Fundamentals of Generalized Recursion Theory (Studies in Logic and the Foundations of Mathematics, Volume 105)

FUNDAMENTALS OF GENERALIZED RECURSION THEORY MELVIN FITTING Herbert H . L e h m a n College C i t y University of N e w...

Author: M. C. Fitting

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FUNDAMENTALS OF GENERALIZED RECURSION THEORY

MELVIN FITTING Herbert H . L e h m a n College C i t y University of N e w Y o r k

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK . OXFORD

@) NORTH-HOLLAND PUBLISHING COMPANY - 1981

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 permission of the copyright owner.

1SBN:O444 X6171 X

Published by: North-Holland Publishing Company - Amsterdam. New York . Oxford Sole distributors for the U.S.A.and Canada: Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017

Library of Congress Cataloging in Publication Data

Fitting, Melvin Chris. Fundamentals of generalized recursion theory. (Studies in logic and the foundations of mathematics ; v. 105) Bibliography: ,p. 1. Recursion theory. I. Title. 11. Series. WY.6 . F57 511.3 81-2145 ISBN 0-444-86171-8 AACW

PRINTED IN T H E NETHERLANDS

To my wife Greer, who heard this book, patiently, many times, and never failed t o make it better

INTRODUCTION I. We describe, briefly, the contents of the book, and the approach to the material that we have elected to follow. We treat ordinary recursion theory and one of its natural generalizations to arbitrary structures, search computability ; and we treat hyperarithmetic theory and one of its abstractions to arbitrary structures, hyperelementary theory. Both are dealt with simultaneously using generalized elementary formal systems, which are simple axiomatic systems of derivation, and which can be thought of as being rules for the generation of a set or relation. Elementary formal systems were announced in Smullyan [1956A], and given in detail in Smullyan [1961]. As presented there, they constitute a mechanism for developing ordinary recursion theory only, thought of as a theory of words over a finite alphabet. Traced further back, elementary formal systems themselves have their origin in Post’s canonical systems (Post [1943]), and can be looked at as a special case, albeit a very elegant and simple special case. The various generalizations of elementary formal systems presented here (to arbitrary structures, allowing inputs, allowing infinitary rules, etc.) are new. It is the elementary formal system mechanism, modified one way or another, that provides the underlying uniformity of treatment in this book. (From now on, we drop the qualifier ‘generalized’; we use ‘elementary formal system’ systematically in the generalized sense.) If one allows elementary formal systems to accept input, one gets natural generalizations of enumeration operators (Rogers [ 19671, pp. 146-147), and these are actually the main objects of study in much of this book. They provide a common generalization of partial recursive functions and of recursively enumerable relations: one can identify functions with operators which, when given a 1-element input, give an output with 0 or 1 elements; one can identify relations with outputs of constant operators. The development is index-free; after the fact we investigate under what conditions an indexing is possible. ix

X

INTRODUCXION

We treat a -recursion theory, using another generalization of elementary formal systems. Here our treatment is more of a survey nature, to show similarities with the preceeding material. Again, operators are brought in, in this case they are the a -enumeration operators. All the foregoing theories have many properties in common; we abstract these properties into an axiomatic concept which we call a production system (no direct relation of Post production systems, though). We derive abstractly, for production systems, under various extra hypotheses, the basic material of elementary recursion theory (up to priority arguments), all of which then applies to many of the theories mentioned above. We develop a theory of embedding between production systems, and of effective embedding. This can be thought of as a generalization of Godel numbering. The approach is algebraic and uses concepts from category theory. Finally we use elementary formal systems, generalized still further, to extend the notion of enumeration operator to that of effective operator of higher type, again for arbitrary structures. The subject matter of this book is becoming standard; the particular development we have chosen is rather individualistic. The reader will see proofs of equivalence between our presentations and some of those more common in the literature, but the development using generalized elementary formal systems will not be found elsewhere. Its advantage is in the resulting uniformity of treatment. A disadvantage, of course, is that the rest of the literature may look unfamiliar. At the least, we can offer as compensation a very brief survey of how recursion theory came to have the particular generalizations we treat, and some references to original papers. 11. In the 1930’s,many people independently tried to formally characterize the intuitive notion of “effectively calculable” for the structure of arithmetic (N,s), where the domain N is the set of natural numbers, and the successor function s is assumed given. As is well known, these attempts all led to the same theory. This theory is ordinary recursion theory and the Church-Turing Thesis asserts that it does capture the intuitive notion it was intended to capture. We are interested in generalizations of ordinary recursion theory; most have arisen by changing the structure involved, by broadening the “allowed machinery of computation” or by some mixture of the two, although it was not always apparent at the start that this is what was happening. In a sense, recursion theory was generalized from its start as a formal mathematical subject. Some people worked with numbers (Church [1936],

INTRODUCTION

xi

the source of Church’s Thesis), while others worked with words (Post [1936], and Turing [1936], the source of Turing’s Thesis). Now the collection of words over a fixed finite alphabet, and the collection of natural numbers, are isomorphic in ways that most people would consider effective. So it was not thought to be important that two rather different structures were in use. Also the techniques available for developing recursion theory using these two structures did not seem to admit of a further generalization to any of the other structures met in mathematics. The study of relative recursion theory also goes back to the beginnings of the subject. It asks, what is ‘effectively calculable’ if, in addition to the usual machinery, one allows the use of some property P of the natural numbers (which need not itself be intuitively effective). It can be thought of as a generalization of ordinary recursion theory to the structure (N, s,P). Again, it was unclear how to generalize further, and vary the domain as well as the given relations. Beginning with the relatively unknown R definability in 1956, several ways were independently invented to develop for arbitrary structures analogs of ordinary recursion theory on the structure of arithmetic, in which the intention was to keep the intuitive notion of ‘finitary effective computation’ as intact as possible. Since it is arguable just what ‘effectively calculable’ should mean for an arbitrary structure, which itself may not be capable of being effectively presented (see Grillot [1974] for more on this) it is not surprising that essentially different theories could be produced. What is surprising, perhaps, is that a large number of the approaches turned out to be equivalent: R -definability (Smullyan [1956]); Frai‘ssecomputability (Fraisse [19611); V-recursiveness, for countable structures (Lacombe [ 1964, 1964193); No-recursiveness (Montague [1968]); search computability (Moschovakis [1969]); and others more recent yet. So whether or not these capture the intuitive notion of ‘effective’ for arbitrary structures in all cases, at least the convergence of so many separate approaches argues for the naturalness of the idea. Also, for the mild generalizations of ordinary recursion theory mentioned in the previous two paragraphs, these approaches simply give an equivalent of what was already accepted. So, picking out the best-known name from the list as a representative, we think of search computability as the primary generalization of ordinary recursion theory to arbitrary structures. We so treat it in this book. In the 1950’s hyperarithmetic theory was invented (Davis [1950]; Mostowski [1951]; and Kleene [1955] from where it gets its name) and refined (Spector [1955]). It studies certain subsets of the natural numbers (those on

xii

INTRODUCTION

the lowest level of the analytic heirarchy) by studying the way those sets can be characterized. It was not meant to be an ‘effective’ theory, indeed the sets involved properly include those definable using the full first order language of arithmetic (the arithmetic sets). But it became apparent that hyperarithmetic theory can be thought of as something like ordinary recursion theory with the computation machinery enlarged. In particular, it allows the asking of certain questions that take infinitely long to answer, but otherwise the machinery is essentially that of ordinary recursion theory. And there are parallels in the theorems provable. Consequently various investigators began thinking of hyperarithmetic theory as a sort of generalized recursion theory. Many of the further developments discussed below arose from attempts to make the analogy between hyperarithmetic theory and ordinary recursion theory into something more than an analogy. The original characterization of the hyperarithmetic sets was a complicated one involving both relative recursion theory and the theory of the recursive ordinals (a recursive ordinal is an ordinal that is the order type of some recursive well-ordering of the natural numbers). As such, it was not clear how to generalize the notion to structures other than that of arithmetic. In the 1970’s an equivalent approach using inductive definitions became central, and this generalizes very well. Hyperelernenfury theory is the name that has been given to this generalization (Moschovakis [1974]). And research has shown parallels between the hyperelementary theory of a structure and the search computability theory of that structure; hyperelementary theory behaves rather like search computability with some infinitary computational machinery allowed. Hyperarithmetic theory is a theory about the natural numbers. As originally developed, it used the recursive ordinals as a tool. In the 1960’s, Kreisel and Sacks suggested reversing things; use hyperarithmetic theory developed by other equivalent means, as a tool to create a theory about the recursive ordinals. The similarities between what resulted and ordinary recursion theory were striking. Indeed, in Rogers [1967] this was simply called the Kreisel-Sacks Analog. Soon, however, it became known as metarecursion theory (Kreisel-Sacks [19651, Sacks [19671). Meanwhile, Takeuti [1960] was working on a recursion theory of the entire class of ordinals, connecting it with the development of L, the universe of constructible sets, first introduced by Godel as a tool in showing the consistency of the continuum hypothesis in axiomatic set theory. Independently, and by different means, Kripke [1964] and Platek [1966] developed an analog of ordinary recursion theory for certain ordinals,

INTRODUCTION

...

XI11

called admissible or recursively regular. Working with the first admissible ordinal yields ordinary recursion theory, while the second gives metarecursion theory. And there are many, many other admissible ordinals, including, but hardly limited to, the cardinals. Now known as a! -recursion theory, this is a rapidly developing area. It can be thought of as being like ordinary recursion theory with the subject matter generalized from the natural numbers to the initial segment a of the ordinals, and with the means of computation also generalized, replacing ‘finite’ by ‘a-finite’, defined in an appropriate way. It has been aptly characterized as the recursion theory of a limited god. The work of Takeuti finds its natural generalization here too, and the connections between recursion theory on the ordinals, and the development of the constructible sets plays a big role in the work on admissible ordinals, making for an interesting relationship between recursion-theoretic and model-theoretic techniques. Search computability generalized ordinary recursion theory to arbitrary structures, while keeping the meaning of ‘finitary’ unchanged, as far as possible. Hyperelementary theory similarly generalized hyperarithmetic theory to arbitrary structures. In a similar way, a-recursion theory, developed via the constructible universe L, has been generalized to admissible sets wirh urelements (Barwise [1975]), in which arbitrary structures can be dealt with, and the notion of finiteness generalized at the same time. This too is a rapidly developing area. Search computability can be looked at as a special case, in fact. Also, the connection between hyperarithmetic theory and the a -recursion theory of the second admissible ordinal (metarecursion theory) has been vastly generalized to apply to the hyperelementary theory of any structure that, like (N, s), has a pairing function that can be generated using the machinery of hyperelementary theory. (This is most easily found in Moschovakis [1974] and Barwise [1975], in the references under HYP.) Relatively early, Kleene introduced into ordinary recursion theory the notion of recursive functional. A functional maps functions to numbers; a recursive functional, as Kleene defined it, is a functional that most people would consider to be ‘effectively calculable’ in an appropriate sense. But it wasn’t until Kleene [1959] that a way was found to continue this up the type hierarchy. In this influential paper, Kleene introduced the notion of recursive functional of finite rype, with type 1 being the usual recursive functions, type 2 being the usual recursive functionals, and with type 3 being ‘effective’ maps from type 2 functions to numbers, and so on up. Soon after, Platek [19661 generalized higher type recursion theory so that it too could be developed for essentially arbitrary structures.

xiv

INTRODUCTION

Once again, hyperarithmetic theory comes in. An early result of Kleene showed that hyperarithmetic theory could be looked at as higher type recursion theory relative to a particular type 2 functional (Kleene [ 19591.) Attempts to extend this have led to major research developments, see Hinman [1978]. Recently, too, partial recursive operators and functionals, familiar from ordinary recursion theory, have been introduced into (Y -recursion theory as well. (See DiPaola [1978], [1978A], [1979], [1981], [I%+].) Search computability, hyperelementary theory, admissible set theory with urelements, and higher type recursion theory all can be looked at as natural and successful generalizations of ordinary recursion theory. Since they all agree in some respects, and differ in others, naturally a kind of axiomatic recursion theory has arisen, to treat of these similarities and differences. In fact, there have been several such. The most successful and most widely adopted are those called ‘Computation Theories’. These had their ancestry in Strong [1968] and Wagner [1969], but were properly born in Moschovakis [1971], and had their descent through Fenstad [1974]. They are summarized in Fenstad [19751, applied rather thoroughly to higher types in Moldestad [1977], and are given a compact presentation in Fenstad [ 19x01. 111. We assume knowledge of a small amount of the material of an elementary logic course, enough to deal comfortably with models, and truth in them. We also assume knowledge of the content of a basic course in set theory. In particular, we freely use elementary facts about the ordinal numbers, transfinite induction, and the like. Many of our examples make use of properties of the constructible sets, but, outside of Chapter 9, no theory depends on this. The material in Chapter 9 supposes a reader more adept at dealing with the formal nuances of axiomatic set theory. Strictly speaking, we do not suppose any prior knowledge of ordinary recursion theory, though anything the reader may already know in this area will certainly be found helpful. We use a small amount of the terminology of category theory. For the most part, nothing beyond the basic definitions is needed; category theory is, here, merely an organi.zing device.

IV. We wish to thank some books, some institutions and some people for contributing to the present work. We begin with the books. (1) “Theory of Formal Systems” by Raymond Smullyan (Smullyan [1961]), in which elementary formal systems first fully appear, and in which an elegant development of ordinary recursian theory, using them, is presented.

INTRODUCTION

xv

(2) “Theory of Recursive Functions and Effective Computability” by Hartley Rogers Jr. (Rogers [1967]), which is an encyclopedic, systematic account of a major portion of recursion theory and its ramifications, as it had developed by the mid 1960’s. (3) “Elementary Induction on Abstract Structures” by Yiannis Moschovakis (Moschovakis [1974]), which presents an elegant generalization of hyperarithmetic theory to arbitrary structures. (4) “Admissible Sets and Structures” by Jon Barwise (Barwise [1975]), which presents the lovely subject of admissible sets with urelements. Next, the institutions. The library at Rensselaer Polytechnic Institute, and the Library at the City University of New York Graduate Center, for being good libraries. The mathematics department at Herbert H. Lehman College (CUNY) for being a place where mathematics happens. Finally, the people: Raymond Smullyan, for inventing elementary formal systems, for telling me about R-definability, and for long discussions about fixed point theorems; Yiannis Moschovakis for helpful suggestions concerning an earlier draft of the manuscript; Einar Fredriksson of North-Holland Publishing Company for making my experience with the company a most pleasant one; and especially Greer Fitting, for expert technical typing, for the secretarial work of keeping track of what was where, and for assistance in matters of style.

CHAPTER O N E

RECURSION AND W - RECURSION THEORIES

1. Introduction It seems best to begin with an informal description of ordinary recursion theory, in which natural numbers are the objects dealt with, and all computations are finite. There are many such informal descriptions. (These days they tend to be based on some sort of notion of ideal computer.) The description we use is directly relevant to the formal notions we will be developing, and we will return to it and elaborate it throughout the book. Suppose you are given a collection of empty boxes, and into these boxes you are to put natural numbers, or ordered pairs of natural numbers, or ordered triples, or the like. Each box is clearly marked: into this box you can only put natural numbers; into this box you can only put ordered pairs of natural numbers; and so on. And suppose you’re given a certain list of instructions as to how to fill those boxes. These instructions are to be what is usually called “effective.” An example of an instruction that is nor effective is: put 7 in box A if the Fermat “last theorem” is true, otherwise put 7 in box B. (If the Fermat conjecture is decided by the time you read this, substitute for it any other unsolved mathematical problem.) An example of an instruction that is effective is: look through box B and see if you put an ordered pair in there whose first component is n ; if you did, then look through box C and see if you put an ordered triple in there whose first component is n + 1; if you did, then put n + 2 in box A. Notice the features of this. You can search through a box to see what is there; you can take successors; and you can put things into boxes (you can not take things out). And the conditions are “positive”; you can act on the basis of what is in a box but not on the basis of what is not. Now, suppose you are given a list of “effective” instructions, and you proceed to follow them. Pick one of the boxes, say box A. As time goes on you put more and more numbers (or pairs, etc.) into box A. In this way a certain set is being generated. Such a set is called recursively enumerable. Much of ordinary recursion theory is the study of the recursively enumerable sets. 1

2

RECURSION AND W-RECURSION THEORIES

[CH. 1, $2

Often other characterizations of ordinary recursion theory are given. One common one is that it is the study of what functions on the natural numbers are “effectively calculable”. It turns out that a function on the natural numbers is effectively calculable just when it is recursively enumerable when thought of as a set of ordered pairs (that is, just when its graph is recursively enumerable.) We find it convenient, for now, to take recursive enurnerability of sets as the basic notion and treat functions via their graphs. Later on we will modify this position. Incidentally, one might ask what would be the effect of allowing, not only adding to boxes, but removing from them as well. As it happens, the effect is great. For one thing, one is never certain of what is going to remain in any given box. You may look in box A after you’ve worked three days and see the number 35 in there, but you have no guarantee that it will still be there in a month; you might find your instructions have forced you to take it out. Even more, the collection of sets and relations that may be generated using this additional freedom properly includes the recursively enumerable ones. It also includes ones which are only “recursively approximable”. We do not develop this notion. And we do not allow instructions to remove things from boxes. Our first goal is to generalize things so that, instead of applying just to the natural numbers, where one is allowed to take successors, one can treat an arbitrary collection of items o n which certain specified operations (more generally, relations) are given to work with. The mechanism we introduce for doing this is that of elementary formal systems. These originated in Smullyan [1961], for the case of words over a finite alphabet. Here they are generalized to apply to arbitrary structures. The connections between them and our informal description above should be obvious.

2. Structures

Let d be a set, and let B,,Bz,. . . , Bk be relations on d.(We allow one-place relations, and generally call them sets.) The k + 1 tuple (d; 9,, %, . . . ,%) is a structure with domain d.We follow the convention that German letters stand for structures, script letters stand for domains and so on. and relations, and the domain of is d,the domain of 23 is 9, We allow trivial structures (a), in which no relations are specified. We say a structure is empty or non-empty according to whether the domain is empty or not. We generally abandon “relation” notation for those relations having a customary symbolism. For example, we may write

CH. 1, §3]

ELEMENTARY FORMAL SYSTEMS

3

+

x y = z instead of + (x, y , z ) or (x, y , z ) E +, and we may write + for the addition relation itself. Certain structures will recur many times in this work, so we introduce special notations for them. N = {0,1,2,. . . } is the set of natural numbers. Let y = x + denote the successor relation on N. The structure (N; y = x’) is the setting for ordinary recursion theory. We denote it by 6(N) = (N; y = x’). A set is called hereditarily finite if it is finite, made up of finite sets, which are made up of finite sets, which etc. The collection of all hereditarily finite sets is denoted L,. Let x U { y } = z denote the relation on L, which holds just when z is the result of adding y as an element to the set x. We denote the structure (L, ;x U { y } = z ) by G(L,). , a,,) be the set of words (finite or empty strings or Let W ( a l , a 2..., sequences) over the alphabet a l ,a2,.. . , a , . Let x * a = y denote the relation on W(a,,a*,. . . ,a , ) which holds just when word y is the result of adding the one letter word a to the end of word x. We denote the structure (W(a1,a?,. . . ,a , ) ; x * a = y ) by G ( u ~ a?, , * . ., a n ) . 3. Elementary formal systems

Suppose a structure ‘2l= ( d ; . . , CRk) has been specified. One can imagine a set of instructions for filling boxes, as in Section 1, but which uses the members of d instead of the natural numbers, and which supposes as “known” the relations %,, . . . ,Sk instead of the successor relation. We now define a simple free-variable logical calculus which should be thought of as the formal counterpart of such a set of instructions. We suppose available an unlimited supply of n -place predicate -symbols for each n > O . We use Latin letters to represent them, P, Q, R, etc. The only other symbols of our alphabet are an arrow and a comma. We will be using axiom schemas, so variables are not needed in the formal language itself. They will, however, be used in the metalanguage; we denote them by x, y , v, v I , v2 etc, By an atomic formula over d we mean an expression of the form P a l , a 2,..., a,, where a l , a 2,...,a,, E d and P is an n-place predicate symbot. For convenience we may write Pa for P a l , .. ., a,. We also define a pseudo-atomic formula over d to be anything of the form Pa,,. . .,a. where each a, is in d or is a variable. Pseudo-atomic formulas are expressions of the metalanguage only. The notion of formula over d is defined by the following rules:

RECURSION AND O-RECURSION THEORIES

4

[CH.

1, 83

(1) an atomic formula over d is a formula over d ; ( 2 ) if X and Y are formulas over d,so is X - , Y . Formulas are to be thought of as being associated to the right. Thus A +B+C+D should be read as if it were A - . ( B - + ( C + D ) ) and thought of as saying A, B and C together imply D. The metalinguistic notion of pseudo-formula over d is defined analogously, being built up from pseudo-atomic formulas over d.And the notion of a (complete) instance of a pseudo-formula over d has an obvious definition. Any instance of a pseudo-formula over d is a formula over d. By the conclusion of a (pseudo) formula we mean the final (pseudo) atomic part of it. Thus if A is (pseudo) atomic, A is the conclusion of X - A , and also of A itself. ; . . .,9ik). Now Recall, we are working with the structure 'II = ( d9?,, suppose we have selected, once and for all, predicate symbols R I ,. . . ,Rk, so that R, is an n-place predicate symbol if 3,is an n-ary relation. We call R I , .. .,R, reserved predicate symbols. Let 9? T consist of all formulas of the form R,a for which 9?,u holds. We say a pseudo-formula X is proper if none of the reserved predicate symbols R I , . . .,R, occurs in the conclusion of X . Let E = {A I , . . . ,A,,,}be some finite set of pseudo-formulas over d,each proper. By a derivation from E over '8 we mean a finite sequence of formulas, X I , . . ,X I , such that each term of the sequence either (1) is a member of 5% T U * . . U 9? *k, or (2) is an instance of some A,, or (3) comes from two earlier terms by the rule

(MP)

x+ Y

provided X is atomic.

