 15.1it follows that v = 0 and @ ( z ;a‘) is defined, or v = 1 and @(z’; a’)is defined. This means that PO‘(%, z’; a)N v , and PO’ is an extension of PO”. The preceding argument proves that PO‘ is an extension of PO. Hence, if PO(%, z’; a)N v and a is total, then PO’(%, z’; a) 2: v and either v = 0 and @ ( z ;a) 0 is defined, or v = 1 and @(z’; a) is defined. We move now to the second relevant property of the functional PO, namely, completeness. We say PO is lefl complete at ( z ;a) if po(z, z’; a) is defined for all z’. Similarly, PO is right complete at (2’; a) if po(z, z’; a) is defined for all z. The completeness properties of a pair ( z ; a )depend to a great extent upon a set of numerical values that are determined by the recursive equation that applies in the evaluation of @ ( z ; a ) . So, if one of the rules RD 1 to RD 9 applies to @ ( z ; a )we shall say that (%;a) depends upon a set of values Dz;a as follows. If

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rule RD 1 or RD 2 applies, then Dzia is empty. If rule RD 3 or RD 4 applies, then Dz;a contains only gl(z,d(z,1,2)). If rule RD 5 applies, then DZiacontains only g l ( t , d ( z , 1,3)). If rule RD 6 applies, then D Z ; , contains all values gz(z,i) for i < d ( r , 1 , 3 , 0 ) , and also the value gS(t). If rule RD 7 applies, then D2;a contains only the value gS(.z). If rule RD 8 applies, then DZ;, contains the value gl(z, d(t,1,3)), and also the value g7(t, y) for all y. If rule RD 9 applies, then D2;a contains the values gs(t, 2). Furthermore, if CP(gs(z, 2)) N 0, then D 2 ; , contains the

value g 8 ( t , 3), and if @(gS(z, 2)) 74 0, then Dz;a contains the value gs(z, 4). If none t ; then D Z i ais undefined. of these rules apply to @ (a), Theorem 15.2 Assume

G is quasi-total and a is a total function. Then

(i) If @(%;a)is defined and PO is left complete at (y;a)whenever y is an element of D,;,, then PO is left complete at ( z ; a ) . (ii) Zf @@’;a) i s defined and PO is right complete at ($;a)whenever y’ is an element of D Z : ; , ,then PO is right complete at (2’; a). . To prove (i) we must show that p o ( z , z ’ ; a ) is defined for any arbitrary t and z’ we have po(z, z’;a) N U where U is an expression in the specification of PO. The expression U involves terms of the form po(y, y’; a) where y is an element of D,;,, so these terms are always defined. The expression U also involves occurrences of the functionals E’ and E”, which are defined because they are applied to total lambda expressions. Finally, U may contain expressions of

PROOF.

2. Depending on

the form O ( g s ( z , 2)) or CP(ga(t’, 2)), but then these expressions are defined because they are preceded (in the chain of cases) by conditions po(gs(z, 2), y;a) N 0 or po(y,gs(r’,2); a)74 0, and consistency applies by Corollary 15.1.1. 0 The proof of part (ii) is similar. Corollary 15.2.1 Assume the functional G i s quasi-total and singular. Then

(i) Zf CP(.z;a) is defined and a is

a

total function, then PO is left complete at

(2;

a).

(ii) If @(%‘;a)is defined and a is a total function, then PO is right complete at (2’;a).

PROOF.

To prove (i) we introduce a (1,l)-ary functional CP’ such that W ( z ; a)N v

if whenever a’is a total extension of a,then O ( z; a’)‘v v and PO is left complete at ( z ; a’),

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231

and @‘(z; a) is undefined otherwise. We shall prove that @‘ is an extension of @ by showing that T@’c1,1@‘, where T is the monotonic transformation induced by the recursive specification of @. we set T@’= @’’and assume @ ” ( z ; a)N v. This means there is a term U in the recursive specification of @, such that @ ( z ;a) N U , and U1 cz v, where U1 is obtained from U by replacing @ with a’. To prove that @’(z; a) N v we take a total extension a’ of a and obtain the term U’ by replacing a with a’ in U , and the term U i by replacing a with a’ in U1. By monotonicity we have Ui N v, and given the form in which 0’ is derived from @, we have also U’ v , hence @ ( z ; a ’ )N v. Noting again the form in which @’ is derived from @, and the assumption that G is singular, we conclude ~ follows that PO is left complete at (y, a‘). that whenever y is an element of D z ; a it By Theorem 15.2 (i) we conclude that PO is left complete at ( z ; a’).Since a‘ is an arbitrary total extension of a,we have a’(”; a) N v. The preceding argument shows that @’ is an extension of @, hence if @ ( z ; a) is defined and a is a total function, then PO is left complete at ( z ; a). The proof of part (ii) is similar. We introduce a functional @’ again, but using 0 right completeness rather than left completeness, and Theorem 15.2 (ii). Theorem 15.3 Assume the functional G is singular and quasi-total. If a is a total function, then PO satisfies the following conditions:

(i) I f p o ( z , z‘; a) i s undefined, then both @ ( z ;a) and @(z’; a) are undefined. (ii) If po(z, z’; a) N 0 , then d ( z ; a) is defined.

(iii) I f p o ( z , z’; a)$ 0 , then

$(z’; a) is

defined.

Part (i) follows from Corollary 15.2.1. Parts (ii) and (iii) follow from 0 Corollary 15.1.1.

PROOF.

Corollary 15.3.1 If G is singular and quasi-total, then PO i s a RC(G), and RC(G) has the p-selector property. PROOF.

Immediate from Theorem 15.3 and Theorem 11.9 (ii).

(a,S)-preorder i n 0

We say that a class 3 of functionals is normal if E is recursive in 3,and the functionals in 3 are singular and quasi-total. For example, the functional E is not singular, so E is not normal. On the other hand, the functionals E‘ and E“ are both normal.

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Theorem 15.4 A s s u m e the class 7 is n o r m a l . T h e n R C ( F ) h a s the p-selector property.

If the predicate P is in R C p ( F ) there is a finite class TO F such that P is in R C p ( F 0 ) and E is recursive in 3 0 , The construction in Chapter 12 provides an interpreter for R C ( F 0 ) and the procedure in this chapter can be easily extended to define a preorder functional for R C ( F 0 ) . It follows that P has a selector functional in R C ( F 0 ) 5 R C ( F ) . 0

PROOF.

EXERCISES

15.1 Assume one of the rules RD 1 to RD 9 applies in the evaluation of @ ( a). t; Assume also that a is a total function and the functional G is singular and quasi-total, Prove that @ ( z ;a) is defined if and only if @(y;a) is defined for every y in the set Dzio. 15.2 Redefine the specification of PO in case (8,6) by setting: p o ( r ;t';a) N [C + B

4

0, D], [ A -+ B , 111.

Prove that Theorem 15.1 and Theorem 15.2 are still valid with this definition. 15.3 Explain why the requirements that G is quasi-total and a is total are necessary in the proofs of Theorems 15.1 and 15.2. 15.4 Explain why the requirement that G is singular is necessary in the proof of Corollary 15.2.1. 15.5 Let f be a normal functional and p a numerical function recursive in f. Prove there is a total numerical function p' recursive in f which is not recursive in

P.

Notes The selector theorem was announced in Gandy [6]. The proof given here is derived from Hinman [lo]. For another application, see Chapter 5, $3, of Sanchis [27].

Chapter 16

Hyperenumeration This chapter is concerned with the properties of quantification over functions, although only a restricted form will be considered, namely, barred quantification. This type of quantification may be existential or universal, and existential barred quantification can be considered a generalization of existential numerical quantification. As the latter was referred to as enumeration, we shall refer to the former as hyperenumeration. In this context bounded numerical quantification loses its significance and is absorbed by unbounded quantification. So, essentially, we shall be dealing with two types of quantifiers: unbounded numerical quantifiers and barred function quantifiers, where each can be existential or universal. Occasionally we may introduce references to bounded quantification. To some extent we shall deal also with unbarred function quantification. General, or unbarred, function quantification is introduced informally. If U is a boolean expression, i.e., a totally defined expression which may take either of the two boolean values, then the expression

means that U holds for all total values of the function variable a. Similarly, the expression

(3a)U 233

L.E. Sanchis

234

means that there is some total function a such that U holds. Note that only total functions are included in the range of both quantifiers. The restriction of function quantification to total functions is another indication of the special role these objects play in the theory of recursive functionals. A similar situation was found in the definition of the characteristic functionals, and in some sense the rationale is the same in both cases. We want predicates to be total operations, well defined even for partial arguments. In representing and manipulating these predicates we must compromise and pay special attention to the total arguments. Partial arguments are essentially approximations to total arguments, and their properties need not be completely determined.

-

16.1 Let U be the expression a(.) N y. Then (Va)(Vc)(3y)U holds, and (3a)(3y)(Vz)Ualso holds. On the other hand, (3a)(3r)(Vy) U does not hold.

EXAMPLE

In order to define the barred quantifiers we need the auxiliary (1, 1)-ary primitive recursive functional br such that

Hence br(0,a) N 0 and br(z+l,a)N . We abbreviate br(z;a)2~ &(X).

Note that the functional br(z;a)may be defined even if a is not a total function. In practice this possibility can be ignored, for the expression E(z) is used when a is in the scope of a function quantifier, hence i t is restricted t o total functions. Existential Barred Quantification. Let Q be a (k 1, m)-ary predicate. We introduce a (k,m)-ary predicate P such that

+

We say that P is obtained from Q by existential barred quantification. Universal Barred Quantification. Let Q be a (k 1,m)-ary predicate. We introduce a ( k , m)-ary predicate P such that

+

We say that P is obtained from Q by universal barred quantification. We consider a barred quantifier, existential or universal, to be just one quantifier, even if we need two quantifiers to express its meaning. The use of "barred" function

Chapter 16. Hyperenumeration

235

quantification is similar to the use of “bounded” numerical quantification. Which one should be called existential and which one universal is a matter of personal preference. We think that by using the expression “barred” we are being explicit about the function quantifier, so we need only to identify the number quantifier. The two bar quantifiers are related by the usual rules that apply to the standard quantifiers. Hence

Furthermore, both can be exported over disjunction and conjunction, provided that collision of variables is avoided.

Theorem 16.1 There are primitive recursive functions ever P is a ( k + 2, m ) - a r y predicate, then

dl

and

d2

such that when-

See proof of this property in Theorem 5.1.1 of [25].

PROOF.

0

16.2 Let p be a total binary function. A descending pchain is a sequence (finite or infinite) to,1 1 , . . ., t,, . . . such that xi # Zi+l, and p(Zi+l, t i ) N 0 holds for i = 0,1, . . ., n,.. .. We can express the property that there is no infinite descending pchain in the form

EXAMPLE

where Q is the unary numerical predicate such that Q(y)

=

There are numbers 1 1 , . . . , t , , , t n + l , n 2 1, such that y = < t 1 , . . . , i , , , x n + 1 > , p ( z i + l , z i ) N 0 whenever 1 5 i < n, and either z, = z,+l or p ( z , + l , t , ) $! 0.

-

Similarly, we can express the property that there is an infinite descending pchain by the expression: (3P)(Vy) Q(p(y)). Let P be a class of predicates. If a predicate P is obtained by existential barred quantification from a predicate Q in P, we say that P is P-hyperenumerable. We denote by Phe the class of all P-hyperenumerable predicates. If P is obtained from

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236

Q by universal barred quantification we say that P is P-cohyperenumerable. We

denote by Phu the class of all P-cohyperenumerable predicates. As usual, these subscripts can be concatenated with the same or different subscripts. For example, we can write P d h c or Phcc. We can also write P h e h e or P h e h u , etc. If 7 is a class of functionals, then we can write F d h e or F p h c . We can also write F d h e g , which is a class of functionals. When the subscript is applied to a class determined by some operator we apply the subscript to the operator itself, which in this way becomes a new operator. For example, if F' = Rc(F), then we write = f?Cdhc(F), and RCdhe is an operator that can be applied to classes of functions to obtain classes of predicates.

Fihc

Theorem 16.2 Let P be a class of predicates closed under substitution with primitive recursive functions. Then,

(i) P

?he and p h c as closed under substitution with primitive recursive functions.

(ii) Phehe (iii) P c

c Phe.

c Phe.

(iv) phe = Pehe = Phee = Phehe. PROOF. Part (i) is trivial, and part (ii) follows from Theorem 16.1. To prove (iii),

assume the predicate P is obtained from Q in the form

and Q is in P . It follows that

so P is in Phc. To prove (iv), note that

phc

Pchc

Pchc

C Phcc

?hee Phchc

(since P C P c ) (since Pe C Phc C ?'he.)

Phchc

(by (iii) applied to Phc)

phc

(by (ii)). 0

Chapter 16. Hyperenumeration

237

Note that the relation Phehe C ?he means that ?he is closed under existential barred quantification. Similarly, the relation Phee = ?he in part (iv) of the preceding theorem means that p h e is closed under existential unbounded quantification.

Theorem 16.3 Let P be a class of predicates closed under substitution with primitive recursive functions, distribution, conjunction, and disjunction. Then (i) ?he

is closed under distribution, conjunction, and disjunction.

(ii) phe is closed under existential unbounded quantification and existential barred quantijcation. (iii) If P contains the primitive recursive numerical predicates, then phe as closed under universal unbounded quantification, and under universal and existential bounded quantification. Part (i) follows by exportation of barred quantifiers, and part (ii) from Theorem 16.2. To prove (iii) we note that PROOF.

P ( z ;a)

= WY)Q(Y,

2; a )

z;a ) A 0

(VP)(3y)(Q([p(y)lij

< [p(~)lo).

Closure under bounded quantification follows from unbounded quantification in combination with conjunction, disjunction, and primitive recursive numerical predicates. 0 EXAMPLE 16.3 Assume the class F is closed under primitive recursive operations. Then Theorem 16.3 applies to F d h e .

16.4 Assume the class T is closed under p-recursive operations. Then Theorem 16.3 applies to Tphe = Fpehe. EXAMPLE

To study the relation between hyperenumeration and recursive operations we introduce the (0, 1)-ary functional H such that:

H ( a ) N v iff

ct

is total, v = 0, and (V@)(3w)a(p(w))N 0.

Note that if P is the (1, 1)-ary predicate and Q is a (0, 1)-ary predicate such

that

P(y;a)

=

a(y)

&(;a)

=

( V P ) ( ~ W ) P ( P (a), ~>;

N

0

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238

then H is a characteristic functional of Q. The predicate P ( y ; a ) may hold even if a is a non-total function and, similarly, &(; a ) may hold even if a is non-total. On the other hand, H(a) is undefined when a is a non-total function. This is perfectly normal, for partial characteristic functionals are not necessarily defined for non-total arguments. Theorem 16.4 The functional H is recursive in the functional E. PROOF.

We recall the functional E” defined in Chapter 15 and introduce the func-

tional T O ( a )N [E”(a)+ O , O ] .

So TO is recursive in E and TO(&) is defined if and only if a is a total function, and if this is the case, then T O ( a )N 0. Next, we introduce a recursive equation with a (1, 1)-ary symbol p :

where cc is a binary primitive recursive function such that if cc(y, z) = z’, y = < y 1 , . . . , yn>,x = <x1,. . . , x,>, then x’ = < y 1 , . . ., yn, 2 1 , . . ., z,>. We assume that in case y is not a sequence number, then cc(y, z) = y for all x. As usual, this recursive equation determines a transformation T p = p’. We shall show that the minimal fixed point of T is the (1, 1)-ary functional H’ such that H’(y; a ) N v iff a is total, v = 0, and ( V ~ ) ( h ) a ( c c ( y , p ( w ) )N) 0.

First we set TH’ = H” and show that H’ is an extension of H”. If H”(y;a ) N v , then v = 0 and a is total, so it is sufficient to show that H’(y; a ) is defined. This is clearly the case if a(y) N 0, for cc(y,p(O)) = y. If a ( y ) $ 0, then from H”(y;a ) N 0 it follows that y is a sequence number, and H’(cc(y, <w>);a ) N 0 for every w , hence H’(y; a ) is defined. Next, we set Tp = p’, and assume p is an extension of p’, to prove that p is an extension of H’. If H‘(y;a) N 0 and y is not a sequence number, this means that a(y) N 0, hence p’(y;a) N 0, and p ( y ; a ) N 0. So we assume that y is a sequence number. To get a contradiction let us assume that p(y; a ) is not defined with value 0. This means a(y) $ 0 and p‘(y;a) is not defined with value 0, hence there is vo such that p(cc(y, ); a ) is not defined with value 0. By the same argument a(cc(y, )) $ 0 and there is v 1 such that p(cc(y, < v o , v l > ) ; a ) is not defined

Chapter 16. Hyperenumeration

239

with value 0. This process continues indefinitely, so there is an infinite sequence 2 1 0 , . . . , ZI,, . . . such that a(cc(y, )) $ 0 for every n. This means there is a function /? such that m(cc(y,p(w))) $i0 for every w , and this contradicts H’(y;a) 21 0. Since H’ is the minimal fixed point of the transformation, it follows that H‘ is 0 recursive in E. Furthermore, H(a) 2: H’(0; a ) , so H is recursive in E. Corollary 16.4.1 Let F be a class closed under recursive operations and assume E is T-computable. Then Fdhe F p .

where Q is in F d . Assume xg is a dual characteristic functional of Q which is T-computable. We get a partial characteristic functional $ p as follows:

This shows that P is p-computable in F.