If there is such a derivation ending with X , we say X is derivable from E over 'II. E determines, relative to ?I, a simple deductive system, called an elementary formal system over 3 . Each A, is an axiom of that elementary formal system. Let P be an n-place predicate symbol, and let 9 be an n-ary relation on d.We say P represents 9 in the elementary formal system determined by E over ?I if a E 9 iff

Pa is derivable from E over 9.

We say 9is representable in the elementary formal system determined by E over 'II if there is some predicate symbol P which represents 9.

CH. 1, 841

EXAMPLES

5

Finally, we say B is recursively enumerable (r.e.) over % if B is representable in some elementary formal system over %. Also B is recursive over % if both 9 and its complement, d"- B7are r.e. over 9.

4. Examples

The relations which are r.e. in our sense over the structure of arithmetic, G(N), are the relations on N usually called r.e. (Rogers [1967]). The relations which are r.e. in our sense over the structure of the hereditarily finite sets, G(L,), are simply the relations C over L, (Barwise [1975]). And the relations which are r.e. in our sense over the word structure, G(u,,. . . u n ) ,are those called formally representable in Smullyan [1961]. This last fact is true essentially by definition (see Smullyan [1961], Ch. 4, §lo). Means of verifying our other assertions will be developed in the course of this book. For now, we simply give a few examples of elementary formal systems in use. First we consider examples involving 6 ( N ) .

I. The addition relation is r.e. over G(N), being represented by P in the elementary formal system with axioms: Px, 0, x ; Px, y , z

-3.

v

=y++

w

= Z+'PX,

v, w.

Just this once we sketch how one shows that these axioms do what we claim. CLAIMA. For each x, y , z E N, if x the above axioms.

+y

= z, then Px, y , z is derivable from

This may be shown by an induction on y .

CLAIMB. For each x , y , z EN, if P x 7 y , z is derivable from the above axioms, then x + y = z. This may be shown by an induction on the number of steps in the derivation of Px, y , z. Specifically, one shows that each line of a derivation is "true" in the sense that it comes out true when Px, y , z is interpreted as x y = z, and v = y + as v = y + 1. (A -3. B is given the usual implication

+

RECURSION AND W-RECURSION THEORIES

6

[CH.

1, $4

interpretation: if A, then B.) It follows that if P x , y , z is derivable, then x+y=z.

From now on we generally skip such details. Also, from now on we write

x

+ y = t instead of P x , y, z. Generally, for the sake of readability, we will

allow such informal notation in our elementary formal system axioms.

11. The “less than” relation is r.e. over G(N), having axioms: the above, and

+ y = 2 +y = w + + x

x

< 2.

111. “Not equal” is r.e. over G(N) having axioms: the above, and

x given by

( @ W P=) @(WP)). PROPOSITION 10.5. The collection of ( w ) enumeration operators over 'II is closed under composition. PROOF.Say @ = [E;] and P = [FZJ where E and F are disjoint. Then @T = [H'o] where H consists of the axioms of E,

Ix 4 Cx,

the axioms of F,

Dy +Ay,

Bz + Oz.

COROLLARY 10.6. If @ is an ( w ) enumeration operator and B is ( w ) r.e., then @(S)is ( w ) r.e. PROOF.Identify ( w ) r.e. relations with constant ( w ) enumeration operators. PROPOSITION 10.7. Each ( w ) enumeration operator @ over 'II is monotone, that is, 9 c 3 j @(B)C@(3). PROOF.If B C 3,every derivation over ('II,3).

(a,9)is also a derivation over

PROPOSITION 10.8. Each enumeration operator @over 'II is compact, that is, v E @(B)

for some finite

6 C 9,v E @(S).

CH. 1, 8121

THELEASTFIXEDPOINTTHEOREM

17

PROOF.Elementary formal system derivations are finite. 11. Recursion and w- recursion theories Let '21=(d;B,,..., 9%) be a structure. In a natural way both the collection of enumeration operators over 2I and the collection of w enumeration operators over 21 can be made into categories. We will study these categories in later chapters. Here we give the definitions. By [a]" ( n > 0) we mean the collection of all n-ary relations on .dThat = power set of d". We generally identify d'and d. is, [d]. The objects of our category are [a]', [&I2, [d]', ... . The morphisms are the ( w ) enumeration operators over 8 . An operator (morphism) of order (n, m ) has domain [d]" and codomain

Composition of morphisms is ordinary function composition. The identity morphisms are the operators I" of the previous section. By rec('21) we mean the category of enumeration operators over 2I as just described. We call it the recursion theory on '21. By ordinary recursion theory, we officially mean the recursion theory on G(N). By w-rec('21) we mean the category of w-enumeration operators over 2 I . We call it the w-recursion theory on 3. For us, hyperarithmetic theory means w-rec(G(N)). Note that by Proposition 9.1, rec('21) is a subcategory of w-rec(2I).

12. The least fixed point theorem Once again, let % = ( d B,, ; . . .,% k ) be a structure, fixed for this section. We prove, for both rec(2I) and w-rec(%), analogs of an important result of ordinary recursion theory, due originally to Kleene. See Rogers [1967], pp. 193-194 for a proof in the ordinary case. Applications of the generalized version will occur throughout this work; the first is in the next section. Let )=

u 6Z dom f.

(2) Let 9 be allowable in A. If the graph o f f is for some @ E A,

@(9x { u } ) =

G

9 (written f

u g dom f .

PROOF.Indentifying f and its graph, we have

f " ( { v } )=

u6Z dom f ,

and the theorem follows easily. DEFINITION. Let f be a partial function from d" to d k .

S

9), then

CH. 2, $61

PRODUCTION SYSTEMSWITH EQUALITY

43

(1) We say f is generated in A if the graph of f is generated in A. (2) We say f is generated pointwise in A if there is an operator @ E A of order (n,k ) such that @ ({u

1) =

D $Z dom f.

COROLLARY 5.5. If a partial function f is generated in A , it is generated pointwise in A . REMARKS.In Chapter 8, we will show there is a large class of production systems in which these two notions coincide. In general, they do not, and for the time being, we give first place to the notion of being generated, identifying f with its graph. In Chapter 10, when we come to discuss higher types, we will take pointwise generation as basic. 6. Production systems with equality

One of the production system axioms says that equality is generated. In this section we consider some effects of requiring that equality be bigenerated. We look specifically at the consequences for finite sets and for functions. DEFINITION. Let A be a production system. We say it is a system with equality if #& is generated in A. DEFINITION. For 1 s i, j s n we define a map of order (n, n ) by

E j ( 9 ) = { ( X I , . . . ,x n ) € 9

I

XI

# x,).

PROPOSITION 6.1. If A is a production system with equality, each

A.

PROOF.Similar to that of Proposition 3.3. PROPOSITION 6.2. Let A be any production system. Let 9C d" be finite. Then 9 is generated in A . PROOF.If a , , . . . , a, E d,each of { a l } ., . ., { a n }is generated, by Axiom (3). Then { ( a , ,. . .,a , ) } is generated, since there is closure under x . Finally, any finite subset of d" can be built up since there is closure under U .

PRODUCTION SYSTEMS

44

[CH.2,§6

COROLLARY 6.3. Let A be any production system, and let 9 be a finite relation on d. (1) 9 s9 for any allowable 9. (2) @(9) is generated for @ E A. (3) B S 9 if and only if 9 is generated in A. LEMMA6.4. Let A be a production system with equality and let 9 E [d]". If 9 has one fewer member than 9,then 9 is allowable in A and 9 S 9. PROOF.For simplicity of notation, say n = 3. Suppose 9 = 9 U {a}. By Proposition 6.2, {a} is generated in A ; let @ E A be constant, of order (n, n ) , and value { a } . Then

9 = P*:P:(E:,4 u E:,s u E:,&13 x @)(9). PROPOSITION 6.5. Let A be a production system with equality, and let 9,9 d " differ by a finite number of elements. If 9is allowable in A then ( 1 ) 9 is allowable in A and 9 S 9. (2) 9 w. (3) If 9 X 9 is allowable in A then 9 and 9 are each bi-generated in the other.

-

PROOF.Let 9 be allowable in A. If 92 has fewer members than 9,9 is allowable in A and 9 9 by repeated application of Lemma 6.4. If 92 has more members than 9,3 = 9 U 9 where 9 is finite. Then 9 s 9 by Corollary 6.3 and 9 S 9,so 9 = 9 U 9s 9 by Proposition 4.7. Thus we have (1). (2) follows by switching the roles of B and 92. Finally we have (3) upon noting that, if 9 and 9 differ by a finite number of elements, the same is true of $ and R. Proposition 4.4 (5) is also needed. COROLLARY 6.6. Let A be a production system with equality, and let 9 be a finite relation on d. (1) 9 s 9 U 9 if and only if 9 s 9. (2) 9 is generated in 9 U 9 if and only if 9 is generated in 9. PROOF.(2) Uses Proposition 4.6.

CH. 2, 871

CLOSURE UNDER LEASTFIXEDPOINT

45

Next we look at the behavior of functions. We assume d has more than one member, so characteristic functions are meaningful.

LEMMA6.7. Let A be a production system with equality. If f is a total function from d"to d mthen , f sf . PROOF.We present the simplest case only: say f maps sd to 1. Now, d is generated; let Qj be constant, of order (2,l) and value d.Then

7 = P:E:,(12 x @)(f).

THEOREM6.8. Let A be a production system with equality, and let f and g be total functions. Then f s g i f and only i f f x f s g x g. PROOF.(1) Suppose f s g. Then g is allowable. Now g s g and by Lemma 6.7, g S g, so g x g is allowable by Proposition 4.3. Further, by that, gsgxg andf~g,sof~gxg.Alsof~f,sof~gxg.ThenfXf~ gx (2) S u p p o s e f x f s g x g . W e l l , f s f x f s o f S g X g . A l s o g S g a n d g s g , so g x g s g . Then f s g .

s.

COROLLARY 6.9. Let A be a production system with equality. I f f and g are total functions, f s g if and only i f f is bi-generated in g. PROOF.By Proposition 4.5. COROLLARY 6.10. Let A be a production system with equality. (1) cp is bi-generated in cw if and only if 9 is bi-generated in 9. (2) If cg is allowable then cp and 9are each bi-generated in the other. PROOF.(1) Follows by the above and Corollary 4.10. For (2), first note that if cp is allowable, so is 9 X 9by Lemma 4.7. Since c9 is total, G s c9. But cp s 9 x 9 by Lemma 4.8. It follows that cp x 6 s 9 x 9. Also 9 x 9s c9 by Lemma 4.7, so 9 x @ s c : x~ G.

7. Closure under least fixed point

We characterize the r.e. and m-r.e. relations directly in terms of production systems, by-passing elementary formal systems. The terminology of inductive definability is from Ch. 1, 613.

46

[CH. 2, $7

PRODUCTION SYSTEMS

PROPOSITION 7.1. Let A be a production system on the structure a = (a;R,, .. . ,w k ) . Let Q be a P-formula (for this structure) where Pis n-place and cp has x , , . . .,x, as free variables. Let r, be the monotone operator that Q defines. Then (1) If cp has no universal quantifiers, r, € A . (2) If A is an w-production system, r, E A. PROOF.By an induction on the degree of

Q,

which we omit.

REMARK.A simple checking of the axioms shows that the least total production system on 52 is precisely the collection of f, defined by P-formulas without universal quantifiers (where, however, to get an operator of order (n, rn) we must allow P to be n-place but Q to have m free variables). Similarly the least total w-production system on is the collection of all defined by arbitrary P-formulas Q. Both these are subsystems of fo (a).

r,

DEFINITION. We say a production system A is closed under the taking of least fixed points if, for every monotone QZ E A of order (n, n), the least fixed point of @ is generated in A. PROPOSITION 7.2. Let A be a production system over the structure 8 with infinite domain, and suppose A is closed under the taking of least fixed points. ( 1 ) Every relation r.e. over is generated in A. (2) If A is an w-production system, every. relation w-r.e. over is generated in A. PROOF.We show (2); (1) is similar. Suppose 2'2 is w-r.e. over 'II.Then by Theorem 1.13(3), 2 is inductive on in %,, . . . , %!k. Thus there is a relation 9' which is a fixed point, and there are a,, . . . , a, ESP so that xE

w

@ ( a , x ) E 9.

9 is the least fixed point of an operator f,,but by Proposition 7.1, I', E A, hence 9' is generated in A since A is closed under the taking of least fixed points. It follows that 59 is generated in A since it may be obtained from 9' by the use of section operators. 7.3. Let be a structure with infinite domain. 3 is r.e. (w-r.e.) COROLLARY over 8 if and only if 2 is generated in every production (w-production) system over 'II which is closed under the taking of least fixed points.

A KLEENE- POSTRESULT

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PROOF.If % is generated in every production system over 2l which is closed under the taking of least fixed points, then 9 2 is generated in rec(2l) so $24 is r.e. over 2l. The converse is the above proposition. 8. A Kleene-Post result

In Kleene-Post [1954] a theorem was shown which implied that in ordinary recursion theory there were two sets, neither recursive in the other. In this section we show an analog of this for a class of total production systems which includes all recursion theories with countable domains. Our proof uses a small amount of the methods of forcing. Even so, it is essentially the original Kleene-Post proof. In Friedberg [19571 and Muchnik [ 19561 the Kleene-Post result was improved to show that in ordinary recursion theory there were two r.e. sets neither recursive in the other. The proof of this introduced what are called priority arguments into the subject, and much research has gone into generalizing the class of structures in which priority arguments can be used. We do not pursue this, quite extensive, topic here. See StoltenbergHansen [1979] for a treatment of priority methods in an axiomatic setting closely related to that of Chapter 8. Since domains are to be countable in this section, we may as well use the natural numbers. We will sometimes make use of special features of N, but from the outside, so to speak, not within our production systems. Let 43 be an operator on N of order (n, k), say. Recall, 43 is compact if x E @(9) x E @(S) for some finite 9C 9.And @ is monotone if 9 c a 3 @ ( S ) Ca)(%).

+

LEMMA8.1. Suppose 43 is an operator on N which is compact and monotone. 9,. Let 9, g2C P3C_ . * * be a chain of n-ary relations, and let 9 = Then

u,

xE

@(9)e for some j , x E 43(9,).

Let 9 be a collection of operators on N. We do not require that 9 be a production system, but we assume operators in 9 are total. We write 9 S 9 % if 9=43(%) for some Q i E 9 . We say 9 and 9 are 9incomparable if neither 9 S9% nor 99 s99. By a 0-1 function we mean a function whose domain is an initial segment of N (not necessarily proper) and whose range is {O,l}. A 0-1 function f

48

[CH.2,§8

PRODUCTION SYSTEMS

whose domain is a proper initial segment of N will be called a finite 0-1 function. If the domain of f is all of N, f is simply a characteristic function.

THEOREM 8.2. Let 9 be a countable set of compact, monotone operators on N, each of order (2,2). There exist 9-incomparable 0-1 functions.

The proof of this, the main result of the section, occupies the next several pages. After it is established, we derive some corollaries. By a forcing condition we mean an ordered pair of finite 0-1 functions, r = ( f r , gr), so that fr and gr have the same domains. Let Ce be the collection of all forcing conditions. If T,A E %, we write r C- A if fr C- fA and gr C g A , and we call A an extension of T.Note that C is reflexive and transitive on forcing conditions. N' is the set N X N. 9 is countable; let &, @,, @*, . . . be an enumeration of it. Next we set up a certain formal first order language. Let f and g be two formal symbols. By an atomic formula we mean any expression of one of the forms (n, k E N are variables). (n,k)Ef,

(n,k)E@jcf);

(n, k ) E g,

(n, k ) E @i (g )-

(Formally, (n, k ) € f is simply some two-place atomic formula, and so is (n, k ) E @jcf) for each j = 0,1,2,. . . .) Formulas are built up from these atomic formulas using A , v, , 3 and 3 in the usual way. Note that we omit V. We define a relation I= between forcing conditions and formulas as follows. On closed atomic formulas: for each T E Ce

-

r l = ( & k )E f

if (n,k ) E f r , if ( n , k ) E g r - ,

t= (n, k ) E g rl=(n,k)E@jcf)

if(n,k)E@jcfr),

rI=(n,k)E@j(g)

if ( n , k ) E @ j ( g r ) .

Next, l= is extended to all closed formulas as follows.

TC(XvY)

e rl=X e TCX

TI=-X

e

T C ( X AY )

and T C Y . or T C Y .

for every A E 59 such that r C A, not-A I= X .

A KLEENE- POSTRESULT

CH. 2, 581

T F ( X 3 Y)

e

49

for every A € 9 such that then t= Y.

rcA, if

Tt=X

r t= ( 3 x ) X ( x ) e r t= X ( c ) for some c E W. This makes (3, C ,F7N2) into a constant domain Kripke Intuitionistic logic model, in the sense of Fitting [1969] (except for the omission of V). LEMMA8.3. Let r E 9 and X be a formula with no free variables. There is an extension A of r, in 9,such that A t= ( X v X ) .

-

PROOF.If some extension 0 of r is such that 0 t=X,set A = 0. If for every extension 0 of r, n o t - 0 t= X, then r t= X , so set A = r.

-

LEMMA8.4. Let f E 9 and X be a formula with no free variables. I f f I= X and A E 3 is an extension of r, then A t=X. PROOF.By induction on the degree of X . % C 9 is called a complete sequence if % is a chain of forcing conditions and, for every closed formula X , there is some T E % such that r t=(xv x).

-

LEMMA8.5. There is a complete sequence. PROOF.Enumerate the closed formulas: XI, Xz, X,, . . . . (It is here that we use the countability of N and 4.)Suppose rl,T z 7 ...,rkhave been defined By so that T , C r , C _ - . . C I ' , and for each l s i s k , Kl=(X,v-Xi). v - X , + , ) ;set Lemma 8.3,there is an extension A of r k such that A I= (xk+I r k + l = A. In this way % is generated. (Note that this is non-constructive.) Let % be a fixed complete sequence. Set

fw=

UU,+-EW,

gw= U { g , ( r E w .

Then both fw and gw are 0-1 functions. LEMMA8.6 (Truth Lemma). Let cp (f, g ) be any closed formula. cp ( f ~g, ~ is) true (over f$) if and only if for some r E %',r t= g).

cpu,

PROOF.By an induction on the degree of cp. Begin with cp atomic. If cp is (n, k)€ f or (n, k)€ g the result is simple. Now suppose cp is (n, k ) E @, (f).

[CH. 2, 88

PRODUC~ION SYSTEMS

50

Case 1. (n, k ) E aj(fw) is true. Then by Lemma 8.1, for some r E %, (n, k ) E Qj ( f r ) . But then r C (n,k ) E @, (f). Case 2. For some r E %, r I= (n, k ) E aj(f).Then (n, k ) E Qj c f r ) but @j is monotone and fr C f%,so (n,k ) E ajc f e ) . Similarly if rp is (n, k ) E Gj(g). Higher degree formulas are left to the reader. (It is here that our omission of V is used. The usual definition of r I= (Vx)X(x) in Kripke models does not allow this theorem to extend to formulas containing V.)

LEMMA8.7 (Combinatorial Lemma). Let r E 9 and j E N. ( 1 ) For some extension A of r, and for some n, A l=:([(n,O>E @ j ( g ) A - ( n , O > E f l v [ - ( n , O ) E @j(g)A(n,O)EflI.

r, and for some n, cf) A - (n,O>E 81 v [ - (n,O)E cf) A (n,O>E gl).

(2) For some extension A of A l=.([(n,O> E @j

@j

PROOF.We show only (1).Let n be an integer new to r (this exists since r is a pair of finite 0-1 functions). Case 1: There is an extension r*of r such that r*!=(n,0 )E @, (8). Now, whether this happens at r* or not depends only on the second component of r*.Let A be the same as r*in the second component but, if (n,O) is the first component of r*,take it out and replace it by (n, 1). (If (n,O) isn't in the first component of r*,(n, 1 ) is, so just set A = r*.) Then also A +(n, 0 )E aj( g ) . But A I= (n, 1) E f , and it follows that no extension of A can force (n,0) E f ; hence A l= (n,0 )E f. Case 2: For every extension r*of r we have not-r* I= (n,0) E @, ( g ) . Then r C (n,0)E ej(g). Let A be some extension of r with (n,0) in the first component. (This exists since n was new to r.)Then A I= (n,0 )E f. But since A extends r,A != (n,0 )E Qj( g ) .

-

-

-

COROLLARY 8.8. For each I'E % and j E N: (1) For some extension A of r and some n,

A t=-[(n,O)Ef=(n,O)E @j(g)].

(2) For some extension A of AC

r and some n,

- [(n,0 )E g = (n,0) E Gjcf)].

PROOF.(Here X = Y abbreviates (X 3 Y) A (Y3 X).) This may be derived directly from Lemma 8.7. Or one may proceed as follows. If X is a

A KLEENE -POSTRESULT

CH. 2, 581

51

theorem of Intuitionistic Logic, r I= X for every r E % (see Fitting [1969]). Also if r I= X and r I= X 3 Y then r I= Y. The result then follows using Lemma 8.7 and the Intuitionistic Logic theorem. [ ( A A - - B ) v ( - A A B ) ] ~ - [ ( A~ B ) A ( B ~ A ) ] .

COROLLARY 8.9 (Key Corollary). For each r E % and each j E N, (1) n o t - r t = - ( 3 x ) - [ x ~ f = €Gj(g)], x (2) n o t - r != ( 3 x ) [x E g = x E Gjcf)].

-

-

PROOF.We show (1). Let r E %. By Corollary 8.8, for some extension A of r and some n, Hence

A I=

- [(n,0) E f = (n,0) E

@J

(g)l.

A !=(3x)-[x E f ~ ExG j ( g ) ] . If we had that r!=-(3x)-[xEf=x

then for every extension not-r* != ( 3 x )

~aj,(g)],

r*of r,

- [x E f = x E aj(g)],

which is a contradiction since A is an extension of

r.

THEOREM8.10. fw and gw are $-incomparable. PROOF.Suppose fw G9gw(the other way round is similar). Say fw Then the following is true: -(3x)-[x

Efw

= Gj(g%).

= x E Gj((g%)].

By the Truth Lemma (8.6), for some

r E %,

~ ~ - ( ~ X ) - [ X E ~ ~ X E E ~ ( ~ ) ]

which contradicts the Key Corollary (8.9). We have completed the proof of Theorem 8.2. We derive some corollaries.

COROLLARY 8.11. Let A be a total production system with a countable collection of operators, in which every operator is compact and monotone. Then, in A, there are two sets, neither bi-generated in the other.

52

PRODUCTION SYSTEMS

[CH. 2,88

PROOF. In a production system, singletons are generated. Since there are countably many operators in A, the domain of A must also be countable. Then by Theorem 8.2, there are two 0-1 functions fw and g , which are A -incomparable. If f, were finite, by Corollary 6.3, we would have fv zs g, which is impossible. So fs, and similarly g,, are not finite. Hence they are characteristic functions, say f, for 9 and g%for 92. Then by Corollary 4.10, neither 9 nor 92 is bi-generated in the other.

COROLLARY 8.12. In any recursion theory with a countable domain, there are two sets, neither recursive in the other.

CHAPTER THREE

EMBEDDINGS

1. Introduction

In group theory one is interested in not only the groups themselves, but also in their relationships with each other: in the homomorphisms. Well, we’re not only interested in production systems, but in their interrelationships. What takes the place of group homomorphism is what we call an embedding. Roughly speading, an embedding is a coding of the domain of one production system into that of another which “carries along” the production system machinery. The collection of all production systems, with embeddings as morphisms, constitutes a category, with the collections of all recursion theories, and of all w-recursion theories as full subcategories. This chapter is the first of a sequence of three which study such categories, both for their general algebraic features, and for features of particular embeddings of interest to us. It is standard practice, in ordinary recursion theory, to discuss effectiveness on structures other than that of arithmetic, by using the device of Godel numbering. Natural numbers are assigned, in some systematic way, to the objects of interest, then one works with the objects indirectly by manipulating these “code” numbers within ordinary recursion theory. Since, in our approach, every structure has its own recursion theory, we do not need such a device here. Instead, standard Godel numbering techniques generally provide us with examples of embeddings between recursion theories. Thus they are still present, but in a different role.

2. Codings

DEFINITIONS. Let 8 be a mapping from d into non-empty subsets of $23 such that, if u # u r then 8 ( u ) fl O ( v r )= 0. We call 8 a coding of d into $23. The members of O ( u ) are codes for u. If, for each u E d,O ( v ) has exactly one 53

EMBEDDINGS

54

[CH. 3, 82

member, we say 8 is a 1-1 coding. If each member of 93 is a code, we say 8 is onto. We generally write 8 : d + 93 to indicate 6 is a coding of d into 93 (sometimes with explanatory words, if confusion with an ordinary function seems likely). Let a : d +- 93 and p : 93 + % be codings. By the composition, pa of a and p we mean the coding satisfying x E(pa)(u)

e

x E p ( w ) for some w E a ( u ) .

Thus x E (Pcx)(u) if x is a “p-code” for some “a-code’’ for u. The composition of 1-1 codings is 1-1. The composition of onto codings is onto. For each set d we define a coding inj, : d +- d by inj, ( u ) = { u } for u E d. inj, is 1-1, onto, and clearly serves as an identity under composition. A (large) category is defined by taking, as objects, all sets and as morphisms, all codings, with composition defined as above, and the inj as identity morphisms. We call this category Dom. In it, 0 is an initial object. But in it the connection between monic, epi, 1-1 and onto is not quite as expected. (See MacLane [1971] for terminology.) PROPOSITION 2.1. In Dom, every morphism is monic. PROOF.Suppose a : d += 93 and p, y :% +d are codings. Suppose also that ap = ay. Let c E %; we show p ( c ) = y ( c ) , and hence p = y. Let x E p ( c ) . Then x E d so a ( x ) is defined. Choose some y E a ( x ) . Then y E a ( x ) for x E p ( c ) so y E ( ap) ( c) . Then y E ( a y ) ( c ) This . means y E a ( z )for some z E y ( c ) . Now y E a ( z )and y E a ( x ) and sets of codes for distinct objects must be disjoint, so x = z. Since z E y ( c ) , x E y ( c ) . Thus p ( c ) y ( c ) . The reverse inclusion is similar, so p = y.

PROPOSITION 2.2. In Dom a morphism a is epi if and only if a is 1-1 and onto. PROOF.First suppose a is epi, say a : d + 93. Let x E d and suppose y , z E a ( x ) . Define codings p, y :93 -+93 by p = injB, and y is inj, except on y and z, where y(y) = { z } and y ( z ) = {y}. Then pa = ya. Since a is epi, p = y and this implies y = z. Thus a is 1-1. Again, suppose a : d + 53 is epi, but that a is not onto. Let y E 93 be not an a-code. Let z$Z 93, and let % = 93 U {z}. Define codings

CH. 3, §3]

PRE-EMBEDDINGS

55

P,y:93+% by P(x) = { x } for all x E 93,and y ( x ) = { x } for all x except y while y ( y ) = { y , z } . Again Pa = ya, so p = y, and this implies y = z, a contradiction. Thus a must be onto. If d is 1-1 and onto it is easy to check that it is epi.

Let 8 : d 493 be a coding. We extend 8 to n-tuples as follows. If a = ( a l , .. . ,a,) and b = ( b l , .. . ,b"),we write b E 8 ( a ) if b , E 8 ( a i )A . * A b. E 8 ( a n ) .Also we define maps associated with 8, but on relations, as

follows. Let 9 C .d".By 0, (9) we mean {b E 93" for some a E %, b E @ ( a ) } . Let Y C 93". By B;'(Y)we mean { a E d" for some b E 9, b E 8(a)}. Then both 8, and 8;' are mappings: 8, :power set (d ")+p o wer set (B"),

I 1

8;'

:power set (93"")ppower

set (d").

The following facts may now be established. Most of them depend on the "disjointness" feature of codings: a # b implies 8 ( a ) and 8 ( b )are disjoint. PROPOSITION 2.3. Let a : d + $33and p : 9 3 + % be codings. Let 9,9C d" and Y C 93". Then (1) (a;' a n ) ( %=) 9. (2) If a is 1-1, ( a n a ; ' ) ( Yc ) 9. (3) If a is onto, Y c ( a n a ; ' ) ( Y ) . (4)

(pa). =

@nun.

(5) (pa)nl = a,'p;'. (6) an (9n 3)= an(9) n a n (3) (7) a n (9u 3)= a n (9) u a n (9) (8) i f cp C S", a n + m (x9cp) = an (9) x am(cp). 3. he-embeddings DEFINITION. Let A and B be production systems, and let 8 :d + $3 be a coding. We call 8 a pre-embedding if (1) for each 9 E [d]" allowable in A, 8, (9) is allowable in B; (2) for each % E[93]" allowable in B, 8 ; ' ( 9 ) is allowable in A.

REMARKS. If A and B are total production systems, recursion or w recursion theories for example, every coding from d to 93 is automatically a pre-embedding. Pie-embeddings will play a non-trivial role in Chapter 9, when many partial production systems will be studied.

56

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[CH. 3, 83

PROPOSITION 3.1. The compositionof pre-embeddings is a pre-embedding. PROOF.Let A, B and C be production systems and a : d + B and p : 9 + (e be pre-embeddings. For 9 d",by Proposition 2.3 (4), (pa), ( 9 )= p. (a,, ($32)). If 9 is allowable in A, a, ( 9 )is allowable in B since a is a pre-embedding; then Pn(a,( 9 ) is ) allowable in C. (Pa);' is treated similarly using 2.3 (5). DEFINITION. Let 8 be a pre-embedding from A to B. We use [d]"for the collection of n-ary relations allowable in A ; and similarly for [9]". If we have a map @ : [d]" + [d]", by Qje we mean the map from to [B]" given by @' = 8,,,@8;'. Similarly, if we have a map 'I' : [ a ] " +[a]"', by I Y H - 'we mean the map = 8 ; 'I'O,. from [ d ] " t o [ d ] " given by 'I'6-'

[a]"

PROPOSITION 3.2. Let 8 be a pre-embedding from A to B. Then for @ E A, (@")"-' = @.

PROOF.Say @ is of order (n,m ) . Then, using Proposition 2.3,

(a'")'' = 8;'@'e8.

= 8i18m@8;18.= @.

PROPOSITION 3.3. Let 8 be a pre-embedding from A to B. (1) 8 preserves composition of operators. (2) 8-l preserves identity operators. PROOF.We only prove (1). . using Say @ E A is of order (n,m ) and P E A is of order ( r n , ~ )Then, Proposition 2.3 again,

'I"@' = 8,1Ye~18m@8;1 = e,IY@e;'= (T@)'. PROPOSITION 3.4. Let 8 be a pie-embedding from A to B which is 1-1 and onto. ( 1 ) 8 preserves identity operators. ( 2 ) 8-' preserves composition of operators. PROOF.Similar to the above. PROPOSITION 3.5. Let 8 be a pre-embedding from A to B, and let @, ?PE A. Then (when defined),

EMBEDDINGS AND CO-EMBEDDINGS

01. 3, 941

(1) (@ n *)* = @* n V, (2) (@ u *)* = @* u ?P*, (3) (@ x qe = @* x

v.

PROOF.We show 1 only. Say @ and ly. are of order ( n , m ) . Then using Proposition 2.3, for 9 E [a]", (@

n T)*(9) = [em(@ n ? ~ ) e ; ' ] ( 9 ) =

e, (CPn v p i l(9))

= 8,

[@(ei'p))n? P ( 8 - ' ( 9 ) ) ]

( @ ( e W ) ) )n 8, (lY(en'(9))) = @* (9) n (9) = (@' n ?P*)(P). =8 ,

?PO

4. Embeddings and co-embeddings

DEFINITION. Let A and B be production systems, and let 8 be a preembedding from A to B. We call 8 an embedding of A in B, and write 8 : A +B, if for each operator @ E A, @' is an operator in B. We call 8-l a co-embedding of B in A, and write 8-' :B +A, if for each operator ly E B, T-' is an operator in A. Embeddings and co-embeddings are 1-1 or onto if the underlying codings are 1-1 or onto. PROPOSITION 4.1. Let CY :A +B and p :B + C be embeddings. Then the composition, Pcy, is an embedding. In fact, @@ = Similarly the composition of co -embeddings is a co -embedding. PROOF. For embeddings only. If we show @* = (@")O, we are done, since both CY and p map operators to operators. Now, say @ E A is of order (n, m ) . We have @ k = (P.)m@(PQ);'

= P,a,@a~'pi' = p,,p,'

= (@")P

58

EMBEDDINGS

[CH. 3, $5

PROPOSITION 4.2. Let 8 be a pre-embedding from A to B. (1) If 8 is an embedding, 0 takes relations generated in A to relations generated in B. ( 2 ) If 8-l is a co-embedding, 0-' takes relations generated in B to relations generated in A. PROOF.We only prove (1). Let 9 E [d]"be generated in A. Then for some @ E A, 0 is constant, of value 9. Then CPo E B, and trivially, Gois constant, of value On(%).

REMARKS. Just because 8 is an embedding, it doesn't follow that 8-' is a co-embedding. The following example illustrates this. Let N be the natural numbers, and let '21 = (N), a trivial structure. Define a coding 8 :N+N by 8 ( n ) = { n } ,that is, 0 = injN.It is easy to see directly, or by use of Theorem 5.1 below, that 6 : rec(%)+ rec(G(N)) is an embedding. Let 8 be the set of even numbers. % is r.e. in rec(G(N)). But 8;'(%) = %, and this is certainly not r.e. in rec('21) since there is not enough machinery there to generate it. Then by the above proposition, 0-' is not a co-embedding. DEFINITION. Let V be a collection of structures. We say T is a theory assignment for V if T assigns to each structure 9 E V some production system T(%) on the structure 8.

5. Theory assignments EXAMPLES. Let V be the entire class of structures. Then rec(-), m-rec(-) and f.o.(-) are theory assignments for V. In Chapter 9 we will consider theory assignments that are meaningful for some, but not all structures, at which time V will play a bigger role. Let T be a theory assignment for V and let '21, 23 E V. If we have a coding 8 : .d + 93, it may be quite difficult to tell whether 8 is an embedding of T ( % )in T(23).But for certain theory assignments we have a simple condition. To keep notation simple, if 8 :9-+ 23, and 3 is an n-ary relation on d, we will often write 92' for 6. (92). DEFINITION. Let V be a collection of structures, let T be a theory assignment for g,and let % be a collection of codings. We call Telementary

CH. 3, 151

THEORY ASSIGNMENTS

59

with respect to % if the following is the case: For any two structures %=(d;%l,...,9?k), and for any coding 8 : d + W which is in '8, 8 is an embedding of T ( % )in T(B) whenever (1) 8 is a pre-embedding of T ( % )in T(B), (2) Each of ( = ,d)', 27,.. . ,5?:is generated in T(B).

%,BE%',where

REMARKS. rec(-), a-rec(-) and f.o.(-) assign total production systems to structures. For such theory assignments clause (1) is automatically satisfied. We will consider partial production systems in some detail in Chapters 8 and 9. THEOREM 5.1. The following are elementary theory assignments for the class of all structures with respect to the class of all codings: (1) rec(-), (2) w-rec(-), f.o.(-). COROLLARY 5.2. If 8 : rec(%)+rec(B) is an embedding, 8 is also an embedding of o-rec(%) in w-rec(B). PROOF.Every r.e. relation on B is also w-r.e.

PROOF OF THEOREM 5.1. We leave the proof of (3) to the reader. Suppose now that T ( - )is either rec(-) or w-rec(-). Suppose % = (d; %, . . .,%k), 8 : d + 93 is a coding (hence automatically a pre-embedding since recursion and w -recursion theories are total), and each of ( = &)',%:,. . ., %: is generated in T(B). We show 8 : T(%)+ T(B) is an embedding. We define a certain translation procedure taking elementary formal systems over % into elementary formal systems over B. The idea is to replace members of d by all codes for them, and the relations . ., 9% by %?,.. ., %:, of course adjoining extra axioms to handle these over %. In order to define our translation procedures more easily, we first introduce some notation. Let A be a pseudo-formula (recall; contains variables) and let C be a one-place predicate symbol. By A (axiom A restricted to C) we mean CX~C +X~+ *.

*+

CX.+ A

where x l , . . .,x , are all the variables of A (in some fixed order). Below C will represent the set of codes. Let = be a two place relation symbol (intended below to represent ( = d ) ' ) . Let P be some n-place predicate symbol. By an equality axiom for P we mean

[CH. 3, 85

E~EDDINGS

60

= x1, y1+ --- x2, y2-

-

*

- += x,,

y. +PX1,.

- 7

Xn + @ I ,

*.

yn

where all variables are distinct. We note that the set of codes, in '3, is the set of first (or second) components of ( = Now, let E be some elementary formal system over 3 (accepting inputs). Let us suppose that the predicate symbols R,, . ..,Rk,have been assigned to W1,. . . ,92k,that the predicate symbols assigned to the given (displayed) relations of B are different, and do not occur in E. Let = (two place) and C (one place) be new predicate symbols. We define an elementary formal system E e over 8 as follows. First, say A is an axiom of E. For each member of d which occurs in A, choose one 8-code for it from B. Let A ' be the result of replacing, in A, the members of d by these codes. Now, the axioms of E e are: (1) For each of i = 1,2,. . ., k, axioms for (94)' in which we use R, again as the representing predicate symbol. (2) Axioms for ( = .d)e in which = is the representing predicate symbol. (3) Equality axioms for each predicate symbol of E. (4) = x, y + cx. (5) (A')Cfor each A in E. Now, if [EL] is an operator in T ( 8 ) of order ( n , m ) , [(E')L] is an operator in T ( 8 ) ,also of order (n,m).And it should be clear that whatever [EL] does, [(E')L] does the corresponding thing to codes. That is, for

&ae.

9 E [&I",

8, [Eb](9) = [(Ee)b]8n (9).

Further, it should be clear that [(E '$1 "ignores" non-codes, and behaves the same whether it has all, or only some codes for a thing. Now, if 92 E [ B ] " 8,8;'(92) , is the result of discarding from 92 all non-codes, then adjoining all other codes for things with any codes there. Hence we may formally state our observation about [ ( E e ) L ]as follows: for 92 E [ a ] " ,

[(E'

( 9 )= [ ( E)LI ~ en8;'

(a1.

Combining our two observations, we have

[ ( E)LI ~ ( 9 )= [(E' ) L I ~8;.' ( 9 ) = 8, [EL]8;'(92)= [ELIe(92).

Since, by construction, [(E')&]is an operator in T(B),so is [EL]'. Since operators map to operators, 8 is an embedding.

CH. 3, 861

EXAMPLES

61

6. Examples We give examples of embeddings between the recursion theories over the structures named in Ch. 1, 92. Since we are working with recursion theories, which are total, every coding is automatically a pre-embedding. Note that by Corollary 5.2, there are similar embeddings between the corresponding w -recursion theories. I. Numbers to sets. Define a coding 8 : N+L, by: 8 ( n) = { n ” } where n # is the Von Neumann ordinal for n. Just this once we go through the details, verifying that 8 is an embedding of rec(G(N)) into rec(G(L-)). The image, under 8, of the successor relation on the natural numbers, is represented in rec(G(L,)) by S using the axioms

s@, {@I7

sx, y + x

u {x} = X I + y u {y} = y ’ +

SX’, y’.

Also, the image of = is represented in rec(G(L,)) by = using the above and

-0,o = x, y + sx, XI-+

sy, y ’ +

= X I , y’.

Then by Theorem 5.1, 8 is an embedding of rec(G(N)) into rec(G(L,)). It is 1-1 but not onto. 11. Sets to words, with an alphabet a l ,a2,. . . ,a. of more than one letter. Let us say two of the letters are } and {. Using these letters, each hereditarily finite set has “names” in a natural way. { } “names” 8. If n l , n 2 , .. . ,nk are “names” for the sets sI,s2,.. .,s k , then {nl nz * * n k } is a “name” for the set {s,,s2, ..., sk}. Now, define a coding 8 :L, + W ( {,}, a?,. . . ,a,) by O(s) is the collection of all “names” for s. 8 is an embedding of rec(G(L,)) into rec(G(a,, . . . , a ) )neither , 1-1 nor onto.

-

111. Words to numbers. Say, for convenience, that our letters are 1,2,. . . , n. Define a coding 8:W(1,2,..., n ) + N by

e(a,a z . .. a , ) = {2“,

.3”2.

where

pi is the j t h

. . . .P ? }

prime,

and

the

a,

are

letters.

Then

EMBEDDINGS

62

[CH.3, P7

8 :rec(G(l,2,. . . ,n))+ rec(G(N)) is an embedding, 1-1 but not onto. (It is usually called a Godel numbering of W ( a l , .. ., a").)

IV. Numbers to words, with an alphabet of one letter. Define a coding 8 :N+ W ( a ) by

e ( n ) = {aa . . . U } Ls---,

n

Then 8 :rec(G(N))+ rec(G(a)) is a 1-1, onto embedding. V. Words to sets. (a) By composing I11 with I, there is an embedding of rec(G(a,, . . . , a , ) ) into rec(G(L,)). (b) A more direct example of such an embedding is the following. Associate with each letter aisome arbitrary "code" set u y . Associate with the word w = w Iw 2 .. . w. (where each w iis a letter) the finite function f w given by fw ( i ) = w y . Set 8 ( w )= cfw}.This too gives an embedding, 1-1, but not onto. VI. Words to words. Clearly, the choice of letters is not important, only the number of them. Define 8 : W ( u l , ..., an)+ W ( a l ,... ,a,, a n + l .r.., ak) by e ( w ) = (w). This is a coding, 1-1 but not onto, and gives an embedding on the recursion theories. Similarly, by composing 111, IV and the above, there are also embeddings that shrink alphabets. There are more directly defined examples of such embeddings in Smullyan [1961]. VII. Sets to numbers. (a) Compose I1 and 111. (b) Another example is the following. Define a coding 8 :L, + N by 8 ( E ) = {n} where F. is the nth constructible set, using the definition in Godel [1939]. This is 1-1 and onto, and 8 : rec(G(L,))+rec(G(N)) is an embedding.

7. The category of production systems

We define a (large) category, of production systems, as follows. The objects are all production systems. The morphisms are all embeddings (domain and co-domain specified, since the same coding may work for many production systems). By Proposition 4.1, the composition of embed-

Ck. 3, 171

THECATEGORY OF PRODUCTION SYSTEMS

63

dings is an embedding. It is easy to see that the injection codings are embeddings, and serve as identity morphisms. Thus we have a category, which we call Prod. In Prod, 0 serves as initial object. Certain subcategories of Prod are of obvious interest to us. The full sub-category having all recursion theories as objects is one; we call it Rec. Likewise there is the full subcategory having all o-recursion theories as objects; we call it o-Rec. And there is the full subcategory of first order theories. We call it F. 0. It is the study of these and related categories that largely occupies the next few chapters. We note that, by taking production systems as objects, but using co-embeddings as morphisms, we also obtain a category. We may call it Co-prod, but we will have little to say about it.