0

Let P be a class of predicates. We set P h # = Phcg, and p h a = P h # d . The functionals in P h # are P-hyperenumeruble, and the predicates in Pha are Phyperarithmetical. If 3 is a class of functionals, the predicates in Fdhe are F-hyperenumerable, the functionals in F d h # are 3-hyperenumerable, and the predicates in Fdha are F-hyperarithmetical. In particular, the predicates in PRdhe are hyperenumerable, the functionals in PRdh# are hyperenumerable, and the predicates in PRdha are hyperarithmetical.

Theorem 16.5 Let P be a class of predicates such that PRd E P and which is closed under substitution with primitive recursive functions, distribution, conjunction, and disjunction. Then

(i) Ph# is closed under p-recursive operations.

(ii) P h a is closed under substitution with quasi-total P-hyperenumberable functionals, distribution, conjunction, disjunction, negation, and universal and existential bounded quantification.

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240

(iii) P h a is closed under existential and universal unbounded quantification. To prove (i) we apply Theorem 13.4 to P h e , noting that the closure requirements follow from Theorem 16.2 and Theorem 16.3. Part (ii) follows from part (i) and Theorems 3.1, 3.3, and 4.7. To prove part (iii), assume a predicate P is obtained in the form PROOF.

where Q is in Pha. Let X Q be a dual characteristic functional of Q in P h # , and GQ the graph predicate of X Q . We introduce a dual characteristic functional x p as follows: x p ( z ;a)21 v G

((Vy)Gg(O,y, S; a)A v = 0) V ((3y)Gg(l, y, 2; a)A v = 1).

From the closure properties in Theorem 16.3 it follows that x p is in Ph#, hence P is in Pha. This proves closure under universal unbounded quantification. Closure under existential unbounded quantification follows using closure under negation. 0 EXAMPLE 16.5 Assume T is a class of functionals closed under primitive recursive operations. Then Theorem 16.5 applies to the class T d h # (the class of Fhyperenumerable functionals) and to the class Tdha (the class of F-hyperarithmetical predicates). In particular, the theorem applies to the classes PRdh# (the class of hyperenumerable functionals), and PRdha (the class of hyperarithmetical predicates).

16.6 Assume T is a class of functionals closed under p-recursive operations. Then Theorem 16.5 applies to the class F p h # and F p h a .

EXAMPLE

We complete this chapter with a few definitions involving predicates and classes of predicates. They will be useful in formulating several results in the next chapter. Let P and P' be (k,m)-ary predicates. We say that P' is a similar extension of P if P' is a boolean extension of P and P is similar to P'. The latter condition means that P ( z ;a)G P ' ( z ;a)when the a are total functions. We write P 2, P' to denote that P' is a similar extension of P . When m = 0 this means, of course, that P = PI. A relation of the same form was defined in Chapter 13 for functionals, and the two are strongly related. For example, i f f Es f' where f and f' are functionals,

Chapter 16. Hyperenumeration

24 1

then G j C_s G j l . On the other hand, the relation general does not imply P CS P'.

$p

cs $ p ,

(or

xp

Gs x p t )

in

This relation is preserved by conjunction and disjunction, but it is not preserved by negation. Hence, if P CS P' and Q CS Q', then P A Q CS P' A Q', and P V Q SS P' V Q'. The relation is also preserved by bounded, unbounded, and barred quantification. Furthermore, it is preserved by general (or unbarred) function quantification, for the range of these quantifiers includes only total functions. Let P and P' be classes of predicates. We say that P' is a similar extension of P if P C P', and whenever P' is a predicate in P' there is a predicate P in P such that P P'. We write P GSP' to denote that P' is a similar extension of P . Note that if P ES P',then Pe CS PL, P, CS PL, P i c GS P i e , Pg ESPk, and P# Ss PB. We denote by PS the class of all predicates that are similar extensions of predicates in P. Clearly, P CSPs, and if P c s PI, then P' C_S Ps.

Theorem 16.6 A s s u m e the class F is closed under primitive recursive operations.

Then Fdh#p = Fdhes.

PROOF. Assume the

(k,m)-ary predicate P is in the class Fdh#p, hence there is

a partial characteristic functional $ p which is in F d h # so G Q is~ in Fdhe. Let P'

be the predicate such that

P'(z; a ) It follows that P' is in Fdhe and P'

G q p ( O , 2; a).

Cs P , hence P

is in the class Tdhes.

Conversely, assume P is in the class Fdhes, so there is P' in the class F d h e and P' cs P . If we set

it follows that Fdh#p.

$p

is a partial characteristic functional of P in F d h # , hence P is in 0

L.E. Sanchis

242 EXERCISES

16.1 Let P be a class of predicates closed under substitution with primitive recursive functions and conjunction. Assume P R d E P. Prove that PS= Pgp. 16.2 Let P be a class of predicates and assume P R d c P , P is closed under conjunction, disjunction, and substitution with primitive recursive functions. Let P be a (k,m)-ary predicate. Prove the following conditions are equivalent: (a) P is P-hyperarithmetical.

(b) There are boolean monotonic P-hyperenumerable (k,m)-ary predicates * PI and P2, such that PI2 s P and P2 c~P . 16.3 Let P and P' be classes of predicates such that P Es P'. Prove that Pg

GS Pi.

16.4 Prove that if H'(cc(y, <w>); a) N 0 holds for all w , and y is a sequence number, then H'(y; a)N 0. 16.5 Let F be a class of functionals closed under recursive operations. Assume the functional E is F-computable. Prove that Fdh#s Fp#. 16.6 Let F be a class of functionals closed under basic operations. Assume PI is Fd-hyperenumerable, and P2 is Fp-hyperenumerable. Prove that PI v Pz is F'p-hyperenumerable.

16.7 Let F be a class of functionals closed under primitive recursive operations, and f a (k,m)-ary functional. Assume f is Td-hyperenumerable and quasi-total. Prove that f is in the class Fdha#.

Notes Hyperenumeration is an operation that is intermediate between number quantification and full function quantification. It can be argued that it is a predicative construction, and as such, relevant in the foundations of mathematics. We give here only a short outline to be used in the next chapter in the proof of Kleene's theorem on hyperarithmetical predicates. For a more extensive presentation, see Sanchis [27], Rogers [24], and Kleene [15]. The relation between enumeration and hyperenumeration is explored in Sanchis [25]. Applications to combinatory logic are given in Sanchis [26].

Chapter 17

Recursion in Normal Classes The main purpose of this chapter is to give a generalization of Corollary 13.5.2 in terms of hyperenumeration. More precisely, we want to determine conditions for a class 3 to satisfy the relation RC(3) cs PRdh#. This can be done in a fairly general context where we can conclude that RC(E) Es PRdh#, and from this it follows that RCd(E) = PRdha = the class of hyperarithmetical predicates. This is a classical result due to Kleene. Our proof of the above relation requires that a number of conditions be satisfied by the class 7 ,in particular that 3 be normal. In fact, we need this condition for just one application, namely, the selector theorem of Chapter 15. Let 3 be a class of functionals and U a basic 3-term in 2; a . We say that U is elemental if U does not contain occurrences of function variables. Note that this does not mean that the list a is empty. Rather, it means that no variable in the list a is used in the construction of U . From this it follows that only numerical functions from the class 3 are used in the construction of U . So U is a pure numerical expression which contains numerical functions, numerical constants, and numerical variables. Note that we do not require that the class 3 contain only numerical functions. On the other hand, in dealing with elemental basic T-terms, only the numerical functions in 3 are relevant. We proceed now to define analytic 3-terms in variables z;a . The definition does not imply any restriction on the class 3,but in applications we shall assume

243

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244

that F is closed under primitive recursive operations and contains only quasi-total functionals. The definition involves five inductive rules AT 1 to AT 5, as follows:

AT 1: If Vl and V2 are elemental basic F-terms in variables x ; a and ai is a variable in the list a , then q ( V 1 ) = V2 and cri(V1) # V2 are analytic F-terms in 2; a . AT 2: If V1 . . . , Vk are elemental basic F-terms in x ; a , and Q is a k-ary numerical predicate in Fd, then Q(V1,.. . , vk) is an analytic F-term in x ; a . AT 3: If U1 and U2 are analytic F-terms in x ; a , then (Ul A U z ) and analytic F-terms in z ; a .

(Ul

V U z ) are

AT 4: If U is an analytic F-term in y , z ; a , then (Vy)U and ( 3 y ) U are analytic 3-terms in x ; a . AT 5: If U is an analytic F-term in x ; P , a , then (VP)U is an analytic 3-term in x ;a.

Clearly, the analytic F-terms are completely determined by the numerical functions in the class F. Still, in general we shall assume that F may contain nonnumerical functionals. Note that an analytic F-term may contain function variables, but they can enter the construction only via rule AT 1. Such variables can be bound later by rule AT 5. The basic terms occurring in an analytic F-term are either elemental, or of the form cr(V), where V is elemental. An analytic term represents a compromise where universal quantification over function variables is allowed, but strong restrictions are imposed on the use of such variables. Still, we shall see that these terms have a considerable expressive power. The semantics for the analytic terms is self-explanatory. We mention only that in rule AT 1 the terms are false if either cq(V1) or V2 is undefined. Similarly, the term in rule AT 2 is false if one of the terms Vl, . . . , v k is undefined. EXAMPLE

17.1 Consider the following two expressions in variables

+

(VP)PY)([Z P ( 4 Z ) ) l 11 = Y 1) ( V P ) ( ~ ~ ) ( ~ ~ J ) ( ~= % v) (ACP(v) X ( Z=) t A +

[Z + t,13

2 ;a:

= y + 1).