CHAPTER FOUR

COMBINING PRODUCTION SYSTEMS

1. Introduction

It is common mathematical practice to combine “like” mathematical objects into another of the same sort. Thus, one can form the direct sum of two rings, and obtain a ring, or the product of two topological spaces, and obtain a topological space. In this chapter we show how two recursion theories may be combined into a recursion theory, using either disjoint union, U ,or Cartesian product, X . Similarly for w-recursion theories, and for first order theories. This will play an important role in the next chapter when it will be necessary to discuss two recursion theories “jointly.” We begin by defining U and x for domains, and for codings. Both U and X have certain common features, and we spend some time discussing them simultaneously. The language of category theory is convenient here. We use no deep results, only the terminology, which is helpful as an organizing device. After treating domains, we define U and x between ‘‘like’’ production systems. It is here that theory assignments take on a major role. If T is a theory assignment which meets certain conditions, we can define U and X for T(%)and T ( 8 ) .Again, nice algebraic structures result, which are best described using category theoretic language. Let us write 0 for either U or X , where it does not matter which. Having defined for the production systems given by the theory assignment T, if A = T ( % ) ,then A O A is meaningful. This production system may or may not have an embedding back into A. If it does, and if such an embedding meets certain conditions, we call A separable (with respect to 0).Ordinary recursion theory is separable, as are many other recursion and w-recursion theories. Being separable with respect to X is a consequence of having an effective pairing function. Being separable with respect to U is a generalization of the feature of ordinary recursion theory that the domain can effectively be split into the even numbers and the odd numbers. Separable production systems are rather well behaved, and the chapter concludes with a study of them. 64

CH.

COMBINING DOMAINS AND CODINGS

4, 821

65

2. Combining domains and codings

We want to combine production systems; here we begin by combining domains. The two techniques we discuss are disjoint union, U, and Cartesian product, x . I. Disjoint union. The disjoint union of two sets is the union of disjoint copies of them. Any systematic way of choosing copies generally will serve. We officially adopt the following: dU 93 = (dx (0)) U (93 x (1)). DEFINITION. Note that, under this definition, dU 93 and 93 U d are, in general, quite distinct. But they are obviously “isomorphic.” Consider the 1-1, onto coding y :dU 93 + B u d given by

r(b,1)) = {(x, 0)).

Y ( ( X ,0)) = {(x, 1)L

Similarly dU (3U V) and (dU 93) U (e are different, but are isomorphic under a 1-1, onto coding, d. The empty domain, 0, acts like a unit for this operation. That is, 0 U d and s4,though different, are isomorphic under an obvious 1-1, onto coding, A. Similarly, there is a 1-1, onto coding p from dU 0 to d. There is also a simple way of defining U on codings so that it “keeps the parts separate.” DEFINITION. Suppose we have codings f : d-d’ and g : 93 f U g : d U % + d ’ U B ’ w e mean thecodinggiven by

+ % I .

By

CfU g)((x,O))={(Y,o)IY E f ( x ) ) ,

Cfu g)((x, 1)) = { ( Y , 1) I y

E g(x)l.

It is easy to check that U has the following pleasant properties. and

cf’ U s’)CfU g ) = ( f ‘ f ) U ( g’g) inj, U inj

= injdua

11. Cartesian product.

I

DEFINITION. d X 93 = {(x, y ) x E d and y E 93). Paralleling the discussion of U we first note that, though d x 93 and 93 X d are generally different, there is a 1-1, onto coding connecting them:

COMEXNING P R O D U ~ ~ ISYSTEMS ON

66

y :d x

[CH.

4, 63

93 + 93 x d where ~ ( ( x y)) , = {(y, x)}.

Similarly d x (93 X C e ) and (dX 93) x Ce are isomorphic, under a, say. One element sets act as units for X . T o be specific, we use (0). Then d x {O), (0) x d and SB are all isomorphic in an obvious way. We also define an operation of x on codings so that it works with each component separately. DEFINITION. Suppose we have codings f : d + d ’and g : 93 + 93’. By 93’ we mean the coding given by

f x g : SB x 93 +d ’x

Cf x g)((x, Y >)= {(a,b ) I a E f ( X I and b E g(Y 1). Then one may check that: and

Cf’xg’)Cfxg)= Cf’f)x(g’g>

inj., x inj, = injdXs.

3. Symmetric monoidal categories

We have defined U and X on Dom, both for objects and for morphisms. As it happens, U is not co-product, nor is X product in this category. Still it is clear that there is much common to what may be said about U and x . In fact, both operations provide us with examples of symmetric monoidal categories, a notion which we define below. We will see several other examples of symmetric monoidal categories in this chapter as well, so we spend some time developing their properties in the abstract. See MacLane [1971], Ch. 7 for a fuller development of these ideas. Let Ce be some category and let 0be some bifunctor (two-place functor) on Ce. Then, in particular, for arbitrary morphisms, and for identity morphisms, 1,n 1, = l d c l * .

DEFINITION. A monoidal category Ce = ( C e ; 0, e, a,A, p ) is: a category Ce, a bifunctor 0: % x Ce + Ce, a (unit) object e, and three isomorphisms. a =a.,.*.w

:dn(BnCe)=(do%)oCe

natural in d,B, and Ce, and

CH. 4,

A FEWTECHNICAL RESULTS

841

67

and p = p d : d 1 3 e = d

A=&:eUd=d

natural in d.Further, the commutativity of enough diagrams is postulated so that the commutativity of “all” diagrams involving these notions follows. See MacLane [1971], Ch. 7, 42 for a proper statement of just what this means. A monoidal category (e is symmetric if it also has isomorphisms y = y,d,,:do93=93od

natural in d and 9, such that y.4*

O

3/*..*

= 1,

= Aya 0 yya,. : 93 Oe

= 93.

And again, ‘‘all’’ diagrams involving these notions commute. I. Disjoint union. Consider the operation U as defined in the last section, along with the isomorphisms (1-1, onto codings) mentioned there. We get a symmetric monoidal category (Dom; U , 0, a,A, p, r) which we denote (Dom, U ) for short. 11. Cartesian product. In working with X we find it desirable to rule out the empty domain. The reason is, there are simple, useful codings of d into d X ’3, except when 93 is empty. So, let Sdom be the full subcategory of Dom in which the objects are the non-empty objects of Dom. Let x be as defined in the last section. Then, using the isomorphisms mentioned there, (Sdom; X ,{0},a,A, p , 7) is also a symmetric monoidal category which we denote (Sdom, X ) for short.

4. A few technical results

Let % be a symmetric monoidal category, with as the monoidal operation. We develop a few simple techniques for working with I7 on the morphisms of %. The first allows the breaking up of f 0g into simpler combinations. We note that we use 1, for the identity morphism on d, which happens to be inj, in our examples.

PROPOSITION 4.1. Let f : d + d‘and g : 93 + 93’ be morphisms in %. The following diagram commutes :

68


[CH. 4, 84

PROOF.0 is a bifunctor, so we have the following calculations: and

(1d' 0g)cf 01 !a)

= (1w f )

( g 1!a) = f

g

cfol W ) ( l d O g ) = Cfld)O(l%g ) = f o g .

Next, for the two examples of Section 3 we define some special codings. DEFINITION. Let d and 93 be objects in %, which is either (Dom, U ) or (Sdom, X ). We define morphisms i = i J o :d+dO93,

j=jd.*:93+d17W

as follows. I. If %' = (Dom, U ) then

i(u) = {(u,O)} u E d, j ( u ) = {(u,l)}

u E 93.

11. If % = (Sdom, X ) then

i ( u ) = { u } x 93 u ~

d

,

j ( u ) = d x { u } u E 93. These codings have the following naturality features. PROPOSITION 4.2. Let %' be one of (Dorn, U ) or (Sdom, X ) . Suppose f : d + d'and g : 93 + 93' are morphisms in %. The following diagrams commute :

CH. 4, 141

69

A FEWTECHNICAL RESULTS

PROOF.Left to the reader. However, in trying to extend the above proposition, U and x separate. PROPOSITION 4.3. In (Dom, U ) the following diagram commutes :

d--!--diKB

-93

i

PROOF.Left to the reader. EXAMPLE.In (Sdom, x ), the following diagram does not, in general, commute:

d

i

i

dxB-93

f \

Consider the following example. Let d = d’= (1); f {2,3); and define g : 93 + 93’ by g ( 2 ) = (2). Then but

(f x g)i.d.e(1) = cf x

g)((L 2)) = HL 3

=

1.d; 93 = {2}; 93’=

1

i . d . a ~ f (=l )i , d . d 1 ) ={(1,2),(1,3)1.

We do, however, have the following awkward substitute.

4.4. In (Sdom, X ): PROPOSITION (1) Consider the diagram (which commutes by Proposition 4.2) &1”.”dXB-B

id.-

70


[CH. 4, 85

Then, for each n and for each 9 C .dn,

[(Id x g)idapIn( 9 )= [id* g l n ( B ” )n [ L a In (9). (2) Consider the commutatiue diagram

5. Combining structures In this section we extend our definitions of U and X to structures. Let % be one of (Dom, U ) or (Sdom, X ). As usual, we use 0 for the monoidal

RECURSION THEORIES, FOR EXAMPLE

Ck. 4, 561

operation of defined.

V. Recall that

i:.d+.dO%

71

for objects d and %, two codings have been

and j : B + d O % .

DEFINITION. Let 2l =(a; . . ,9&)and B = (%; YI,. . . , Y m )be two structures, with d and % objects in %. By 2l 0B we mean the structure

(.dnB;(=dy,%;,..., %e;,(=,y,Y: ,..., YA). 6. Recursion theories, for example

This section is intended to motivate what follows. We describe some very pleasant aspects of the theory assignment rec(-) and then discuss what complications must be introduced to handle other theory assignments in a similar way. The actual work involved is carried out in the next few sections. Let %’ be one of (Dom, U ) or (Sdom, X ), with 0 as the monoidal operation. T h e theory assignment rec(-) assigns to each structure % (with its domain d in V) a production system, namely the recursion theory of 8. Now, we can define 0 on these recursion theories as follows: rec(%)Orec(B) is to be rec(2l 023) where 2l 0B was defined in the previous section. It can be shown that the 0 operation is well-defined in the sense that, if rec(2l) = rec(2l’) (as production systems) and rec(23) = rec(B’), then rec(%0B)= rec (2l’ B‘)also. Suppose 8 and 8’ are embeddings between recursion theories, say 8 : rec(%)+rec(B) and 8’ : rec(21’)+rec(Br). Then 8 and 8’ are codings, so 8 08’ has been defined as a coding. It can be shown that 8 08’ is actually an embedding, 8 08 ’ : rec(%)Orec(%’)-,rec(B)Orec(B’). Thus we have a notion of both for recursion theories and for embeddings. It can be shown that the category Rec becomes a symmetric monoidal category, using 0, in a natural way. All this is a rather pleasant algebraic state of affairs. Now we ask, to what extent can this be duplicated for other theory assignments T. In Chapter 9 we will consider theory assignments that don’t make sense for all structures, but only for a restricted class of them. And we will not be considering all codings but only certain special ones. So our first modification is to work not with the objects of V itself, but with a subcollection 9, and likewise to restrict our attention to a subcollection 8 of all codings. We

72


[CH. 4,87

must postulate certain items about T, 9 and 8, of course, in order to get decently coherent results. In general, these postulates will be obvious for the case we begin with, rec(-) on % allowing all codings.

7. Combining production systems

Let % be one of (Dom, U ) or (Sdom, x ) and as usual 0is the monoidal operation of %. Also is an operation on structures whose domains are in %, as defined in Section 5. We give the following definition, which is really part of a longer definition, but this is enough to consider for the time being. The other parts will be given in the next few sections. DEFINITION. Let 9 be a collection of structures with domains in %‘,let T be a theory assignment for 9, and let 8 be a collection of codings. We say (T,9, %) is object-faithful (with respect to O ) , if: (1) T is elementary on 9 with respect to 8. (2) For 8 , B E 9 and for 8 E 8, if 8 is a coding from d to B then 8 is a pre-embedding from T(8)to T(B). ( 3 ) For 8 E 9, inj, E ‘8‘. (4) 9 is closed under 0. ( 5 ) For any 8,B E 9, id,, and jd.& are in 8. REMARK.rec(-), w-rec(-) and f.o.(-) trivially meet these conditions if 9 is the class of all structures (all non-empty structures if 0is x ), and 8 is the class of all codings.

a ) be object faithful. Then for any 8 ,B E 9,both LEMMA7.1. Let ( T , 9, and j d a are actually embeddings, i, iaa : T(8)- T ( 8

8) and j i 4a : T(B)- T ( %023).

PROOF.id.$ and are in 8 by clause (3, hence they are pre-embeddings by clause (2). The result follows from the fact that T is elementary and from the definition of 0 on structures.

LEMMA7.2. Let (T,’24%) be object-faithful. Let $, %‘, B, B’ E 9.If T(8)= T(%’)and T(B) = T(B’) then T(%08 )= T(a’0B’). PROOF.If T ( % )= T(%’)then d = d‘. Similarly B = 3’. Then d 0B =

CH. 4, 871


73

d’U93’. Now, let 8 = injdnse. We will show 0 is an embedding, 8 : T(9 B)+ T(9’ 0B’),then since 8 carries each operator to itself (and since a similar thing can be done with 9 and 9’, B and 8’ switched around) this will establish the lemma. By clauses (4) and (3), 8 E 8, and hence is a pre-embedding from T ( 9 0 B ) to T(9’OB’) by clause (2). Say i?I = ( d 9,, ; . . . ,%!k) and ‘23 = (93; Y,, . . . ,Y,,,). Then

Let 9 be one of = d , 9 , , . . . , we 9 kshow ; (9‘)@ is generated in T(9’UB’). A similar thing can be shown for (=a)’, Y:, . . . ,Y!,,. Now, 9 is generated in T(9) (since it is a production system on 3). Then 9 is generated in T ( 9 ’ )since T ( 3 )= T(9’).So by Lemma 7.1, 3‘is generated in T(9’OB’). But (9’)” = 9‘. So 8 takes the “given” relations of “2l B to generated relations of T(9’023’).Also 0 takes =.dnl to = which is also the equality relation for T(a’UB’), hence is generated. Then, since T is elementary, 8 is an embedding.

DEFINITION. Let (T,9, 8 ) be object-faithful. T ( 3 )0T(B) we mean T(9 ‘23).

For

9,‘23 E 9, by

We conclude with some examples. In these, T(-) is rec(-), 9 is the collection of all structures, and 8 is the collection of all embeddings.

EXAMPLE I. Let N’ be the natural numbers without 0. By G(N’) we mean (N’, S) where S is the successor relation on N’. Now, rec(G(N)) X rec(G(N’)) is defined, and can be thought of as a recursion theory of fractions. The domain is the collection of ordered pairs (x, y ) where x and y are natural numbers with y#O. The “given” relations are: successor in the first component; and successor in the second component. EXAMPLE 11. G(N’) is as in the previous example. rec(G(N)) U rec(G(N’)) can be thought of as a recursion theory on the signed integers. EXAMPLE 111. Let W be the real numbers. By G(R) we mean the structure (W;+ , x ,> ). Then rec(G(R)) X rec(G(R)) may be thought of as a recursion theory on the complex numbers.

74

COM~INING P R O D U ~ ~ ISYSTEMS ON

[CH.4,§8

8. Combining embeddings

We have now defined U and x between certain production systems. Our goal in the present section is to define related notions for embeddings. Recall that in Section 2, U and X were defined for codings; what we show now is that, at least in some circumstances, these same definitions work for embeddings. Once again, % is one of (Dom, U ) or (Sdom, X ) and 0is U or X as the case may be. DEFINITION. Let 9be a collection of structures with domains in %, let T be a theory assignment for 9, and let 8 be a collection of codings. We say (T, 9, %) is morphism faithful (with respect to 0),if: (1) ( T , 9, 8) is object-faithful. (2) 8 is closed under 0, as defined in Section 2. (3) 8 is closed under composition. REMARK.Again, trivially, rec(-), w-rec(-) and f.o.(-) are all morphismfaithful using the collection of all structures (all non-empty structures if 0 is X ) and the collection of all codings.

LEMMA8.1. Let ( T , 9, %) be object faithful. Let a, 8, K E 9and suppose there is a coding 8 : d 0$23 + % with 8 E 8. Then 8 is actually an embedding, 8 : T(%023) + T ( O ) provided each of the following holds : (1) For each relation 3 E [d]“ which is generated in T(%), 8, (3’) is generated in T(Q). (2) For each relation Y E [a]” which is generated in T ( B ) , 8. (Yj)is generated in T ( Q ) . PROOF.Suppose 8 meets the conditions. First we show 8 2 ( = d m a ) is generated in T(Q). is = d u o which is (=d)’ U(=,)’, so by Case 1: 0 is U . Then Proposition 3.2.3 &(=duo) is € J 2 ( ( = d ) ’ ) U &((=,)’). But by hypothesis, each part of this union must be generated in T(O),hence the union itself is. Case 2: Clis then=^^* i s = d x , whichis(=d)’fl(=B)’.Thenby Proposition 3.2.3 again, 0 2 ( = d x B )is 8 z ( ( = d ) ‘ )n &((=,)’) which must be generated in T((5). The lemma now follows from the definition of 0on structures, and the fact that T is elementary.

COM~INING EMBEDDINGS

CH.4, §8]

75

PROPOSITION 8.2. Let ( T,9, S) be morphism-faithful. Let a, a', B, B' E 9, let a, p E S, and suppose a : T(%)+ T ( a ' ) and p : T ( B ) + T ( B ' ) are embeddings. Then the coding a O p is actually an embedding, a O p : T(%OB)-+T( % 'U B') . PROOF.a,p E S, hence a 0p E %, hence a 0p is a pre-embedding. We use Lemma 8.1 to show it is an embedding. We treat U and x separately. Case 1: 0 is U. Let $33 E [a]"be generated in T ( 8 ) ; we show (aU p)"($2') is generated in T(%'UB'). But, by Proposition 4.3, (aU p ) n ( W ' ) = [id,.,.a]. ($33). Now, a is an embedding, and by Lemma 7.1, so is id),,,, hence so is the composition. But by Proposition 3.4.2, embeddings take generated relations to generated relations. Thus (aU p). ($33') is generated in T ( 8 ' U 58').There is a similar argument for Y E [ B ] "generated in T ( B ) .Then, by Lemma 8.1, a U p is an embedding. Case 2: 0is X . We are to show a X p is an embedding. By Proposition 4.1, a x p = (injd~X @ ) ( ax inj,). If we can show each of these is an embedding, the result will follow. We do this for a Xinj,, the other is similar. As a coding, a x inj, : x 93 +d' X $%.Suppose 9 E [a]" is generated in T(%).Then by Proposition 4.2, (a X inj,), ($2') = [i.,..,a].($2) and this must be generated in T ( a ' X B) since a and i are embeddings. Suppose 9 E [B]]"is generated in T ( B ) .This time, by Proposition 4.4 (2), (a x inj,),

(Y') = [ i , . . , ~ ] ,(a") n [j..d..tftIn(Y).

Now both parts of this intersection must be generated in T ( a ' X B) since i, j and a are embeddings, and 9' and d" are generated. And generated relations are closed under n . Now a X inj, is an embedding, a x inj, : T ( a x B)+ T(8' x B), by Lemma 8.1. This concludes the proof. If (T,9, S) is morphism-faithful, we have now defined both for the production systems T yields on 9, and for the embeddings between them which, as codings, are in S. Our definitions are such that, for 3 ,a', @,a' E 9 and a,p E S, if a : T ( S ) + T('2I')and p : T(B)+ T(B') are embeddings, then so is a 0p : T ( 9 )0T(B)+ T(%')0T ( 8 ' ) .


76

[CH.4, §9

9. Monoidal subcategories in Prod

As usual, % is one of (Dom, U ) or (Sdom, x ), and 0 is the monoidal operation. It is in this section that we “complete” the sequence of definitions started in Section 7. Let (T,9, 8) be morphism-faithful, as in Section 8. We define an associated category as follows.

C (T, 9, 8 ) is the category with objects all production systems DEFINITION. T(%) where YI E 9, and as morphisms, those embeddings between the objects which, as codings, are in 8. It is easy to check that C(T, 9, $) is a category, in fact, a subcategory of Prod. What we do in the present section is postulate enough about (T,9,8) to make C(T,9,8) into a symmetric monoidal category. First, some preliminaries. Let e be the unit object of %. That is, if % is (Dom, U), e = 0, and if % is (Sdom, X ), e = {O}. Then ( e ) is a structure which will serve as a unit structure. Recall, there are certain natural isomorphisms in %: (a) For each d,93,% E % a reassociating map ff.d.*.%

:d

0(a %)+ (d0a)0%.

(b) For each d,93 E % a commuting map y.a.a : d

09 34 93 0d.

(c) For each d E %, maps A,* : e l 7 d - + d and

pd : d O e + d .

DEFINITION. Let 9be a collection of structures with domains in %, let T be a theory assignment for 9, and let 8 be a collection of codings. We say (T,9, 8 ) is faithful, if: (1) (T,9,8) is morphism-faithful. (2) The unit structure, ( e ) is in 9. (3) For a, ‘23, (5 E 9 we have ff&#.a.uE 8,

Y.d.9 E 8,

A,*, p.d E 8.

THEOREM 9.1. Let (T,9, 8 ) be faithful. Then C(T,9, 8 ) is a symmetric monoidal category.