The first expression is not an analytic PR-term, but the second is. Note that the applications of equality correspond to two different rules: AT 1 (for a(.) = t~ and

Chapter 17. Recursion in Normal Classes

245

B(v) = z ) and AT 2 (for [x --t z , 13 = y + 1, where Q is the primitive recursive equality predicate). Note also that the two expressions are equivalent if a is a total function. If x # 0 and a(.) is undefined, then they are not equivalent.

A (k,m)-ary predicate P is F-analytic if there is an analytic 3-term U in variables z;Q such that P ( z ;a) U . The class of all F-analytic predicates is denoted by 3 a . Theorem 17.1 Assume the class F is closed under primitive recursive operations. Then,

(i) The class 3 a is closed under conjunction, disjunction, universal and existential unbounded quantification, and universal function quantification. (ii) The class F a contains all the numerical predicates in 3 d and is closed under distribution. (iii) The class F a is closed under substitution with numerical functions in the class

F. (iv) The class 3 a is closed under universal and existential bounded quantification. (v) The class F a is closed under existential barred quantification. PROOF. Parts (i) and (ii) are clear from the definitions. Part (iii) is not completely obvious because we are not assuming the numerical functions in the class 3are total, and the substitution of a non-total function in an elemental term may be defined. In general, we have a substitution

where Q is in the class 3 a and f i , . . .,f., are numerical functions in 3. This substitution can be expressed as

Part (iv) is clear, for bounded quantification can be expressed using unbounded quantification with conjunction, disjunction, and primitive recursive numerical predicates.

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246

To complete the proof we consider part (v) where a predicate P is obtained by existential barred quantification in the form

P ( z ;a)= (VP)(~Y)Q(P(Y), z; a). An equivalent expression can be obtained as follows:

where SQ is a primitive recursive predicate such that SQ(y) holds if and only if 0 y = < X I , . . .,xn>,n2 0. A class F of functionals induces the class Fag which also contains functionals. Note that if F is closed under primitive recursive operations, then Fa = Fae,hence Fag = Faeg = Fa#. The functionals in the class Fag are called F-analytic. We show below that classes of the form Fa# satisfy important closure properties.

Corollary 17.1.1 Assume the class 3 is closed under primitive recursive operations. Then Fa# is closed under p-recursive operations. PROOF.

Immediate from Theorem 13.4 and the closure properties of Theorem 17.1.

Theorem 17.2 Assume the class F is closed under primitive recursive operations. Then Fa# is closed under functional substitution. PROOF.

Consider a substitution of the form:

where the functionals g, h l , . . . , ha are in the class Fa#.This means we can write

Chapter 17. Recursion in Normal Classes

247

We assume p = P I , . . . ,p3 and note that an occurrence of P; in Uo is either of the form Pi(V1) = V2 or of the form Pi(&) # V-. In each case the substitution of is equivalent to (Xy)hi(y,x; a ) for

= y A V2 = W) (3W)(3Y)(ui A VI = Y A VZ # w). (3~)(3y)(ui A

b1 '

So we obtain the graph predicate GI by performing these replacements in the term

uo .

0

EXAMPLE

17.2 The graph predicate of the functional E can be expressed as follows:

G E ( w ;a)

f

( ~ z ) ( c Y= ( ~0 )A (vy < z)a(y) # 0 A v = 0) = y 1 A w = 1).

V (b'z)(3y)(a(z)

+

It follows that the functional E is PR-analytic. 17.3 The graph predicate of the functional E' defined in Chapter 14 can be expressed as follows:

EXAMPLE

G E ( w ;a) f G E ( v ; a ) A (Vz)(3y)~~(z) = y, so E' is also PR-analytic. EXAMPLE 17.4 The graph predicate of the functional H defined in Chapter 16 is also PR-analytic. EXAMPLE 17.5 We prove later in this chapter that any functional which is recursive in E is PR-analytic.

Theorem 17.3 Assume the class 3 is closed under primitive recursive operalions and contains only quasi-total functionals. Then,

(i) If P ( x ;a) 3 U where U is an analytic 3 - t e r m in x;a which does not contain quantifiers, then P is in 3 d .

(ii) If U i s an analytic 3 - t e r m in x;a , then there is an analytic 3 - t e r m U' in y,x;P,a such that U (Vp)(3y)U', and U' does not contain quantifiers.

(iii) If P is an 3-analytic predicate, then P can be obtained b y existential barred quantification from a numerical predicate in 3 d .

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248

The assumption that F contains only quasi-total functionals is too strong. We need only that the numerical functions in F are total. The proof of (i) is straightforward by induction on the construction of the term U , using Theorem 3.1 (i) and (ii), and Theorem 3.3 (ii). To prove part (ii) we perform a number of transformations on the term U , each of them producing a term equivalent to U . The final term will be of the form (Vp)(3y)U‘ where U’ does not contain quantifiers. The term U may contain unbounded quantifiers and universal function quantifiers. First, we show that universal unbounded quantifiers can be replaced by universal function quantifiers. In fact, given a term of the form (Vy)U’, this can be replaced by the equivalent term (Va)(3y)(a(O) = y A V ’ ) , where a is a new function variable. With this understanding, we assume U contains only existential unbounded quantifiers and universal function quantifiers. Since quantifiers can be exported over disjunction and conjunction, we assume U is in prenex normal form where all quantifiers precede a quantifier-free term. We show now that the prefix of the prenex normal term can be arranged in such a way that all universal function quantifiers precede the existential unbounded quantifiers. For this purpose it is sufficient to show that any combination (3z)(Va) in the prefix can be replaced by (Va)(3z). For the permutation to be valid we need to execute some substitution in the body of the term. The general situation can be described as follows:

PROOF.

(3z)(Va). . . z . . . a ( V )= V‘ . . . a ( W )# W ’ . . . with the understanding that the term may contain several occurrences of the function variable a. We execute the following transformation: (Va)(32). . .2... a ( < t , V > ) = V’ . . . a ( < t ,W > ) # W’. .. It is clear that if the original expression holds, then the new expression also holds. On the other hand, if the original expression fails it means that for every z there is a function a, that falsifies the body of the term. If we set a ( y ) = a[,ll([y]z) we obtain a function a for which there is no I that satisfies the body of the new expression. So both expressions are equivalent. Note that the new expression after the permutation of quantifiers is again an analytic F-term. We have now an analytic F-term in prenex normal form where all universal function quantifiers precede the existential unbounded quantifiers. We proceed to

Chapter 17. Recursion in Normal Classes

249

contract all the universal function quantifiers into just one quantifier followed by a number of existential unbounded quantifiers. To explain the construction we assume there are two function variables, each occurring twice in the body. The given term has the form:

By contracting the function quantifiers we get the expression:

At this stage the term U is in prenex normal form with one function quantifier preceding a sequence of existential unbounded quantifiers: ( V P ) ( 3 V I ) . * . ( 3 v m ) . . .Ul . . .Vz * . . o m . ..

We contract the existential quantifiers as follows: (VP)(3V). . . [ V ] 1 . . .[V]Z... [Vim * ..

This completes the proof of part (ii). To prove (iii) we assume the predicate P can be expressed in the form P(x;a ) (VB)(3v)U’, where U’is an analytic F-term which contains no quantifier, and execute a number of substitutions with primitive recursive functions until we obtain a term U”. Here we use again the notation d(y, z) = [y]$ and substitute d(d(y, 0)O) for v in U’. We call this term Uo. We list all occurrences of p in UO,which are of the form P ( K ) = V;, . . . ,P(V8)= V:,P(Wl) #

+

W ; ,. . . , p (W,) # W,!, and replace P(&) = 4’with d(y,K 1) = K‘,i = 1,.‘.,S, and p(Wj) # Wj’ with d(y, W, 1) # W i , j = 1,. . . , r . This term we call U1. The term U” is a conjunction of the form

+

ui A vi < d(y, 0) A . . . A < d(y, 0) A Wi < d(y, 0) A .. . A W, < d(y,O). Noting that U” is an analytic F-term in y,x;a , we introduce a predicate Q such that Q(y, s;a )

U”.