MONOIDAL SUBCATEGORIES IN Prod

CH. 4, $9)

77

PROOF.Actually, all that is left to show is that a , A , p and y are embeddings, hence morphisms of the category in question. We show this for y, and sketch it for the rest. Let ' % , B €9. Then yd,B : d U B +93 Od is in 8. We use Lemma 8.1 to show it is an embedding. Let 92 E [ d ] "be generated in T('%); we must show yn( 9 2 ' ) is generated in T(BO0'21). Now it is easy to check that, whether 0 is U or X , 'Yd.3ld.B

= 1B.d.

Then yn( 9 2 ' ) = 92'a.d. This is one of the given relations of the structure B 03 , hence is generated in T(%Cia)since this is a production system on the structure B Cia. If Y E is generated in T(B) we proceed similarly, using

[a]"

y.d.B]d.ya

= ZB,.d.

Then by Lemma 8.1, y must be an embedding. To show cy is an embedding, use

and

ad,~,wJ.d.a.yeoe 1a.ye.w = Z d m a . w ] .rp.o

a ..s.a.u].d,auwJ 3.v = ] dna.v.

A is similar, but simpler. Finally, py = A * Y ~ . ~so , p is an embedding too.

SUMMARY. For convenience, we collect together the definition of faithful. DEFINITION. Let 9be a collection of structures with domains in %, let T be a theory assignment for 9 and let 8 be a collection of codings. (T,9,8)is faithful (with respect to 0),if: (1) T is elementary on 9 with respect to 8. (2) For '%,BE9 and 8 E 8,if 8 : d + 93 is a coding, then 8 is a pre-embedding from T ( 3 ) to T(B). (3) For 0'21 E 9, inj.&E 8. (4) 9 is closed under U. (5) For 3 ,B E '3, both id.* and j.d.!.csare in 8. (6) 8 is closed under 0. (7) 8 is closed under composition. (8) The unit structure, (e), is in 9. (9) For 3,8, BE 9

COMBINING PRODUC~ION SYSTEMS

78

E 8,

~~d.9.w

7/.wz.aE 8,

[CH. 4,

$10

Ast, pwz E 8.

Then, if (T, 9,8) is faithful, C (T, 9, a )is a symmetric monoidal category. 10. Separability

Let Ce be one of (Dom, U ) o r (Sdom, X ) . Let 9 be a collection of structures with domains in Ce, let T be a theory assignment for 9, and let 8 be a collection of codings. For this section, we suppose (T,9,8 )is faithful and we investigate the symmetric monoidal category C(T,9, 8). In monoidal categories, objects called monoids are often singled out for study. See MacLane [1971], pp. 166-167. Here, however, that notion is too narrow to be of interest. We replace it by a weaker one, that of being a separable production system in C(T,9, '8).

DEFINITION. Let A be a production system in C(T,9, 8). We call f a separation morphism for A , if: (1) f is a morphism, f : A 0A +A. (2) The relations y E ( f i ) ( x ) and y E Cfi)(x) are both generated in A, where A-AAOA-A

i

A We call A separable in C ( T,9,8) if A has a separation morphism. In the next few sections we establish some general results about separability. Here we prove a few theorems which make it easy to give examples.

DEFINITION. A has an effective pairing function if there is a 1-1 function J : d X d -+I which is generated in A. REMARK. In the proof below, and in several others in this chapter, we assume the generated relations are closed under the logical operations A , v and 3. This follows easily from the production system axioms, or see Ch. 7, $1.

CH.4, 8101

79

SEPARABILITY

THEOREM10.1. Suppose 0 is X , T assigns total production systems to structures, and 8 contains all codings. Then, for %€9,if T(%) has an effective pairing function, then T(%) is separable. PROOF.Define f by f ( ( x ,y ) ) = { J ( x ,y ) } . Since J is 1-1, f is a coding from d x d to d.Since T assigns total production systems, f is automatically a pre-embedding from T(%X 8)= T(%)X T(%) to T(%).

We next show f is an embedding, f : T(%X %)+ T(%). Suppose 9 C d"is generated in T(%);we show f. (3') is generated in T(%)also. Similarly for f. (9').Then, that f is an embedding follows from Lemma 8.1. To keep notation simple, say n = 2 . Then

so

3 ~ = { ( ( x , Y ) , ( z , w ) ) l ( x , z ) €and 9

Y 9 W E . 4 .

f 2 ( 9 e ' ) = { ( a , b > =l aJ ( x , Y ) b = J ( z , w ) , and ( ( x , Y ) , ( ~ , w ) ) E ~ ' } ={(a,b)la=J(x,Y), b = J ( z , w ) , ( x , y ) E % and y , w E d } .

Thus

(a, b ) E f 2 ( 9 ' )

e ( 3 x , y , z , w ) [ ( x , z ) € 9 ~ =a J ( x , y ) A b = J ( z , w ) ] . Now, 3 and J are both generated in T(%),and it follows that this is also generated in T(%). Thus f is an embedding. Finally we show fi and fj are both generated in T(%). We show this for fi, the other is similar. Now,

Y E(fi)(x)@ @

y E f ( z ) for some z E i ( x )

y E f ( ( x ,a ) ) for some (x, a ) E i ( x ) @

y = J(x, a ) for some (x, a ) E i ( x )

@

y = J(x, a ) for some a E d

@

( 3 a ) [ y= J ( x , a ) ] .

But J is generated in T(%)and we have closure under 3, that is, under projections.

REMARK.Actually, all that is needed for the above proof to work is that f ( ( x ,y ) ) = { J ( x , y ) } should be in 8.

80


[CH.4,510

EXAMPLES. We give three examples of recursion theories having effective pairing functions, and one of an w-recursion theory. Then, by the above theorem, these are separable production systems (under X ). I. There are many well-known pairing functions on the natural numbers, N. The oldest occurs in Georg Cantor “951, Section 6 and is (correcting for the difference that his natural numbers start at 1while ours start at 0) J(x,y)=x+f(x+y+l)(x+y). It is easy to check that this is generated in rec(G(N)). 11. There are also several pairing functions on the hereditarily finite sets, L,. The oldest is due to Norbert Wiener in Wiener [1912], and is

J ( x , y ) = {{{XI, 01, {{Y HI. A more common one today is due to Casimir Kuratowski in Kuratowski [1921] and is Either of these is easily shown to be generated in rec(G(L,)). 111. There are also pairing functions on words, W ( a , ,. . . ,an).They tend to be more complicated to state. We will develop one in Ch. 6, 06, based on Quine [1946]. Another example, coming out of computer science, may be found in Even and Rodeh [1978]. Relying on these, rec(G(a,, . .. ,a n ) )is also an example of a production system with an effective pairing function.

IV. Let G(W) be the structure (W; + , x , > ) where W is the set of real numbers and + , X and > are the usual addition, multiplication and order for them. There are many effective pairing functions in w-rec(G(W)). Once again, the oldest is due to Cantor, see Cantor [1878]; his example of a map between the line and the plane, is, in fact, w-r.e. in w-rec(Q(W)). His map is not the one customary today; the modern one “meshes” decimal representations. It is somewhat easier to check that it, too, is effective in w-rec (G(R)). Finally, the various space-filling curves are generally effective pairing functions; the first such may be found in Peano [1890].

is U, T assigns total production systems to THEOREM10.2. Suppose structures, and 8 contains all codings. Then, for Yl E 9, T ( 8 )is separable if and only if there are two codings p, u : d + d with d ”and d disjoint, and the relations y E p ( x ) and y E u ( x ) both generated in T(Yl).

CH. 4, $101

SEPARABILITY

81

PROOF.First, suppose T ( % )is separable, with f as separation morphism. Set p = f i and v = fi, and the appropriate conditions are satisfied. Second, suppose there are codings p, v : d + d,both generated in T(%),with d" and d "disjoint. We produce a separation morphism f for

T(W.

Define f on dU d by f((x,O)) = p ( x ) andf((x, 1))= v ( x ) .Since d' and a" are disjoint, f is a coding. By definition of f, we have p ( x ) = f i ( x ) and v ( x ) = fi(x), so both f i and fj are generated in T(%).

Since T assigns total production systems, f is a pre-embedding from T(%US) to T(%).All that remains is to show f is an embedding. Suppose 9 d"is generated in T(%).Then f. (9') = fni. (9) = pn(%), and this is generated in T ( % ) since (yl, . . . , y , ) E p n ( 9 ) ( 3 x ,,..., x , ) [ ( x , ,..., x , ) E ~ ~ y ~ E p ( ~ , ) ~ . ~ . ~ yand . € p p (and x . )%] are generated. Similarly f.(9') is generated in T(%). That f : T('21)U T(%)+ T ( % )is an embedding now follows from Lemma 8.1.

REMARK. Again, all that is needed for the above proof to work is that the map f defined from p and v be in $. EXAMPLES. The recursion theories of the following structures are separable under U . I. Numbers G(N). Take p ( x ) = {2x} and v(x) = {2x + 1). 11. Sets G(L,). Take p ( x ) = {(x,0)) and v(x) = { ( x , 1)). 111. Words, two or more letters G ( a l , .. .,a,,). Take p ( x ) = { x * a l }and v(x) = {x * az}. IV. Words, one letter G(a).Take p ( x ) = {x * x} and v(x) = {x * x * a}. Actually, separability under U and x are not independent, and the above examples are consequences of the one earlier in the section, using the following. THEOREM10.3. Suppose T assigns total production systems to structures, and % contains all codings. Then for '21 E $, if T(%) is separable using x then T ( % )is separable using U .

PROOF.Suppose f is a separation morphism for T(%),under x


82

[CH. 4, 811

fi and fj both generated in T(%). Since f : d x d + d is a coding, cardinality considerations give us that d can not be finite. Choose two members of d,say a and b. Define maps p and v by

Y E P ( x ) @ Y EcF)(x)

and

Y ECfjNa),

YE

and

Y E(fj)(b).

4 X )

Y E Cfi)(x)

Since fi is a coding of d into d,it follows that p and v are also codings of d into d.Since f i is a coding of d into d,it follows that d” and d ”are disjoint. Finally, fi and f j are generated in T(%)and it follows that p and v are also. Now use Theorem 10.2. 11. A few results about separability

As usual, 0 is U or

X

and ( T ,9, 8) is faithful.

PROPOSITION 11.l. Let A, B, C E C(T, 9, 8), and suppose C is separable. If each of A and B has a morphism to C, so does A 0B. PROOF.Let f : C U C + C be a separation morphism for C, and let g : A + C and h : B -+ C be embeddings. Thenf(g h ) :A 0B + C is an embedding.

11.2. Let A, B E C(T,9, 8).If A and B are both separable, so PROPOSITION is A 0B. PROOF.Suppose f is a separation morphism for A , and g for B. Then we have

A

fk.4 and fjYg.* both generated in A, and

A FEWRESULTS ABOUTSEPARABILITY

CH. 4, $111

83

g k B and gj98,B both generated in B. We want an appropriate

embedding h :

(where we omit subscripts on i and j , which are horrifying). Let rr be the obvious isomorphism 7r

: (se

93)0(% 09) = (Jdz 0%)0(90 39)

compounded appropriately out of the reassociating maps and the commuting maps. Now set

h =(fOg)rr. Then h is a composition of embeddings from % (Theorem 9.1 enters here); hence h is an embedding from 8. We must show hi and hj are generated. But it is easy to see that

so

i

= 7r(i.d.,d 0i*.*)

and

rrrr = 1.

hi = (f0 g)rrrr(i.d,,d0iYe.%) = (f 0g ) (i.d.d

0iB,*) = (fi.d..d)0( g k 9).

is generated in A. Similarly for gi,.s in B. We leave it to the reader to verify, using (*), that (1) if 0 is u , y E ( h i ) ( x ) (y, x ) E (fid~d)’d% U (gi9.*)’d*, (2) if 0 is x , y E ( h i ) ( x ) e ( y , x ) E ( f i d ~ d ) ’ d * n(gi9B.9)’da. Now, ( f i d , d ) ’ d * and (giB,9)’d*are both generated in A O B by Lemma 7.1, and the generated relations are closed under r l and U . Thus hi is generated, as a relation, in A O B . Similarly for hj, so h is a separation aorphism.

COMBINING PRODUC~ION SYSTEMS

&I

[CH. 4,812

12. Injections and co-embeddings

We follow our usual conventions: 0is one of U or X , and (T, 9, %) is faithful. The entire of this section is devoted to a proof of the following theorem, our first result about the existence of co-embeddings. This will be developed further in Theorem 5.3.4.

THEOREM 12.1. Let A, B E C(T, 9, $), suppose B is separable, and A has a morphism to B. Let j,d.p : B +A 0B be the usual injection embedding. Then ( j d , B ) - ' is a co-embedding. The proof of this theorem is different if is U o r X (because of the failure of the analog of Proposition 4.3 for x ). We begin with a useful result, independent of 0, then turn to U and X separately. PROPOSITION 12.2. Let 6 : d + d be a coding such that the relation y E S(x) is generated in A. Then, for each n, both S, and 6;' are operators of order ( n , n >in A. PROOF.We show this for S., the proof fdr 6;' is similar. Since y E S(x) is generated in A, so is the relation

9 ( x I,...,xn,yl,...,y.)

e

y l ~ 6 ( ~ , ) ~ . E. 6- (~x ny) n.

But then, for 9 E [d]", &(P) = %"(S),which is an operator in A by Corollary 2.5.2. Now we prove a sequence of lemmas which establish Theorem 12.1.

LEMMA12.3. For each n, and for any coding f : d + B , ( j ~ . ~ ) ; ' c0 f inj,),, where

f

1

id.-

,

B-BmB?B 1S.S

I

=

Id.fninj,

la.-

PROOF.To keep the notation simple, we take n = 2. We give separate arguments for U and X .

CH. 4,

$121

INJECTIONS AND CO-EMBEDDINGS

85

LEMMA12.4. For each n, and for any coding f : d + 3, (i.d,B)n

= (fU

injB);'(-is,*)n.

PROOF.By Proposition 4.3, the following commutes,

Hence cfU inja)(j.d,a) = j3,%. Then, using Proposition 3.2.3, cfU inj9)i'Cfljinj,),

(j.d.*).

=c ~ U inj,)i'(ja.a>n,

(jd.*ln = (fU inja)il(ja.a)n. 12.5 (Theorem 12.1 for U). Suppose 0 is U, A , B E PROPOSITION C(T, 9, %), B is separable, and f :A +B is a morphism. Then (jd.9)-'is a co -embedding. PROOF.Since (T, '3, 8 )is faithful, j,d.a is an embedding. We show (j.d.a)-'is a co-embedding using the definition of co-embedding directly (Ch. 3, 94). Let g be a separation morphism for B. Then, using Proposition 4.3, the following commutes.

86

COMBINING PRODUCIION SYSTEMS

B Also, set h = gcfU inj,) :A U B +B. Now, let @ be an operator in A U B of order (n, m). We show @"--' is an operator in B, which establishes that (jd.!*)-' is a co-embedding. Set lp = (@k)(a*.~l-l.We first show that lp is an operator in B, and second, that !P = @('*.*I-'. (1) @ is an operator in A U B and h is the composition of two embeddings, hence an embedding, so O h is an operator in B. g is a separation morphism, so gjB.B is generated in B. Then by Proposition 12.2, both (gjB,B))n and (gjB.,)il are operators in B. Then the following is an operator in B: (gjB.B),'@"(gjB,,),

= (@k)(=*

(2) To see that lp = @"*.:e)-',

sf

' = P.

we make the following computation.

lp = ( @ h ) ( d . B . J ' = (gjB.3 )my @ ) (gja.9)" = (gj*.*)Z'hm@hi'(gj,.a)n

= (gj*.ya)i'[g(fU inj, )]rn@[guU inja)]i'(gjB.a)n = (i~,,)Z'gi'gm

(fir inj,),,,@(fU

inj9)i'g;'g.

(i,.U),,

= (ia.ir8)m'~fUinj3)m@CfUinja);'(is,B)n = Cjd.,);'@(i,d,B))n (by Lemmas 12.3 and 12.4) = @%Lsl)

1.

This completes the proof for U .

LEMMA12.6. Suppose f : d + 93 is a coding, so the following commutes (by Proposition 4.2):

INJECTIONS AND CO-EMBEDDINGS

CH. 4, 1121

87

Then (1) for 8 c a", (2) For 22

c (dx a),,

PROOF.(1) Use Proposition 4.4 (2), and take (f X inj,);' of both sides. (2) By Proposition 4.2

(ia,s)i'[(ige.af),( d mn)(f x injge), (921 = (ia.sg)i'[(f x inj,),

( k e ) m

n

[ ( i . d . s ) , (d"')91

= (ia,s)i'(f

x inj,),

= (ia,a)i'(f

x injS), ( 9 )

= (i,d.S)i'(9) (by

(d") n (f x inj,), ( S ) ]

Lemma 12.3).

PROPOSITION 12.7 (Theorem 12.1 for x ) . Suppose 0 is X , A, B E C(T,9, S),B is separable, and f : A +B is a morphism. Then (id.s)-' is a co -embedding. is an embedding. PROOF.Again, since (T,9, S)is faithful, Let g be a separation morphism for B. Then, by Proposition 4.2, the following commutes.

B

Also, set h = g ( f x injS), so h is an embedding, h : A x B +B.


88

[CH.4, $12

Let @ be an operator in A X B of order (n, rn). We show @('d ' is an operator in B. Let d'= (gi.uJ.B.sf)m (d"') and d"= (gira.%f), (a"). Define a map ly. of order (n, r n ) by: for 9 C g",

~(9 = (gja ) uB )rnl[dn' ah (,ti

n ( g j B 1. (9))l.

Claim 1. ly. is an operator in B. This follows since d ' and d" are are operators in B, Qjh is an operator generated in B, and in B, and production systems have appropriate closure properties. Claim 2. ly. = @ O d B~ One may show, for 9 C g",that

(aa

(a3

I.

V(9)=

W

J

B

)

'(9)

by a computation similar to that in the proof of Proposition 12.5, but using Lemma 12.6 in place of Lemma 12.4. We omit the steps.

CHAPTER FIVE

EFFECTIVE EMBEDDINGS

1. Introduction

In proving the incompleteness of elementary arithmetic, Godel introduced the device now called Godel numbering. It can be looked at as a coding, of words by numbers, and is, essentially, Example I11 in Ch. 3, 06. Often the “effectiveness” of the Godel numbering procedure is emphasized, for example, Beth [1964], p. 297 says, “It will be clear that the Godel number g ( U ) of any expression U is uniquely defined and effectively calculable; and conversely, any natural number n can be the Godel number of at most one expression and if g ( U ) is given the U can be effectively determined.” But then Beth continues with the following interesting statement. “So we have established, by a recursive definition, a one-to-one correspondence between the expressions U of sentential logic and the elements g(V) of a certain set G of natural numbers; this set G also admits of a certain recursive definition.” Now when Beth says the Godel numbering is given by a recursive definition he is speaking informally, since he only discusses recursive definitions in arithmetic, while the Godel numbering g is between words and numbers. Nevertheless it is clear that his statement is meaningful (and true) and it is worthwhile to introduce the technical machinery to deal with such concepts. Prof. Smullyan informs us that in that lost paper of 1957 (see Ch. 1, 914) he gave a technical definition of an effective Godel numbering along the following lines. He worked in a recursion theory of words, and identified the number n with the word (numeral) consisting of n 1’s; there were other letters present besides 1, and hence other words besides numerals. Then, a Godel numbering, being a function g from words to numbers, could be thought of as a function from words to numerals, and as such could be dealt with by the recursion theory of words. Prof. Smullyan called g an effective Godel numbering if, in his recursion theory of words, g was a recursive function. 89

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[CH.5,62

For us, Godel numbering is one kind of embedding, and we would like to say, in general, when an embedding is effective. For Godel numbering, Prof. Smullyan’s approach was available because there is a single recursion theory that “discusses” both words and numerals. Now, in general, if 8 : rec(8) +rec(B) is an embedding, we must have a recursion theory that can “discuss” things pertinent to 8 and also to B, if we are to give a meaning to “effectiveness” for 8. Well, rec(8)O rec(B) is precisely such a recursion theory (for either U or X ). In brief, we will say 8 is effective (relative to 0) if 8 is generated in rec(%)Orec(B). The only, minor, difficulty is that 8 itself is not quite the right thing to look at. We replace 8 by en,where, if y E @(x), then y r E8O(x’) where x ‘ E i ( x ) and y ’ E j ( y ) . That is, On behaves like 8, but on the “copies” of x and y that are present in d 9. Then, 8 is effective if On is generated. In this chapter we develop basic properties of effective embeddings, not only for recursion theories, but more generally. Let (T,9,8) be faithful, and A = T(%)for B E 9. Now, inj, :A + A is a trivial example of an embedding, but curiously enough, it need not be effective. We call A reflexive (relative to 0)if inj, is an effective embedding. Reflexivity seems to capture, at least in part, the idea that a production system domain should be capable of being “built up” using the given machinery. We do not investigate this connection however. Ordinary recursion theory is reflexive, as are many others. We investigate the properties of production systems that are both reflexive and separable. They are quite pleasant. We conclude with some general theorems that imply that all the embeddings of Ch. 3,§6 can be replaced by isomorphisms. It should be understood that when T(-) is rec(-) or a-sec(-) we automatically take 8 to be all codings and 9 to be all structures (all non-empty structures, if 0is X ). Thus mention of 9 and 8 is suppressed in the examples of this chapter. For the entire of this chapter, is either U or X ,and (T,9, 8 )is faithful.