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250

T h e predicate Q is T-analytic. Furthermore, since U“ contains no quantifiers, it follows that is in the class T d . We prove now t h a t

, there is v such t h a t U’ holds. Since U’ Assume P ( z ; a ) holds. For a given O contains no quantifier the evaluation of U‘ requires a finite number of values of p, which can be encoded in a number P(y0) for some yo. We can assume also t h a t v; < y0,i = 1 , . . . , s , and Wj < yo, j = 1,.. .,P. And we can assume that d(y0,O) = v . In these conditions the substitution of P(y0) for y in U” holds, since

+

+

YO), 0)O) = d(yo,O) = 21, @(yo), K 1) = P(K), a n d @(YO), Wj 1) = p(Wj). This proves t h a t for every p there is y such t h a t &(p(y),z; a)holds. Conversely, assume that (Vp)(3y)Q(P(y),z; a).Hence, given p, there is yo such t h a t U” holds when we set y = p(y0). If we take v = d(y0,O) it is clear t h a t U’ 0 holds. Hence for every P there is v such that U’ holds, so P ( z ;a)holds. Corollary 17.3.1 Assume the class 7 i s closed under primitive recursive operations and contains only quasi-total functionals. Then E Tdhe, and Fa# 5 Tdh#. PROOF. Immediate from Theorem 17.3.

0

A class of the form Fahas strong closure properties t h a t are simply derived from the rules that are allowed in t h e construction of analytic T-terms. In particular, closure under function quantification is a very powerful construction, although u p t o this point we have had no opportunity to use it. Now we shall show t h a t closure under universal function quantification implies closure under a form of non-finitary induction. We have used non-finitary induction before in this work, to define the reductional semantics in Chapter 10. T h e treatment there was t o some extent informal, and we assumed the reader was familiar with this type of construction, in particular with proofs by induction. Here we must be more cautious, for we want to show that whenevei. a predicate is introduced by an induction involving analytic terms, then the predicate is analytic. To carry out the proof in complete generality we have t o provide an adequate theory of induction. We derive a theory of induction from the fixed point Theorem 1.2. In this way we look a t recursion and induction as two aspects of the same phenomenon. Still, there

Chapter 17. Recursion in Normal Classes

251

are important differences that are not discussed here, for the domain implicit in a definition by induction is a complete lattice, and this has substantial consequences. We start by extending the system of analytic 3-terms and define inductive analytic F-terms in variables z;a . This requires the introduction of a new uninterpreted predicate symbol I’, of some fixed arity, say k. We use I’ as a k-ary numerical predicate, but the induction introduces rather a (k, m)-ary predicate. We define inductive analytic F-terms in variables z;a by six inductive rules IAT 1 to IAT 6. Rules IAT 1 to IAT 5 are derived from rules AT 1 to AT 5, by changing “analytic” to “inductive analytic.” Rule IAT 6 is as follows: IAT 6: If V1, . . . , V k are elemental basic F-terms in z;a , then I’(V1,. inductive analytic F-term in z; a .

,,

, Vk) is an

An inductive analytic F-term U in variables z;a is regular if the arity of z is also k, so the expression r(z)is well formed. If U is a regular inductive analytic F-term in z;a we introduce an inductive condition in the form If U then

ra(z).

The meaning of this condition is determined in the following way. If we fix the functions a we can define a transformation Ta(r)= I’L, where I’ and I?& are k-ary numerical predicates and I’&(z) G U . Note that here I‘ denotes a k-ary numerical predicate independent of the functions a,and I’L denotes also a k-ary numerical predicate which is dependent on I’ and a . For each a the transformation Ta is boolean monotonic on the domain PREk of all k-ary numerical predicates, so there is a minimal fixed point I’a which satisfies the inductive condition and is minimal with that property. This means that if we put U a the result of replacing I’ in U with then the relation

ra,

If Ua then

ra(z)

holds for all values of z.The mininiality condition means that if predicate which satisfies the condition

I” is another k-ary

If Ub, then P(z), where Ub, is obtained by replacing I’ with I“, then I” is a boolean extension of ra. We complete the definition by setting r(z;a ) G I’a(z). We say that the (k,m)ary predicate r is specified by F-analytic induction with basis the term U .

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252

Theorem 17.4 If r i s a ( k , m ) - a r y predicate specified by 3-analytic induction, and F is closed under primitive recursive operations, then r is F-analytic. PROOF.

We consider an inductive condition of the form If U then

ra(z)

and change U to U' by replacing all variables z with new variables y, and replacing all occurrences of r(V1,. . ., Vk) with p() = 0, where p is a new variable. Let P be the (k,m)-ary predicate such that

P ( z ;a) (Vp)(p() = 0 V (3y)(U' A p()

# 0)).

Clearly, the predicate P is F-analytic. We shall show that P = r is the minimal solution of the inductive condition. P ( z ; a ) holds.. Let /3 be the We fix the functions a and assume Pa(.) function such that:

From Pa(.) it follows that I'a(z)holds, for otherwise there are y such that U' and I'a(y1, . . ., y k ) , and this is impossible because ra is a solution of the specification. Conversely, if ra(z)holds, let p be an arbitrary function and assume that whenever U' holds, then p() = 0. If we put

-

it follows that r&satisfies the inductive condition, so I'd is a boolean extension of ra. Since ra(z) holds, it follows that r&(z)also holds, hence p() = 0, hence Pa(.) holds. Technically we define induction via transformations Ta and use Theorem 1.2 to identify the minimal fixed point of the transformation. On the other hand, in the proof of Theorem 17.4 we identify the minimal fixed point using universal function 'quantification and prove it is, in fact, a minimal fixed point. It follows that Theorem 1.2 is not necessary to handle this type of induction. Note that Theorem 1.2 may be necessary to handle inductions that cannot be expressed using universal function quantification.

Chapter 17. Recursion in Normal Classes

253

The preceding result shows that whenever F is closed under primitive recursive operations, then the class Fa is closed under F-analytic induction. We prove next that the class Fa# is closed under recursive operations. To prove this relation we must assume again that the class F is closed under primitive recursive operations. Note that we know already that Fa# is closed under p-recursive operations and functional substitution. Theorem 17.5 If 3 is closed under primitive recursive operations, then closed under recursive operalions.

Fa# is

It is sufficient to prove that if 3‘ is a finite subset of Fa#,then RC(7’) is also a subset of .Fa#. This will follow if we show that the universal interpreter for RC(F’) defined in Chapter 12 is in the class .’Fa#. Essentially, we transform the recursive specification of the interpreter in an F-analytic induction of the graph predicate and apply Theorem 17.4. To simplify the presentation we shall assume that the class F’contains exactly one (1, 1)-ary functional which, departing from the notation in Chapter 12, we denote with the letter f . We want to prove that the functional Qrn is in the class Fa# for every m 2 0. In the same spirit we assume that m = 1 and consider only the functional cP1. Since this is a (1,1)-ary functional the graph predicate is (2,l)-ary, and it is represented by the 2-ary symbol r. We must write an inductive condition PROOF.

If U then r a ( v ,z ) , where U is an inductive analytic F-term in v , z ; a . Actually, the term U is a disjunction of nine subterms U1,. . . , Us,each derived from the corresponding equation in the recursive specification of @ I . The derivation is in most cases trivial. Only in the case of the term U S ,derived from equation RD 8, must we refine the analysis. We start with equation RD 1 where the term U1 is rpi(z, 0) = 0 A v = d ( z , 2, d(z, 1,2)).

The translation of equation RD 2 into the term Uz is entirely similar, although here we impose rp;(z,2) = 0. The translation of equation RD 3 is the following analytic term U3:

254

L.E. Sanchis

The translations of equations RD 4 and RD 7 into terms U4 and U7 are similar. The term Us which translates equation RD 5 is the only one that explicitly involves the variable a , and it is

In the case of equation RD 6 we note that the only purpose of the bounded product is to make sure the terms are defined, and here we express the condition using the universal bounded quantifier (which is, in fact, an abbreviation of a combination using the universal unbounded quantifier with disjunction and a primitive recursive numerical predicate). The term Us is a conjunction of e q l ( z , 6) = 0 and the term (VW

< d(z,1,0))(3Y)(r(Y,

92(2, w))A q v , g3(z))).

Equation RD 9 translates into the term Us, which is a conjunction of rp:(z, 9) = 0 and the term

Finally, we consider equation RD 8 where the given functional f is F-analytic, so there is an analytic F-term U’ in v, x;/3 such that

G J ( v , X ;p)

U’

We know the equation requires the substitution of (Xy)@1(g7(zIy); a ) for p in the term U’.Since the occurrences of /3 in U’ are of the form p ( V ) = V‘ or p ( V ) # V’, it is sufficient to replace these expressions with

v)) v’

(gw)(r(W, g7(2, A = w) (3w)(r(W, g7(z, V ) )A V’ # w), depending on the type of expression. We call U” the term obtained by executing these replacements in U’ and set the term Us as r g i ( z , 1) = 0 A ( 3 z ) ( r ( T , g i ( z ,d ( z , 1,3))) A U”).

This completes the description of the analytic terms U1,. . . , Us, so we put U = U1 V . . . V Us and write the inductive condition

If U , then r 0 ( v , 2).