2. Effective embeddings

we mean the binary relation DEFINITION. Let 8 :.d+ $23 be a coding. By on d 09 given by (x’, y’) E e for some x, y, y E 8(x), x ’ E laB (x) and y ’ E j d , ~ ( y ) . PROPOSITION 2.1. Let 8 : d + $23 be a coding. Suppose x ’ Eid.B (x) and e y E 8(x). y r~ j(y). Then ~ . y‘) ~E (XI,

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91

PROOF.If y E O ( x ) then ( x ’ , y ‘ ) E flu by definition. Suppose ( x ’ , y ’ ) E 8O. Then by definition there are a, b, with b E 8 ( a ) , x ’ E i.rP.4B(a) and y ’ E j d , 8 ( b ) . But i is a coding, so if x ’ E i.d.B(a)and x ’ E i d , * ( x ) then a = x . Similarly b = y , so y E 8 ( x ) since b E 8 ( a ) .

DEFINITION. Let 3 , ‘23 E 9. If 8 : T(%)+ T ( 8 )is an embedding, we say 8 is an effective embedding (relative to 0) if the relation On is generated in

T(%)OT ( B ) .

EXAMPLES. In Ch. 3, 96 we defined (Example I) a coding 8 : N-L, by: 8(N) = { n “ } where 11” is the Von Neumann ordinal for n. And we showed it gave an embedding of the corresponding recursion theories, 8 : rec(G(N))-rec(G(L,)). In fact, 8 is effective (for both U and x ). We show it is effective relative to U ;the proof that it is effective relative to x is quite similar, or see Theorem 3.5 below. rec(G(N)) Urec(G(L,))

= rec((N U L,

;(=,J, A ’ ,(=,J,

B’))

where A is the successor relation on N, A ( x , y ) y = x ’ ; and B is the relation on L,, B(x, y, z ) x U { y } = z. Now, for this recursion theory, 8‘ has the following elementary formal system axioms: (we use 8’ as a predicate symbol too, for convenience)

e w , o), ( 0 , ~ ~ 8’(x, y ) + A ’ ( x , z ) + B ’ ( y , y , w ) + 8”(z, w ) . As a matter of fact, every embedding given in Ch. 3, 06 is effective, both for U and x . This is important, and the reader should convince himself that it is true.

3. Results about effective embeddings

THEOREM3.1. Let A, B, C E C(T,9,8) and suppose A and C are separable. If f : A +B and g : B + C are morphisms of C (T, 9, E p ) which are effective embeddings, then (gf) : A + C is also an effectiveembedding (and of course a morphism).

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PROOF.By Proposition 3.4.1, gf is an embedding. We show it is effective, that is, (gf)" is generated in A OC. f :A +B is effective so f" is generated in A 0B. Also g :B -+ C is effective so go is generated in B 0C. Now C(T ,9, $) is a symmetric monoidal category (Theorem 4.9.1) so we have

AOB and

B

-

(A O B ) O c

c -LA

(B

Y 3( B O A ) U c 5B U(A

C)

C )S(A U B ) O cYOly. (B

A)

c4B

(A

c).

So, let /. = la

- y y c3 I%),

u = a(y 0lq)a-'j.

Then these are both embeddings,

p:AOB+BO(AOC), u:BOC+BO(AOC). Then, since embeddings take generated relations to generated relations, both cf")" and (g")" are generated in B 0(A 0C ) . Now define a relation h in B 0(A C ) by: (x, y ) E h

for some z, (x, z ) E cf")"

and (z, y ) E (go)".

Using the closure properties of production systems, h is generated in B O(A 0C ) . We leave it to the reader to verify (treating U and x separately is easiest) that

h = ((gf)"Y where j = j 4 e . ~ ~ ~ : d U O ~ 9 c 3 ( d O O ) . Thus ((gf)"y is generated in B 0(A 0C ) . Now, both A and C are separable, hence so is A 0C by Proposition 4.11.2. Also B has a morphism to C (namely g ) and C has a morphism to A 0C (injection), so B has a morphism to A 0C. Then by Theorem 4.12.1, j-' is a co-embedding. But co-embeddings also take generated relations to generated relations, hence (((gf)")'Y-'is generated in A 0C. But this is j;ljZ((gf)") = (gf)". And thus gf is effective.

DEFINITION. Let 8 : 9 +d be a coding. For each n we define maps (On), and (6");' by: for 9 C (9O d ) "

RESULTS ABOUT EFFECTIVE EMBEDDINGS

CH. 5, 131

93

LEMMA3.2. Let f : 93 + SB be a coding. For each n, fn

fnl

= (i.d,a)i'(fu)n (i.p..a)n, = Ci.~,P.W)n'Cfn)n'(i.~..a)".

PROOF.Left to the reader. LEMMA3.3. Let f : B + A be an effective embedding. Then for each n, Cfo). and (f");' are operators in A 0B (note order of A and B ) . PROOF.f is effective, so f" is generated in B O A . But there is an isomorphism y : B 0A + A 0B, so y z (fo ) is generated in A 0B. Define a relation R by R ( ~,..., I Xnryl,..., y.)

@

(yi,xi)E

rzCf")A.'.h(Y",X")Ey~(fn).

Then R is also generated in A 0B. Also, for 8 (SB 0B)",(fu).( 8 )= R"(8),and we have Corollary 2.5.2. Similarly for (f");'. THEOREM3.4. Suppose, in C(T,9, %), (1) A has u morphism to B, (2) B is separable, (3) f : B + A is an effective embedding. Then f is a co -embedding. PROOF.Let @ be an operator in A of order (n, m).We show operator in B. But, by Lemma 3.2, @'-I

=

@'-I

is an

frn'@f"

= u . ~I;) . (f ~ ");I

(i.&,*) m @ (i.d.9 );

= cj,,>m'(fo)ml@'d.."

(f").

l(f0).

(jd.9

)"

(L4.53)".

is an operator in Set VI = (f");'@ '+f").. Since @ is an operator in A, A O B . Then, using Lemma 3.3, so is VI. Then, continuing the above, @ f - ' = (j.d.Ya);lVI(jd.a). =


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[CH. 5, $3

But this is an operator in B since j is a co-embedding by Theorem 4.12.1. REMARKS. In Ch. 3, $6 we gave an example of an embedding from rec(G(N)) to rec(G(L,)) [Example I] and the other way around [Example VII]. Both are effective embeddings, and both recursion theories are separable. Hence both are also co-embeddings, by the above theorem. It follows that a set of numbers is r.e. in rec(G(N)) iff the corresponding set of ordinals is r.e. in rec(G(L,)). Similarly, all the other embeddings of Ch. 3, P6 are also co-embeddings. Actually we have been discussing two notions of effective embeddings, depending on whether 0is U or X . The following relates them somewhat. THEOREM3.5. Suppose T assigns total production systems to structures, and 8 contains all codings. Let A, B E C(T,9, 8 ) . Suppose A and B are both separable using x . Iff : A +B is effective relative to U , it is also effective relative to x . PROOF.In this proof we need to work with both U and X . T o keep things straight, we use i, j for the injection maps where U is concerned, and I, J for the injection maps where x is concerned. A and B are both separable using X ; then by Proposition 4.11.2, A X B is also separable using X ; and then by Theorem 4.10.3, A X B is separable using U . Let g be a separation morphism for it. Then (using Proposition 4.4.3) the following commutes. A

1'9 B

AUB Id,U

AOA

,I

1,

inj.d,s = e

AOB

Now, A is reflexive, so (inj,)" is generated in A CIA. Then ((inj,)O)' generated in A O B . It is (injd,B)n.

is

THEOREM6.4. Let A, B E C(T,a,%). Suppose B is a n extension of A, A is reflexive and separable, and B has some morphism to A. Then B is a conservative extension of A. PROOF.By the lemma, inj.d.a.ia, is effective. The result now follows by Theorem 3.4. Suppose B is an extension of A and h : B + A is a morphism of C(T,9,%) Let . h d be the coding h restricted to members of d (recall, if B extends A, d C 9). Clearly h r d is a coding of d to d.But also, it is easy to see that h r d = h inj,,,, hence it is a morphism of A to A. DEFINITION. Let B be an extension of A and h : B -+ A be a morphism. If the relation y E (h t d ) ( x ) is generated in A we call B an inessential extension of A (via h). THEOREM6.5. If B is an inessential extension of A then B is a conservative extension of A. PROOF.Say B is an inessential extension of A via h. Let @ be an operator in B of order (n, m ) . We show @""' is an operator in A to establish directly that inj-' is a co-embedding (we are writing inj for inj,.,). h :B + A is a morphism, so @ h is an operator in A . Also y E ( h a)is and (h ld);' are generated in A so by Proposition 4.12.2, both (h rd),, operators in A. Then the following is an operator in A : (h I d ) i l Q h ( h r d)"= (h inj);'h,@h ;'(h inj), = inj;'h;'hm@hi1hn

= inj;'@ inj, =

@'"J-'.

inj,

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7. Isomorphisms

Let A be a production system. It can be viewed as a category, with objects the various [a]“, and with operators as morphisms. It is the 1-1, onto embeddings that can serve as functors between such categories (see Propositions 3.3.3 and 3.3.4) so it is natural to ask when 1-1, onto embeddings exist, and when they are unique. We begin with uniqueness. We will see roughly that effective 1-1, onto embeddings between reflexive, separable production systems are unique up to natural equivalence. Then we give some conditions for existence, which are enough to guarantee that there are 1-1, onto, effective embeddings between rec(G(N)), rec(G(L,)) and rec(G(a,, . . . , a,)), our “standard” recursion theories. DEFINITION. Let A, B E C(T, 9, $), and let f : A +B be a morphism. If f is 1-1 and onto, we call f an isomorphism. Iff is also effective, we call it an effective isomorphism.

REMARKS. If f : A + B is an isomorphism, then as a coding, it has an inverse. The symbol f-’ is already in use for the co-embedding, so we use f‘ for the inverse coding. Specifically, f : W 4 is given by Y Ef’(x)

@

x Ef(Y).

Clearly ff‘= le and f’f = 1,. Also f ‘ and f-’ are closely related: if @ is an operator in B, @” = Qf-’. DEFINITION. We call ( T,9, a ) full,relative to 0, if, for any objects A, B of C (T, 9, $), and embedding 8 : A +B, if 8 is effective (relative to then 8 is a morphism of C ( T ,9,8).

a),

PROPOSITION 7.1. Suppose (T, 9,8) is full, A, B E C(T,9, $), f : A +B is an effective isomorphism, A is separable, and B has some morphism to A. Then f’ :B + A is also an effective isomorphism. PROOF.By Theorem 3.4, f-’ is a co-embedding. For each operator @ E B, @” = @ j - ’ and it follows that f ’ is an embedding. It is trivially 1-1 and onto. Let L be the logical operator in B O A such that L : (u, v ) + ( v , u ) . Also let y be the “commuting map” of C ( T ,9,8) Then . f’ is effective, since Cf‘)O

=L

(Cf”)’).

Since (T,9, %) is full, f’ is a morphism.

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[CH. 5, $7

COROLLARY 7.2. Suppose (T, 9, S ) is full, h : B +B is an effective isomorphism and B is separable. Then h' is also an effective isomorphism. PROOF.Take A = B above.

THEOREM 7.3. Let A, B E C( T, 9, S ) where (T,9, '%) is full. Suppose B is separable and reflexive. Let f , g : A -+B be two effective isomorphisms. Then there is a coding h : 93 + 93 such that: ( 1 ) The relation y E h ( x ) is generated in B. (2) h :B + B is an effective isomorphism. (3) g = hf. PROOF.f and g are effective, so both f" and go are generated in A 0B. Define a relation R by

R(u,v)

e

( 3 x ) [ ( x , u ) E f nand ( x , v ) E g 0 ] .

Then R is also generated in A 0B. By Theorem 4.12.1, Rod,@)-' = S is generated in B. And it is easy to see that

S(q, r )

e

q E f ( x ) and r E g(x) for some x E d

Now define a mapping h by r E h(q)

s(q,r).

Since f and g are 1-1 and onto, it follows that h : 93 + 93 is a 1-1, onto coding. And the relation y E h ( x ) is generated in B. By Proposition 4.12.2, for each n, both h, and h i ' are operators in B. Since operators take allowable relations to allowable relations, h is a pre-embedding. Let @ be an operator in B of order (n, m ) . Then Q h = h,@h;'. But this is the composition of three operators in B, hence is an operator in B. Thus h is an embedding. h is effective by Proposition 4.2, hence also an isomorphism since ( T ,9, 8 ) is full. Finally, that g = hf is a straightforward calculation. COROLLARY 7.4. Let A, B E C(T, 9,a) where (T,9, %') is full. Suppose B is separable and reflexive. A n y two effective isomorphisms from A to B are naturally equivalent (thought of as functors). PROOF.Let f, g : A + B both be effective isomorphisms. Let h be as in Theorem 7.3. We use it to define a natural transformation

ISOMORPHISMS

CH.5, $71

103

'T:f+g. Since y E h ( x ) is generated in B, for each n, h. is an operator in B. Let

[d]" be an object in A ; we let T assign to it the morphism h, in B. Now, let Qi : [d]" + [&Im be an operator in A. We show the following diagram commutes, which says

T

is a natural transformation.

= hm@' or That is, we must show QiPBh,

gm@gi'hn = h m f m @ f n l * Now, hf = g,

SO

h.fn = g.,

SO

= 1.

g,lh.f,

And since f is 1-1 and onto,

gi'h,

=f;'.

Also hmfm= g,,,, and these facts give (*). Thus T is a natural transformation. And each component of T is invertible; specifically, the inverse of h, is h i ' , also an operator in B. Thus we have a natural equivalence. Now we turn to the question of the existence of effective isomorphisms. We begin with replacing embeddings by 1-1 embeddings.

THEOREM 7.5. Let A, B E C(T,9, %) and let f :A +B be an embedding. Suppose the following conditions hold : ( 1 ) f2(=.d) is bi-generated in B. ( 2 ) The domain of B is an ordinal. (3) If R ( x , y ) and its complement are both generated in B, then S ( z , y ) is generated where S(z,y)

e

z = p x R (x, y ) @ z = the least x such that R (x, y).

Then there is a 1-1 embedding g :A +B. Further, i f f is effective, so is g.

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[CH. 5,87

PROOF.f may not be 1-1. We define a new coding g so that g ( x ) = { y } where y is the least member of f ( x ) . Then we show g is an embedding. Let w be some member of f ( x ) . Then in fact,

1

f(x) = ( 2 w(=spyz). So, our definition of g is:

y Eg(X)

e

(IW)[W

E f ( x )A y

= Pf(W(=.dYf)].

It should be clear that g is a 1-1 coding, g : d + 3. ( = d y is bi-generated in B, so S ( y , w ) e y = ~ z [ w ( = ~ ) ' isz ]generated in B. For each n, we define relations S , and S;' by S n ( y l , . . . ,ynj w I ,

. . ., wn)

S,'(w,y)

(Y, w ) .

s n

S(y,,~

rA)

. . . A S(Y., wn),

Then S,, and S ; ' are also generated in B. For 9 C 3", S:(9') is the result of enlarging 9 by throwing in all f-codes while ( S ; ' ) ' ' ( 9 ) cuts for members of d " which have minimal f-codes in 9, things down to just minimal f-codes. It is not hard to see, then, that

g. = (S;')"f,,

and

g;' =f;'S:.

Further, by Corollary 2.5.2, both S : and (S;')'' are operators in B. Now, if 9' C d" is allowable in A, g, (9')= (S;')"fn(9'),which is allowable in B since f is an embedding and (,;')' is an operator. Similarly for g ; ' . Thus g is a pre-embedding. Let CP E A be of order (n,m).Then CPs = g,@g,' = (sm')"fmCPf"'s: = (Sm'yQifs:.

Since f is an embedding, CPf E B, hence CPg E B since B is closed under composition. Thus g is an embedding. Now suppose f is effective; we show g also is. S is generated in B, so S' is generated in A B. Then Q is also generated in A 0B where

W , y )

e

( ~ w ) [ ( xw, > E f D and S ' ( y , w ) ] .

We leave it to the reader to verify that

(x,y)Ego

Q(x,Y)

which concludes the proof.

REMARKS. B = rec(G(N)) meets conditions ( 2 ) and (3) above. (2) is obvious,

ISOMORPHISMS

01. 5, 571

105

while (3) is a standard result of ordinary recursion theory. Now, in Ch. 3,06 we gave embeddings of rec(G(Lw)) [VIIa] and rec(Q(al, ...,a,)) [111] into rec(G(N)). In the present chapter we remarked that these embeddings are effective. It is easy to check that they carry the respective equality relations to recursive relations on N. Then, by the above theorem, there are effective 1-1 embeddings of rec(Q(L,)) and rec(G(al,. . . ,a,)) into rec(G(N)). Next we turn to the existence of onto effective embeddings.

THEOREM7.6. Let A, B E C(T,9,S) where (T,9,S ) is full, and suppose

each has an effective 1-1 embedding in the other. Suppose the following additional conditions hold : (1) B is reflexive. (2) B is separable. (3) The domain of B is an ordinal, p. (4) Every generated set in B is the range of a 1-1 generated function whose domain is an initial segment of p. ( 5 ) There is no 1-1 generated function from 0 to a proper initial segment. Then A and B are effectively isomorphic.

PROOF.Let f : A +B and g :B -+A be effective, 1-1 embeddings, hence morphisms. d is generated in A, so d' is generated in B. Then there must be a 1-1 generated function, say h, whose domain is an initial segment of p, say it is the ordinal CY C p, and whose range is d'.We first show CY = p. Both f and g are effective, hence the composition f g :B +B is also effective, by Theorem 3.1, and trivially 1-1. By Proposition 4.2, the relation y E cfg)(x) is generated in B. Now define a map k by

y

=k(x) G

( 3 z ) [ h ( y= ) z and z E cfg)(x)].

k is well-defined, is 1-1, and k : p + a.Further, k is generated in B. Then by condition (5), CY = p. So h : p -+ d f . Now we define a coding p : d + 9 3 as follows. y E p ( x ) G ( 3 z ) [ zE f ( x ) and y

= h-'(z)].

It should be clear that p is a 1-1 onto coding. That p is an embedding, and is effective, are both routine.

REMARKS. B = rec(G(N)) meets the five conditions of the above theorem. The only one that is not immediate at this point is (4), which is a standard result of ordinary recursion theory not proved here.

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[CH.5,87

In Ch. 3, $6 we gave 1-1 embeddings of rec(G(N)) into rec(G(L,)) [I] and rec(G(alr... ,a,)) [IV]. These embeddings are effective. By the remarks above, there are effective 1-1 embeddings the other way around. It follows that rec(E(N)), rec(G(Lw)) and rec(Q(a,, . . . ,a,)) are all effectively isomorphic. In Example 7 of Ch. 3,$6, two embeddings from rec(G(L,)) to rec(G(N)) were given. This gives two effective isomorphisms between these recursion theories. They are not the same, but they are naturally equivalent.

CHAPTER SIX

INDEXED RECURSION AND W - RECURSION THEORIES

1. Introduction

In any recursion theory the r.e. sets are the ones that have elementary formal systems for generating them. In Ch. 1, §lo we proved a number of facts about r.e. sets by manipulating elementary formal systems. Now, in ordinary recursion theory it is possible to “code” elementary formal systems by natural numbers in a way that captures all the pertinent information. A n index of an r.e. set 9 is a natural number that thus “codes” instructions for generating 9. In ordinary recursion theory every r.e. relation has an index. (Kleene’s Enumeration Theorem.) These indexes, being natural numbers, are subjects to be worked on within ordinary recursion theory. It turns out that the indexing can be done in such a way that the manipulations we do on the r.e. sets (forming f l , U , etc.) correspond to recursive functions on their indexes. In effect, ordinary recursion theory can “talk about itself” via this indexing. All this is equally true of hyperarithmetic theory, but it is not the case for every recursion and w-recursion theory. In this chapter we investigate which ones can be indexed in a way that makes “talking about itself” possible. Then in the next two chapters we examine the consequences of this ability. Our plan of procedure is quite direct. We “talk about” the r.e. sets of rec(%) by using elementary formal systems. An elementary formal system axiom is a word, whose letters are members of .d and various formal symbols. So we set up another recursion theory, rec(%’), which has these words as members of its domain. We construct it in such a way that the sorts of things we do with the elementary formal systems of rec(%), from the outside, can be done within rec(%’), and in pretty much the way we were doing them. Thus for every recursion theory rec(%), there is another recursion theory, rec(%’), that can “talk about it” in essentially the same way we can. (The same is true for w-recursion theories as well.) Then we show that if rec(%) has the right properties, rec(%’) will be an 107

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[CH.6,82

inessential extension of it (Ch. 5, §6), and this has the consequence that rec(%) can “talk about” itself. As described above, for each %, rec(8’) is a recursion theory of words, and may be thought of as generalizing rec(G(a,, .. , a,)). It is possible to create a recursion theory of sets that also can “talk about” rec(%). It may be thought of as generalizing rec(G(Lu)).We find the word approach more natural, but we define the basics of the set approach as well, partly because it allows us to make connections between our work and other notions of generalized recursion theory in the literature, notably search computability. See Moschovakis [1969]. Finally, by techniques similar to those described above, we show that the finite sets can often be given codes, so that they can be “discussed” within our theory. This is important for recursion theories, at least, since enumeration operators are compact, and so are completely determined by their behavior on the finite sets. It is not the case for w-enumeration operators, however. In Chapter 8, we will produce sets that play the role, for w-enumeration operators, that the finite sets play for enumeration operators. But the situation is much more complicated, and we will find partial production systems emerging then. IWORTANT NOTE. For the entire of this chapter, T ( - ) is either rec(-) or w-rec(-). No other theory assignments are considered.