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255

The minimal solution of this condition is a (2,l)-ary predicate r, and we claim that r =G~,. To prove the claim we note that Gal satisfies clearly the inductive condition, so it is a solution, and this means that Gal is an extension of I?. It follows that r is single-valued and there is a ( l , l ) - a r y functional cb’ such that r = Get. We must prove that cb‘ is an extension of cb1. We recall that cb1 is the minimal fixed point of a transformation T induced by the recursive equations. We set Tcb‘ = cb”, and it is sufficient to prove that cb’ is an extension of cb”. To prove this relation we must show that whenever cP”(z; a) N v, then cb’(z; a) N v by considering all possible cases in the specification of cP” by the equations RD 1 to RD 9. We discuss a few cases. The general idea is that whenever cP”(z;a)N v holds by equation RD i, then the term Ui holds, hence r a ( v , z ) also holds. This a)N v. means, of course, that a’(%; If @’’(z;a)N v by equation RD 1 we have rp:(z,O) = 0, and also v = d ( z , 2, d(z, 1,2)), hence the term U1 holds. If W ’ ( z ; a ) N v by equation RD 6, then cb’(gz(z,i);a) is defined for every i < d ( z , l , O ) , and cb‘(gg(z); a) 2: v. Since we are assuming r = Gat, it follows that equation us holds. If W ( z ;a)N v by equation RD 8, f(cP’(gl(z, dl(t,1,3));a);(Xy)cb’(g,(z, y); a)) N v. Since r = Gat, there is an z such that I ’ ( z , g l ( z , d ( z , 1 , 3 ) ) ; a ) holds, and f ( z ; a ) N v , where p = (Xy)cb’(gT(z,y);a). It follows that U’ holds with this 0 particular p, hence U” holds. We conclude that Us holds. Corollary 17.5.1 Iff is recursive in E, then it is PR-analytic.

From Example 17.2 we know that E is in the class PRa#, hence from 0 Theorem 17.5 it follows that f is also in the class PRB#. PROOF.

Theorem 17.6 Assume the class 3’is normal, 3 contains only quasi-total functionals, 3’ C PRa#(F), and P R ( 3 ) E RC(3’). Then,

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From Theorem 17.5 we know that PRa#(F) is closed under recursive operations, hence RC(F’) PRa#(F), and from Corollary 17.3.1 we know that PRdh#(F). It follows that RC(F‘) 2 PRdh#(F). To complete the PRa#(F) proof of (i), assume f is a functional in the class PRdh#(F), hence G j is a predicate in PRdhe(F). Noting that PRdhe(F) RCdhe(F’) follows from the assumptions, RCp(F’) follows from Corollary 16.4.1, it follows that G f is in and RCdhe(F’) the class RCp(F‘). Since RC(F’) has the p-selector property there is a functional f’ in RC(F’) which is a selector for G,. It follows that f’ Es f . This completes the proof of (i). Part (ii) follows immediatelyfrom (i), noting the definition PRdha(F) = PRdh#d (F).To prove (iii) we note that from (i) it follows that RCp(F’) = PRdh#p(F), 0 and we have also PRdh#p(F) = PRdhes(F) by Theorem 16.6. PROOF.

s

c

c

Corollary 17.6.1 Let 3 be a class which contains only quasi-total funclionals, and F RC(E). Then,

c

(iii) RCp(E) = PRdhes(F) We apply Theorem 17.6 to 3’ = {E’}, which is a normal class, noting that RC(E) = RC(E’). 0

PROOF.

EXAMPLE

1 7 . 6 If we apply Corollary 17.6.1 with 3 = 0 we get the following rela-

tions:

EXAMPLE

relations:

17.7 If we apply Corollary 17.6.1 with F = {E} we get the following

Chapter 17. Recursion in Normal Classes

257

EXERCISES

17.1 Prove explicitly (i.e., without using Corollary 17.5.1) that the functional H is PR-analytic.

17.2 Explain where the requirement that all numerical functions in 7 are total is necessary in the proof of Theorem 17.3. 17.3 Complete the proof of Theorem 17.3 (ii) and show that the replacement of (31)(Va) by (Vcr)(3z) in the prefix produces an equivalent term. 17.4 Let P be an 7-analytic predicate, where 7 is a class of functionals. Prove that P is boolean monotonic.

17.5 Consider the following (2,l)-ary predicate P such that

P ( v ,t;a )

21,. . . , zn, n > 1, such that = v , and a ( q ) = q + l for 1 5 i < n.

There is a sequence I

= 11,

I,

Show that P can be specified by PR-analytic induction.

The crucial result in this chapter is Theorem 17.6, particularly in the form given in Example 17.6 where it is shown that recursion in the functional E is equivalent to hyperenumeration. This result was proved in Kleene [16]. Here we follow the approach of Hinman [lo]. The main tool in the proof is the restriction to PRanalytic predicates, which allows for considerable latitude in the use of universal function quantification and at the same time imposes rigid conditions in the use of function variables, which are allowed only via application. The conclusion of this combination is that PR-analytic terms can be expressed via hyperenumeration. The transition from recursion to induction, and vice versa, is a peculiar situation that we have found before in the discussion of reductional semantics. In general, a recursive specification can be translated into an inductive specification, and in many cases inductive specification can be seen to be essentially recursive. Still, we cannot identify recursion and induction, for the former is deterministic, so belongs properly to computability theory, while the latter is non-deterministic and belongs properly to definability theory.

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For a general theory of inductive definability, see Moschovakis [21] and Hinman [lo]. The presentation in Sanchis [27] is concerned primarily with the formalization of proofs by induction.

Appendix

Recursion and Church’s Thesis Church’s thesis has been discussed a few times in this work, mainly in Chapter 6, where the original standard form was formulated, and in Chapter 8, where a more general extended form was introduced. We want to give here a more comprehensive presentation, with emphasis on the role of recursion. It is a well known situation that Church’s thesis is not a mathematicalstatement, so it cannot be proved in the usual way. This applies also to the extended form introduced in Chapter 8. Essentially, the thesis relates a general but informal notion of computable function (or functional) to another notion which is precise and formal. The thesis asserts that any instance of the informal notion can be reduced to an instance of the formal notion. If this is the case, and Church’s thesis has no precise meaning and cannot be proved, we should ask what is the reason for the inclusion in this work. In fact, the thesis is never applied in a formal sense, and could be excluded from the text without affecting the remaining results. Our position is that Church’s thesis is relevant for the foundations of mathematics, for it provides a formal description of a fundamental type of mathematical activity. There are other theories that attempt to formalize different realms of mathematical activity. In particular, set theory attempts a universal formalization. There is another dimension where Church’s thesis is relevant, namely, in the domain of practical computing. By using Church’s thesis we can prove that some problems are algorithmically unsolvable. Obviously, this information can be critical in some real world situations. 259

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We are interested here in the foundational approach to Church’s thesis, particularly in the role of recursion in computability theory. This theory was identified from the beginning as a theory of recursive functions and functionals, so we may expect that the role of recursion must be critical, even at the lowest level which is concerned only with the computation of numerical functions. We shall deal first with this level, noting the peculiarity that recursion is not explicitly mentioned in the standard Church’s thesis. The standard form of Church’s thesis usually applies to the computation of numerical functions, including the relativization t o a given class F of functions. In Chapter 6 the thesis is given a more general application, for it covers the computation of functionals relative to a class F of functionals. Actually, only some types of computations are considered, namely, the explicit computations. Since explicit computations are non-recursive, it appears that recursion is, in fact, excluded from the standard Church’s thesis. The standard Church’s thesis reduces explicit computations relative to a class F of functionals to p-recursive operations relative t o the class F.Note that p-recursive operations essentially involve primitive recursion and unbounded minimalization. At this stage we may expect objections to be raised, in the sense that the restriction to explicit algorithms is arbitrary, and it is not a part of the usual formulation of Church’s thesis (for example, as given in Kleene [14]). We answer that the restriction is meaningful, for in dealing with computable functionals in general, the thesis is not true unless the restriction to explicit algorithms is invoked. In fact, it is fairly clear that primitive recursion and unbounded minimalization are not sufficient for the reduction of non-explicit functional computations. At the numerical level, where Church’s thesis is concerned with the computation of numerical functions, the situation appears to be different. The thesis is usually asserted for computations in general, and recursive computations are not explicitly excluded (see again Kleene [14]). It appears to us that some amount of ambiguity has been generated on this matter, which is harmless a t the numerical level, but may be confusing a t a higher level. We note first that although we cannot give a formal proof of Church’s thesis, there are some arguments that make it extremely convincing. In particular, a crucial argument is the existence of a canonical procedure where a given computation can be described in terms of primitive recursion and unbounded minimalization. In this

Appendix

26 1

procedure, which is explained in Chapter 6, the computation itself (as a sequence of steps) is expressed by primitive recursion, and the determination of the input for a halting computation is expressed by unbounded minimalization. The existence of this canonical reduction is in our opinion a fundamental element in the formulation of the standard Church’s thesis. On the other hand, it is obvious that the procedure works properly only in dealing with explicit computations. From this point of view it appears that the restriction to explicit computations is an implicit part of the standard Church’s thesis. This means that, in principle, recursive computations are excluded from the thesis. The catch is that, in practice, recursive computations can be reduced to explicit computations, so they are actually in the scope of the thesis. In fact, a general argument can be given in the sense that such a reduction is always possible, as explained in Example 6.3. This argument applies only to special forms of recursion where relativized functionals are involved, but it is fairly general for numerical functions. It appears entirely possible to give a more general formulation of the standard Church’s thesis where recursive algorithms of some form can be reduced to primitive recursion and unbounded minimalization. In the presence of the full notion of recursive algorithms given in Chapter 8, which includes functional substitution, such a formulation would be mostly negative. The full notion provides an adequate frame where special forms of recursion can be reduced via the extended Church’s thesis. For these reasons we have formulated the standard Church’s thesis in terms of explicit computations, and reserved the analysis of recursive computations to the extended Church’s thesis. Concerning the extended Church’s thesis of Chapter 8, the situation is quite different. Here we deal with recursive algorithms which are characterized via functional recursion. Recursion, which was not a part of the standard thesis, now permeates both the computational process and the mathematical formalization. We note first that the formulation of recursive algorithms given in Chapter 8, essentially via direct and indirect calls, is strongly conservative. The main purpose there was to outline a structure where computations were transparent enough, and at the same time sufficiently strong for the computation of some functionals, particularly the interpreter of Chapter 12,and the preorder functional of Chapter 15. Some possible extensions-for example, allowing functional substitution in direct

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calls-were excluded from the model, as we found that the execution of such calls offered some difficulties. We wanted to have a model that was convincing, in terms of computability, and sufficiently general for the formulation of Church’s thesis. This means that some extensions of the model are possible, within the frame given by functional recursion. In particular, the formalism of Chapter 9, via the lambda calculus, provides a generous structure for the definition of computable functionals. From this structure, it appears feasible to derive a more comprehensive model for recursive algorithms. The extended Church’s thesis provides a mathematical characterization where recursive algorithms are reduced to recursive operations. The formalization of functional recursion requires the application of Theorem 1.2, which depends strongly on the axiom of set theory. We follow here the traditional approach, which we did not want to question in the text. At this stage we would like to formulate some qualms, and suggest that dependence on impredicative set theory can be avoided. T h e point we want to make is that the extended Church’s thesis proposes a char-

acterization of computational processes, which are intrinsically operational. This being the case we should expect the characterization to be free from extreme set theoretical assumptions. The characterization given in Chapter 8 involves functional recursion, which depends on Theorem 1.2, hence on transfinite recursion. The proof of the latter usually requires strong set theoretical axioms. We do not propose to question here the validity of set theory. We complain only that since computability is essentially operational we should look for an operational mathematical characterization. On the other hand, we cannot help mentioning that many persons have raised questions about the significance and validity of the axioms of set theory. In a number of cases such questions do not originate from an ideological commitment to an intuitionistic philosophy of mathematics. The extended Church’s thesis involves recursive algorithms, and also functional recursion. In both situations we invoke recursion. Apparently, in both cases, the same notibn of recursion is involved. Still, in the case of recursive algorithms recursion enters as an operational structure, and in the case of functional recursion as a set theoretical construction. We propose a more restrictive frame which is essentially operational and covers both recursive algorithms and functional recursion. The basic approach w a s already clear in the discussion of recursive algorithms, which are obtained by adding recursive calls to explicit algorithms. The latter

263

Appendax

are characterized via the standard Church’s thesis as finitary processes determined by formal rules. Obviously, we can adjoin recursive calls to non-finitary processes determined by arbitrary rules. The validity of such calls is not affected by the nature of the underlying processes, and as long as these are free from strong set theoretical assumptions we can claim that the adjoining of recursion does not involve further set theoretical commitments. We understand that the preceding claim is controversial, and open to serious objections. Still, we do not think the claim is more controversial than the usual separation and replacement axioms in set theory. At any rate, we give a short outline of the manner in which we think these ideas should be developed. Let U be an expression or program that defines a process that can be performed with an input denoted by X. This process is well defined in the context of some mathematical principles and definitions, and it is not assumed to be finitary. For example, it may involve the evaluation of quantifiers ranging over infinite sets. Being a process, it may halt or it may continue indefinitely. In the first case some output is produced. If the process does not halt, this is a purely negative situation that is never available during the process. In the conditions described above we say that U explicitly defines an operation f such that

f ( X ) = u, which means that f ( X ) is defined if and only if the process defined by U halts with input X , and the value of f ( X ) is the output of such a process. We can see that the preceding assumptions force us to deal with partially defined operations. In fact, this is a characteristic feature of operational mathematics. Note that we do not advance any assumption in the sense that f is being computed by the process. To have a computation the process should be finitary, and this is not one of our assumptions. Concerning the process itself, we shall say only that we assume it consists of a number of sub-processes, even an infinite number, and each one of the sub-processes may generate new sub-sub-processes, etc. In general, for a process to halt and produce output it will be necessary that some of the sub-processes halt, but not necessarily all of them. If we arrange this structure in the form of a tree we shall find that the process may halt even if some branches of the tree are infinite, which corresponds to partial non-halting processes. We can eliminate those branches, but

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only by looking at the process as a complete object. In this way we obtain a tree where all the branches are finite. The next step is familiar from the theory of recursion. We may allow the symbol f to occur in the expression U , and write the recursive equation

f ( X )N

u.

The process to evaluate f ( X ) in this situation is the normal process defined by U , with the understanding that calls to f inside U are evaluated by the same procedure using U again with the input determined by the call. In these conditions we say that the operation f is recursively defined by U . In fact the operation f defined in this way satisfies the equation above, so f ( X ) N U holds for all values of X , as long as f is the operation defined by the recursion. In practice we need more, namely, that f is the minimal operation that satisfies the equation. To prove this we must assume that the expression U is monotonic in f, which means simply that the more defined is f , then the more defined is U . The proof uses a form of induction given by the halting derivation tree explained above. Clearly, a considerable amount of work will be necessary to transform these rough ideas into a workable structure. At this stage we want only to make clear that the theory of recursive functionals is by no means committed to a rigid impredicative system of set theory.

References [l] Barwise, K. J . Handbook of Mathematical Logic. Amsterdam: North Holland (1977). [2] Crossley, J. N. Sets, Models and Recursion Theory (Proceedings of the Summer School in Mathematical Logic and Tenth Logic Colloquium, Leicester 1964, editor). Amsterdam: North Holland (1967). [3] Fenstad, J. E., Gandy, R. O., Sacks, G. E. Generalized Recursion Theory I1 (Proceedings of the 1977 Oslo Symposium, editors). Amsterdam: North Holland (1978). [4] Fenstad, J. E. General Recursion Theory. Berlin, Heidelberg, New York: Springer Verlag (1980). [5] Gandy, R. 0. Computable functionals of finite type I , in Crossley (1967), 202-242. [6] Gandy, R. 0. General recursive functionals of finite t y p e and hierarchies offunclions, Ann. Fac. Sci. Univ. Clermont Ferrand 35:5-24 (1967). [7] Grilliot, T. J . O n eflectively discontinuous type-& objects, Jour. Symb. Log. 36:245-248 (1971). [8] Heyting, A. Constructivity in Mathematics (Proceedings of the Colloquium held at Amsterdam 1957, editor). Amsterdam: North Holland (1959). [9] Hindley, J . R., and Seldin, J. P. Introduction to Combinators and A-Calculus. Cambridge: Cambridge University Press (1986). 265

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[lo] Hinman, P. G. Recursion-Theoretic Hierarchies. Berlin, Heidelberg,

New York: Springer Verlag (1978). [ll] Hinman, P. G . Hierarchies of eflective descriptive set theory, Trans.

Amer. Math. SOC.142:lll-140 (1969).

1121 Hinman, P. G . Degrees of continuous functionals, Jour. Symb. Log. 38:393-395 (1973). [13] Kechris, A. S., Moschovakis, Y. N . Recursion in higher types, in Barwise (1977), 681-737. [I41 Kleene, S. C. Introduction to Metamathematics. Amsterdam: North Holland; Groningen: P. Noordhoff New York: van Nostrand Co. (1952). [15] Kleene, S. C. On the forms of the predicates in the theory of constructive ordinals (second paper) Amer. J. Math (1955), 77:405-428. [16] Kleene, S. C. Recursive functionals and quantifiers offinite type I, Trans. Amer. Math. SOC.(1959), 61:193-213. [17] Kleene, S. C. Countable Functionals, in Heyting, (1959), 81-100. [18] Kleene, S. C. Recursive functionals and quantifiers of finite type revisited I, in Fenstad-Gandy-Sacks (1978), 185-222. [I91 Kreisel, G. Interpretation of Analysis b y means of functionals offinite type, in Heyting (1959), 101-128. [20] Moldestad, J. Computations on Higher-Types, Lecture Notes in Mathematics 574. Berlin, Heidelberg, New York: Springer Verlag (1977). [21] Moschovakis, Y. N. Elementary Induction on Abstract Structures. Amsterdam: North Holland (1974). [22] Norman, D. Recursion on the Countable Functionals, Lecture Notes in Mathematics 811. Berlin, Heidelberg, New York: Springer-Verlag (1980). [23] Platek, R. A. Foundations of Recursion Theory, Ph. D. thesis and supplement, Stanford University (1966).