2. Elaborating a structure

Let % = (a; %,, .. .,S e ,be ) a structure. It may happen that ‘21 itself is not rich enough for our purposes, and so we extend % to a structure 8* which is more elaborate, and we do our work there instead. For example, in Montague [1968], A! is enlarged to a set closed under the formation of subsets bounded in size by a given cardinal. In Moschovakis [1969], closure under a pairing function is similarly introduced. In this section we describe three methods of elaborating a structure: (1) closing it under a pairing function, as in Moschovakis [1969]; (2) closing it under the formation of finite words; and (3) closing it under the formation of finite subsets, which is essentially the Nocase of Montague [1968]. In the next few sections we develop methods (2) and (3) in some detail, then concentrate on method (2) for the rest of the chapter. Any of the three methods can be used to derive similar results; we simply find the word approach most congenial.

CH. 6, $21

ELABORATING A STRUCI'URE

109

(1) Closure under pairs ; Moschovakis [1969]. Choose some object not in d,call it 0. Let do= d U (0). Choose some pairing function J having the property that no member of d"is a pair under J. Form the least set, d p ,which (a) extends do,and (b) is closed under J. Now, by aPwe mean the structure ( d pd, ; J Idp, !%,. . .,$Rk). REMARKS. It is essential that no member of d obe a pair under J. We often wish to extend a function f from d oto d pand , we do this inductively; we define f on J ( x , y ) in terms of its values on x and y . If some member a of d owere itself a pair under J, an ambiguity could arise, since f has its original value on a, and may get a different value under the process of extending f to pairs. It is an important point where the pairing function J is to come from. We assume that all the mathematics in this book is being carried out in the framework of set theory as a metatheory (ZFC if you like). In set theory it is easy to construct many examples of pairing functions, and to produce one, J, meeting the condition that no member of d ois a pair under J. (As one example, let J ( x , y ) = ( ( x , d o ) , ( y , d o ) ) where ( , ) is the usual Kuratowski pairing function. If some member of d"were a pair under J the axiom of regularity would be contradicted.) This section is one of the few places we need to be conscious of our metatheory.

(2) Closure under finite words. Choose some notion of finite sequence (word, string) having the property that no member of d is a finite sequence. By d" we mean the collection of all words over d.We identify a word of length one with its only letter, so d C d".We allow the empty word, of length 0. Let * be a notion of concatenation appropriate to the notion of finite sequence chosen. By 8'"we mean the structure ( a w d, ;* I d",B1,.. .,a,).We call a" the word structure over 3. Using the notation of Ch. 1, 02, {al,. . .,a,}" and W ( a l , .. . , a , ) are essentially the same, so any results we derive about our present word structures apply to 6(al,. . . , a").If d is infinite, 8" can be thought of as a structure of words on an infinite alphabet, with some relations given on the letters. REMARKS. Again it is essential that a notion of finite sequence is chosen so

110

INDEXEDRECURSION AND w -RECURSION THEORIES

[CH.6,53

that no member of d is also a finite sequence. In the framework of our set-metatheory there are many ways this can be done. (3) Closure under finite subsets ; Montague [19681. Choose some notion of “set”, a corresponding notion, “ E ” , of “member-of”, so that no member of d is a “set”. Define a sequence by

d(’ = 0, d“+’ = the collection of all finite “subsets” of d ”U d. Let dS8” = U,,,,dn. The members of d are called urelernents and the members of dr8” are called hereditarity finite sets over d.Let d’= d”U d.Let us denote by x U { y } = z the relation: “set” z is “set” x with y added as “member”. By 3’ we mean the structure (d’; d,(x U { y } = z ) Id‘,B,,. . . ,$Be,).We call 3’ the set structure uuer 3 . We note that it is an obvious generalization of ,S(L,) of Ch. 1, 92.

REMARKS. It is essential that no member of d be a “set”, just as above, no member of d was to be a pair, or a finite sequence. One way to do this is as follows. Call something a “set” if it is of the form ( x , d ) [( , ) is the usual pairing function of set theory]. Say ( x , d ) “ € ” ( y , d ) if x E y. This produces, within set theory, an isomorphic copy of the universe of sets, but in which no member of d is a “set”. One might feel a certain discomfort with this third method. The problem is psychological, rather than logical, but nonetheless real. Within our metatheory, which is set theory, one is shifting the meaning of set. There is no comparable problem with finite sequence, or ordered pair, since these are not primitives of set theory. And there would be no psychological difficulty at all if our metatheory were something other than set theory. The unfortunate fact is that for most of us, set theory, as a foundation for all mathematics, is the only game in town. This, surely, is an accident of history, and will not continue. On the other hand, whatever foundation of mathematics is accepted, a similar sense of uneasiness may arise when its primitives must be shifted. 3. Recursion and w- recursion theories

Recall the definition of extension, in Ch. 5 , $6. Also recall, T ( - )is one of rec(-) or w-rec(-) for this chapter.

CH. 6, 831

RECURSION AND WRECURSION THEORIES

111

PROPOSITION 3.1. T ( W ) , T ( a w )and T ( W ) are all extensions of T ( a ) . PROOF.We state the proof for T ( a p ) ;the others are similar. Define inj(u) = { u } for u E d.We must show inj : T(%)+ T ( W ) is an embedding. Now, inj is a pre-embedding since T assigns total production systems to structures. Observe that (=d)ml is simply the equality relation on d p , intersected with d X d.Since d is “given” in aP,it follows from the closure features of production systems that (=&)’”’ is generated in T ( a P ) . The proposition now follows by the definition of and the fact that T is an elementary theory assignment. The question of when T ( B P )or T(%”) or T(’2l’)is an inessenrial extension of T(%)is an important one, and we will turn to it in Section 8. We devote the rest of this section to extending embeddings to the elaborated structures we have introduced. DEFINITION. Let d and 93 be sets and consider d pand LBp. We denote the extra object added to d by O.d, and the pairing function for d Pby J,. Similarly for W. Let 8 : d + 24 be a coding. We extend it to a map O p on d pas follows: (1) if a E d , 8’(a) = 8 ( a ) , ( 2 ) ep(0,) = {Oeal, ( 3 ) 8 p ( s d ( x , y ) ) = { J ~ ( U , 2 , ) I U E ~ P ( xand ) 0 E8P(y)l. It should be clear that 8” is also a coding, O p :d p+ LBp, that agrees with 8 on d. DEFINITION. Again let d and W be sets, and consider d ” and 9”. We denote by juxtaposition the concatenation operations of each. Let 8 : d + 24 be a coding. We extend it to a map 8“ on d“ as follows. Let a , a z .. a, be a word over d,where each a, is a letter (member of d). By 8 ” ( a , a z* . a,) we mean

-

{blbz-

*

- b, I bl E @(aI)A . . - A b, E 8(a,)}.

Again, it should be clear that 8 ” is a coding, 8” : d“+ 93 ”, that agrees with 8 on d. DEFINITION. Finally, let d and W be sets again, and consider d “and 24’. We use { x , , . . .,x,} to indicate the “set” with xI,..., x, as “members”, both in d 5and 93’. Let 8 : d + 93 be a coding. We extend it to a map 8“ on d sas follows. (1) if a ~ dO S,( a = ) 8(a),

INDEXED RECURSION AND U-RECURSION THEORIES

112

[CH.

6,84

(2) if (aI,.. . ,a n }is a "set" in d',

I

8'({a,,..., a , } ) = { { b l ,..., b , } E 9' ~ I 8 "E( U l ) A * * * A b. E 8s(am)}. Once again, it should be clear that 8" is a coding, 8" : d s+ W",that agrees with 8 on d.

PROPOSITION 3.2. Let * be either p or w or s. If 8 : T ( 8 ) - , T(B) is an embedding, then 8* : T(8*)+ T(B*) is also an embedding. PROOF.Left to the reader.

4.

C

completeness

This section is the first of several in which we study word and set structures in some detail, dropping the pairing function elaboration. In word structures we use * for concatenation. In set structures we use the customary E symbol for membership. No confusion should result. We define two notions of C formula, one for set structures, one for word structures. We call production systems Z complete if every relation definable by the appropriate version of 2 formula is generated. In the next section we investigate the 2 completeness of recursion and w-recursion theories. Let 8 = (d; %,, . . . ,!?Be,)be a structure. By an atomic set formula over 8 we mean one of (x = y), ( x # y ) , (x E y ) , ( x e y), R l x , . . . ,Rkz, where x, y,x,. . . , z are variables or members of d'. By a AS, formula over $ we ? mean I any formula built up from atomic set formulas over 8 using: conjunction, A , disjunction, v, and bounded quantification, (Vx E y ) and ( 3 x E y), where y is a variable or a member of d'.(This notion comes from Levy [1965].) By a Z sformula over 8 we mean a formula built up from atomic set formulas over 8,using the machinery of A; formulas, and also unbounded existential quantification, ( 3 x ) . When we talk about the truth of a 2' formula over 8,we mean, in the structure where 9, interprets R i , and E and = are given the obvious interpretations. We call a relation % on d"a Z srelation (a AS, relation) over 8"if there is some formula q ( x ) which is C' over % (AS, over 8) and

as,

%(x)

e

q ( x ) is true.

CH.

ELEMENTARY FORMAL SYSTEMS AND P COMPLETENESS

6, §S]

113

Let A be a production system on '21'. We call A 2' complete (with respect to '21') if every relation which is 2' over '21" is generated in A. Next we set up similar notions for word structures. By an atomic word formula over 9 we mean one of (x = y), ( x # y ) , (x * y = z ) , (x * y # z ) , R,x, . . . , Rez, where x, y, z, x, . . ., z are variables or members of d". If u and u are words over d we say u is part of u if there are words x and y (possibly empty) so that x * u * y = u. We write u pt u to indicate that u is part of u. By a A ," formula over '21 we mean any formula built up from atomic word formulas over $?I using: conjunction, A , disjunction, v, and bounded quantification, (Vx pt y ) and ( 3 x pt y), where y is a variable or a member of d W . By a 2 " formula over '21 we mean any formula built up from atomic word formulas over '21, using the machinery of A ," formulas, and also unbounded existential quantification, ( 3 x ) . Truth of a 2'" formula over $?I means, in the structure a", in the obvious way. A relation 93 on d" is a 2" relation (a A," relation) if there is some formula q ( x ) which is 2'" over 9 (A," over 8)and %(x)

e

p(x) is true.

NOTE. A," relations are esentially a generalization of the rudimentary relations of Smullyan [1961]. Let A be a production system over '21". We call A 2'" complete (with respect to '21") if every relation which is Z Wover '21'" is generated in A.

5. Elementary formal systems and 2 completeness In this section we show that both recursion and w-recursion theories over word or set structures have the appropriate version of 2 completeness (provided equality is well behaved). Also, for recursion theories, we sketch a converse result, that allows us to relate our notion of recursion theory with certain other generalizations in the literature.

LEMMA5.1. Suppose rec('2l) (w-rec(U)) is a theory with equality. Then (1) In rec(%') ( i n w-rec($?I*))thefoilowing are r.e. ( u - r x . ) : = , # , E , (2) In rec('21") ( i n w-rec($?I")) the following are r.e. (w-r.e.): = , # ,pt.

e.

INDEXED RECURSION AND w -RECURSION THEORIES

114

[CH.6,§5

PROOF.(1) = is always generated. The following axioms generate, in both rec(%') and w-rec(%I"), = d",E, and # :

e

a

axioms for

#.d,

xU{y}=z'yEz, 20, y Ex+dx, d x

'd y

+

x

#d

y

+

x # y,

'a y x # y, a x 'a y +z E x + ze y + x # x# y ' y # x,

d x

y,

dx+y!zx, ye01 x!zy+x#

w -+y

u{w)=2 + x e z

[here we used 0 for the "empty set" of d " ] . (2) Again = is generated. The following axioms generate, in both rec(8") and w-rec(%"), the relation # (we begin with an iterated concatenation, for convenience): x * a = b+ b

axioms for

d c+x

*u =y+x

* a * u = y,

Zd,

* c = y +nonempty

y,

nonempty y - + x * y = z + x # z , nonempty y + x * y

a #&b +x

*a *u

=z+z#x, = q +x

*b*v

r + q # r.

And the following generate the relation pt: x * u * y = u'uptv.

LEMMA5.2. Suppose rec('2l) (w-rec('2l)) is a theory with equality. Then (1) the r.e. relations (the w-r.e. relations) of rec('2l') (of w-rec(Bs)) are closed under bounded quantification, ( V x E y ) and ( 3 x E y ) ; (2) the r.e. relations (the w-r.e. refations) of rec(B") (of w-rec(%")) are closed under bounded quantification, ( V x pt y ) and ( 3 x pt y ).

ELEMENTARY FORMAL SYSTEMS AND 8 COMPLETENESS

CH. 6, §S]

115

PROOF.(1) We have closure under 3 and A , and E is generated by the previous lemma, thus we have closure under ( 3 x E y ) since (3xEy)RX,z

( g X ) [ x E yA R X , Z ] .

We show closure under (Vx E y ) . Suppose we have axioms for Rx, z. To get axioms for (Vx E y)Rx, z add the following:

d y +(Vx E y)Rx, z, (Vx E 0)Rx, z, (Vx E a)Rx, z + Rb, z + a U { b }= c -+ (VX E c)Rx, z.

(2) We have closure under ( 3 x pt y ) , using Lemma 5.1, since

(3x)[xpty ~ R x , z ] .

(3xpty)Rx,z

Suppose we have axioms for Rx, z. To get axioms for (Vx pt y)Rx, z add the following (where 0 is the empty word):

R 0, z + (Vx pt O)Rx, z,

d m + RO, z

+ Rm, z

d m + d n +m * v + (Vx

=

+(Vx pt m)Rx, z ,

b+ v * n = e + b * n = w

pt b)Rx, z + (Vx pt e)Rx, z

+ Rw, z + (Vx

-

pt w)Rx, z.

In following the last axiom the reader should have in mind the picture e

m*u*n=w w

b

where m and n are letters, i.e. members of d.

THEOREM 5.3. If rec(?l) is a theory wirh equality, then rec(?I'") is Z w complete, and rec(3") is 2' complete. Similarly for w-rec(-). PROOF.Immediate from the above lemmas and the closure properties of production systems.

In a sense, recursion theories are minimal among 2 complete production systems. More precisely,

116

INDEXED RECURSION AND IAJ-RECURSIONTHEORIES

[CH.

6, $6

THEOREM5.4. If rec(?l) is a theory with equality, then (1) If 9 is a relation on d", 9 is Zsover %' if and only if 9 is r.e. in rec('2l"). ( 2 ) If 9 is a relation on d w9 , is X W over 'u" if and only if 9 is r.e. in rec(?lw). We do not need this result in this book, so we merely suggest the proof, leaving the details to the reader. Half of (1) is from the previous theorem. For the other half, (if 9 is r.e., then 9 is Zs), elementary formal systems for 9 may be coded into and derivations from them shown to be 2' (indeed A i)using the chart on p. 14 (30)[ D is a derivation of R x ] makes in Barwise [1975]. Then x E 9 9 2'. (2) has a parallel development. In Montague [1968], in effect, a relation 9 on d is called Norecursively enumerable if 9 is Zsover 'us(this is not his terminology). By the above theorem, then, if rec('3) is a theory with equality, 9 is No recursively enumerable over ?l if and only if 9 is r.e., in our sense, in rec(?l"). Indeed, we will see later in the Chapter, that if rec(?l) has an effective pairing function, this is further equivalent to: 3 is r.e. in rec(2l) in our sense. In Moschovakis [1969] a definition of search computability is given, by using an inductive definition on ?lp,involving indexes. This is yet another generalization of ordinary recursion theory, and what he calls the a:, relations are intended to be generalizations of the r.e. relations. Gordon [1970] shows that the a: relations on a structure ( d s ; l ,..,9 k ) are exactly the No recursively enumerable relations on the structure

as,

(d;

9 1 9 . .

. %, Bl, 9

* *.

9

g k , =.d,

fd).

Further, in Moschovakis [1969A], the search computability approach is shown to be equivalent to two other generalizations of ordinary recursion theory, due to Fraisse [1961] and Lacombe [1964], [1964A]. In Ch. 1, §13,14 we also established equivalences with inductive definability without universal quantifiers, and to R -definability. Clearly the notions involved possess great stability.

6. Separability The entire of this section is devoted to a proof of the following.

THEOREM 6.1. If T(%)is a theory with equafity, then both T(Ss) and T(?lw) have effective pairing functions, and are separable with respect to both x and U .

CH. 6, 061

SEPARABILITY

117

PROOF.If we show the existence of an effective pairing function, then separability under X follows by Theorem 4.10.1, and separability under U by Theorem 4.10.3. We show the existence of an effective pairing function by producing an appropriate Z formula and using Theorem 5.3. Case 1: sets. By (x, y) we mean the usual {{x}, {x, y}}. The relation z = (x, y ) is 2' (indeed A:) over %?Is by the following.

w ={x,y} Z=(X,y)

e e

xEw

A

y E w A ( V ZE W ) ( Z = x v z

= y),

(3UEZ)(3UEZ)(U={X,X}AU={X,y} A 2 = {U,U}).

Then J(x, y ) = (x, y ) will serve as an effective pairing function for both rec(%") and w-rec(%'). Case 2: words. Actually, we must consider two cases depending on whether d has one member or more than one. Suppose d has one member, say d = { 1). We can "identify" a string of n 1's with the number n and it is easy to see that %" is rather like G(N) in disguise. Then any of the pairing functions from ordinary recursion theory (say Example I in Ch. 4, §lo) can be "transferred" to both rec(%") and w-rec(8'"). We leave details to the reader. Now suppose d has more than one member. We produce a pairing function for d", which is X'",which completes the proof. The method we give is from Quine [1946]. See Even and Rodeh [1978] for another. Say 1,2 E d (we use these without numerical significance). By a tally we mean a non-empty string of 1's. If r is a tally, let = 2 * r * 2. If r is the shortest tally not occurring as part of x or y, set

+

J(x, y ) = 7? * x * it * y

* +.

We leave it to the reader to verify that this serves as a pairing function for d'". We show it is 2" (in fact, A,"). Below is a list of A," relations, ending with z = J(x, y). (1) For each n, the relation

y =xl*-"*x"

e

(3yI pt y ) . . *(3y,-zpt y ) [)'I= X I *X2A Y z = Y I *X3 A

.. . A y

= y.

-2

* X, ]

(2) y p t x e ( 3 u p t x ) ( 3 u p t x ) [ x = u * y * u ] . (3) ux e x i s a t a l l y e x # O ~ ( V y p t x ) [ y = O v l p t y ] ( h e r e O i s t h e empty word).

INDEXED RECURSION AND W-RECURSION THEORIES

118

[CH. 687

(4) uMy e u is the longest tally in word y e (TD A U pt y A (VX Pt y )[X # U * 11. ( 5 ) UBX,y

u is the longer of tallys x and y

e

Oy A [(X Pt y A 0 = y ) V ( y Pt X A U = X)]. ( 3 v pt z ) ( 3 qpt z ) ( 3 r p t z ) ( 3.rr pt z ) ( 3 f i p t z ) [qMx A rMy A uBq, r A T = u * 1 A ii = 2* IT * 2

OX A

(6) z = J(x, y )

A 2 =fi*X*7?*y*7?].

7. Reflexivity

Again the entire section is devoted to a single result.

THEOREM 7.1. If T(%) is reflexive with respect to U , so are T(%') and T(8"). PROOF.We give the proof for rec(-); that for o-rec(-) is similar. Suppose rec(%) is reflexive with respect to U . As usual, let inj : d + d be defined by inj(x)={x}. Then by definition, inj" is r.e. in rec(8) U rec(8). It is immediate that inj" is also r.e. in rec(%I")U rec(%") and in rec(8") U rec(%"). Now we treat these cases separately. Case 1: sets. Define f : d s + d s by f(x)={x}. We show f " is r.e. in rec(8') U rec(81E),which establishes reflexivity of rec(%"). Now, u = u U { w } is r.e. in rec(8") hence [ u = u U { w } ] ' and [ u = u u {w}]' are r.e. in rec(%') U rec(%'). Then, as elementary formal system axioms for f", the following will serve. y E inj'x + y E f'x, y E f U x+ b E f " a + [ y ' = y U { b } ] '

+[x'= x

u { a } ] '+ y ' =

tux'.

Case 2: words. Define g :d'"-+d"by g(x)={x}. We show g " is r.e. in rec(8") U rec(%"). Now, u = u * w is r.e. in rec(%"), hence [ u = u * w J and [ u = u * w]' are all r.e. in rec(8'") U rec(%"). As axioms for g u we take y E inj'x

+ y E gux,

y E gun + b E inj"a + [x' = x

+[

y' = y

* a ] '+y ' E g"x'.

* b]'

CH. 6, §8]

SEQUENCE CODINGS

119

8. Sequence codings

We have carried our parallel discussion of set structures and word structures far enough. From now on we concentrate on the details of word structures only. This will lead to some general results that incidentally give us information about set structures, however. Alternately, the entire development from here on could be modified to fit set structures (or pairing function structures) directly. We leave that to the dedicated reader. Since we are concentrating on word structures, this section concerns itself with the possibility of “coding” words over a structure back into the original structure. It is the first of a series of sections ending in Section 14, which combine to show there are many recursion and w -recursion theories that can “talk about” themselves. DEFINITION. We say T(%)has a sequence coding if there is a way of assigning to every word x in d”, some distinct sequence code x # for it, where x # E d, so that (1) the function c, given by c ( u #, v # ) = ( u * u ) # , is generated in T(%); (2) the function d, given by d ( a ) = a # for all a E d,is generated in

T(W. Primarily, in this section, we show T ( % )has a sequence coding if and only if T(%)has an effective pairing function. THEOREM8.1. If T(%)has a sequence coding, then T(%)has an effective pairing function (and hence is separable with respect to both x and U).