REFERENCES

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[24] Rogers, H . , J r . Theory of Recursive Functions and Effective Computability. New York: McGraw-Hill (1967). [25] Sanchis, L. E. Hyperenumeration reducibility, Notre Dame Journal of Formal Logic (1978), 19:405-415. [26] Sanchis, L. E. Reducibilities i n t w o models for combinatory logic, Jour. Symb. Log., (1979), 44:221-234. [27] Sanchis, L. E. Reflexive Structures. An Introduction to Computability Theory. Berlin, Heidelberg, New York: Springer-Verlag (1988). [28] Tourlakis, G . J . Computability. Reston: Reston Publishing (1984). [29] Tuguk, T. Predicates recursive i n a t y p e - 2 object and KIeene Hierarchies, Comment. Math. Univ. St. Paul (1960), 16:115-127.

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Index Abstraction, 161 Algorithm, 86, 181 deterministic, 74 explicit, 50, 86 recursive, 110 Analytic, 243 Anti-symmetric, 5 Application, 1 functional, 37 Assignment, 129 Basic, 37 functional, 37 operation, 39 primitive recursion, 52 recursive term, 69 term, 47 Boolean, 20 values, 20 extension, 23 monotonic, 23 Bottom, 6 Bounded, 56 minimalization, 58 product, 57 quantification, 60 sum, 56

Call, 111 direct, 113 indirect, 114 Cartesian product, 19 Cases, 38 definition, 38 general, 40 Characteristic functional, 25,35 dual, 25, 35 partial, 25, 35 Church’s thesis, 67, 88 Closed, 128 Coenumerable, 193 Cohyperenumerable, 236 Complete, 17, 229 Computable, 36 recursively, 117 Computation, 86, 110 rules, 86, 110 Condition, 4 inductive, 251 single-valued, 4 total, 4 Configuration, 86, 110 Conjunction, 21, 24 Consistent, 7, 172, 227 set, 7 Constant functional, 37

269

270

Continuous, 206 Cover, 142 Definable, 136 formally, 136 Defined, 1 mapping, 1 well, 4 Definition, 3 mapping, 3 predicate, 26 Deflector, 210 Denotation, 129 Discontinuous, 209 effectively, 213 Discrete, 5 poset, 5 Disjunction, 21,24 Distribution, 38 predicate, 42 Domain, 8 Dual, 26, 35 Effectively discontinuous, 213 Elemental, 243 Enumerable, 193, 197 Existential quantification, 19 barred, 234 bounded, 50 functional, 233 unbounded, 190 Explicit, 3 algorithm, 50 specification, 3, 50 Extension, 4

L. E. Sanchis boolean, 23 mapping, 4 Factorization, 209 Fenstad, J. E., 14, 170 Fixed point, 9 Formalization, 125 closed, 128 language, 126 linear, 128 regular, 128 term, 126 variable, 126 Function, 30 number encoding, 59 number decoding, 59 Functional, 16, 30 application, 37 characteristic, 26 complete, 17 constant, 37 continuous, 206 discontinuous, 209 effectively discontinuous, 213 preorder, 165 primitive recursive, 54 projection, 37 quasi-total, 17, 55 recursive, 110 similar, 18 singular, 18 successor, 5 1 weakly normal, 213 Gandy, R. O., 124, 232

Index

Graph predicate, 26, 35 Greatest lower bound, 8 Grilliot, T. J . , 205, 206, 211, 212, 217 Hindley, J . R., 139, 156 Hinman, P. G., 14, 45, 170, 187, 217, 232, 258 Hyperarithmetical, 239 Hyperenumerable, 235, 239 Inconsistent, 7 Index, 159, 174 Induction, 251 Inductive, 251 Interpreter, 158 Intersection, 34 Jump, 184 Kechris, A. S., 108, 124 Kleene, S. C., 45, 95, 187, 200, 217, 242, 258, 260 Kreisel, G., 217 Least upper bound, 7 Linear, 128 Lower bound, 6 greatest, 8 Machine, 111 Mapping, 1 assignment, 129 defined, 1 definition, 3 monotonic, 15

271 partial, 2 specification, 2 total, 2 undefined, 1 Maximal, 6 Minimal fixed point, 10 Minimalization, 58, 68 bounded, 58 unbounded, 68 Moldestad, J., 124 Monotonic, 15 boolean, 23 extension, 23 Moschovakis, Y. N., 108, 124, 258 Negation, 21, 24 Normal, 231 weakly, 213 Norman, D., 217 Numerical, 30 function, 30 predicate, 34 substitution, 37, 40 Operations, 39 basic, 39 p-recursive, 68 primitive recursive, 52 recursive, 110 Oracle, 112 Partial characteristic functional, 25, 35 Partial ordered set, 5 Partial ordering, 5 Platek, R. A,, 124

272 Point, 9 minimal fixed, 9 Polish normal form, 75 Poset, 5 discrete, 5 Predicate, 22, 33 coenumerable, 193 distribution, 42 enumerable, 193 numerical, 34 partially recursive, 192 primitive recursive, 192 recursive, 192 Prefix, 127 p-prefix, 127 A-prefix, 127 Preorder functional, 165 Preordered, 168 Prime number enumeration, 58 Primitive recursion, 51 basic, 52 Projection functional, 37 Quantifier, 21 barred, 234 bounded, 60 unbounded, 190 Quasi-similar, 153 Quasi-total, 17 Range, 27 Recursion, 51 basic, 71 functional, 102 operational, 110

L. E. Sanchis primitive, 51 simultaneous, 106 Recursive, 69 algorithm, 110 call, 77 functional, 110 machine, 112 operation, 110 Represent at ion, 172 Reflexive structure, 158 Regular, 6 element, 6 poset, 6 set, 149 term, 101 Restriction assumption, 145 Rogers, H., 242 Sanchis, L. E., 13, 29, 47, 58, 67, 69, 71, 84, 89, 95, 162, 167, 170, 187, 189, 232, 258 Seldin, J. P., 139, 156 Selector property, 168 Semantics, 127 reductional, 127 structural, 127, 131 Similar, 18 extension, 191, 240 Single-valued, 4 Singular, 17 Specification, 2 explicit, 3 mapping, 2 relational, 3 Stack, 74

Index

algorithm, 74 Structure, 158 Substitution, 37 formal, 134 free, 134 functional, 62 numerical, 37 predicate, 43 Support, 113 System, 110 Term, 47 analytic, 243 basic, 47 elemental, 243 functional, 98 inductive, 251 regular, 251 TOP, 6 Total, 2 condition, 4 mapping, 2 Tourlakis, G. J., 45, 124, 187 Transformation, 9, 70 TuguC, T., 187

273 Tuples, 31 Type-0, 29 Type-1 , 126 numerical, 126 functional, 126 Type-k, 126 Type-(k, m),126 Union, 34 Universal quantification, 19 barred, 234 bounded, 49 functional, 233 unbounded, 190 Upper bound, 6 Variable, 126 bound , 128 formal, 126 free, 128 renamed, 145 renaming, 145 Variant, 146 Weakly normal, 213

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List of Symbols fx 2 Y fx 1 fx t UDA UDAX UEV

u?4v fx

N

gx

f=s fx

N

u

N MAP(A, B ) MAP CP

fcMAPg

x

c A

Y

CO 2 Cp

Y

c P(A) Z MAP(N, N) LUBP(M) GLBP(M) MON(A, B ) P(A) FUN(A, B )

CFUN

1,4 1 1 1 2 2 2 2 2 3 3, 29 4 4 5 5 5 6 6 6 6 6 7 8 8 11 12 16 16 275

19, 30 19 19, 31 19 19 20 20 20 20 21 21 21 22 22 23 23 23, 34 24, 34 24 24 24 24 24 26, 35 26, 35 26, 35 26, 35 27

L. E. Sanchis

276

...

29 30 30 30 31 31 31 31 31 31 32 32 33 33 36 36 36 37 37 37 37 38 38 48 51 53 53 53 53 53 55 55 55 56 56

57 57 58 58 58 59 59 59 59 59 59 59 59 63 68 68 68 81 81 81 81, 89 81, 90 87 87 110 110 117 117 126 126

List of Symbols

277

1 4 ! J

130 130 131 134 136 136 136 137 137 141 145 152 153 157, 158, 180 158 158 163 163 165 171 174 174 174 175 175 175 175 176 178 184 191 191 191 192 192

192 192 192 192 192 192 192 192 193 193 193 197 197 206 207 210 210 210 212 213 213 214 214 214 220 234 236 236 237 239 239 24 1 245 246

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