PROOF.Suppose T(%)has a sequence coding. For x , y J ( x , Y 1= ( x * Y

E d , set

= c ( d ( x ) ,d(Y 1).

This is generated in T(%)since

J(x,y) = z

e ( 3 q ) (3 r ) [ q = d ( x ) A r = d ( y ) A z

= c(q, r ) ]

and c and d are generated. And it is trivial that J serves as a pairing function. THEOREM 8.2. If T(%)has a sequence coding, then T(%”)is an inessential extension of T(%), via a 1-1 embedding. PROOF.Let T(%)have a sequence coding. Define h by: for a word x E d”,

INDEXEDRECURSION AND WRECURSION THEORIES

120

[CH.6,§8

h ( x ) = { x # } . h is a coding of d" in d,and is 1-1. It is trivially a pre-embedding of T ( % " )in T ( % ) we ; use the fact that T is an elementary theory assignment to show h is an embedding. Let % = ( d ;,..., 9 , gk). Then %'"=(d";d,*,9, ,..., %k). First, d h= {y ( 3 x ) d ( x )= y } and hence is generated in T(%). Next, (a")" = {y y E d"v ( 3 q ) ( 3 r ) c ( q ,r ) = y } , so it is generated in T(%).Then ( = d w ) h is generated, since it is ( = d ) n [ ( d " )xh (dw)h]. *' is simply the function c, which is generated. For each i = 1,2,..., k, (3,)' is {(yl ,..., yn,)1(3x~)*..(3x,) [ ( x l , .. .,x , ) E 9, A y , = d ( x , ) A . . * A yn, = d(x,,,)]}. Thus it is generated in

I

I

T(W-

Since T is elementary, h is an embedding. Finally, to show T(%"')is an inessential extension of T ( % )we need that y E ( h Id ) ( x ) is generated in y = d ( x ) .This completes the proof. T(%).But in fact, y E ( h Id ) ( x ) 8.3. Suppose d is infinite and T ( % )has an effective pairing PROPOSITION function. Then rec(G(N)) has a 1-1 embedding into T(%).

REMARK. If T(%)has a pairing function at all, cardinality considerations show d must have 0 or 1 members, or be infinite. Thus the hypotheses are

stated in a stronger form than strictly necessary. The conclusion is, of course, that a copy of ordinary recursion theory will be present.

PROOF.Let J be an effective pairing function in T(%).Pick some xoE d and define J'(x, y ) = J(J(x,, x), J(xo,y)). Then J' is also an effective pairing function, and its range omits infinitely many members of d (in particular, if x l # xo, J ( J ( x l ,x ) , J ( x l ,y)) is not in the range of J'). Now pick a, b E d with a # b, and a and b not in the range of J'. For each natural number n we define a corresponding member n' E d as follows. 0' = a and 1'= b. Now suppose 0', l', . . . ,( n - I)', n' have been defined, and are all distinct. Set ( n + 1)' = J'((n - 1)',n'). With this definition, ( n 1)' must be new, for the following reasons. ( n + 1)'cannot be 0' or 1' since a and b are not in the range of J'. And if (n + 1)' = k' where 2 s k s n, then J'((n - l)', n') = J'((k - 2)', (k - 1')), and SO n' = (k - 1)' contradicting the supposition that all of 0', l', . . .,n' were distinct. Now define a coding 6 : N+ d by @ ( n ) = {n'}. We omit the routine verification that 8 :rec(G(N))+ T ( % )is an embedding.

+

THEOREM 8.4. Suppose d is infinite and T(%) has an effective pairing function. Then T ( % )has a sequence coding.

ADDING LETTERS

CH.6, §9]

121

PROOF.Let J be an effective pairing function for T('21).We write y = (xl, x2, xs) for y = J ( x l ,J ( x z , x3)). Then this too is generated; an effective tripling function. By the previous theorem there is a 1-1 embedding 8 :rec(G(N))+ T('21).For simple notation, we write n for the only member of 8 ( n ) , thus identifying some members of d with the natural numbers. Also we write + for which is generated in T('21)of course. Now, the idea is to code the empty sequence by (O,O,O), and the sequence abcd, for example, by (a,4, (b, 3, (c, 2, ( d , 1, (O,O, 0))))).We give the following elementary formal systems which establish that appropriate c and d functions are generated in T('21). Axioms for the d function:

+',

Y

= (x,

1,(0,0,0))-+ Y = d ( x ) .

x is a sequence code for, say, xo, and y Let L be the relation Lx, a, y is a sequence code for xo * a, where a E d.L has axioms:

Y = d(a)+L(O,O,O),a,y, Lx, a, y + x

= (u, u, w ) + u = u

+ y = (e,f, g ) +f '

=f

+ 1 +y

+ 1+x = (4, f',

= (4,u

f ,

x)

y ) +L x ', a, y '.

Cx G x is a sequence code. C has axioms

C(O,O,O), c x +Lx, a, y

+

cy.

Finally, axioms for the c function c x + x = c(x, (O,O, 0 ) )

y = c(u, u)+ Lu, a, uf+Ly, a, y'+ y f = c(u, u f ) . COROLLARY 8.5. Suppose d is infinite and T('21)has an effective pairing function. Then T('21")is an inessential extension of T('21), via a 1-1 embedding. PROOF.Theorems 8.2 and 8.4. 9. Adding letters

An elementary formal system axiom over '21 is a word, made up of letters which are members of d and various formal symbols (predicate letters,

122


[CH.($59

arrow, comma). We will be treating such an axiom as a word in some word structure. Now, '21w would do if we only needed letters which are members of d , but we also need some letters to play the role of the formal symbols. Additional letters are called for. In this section we show that ?I can be expanded by adding what amounts to countably many new letters, so that the ( w ) recursion theory of the word structure over this expansion is an inessential extension of the ( w ) recursion theory of 3". (Ch. 5 , 96.) Recall, G(N) is the structure of arithmetic, (N; y = x+). We use 0 for the trivial, empty structure. THEOREM9.1. Let % be a structure such that T(%)is a theory with equality. Then ~ ( 1 % U G(N)J") is a n inessential extension of T([%U 0 l W )via a 1-1 embedding.

REMARKS. We are forced to use ?I U !d rather than % itself for the technical reason that dU 0 C dU N but not d dU N. We have more to say about this in the next section. By Theorem 5.6.5, inessential extensions are also conservative ones, so no new sets are generated on the common domain. PROOF.The plan of the proof is as follows. In part I we define a 1-1 coding 8 : [dU N]" + [dU 01'". In part I1 we show 8 is an embedding. Finally in part I11 we show the relation y E [6 1 (dU0)"](x) is generated in T([%u 01"). Part I: I f x E d , t h e n ( x , O ) E d U N a n d ( x , O ) E d U 0 . W e w r i t e x ' for (x,O). Similarly, if n E N then (n, l ) € dU N. We write n' for (n, 1). T(%) is a theory with equality so trivially T(Y1 U 0) also is. Then by Theorem 6.1, T([?l U 01") has an effective pairing function J, and then by Theorem 8.4 it also has a sequence coding # , with generated c and d functions as in Section 8. We make use of this machinery to define our coding 8. Choose some member c E d,fixed for the rest of this section. We first define an auxiliary map, on letters, d U N + ( d U ! d ) ' " as follows. (1) If x E d,by (x')" we mean J ( x ' , c'). (2) If n 63 N, by (n'y we mean J ( c ' c ' . . . c', c ' c ' ) .

-

0 :

n

(Here juxtaposition denotes concatenation in (dU g)'".)

(h.6 , §9]

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123

Then z - has been defined on every member of dU N, and clearly the mapping is 1-1. Now, let s be word in (dUN)". (1) If s is the empty word of ( d U N ) " , s is also the empty word of (dU0)". Set 8(s) = {s#}. Briefly, 8(0) = {0#}. (2) If x = zIzz. . . z,, where each z, is a letter, i.e. a member of d U N, then for each i, z ? is a member of ( d U 0 ) " . Then Z ? * Z ~ * - - . * Z : is a sequence over (dU0)" (caution: over, not in). Set z;*.

e(s) = ((2:"

. . * z:)#}.

It is straightforward that 8 is a 1-1 coding, 8 : ( d U N)" +(.dU 0)". This ends part I. Part 11: We show 8 is an embedding, 8 : T ( [ %U G(N)]") + T ( [ %U 01'") by making use of the fact that T is an elementary theory assignment. Since T ( - ) assigns total production systems to structures, 8 is automatically a pre-embedding, so we need only show that certain relations are taken over to generated relations under 8. This part of the proof is largely bookkeeping. We begin by carefully saying what relations we must be concerned with. Let 8 = ( d %,, ; . . .,se,). Then

so

a 6c ( N ) = ( d U

N ; ( = d ) ' , %!;,. . . 7

%L;,(=Ny,(y

= x'y)

[a U G(N)]" =((auN)";dUN,*,(=d)',%~,

...,%~,(=Ny,(y

=x'y).

Likewise, but slightly simpler since d U 0 = d',

[a u 01" =

;d', *, (=rP)1,9i,

. . ., 9 i ) .

We have denoted concatenation by * in both structures. Now, to show 8 is an embedding, we must show 8 takes each of the following relations to a relation generated in T([%U 01'") (see Ch. 3, S.5): -

-(4UN)Y)

~ U N N*,,

%;,-.-7%L,

(=d)',

(=NY)

(y=x'y.

We first show some auxilary relations are generated, then proceed to these. is generated in T([%UO]"): (1) By definition of 8,

(a')' = { t I for some z E ={t = {r

t = (zO)#)

I for some z E d',t

= ( ~ ( zc'))#) ,

1 for some z E d',r = ~ ( J ( zc'))}. ,


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[CH.6,§9

Now, d'is a "given" relation of T ( [ 8U 0]"), and d and J are generated in it. It follows from the closure properties of production systems that (d')" is generated in T ( [ %U 01"). (2) Let A be the set of words in ( d U 0 ) " of the form c'c' * - . c ' (including the empty word 0). A is generated in T ( [ %U 01"). Indeed, by the elementary formal system

A 0, Ax-y

=xc'+Ay

(juxtaposition denotes concatenation). (3) (N')" is generated in T([%UO]"): As in item (l), (N')" = { t for some z E A, t = d ( J ( z ,c'c'))}and hence is generated since A, t and d are. Now we turn to some of the relations in our list earlier. (4) ( d U N)" is generated in T ( [ %U @I"), since it is in fact, U (N')" and we have closure under unions, and items (1) and (3). (5) ((= is generated in T([%U 01"). Since 6 is 1-1, this relation is simply the equality relation of (% U 0)" restricted to (d')', that is,

1

(&I)"

((=.9)')H

is

=.dU(I

n [(d')" x (d')"].

This is generated by item (1) and the properties of production systems. (6) ((=N)')@ is generated in T ( [ 8UO]"). This is similar to item (5). (7) Each of ($%E;)", . . . ,($%E)" is generated in T ( 1% U 01"). This is like item (l),taking into account the fact that gPmay be more than one-place. (8) Another auxiliary relation, ((dUN)")' is generated in T([%U 01'"). It is not hard to see, directly from our definition of 0, that it is represented by H using the following:

(0)"

E H,

xE(d')a+XEH,

x E (N')" + x E H,

xEH+yEH+z

= c ( x , y ) + z EH,

(here we have written x E H for Hx, in keeping with our other informal notation). (9) ( = ( d U N ) Y ) e is generated in T ( 1%U 01 "). This follows from item (8) by an argument like that in (5). (10) (*)" is generated in T ( 1% U0J"). It is simply the c function restricted in ((d U N)" )@.

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1

(11) ( ( y = xi)')e is generated in T 191 U 01"). It is, in fact, {(u, u ) for some x E A, u = d ( J ( x ,c ' c ' ) )and v = d ( J ( x c ' ,c ' c ' ) ) } . This concludes Part 11. Part 111: We show the relation y E [ 8 r ( d U 0 ) " ] ( x ) is generated in T([Y1 6 01'"). Well, it is represented by F using the following axioms: x E d ' + t = J(x,c')+y = d ( t ) + y E F(x),

~ ~ E F ( x ~ ) + ~ ~ E F ( x = Zx I x) ~ ++ Y X

=c(yi,yz)+y €F(x).

This completes the proof.

10. Copies and extensions

We have discussed, in Chapter 5 , isomorphisms between productions systems, in particular between recursion and w-recursion theories. There is a stronger notion we introduce now, which we call being a copy. Let 81 = ( d 9,, ; . . . , 92,)and 8 = ( 3 Y ;, , .. . ,92,)be two structures. Suppose we have a 1-1, onto coding O : . d - + 3 such that, for each i, 9, = 9y. Then we'll call ,H a copy of Yl. For example, 91 U 0 is a copy of 91. Now if B is a copy of 91, we'll also call rec(B) a copy of rec(%), and likewise for their w -recursion theories. For all reasonable purposes, we need not distinguish between copies; they are, after all, direct renamings of things. So, from now on, unless there is a strong reason to do otherwise, we'll identify rec(Y1 U 0) with rec(81). With this in mind, we can restate the result of the last section more simply, as follows. THEOREM 10.1 (Theorem 9.1 restated). Let 91 be a structure such that T(9I) is a theory with equality. Then T([81 U 8G(N)]") is an inessential extension of T(8f") via a 1-1 embedding.

We would like to iterate this result, so that we can throw in many copies of 6 ( N ) , not just one. To do this in the simplest way, we list two independent results. PROPOSITION 10.2. If T(B) is an inessential extension of T(81) via g, and T(G)is an inessential extension of T ( 8 ) via f, then T ( 6 ) is an inessential extension of T ( % )via gf.

PROOF.Both f : T(O)-+ T(B) and g : T(\P)+ T(8I) are embeddings,

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6, $10

hence gf : T(Q)+ T ( % )is an embedding. All that needs to be shown is that the relation y E (gf r d ) ( x ) is generated in T(’2I). Now, y E (f r 9 3 ) ( x ) is generated in T(B), and g is an embedding, so g carries this to a relation generated in T(!?l);let us call it R. Then

R ( y ,x )

e for some

a, b E 93, y E g ( b ) ,

x E g ( a ) and b E cf B ) ( a ) . Recall from Ch. 5, §5 that f 193 = f inja.,: (we will make use of this and similar things quite a bit). Hence

R ( y ,x )

e for some a, b E 93, y

Eg(b),

x E g ( a ) and b E f inja.w( a ) . But then, R ( y , x ) a for some a E 93, x E g ( a ) and y EgfinjB.%(a). Also the relation y E ( g d ) ( x ) is generated in T(’2I),hence so is the following:

r

for some x E d , R ( y , x ) and x E ( g d ) ( z ) .

S(y,z )

Writing out the characterization of R, and rewriting g r d, S(y, z )

e for some a E 93 and some x E d , x

Eg(a),

y E gf injB.u ( a ) and x E g inj.a,a (2). Now, if x E g ( a ) and x E g(inj,a,B( z ) ) ,since g is a coding, it follows that a E injd.* (2). It further follows that

S(y,z )

e

y E gf injs.%inj.d,9(2) y f gfinj,d,q (2)

y E ((gf) t d ) ( z ) .

Since S is generated in T(%),we are done.

PROPOSITION 10.3. If both T(!?l)and T ( 8 ) are theories with equaliry, so is T ( 8 )u T ( 8 )= T ( %u 8). PROOF.By hypothesis, # a is generated in T ( 8 ) and # %is generated in are T ( 8 ) . Now, 8U8=(dU93;d’,BJ, ...) and both (Zd)’ and generated in T ( 8 U 8)since i and j are embeddings. Then # a u 9 is also generated, since it is the relation (# a)I U (fa)’ U (d‘ x 93’)U (93’ x d’). Now we can give our “iterated” version of Theorem 10.1. THEOREM 10.4. Let 8 be a structure such that T(’2I) is a theory with

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PROOF.Line (I) is a restatement of Theorem 10.1. T(%)is a theory with equality, by hypothesis, and we know T(G(N))is a theory with equality too, so by Proposition 10.3, T(%UG(N)) is a theory with equality. Applying Theorem 10.1 to it we get that T([(%U G(N)) U ~ ( ~ 1is1an~inessential ) extension of T([%U G ( N ) ] ~via ) a 1-1 embedding. Combining this with the fact of line (l), using Proposition 10.2, gives us line (2). And we may continue in this fashion.

11. Universal machines

For this section % is a structure such that T(%)is a theory with equality. We use 9 ( n , m ) here to denote the collection of all operators in T(%")of order (n, m ) . What we show in this section is that % has an extension %*, such that each class 9 ( n , m ) can be "mimicked" by a single operator in T(%*").Precisely what this means will become clear as we proceed. To begin with, we need a context in which T(%I")can formally be "talked about" in essentially the same way we do. We "talk about" T(%'") by working with ( w ) elementary formal systems. So, we want a setting in which (a)elementary formal systems, and derivations, are objects which can be formally manipulated. As a start, we need variables, relation symbols, and punctuation, as well as the members of d".Well, let %* be the structure ((aU G(N)) U G(N)) U G(N). In this structure there are three copies of N; we simply identify one copy of N with the variables [and write var(x) if x is in that copy], we identify a second copy of N with the relation symbols [and write rel(x) if x is in that copy], and we identify as much of the third copy of N as is needed with the punctuation symbols [and write punct(x) if x is in that copy]. Also there is a copy of d "given" in this structure; we simply use d to represent it. Thus the domain of %* has all the "letters" we need to construct elementary formal systems over a"', and hence actually has, in its domain, each elementary formal system axiom over Yl'". We also have that in T ( g * " )each of var, rel, punct and d is generated (in fact, each is one of

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[CH.6,811

the “given” relations). T ( a * ” )is an excellent place to try “talking about” T ( 8 ” ) .Incidentally, by Theorem 10.4, T(%*”)is an inessential extension of T(’ZI”),but this will play n o role until Section 14. We use d *as the domain of a*; thus d *= ((dUN) U N) U N. We take as one of our formal punctuation symbols a space, and use # to denote it. Suppose we have an ( w ) elementary formal system over a”’ with axioms A , , A:, . . . ,A,. Then, in accordance with the above discussion, we can consider each A, to be a word in d*”. Also # A I # A 2 # . . . #A,# is in d*”. Such a word is called a base, with components A , , A z , .. . , A , . In a sense, it codifies the ( w ) elementary formal system with axioms Ai,Az,. . . , A , . 9 ( n , m ) is the collection of all ( w ) enumeration operators in T ( a ” )of order (n, m ) . By a universal machine for the class 9 ( n , rn) we mean a “black box” (operator) Du of order (n, m + l),whose behavior is as follows. Let @ E $ ( n , m ) ; let A , , A z ,..., A, be axioms for an ( w ) elementary formal system for @, and let a = # A l # A z # . * A, f . Then, if 9 is any we should have n-ary relation on d”, (XI,..

.,x,)E

@(9) e

( 4 x 1 , . . . , x , ) E Du(8).

Thus Du “mimics” each operator in 9 ( n , m ) , keeping track of each by “tagging” outputs by, essentially, the elementary formal system axioms used to generate it. T h e result promised at the beginning of this section, that in T(%*”)there was an operator that could “mimic” the entire class 9 ( n , m ) , can now be stated properly. THEOREM11.1. Let Du be a universal machine for the class 9(n, m). Then Du is an ( w ) enumeration operator in T(%*“). Thus the behavior of all members of 9 ( n , m ) is captured by Du. But note, members of 9 ( n , m ) are operators in T ( W ) ,while Du is an operator in T(%*”).We will simplify this later on. We devote the rest of this section to a proof of the above theorem, which (simply) amounts to writing an ( w ) elementary formal system for Du. And to do this, we give formal counterparts of the definitions in Chapter 1, through that of enumeration operator. Before doing so, however, we have some remarks. First, there is no harm if we use, say, P as both an n-place and a k-place relation symbol. We can tell the two uses apart by looking at the number of terms which follow P. This means we can make do with one list of relation

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symbols, instead of separate lists for 1-place, 2-place, etc., and this is a technical convenience. T(G(N))is a theory with equality. It follows that in T(YI*"),the unequals relation for each copy of N is generated. We write Z,,, and Z,,, for those on the variables and on the relation symbols respectively. As punctuation symbols, in addition to # , we need an arrow, a comma, and (if we are talking w-recursion theory) a universal quantifier. T o tell the symbols of d * wfrom the symbols we use, we write

3 0

T

for the arrow in d*", for the comma in d*", for the universal quantifier in d*".

The symbols we use, from the outside, will be as usual: the above are in punct. There is a similar problem with relation symbols. Now, YI = ( d Bl,. ; . . ,s k ) . For use in our elementary formal systems, certain predicate symbols have been assigned to B,,. . . ,9L; we have been designating them R , , . . . , Rk. Well, certain members of re1 in d*"will also have to be assigned to Bl,.. . , Bk. Let us designate them by R , , . . . , R,. Thus R, is a symbol we use, while R, is a member of d*". Similarly for input and output symbols. We will continue to use I and 0 for the purpose. We assume two symbols of re1 have been designated to play their role "inside" the theory; we denote them and 0. % embodies many different enumeration operators. In giving axioms for % we will need some way of keeping track of these various operators. More precisely, if X is derivable over T('21")using axioms A ,,A,, . . . ,A,, we need some simple way of "remembering" this. Well, # A I # A, # . . . # A, # X is a word in d*" ;we work with it, thus keeping track of the axioms used, and X at the same time. Finally, T is either rec(-) or o-rec(-). In the one case it is an elementary formal system we must construct; in the other, an o-elementary formal system. The two are very similar, so we list the axioms assuming T ( - )is rec(-) and then we indicate the changes necessary if T ( - )is w-rec(-). Now we list our axioms for %, in rec(%*"). As we introduce each new relation symbol we say what it is intended to represent before giving axioms for it. 0 is the empty word here. (1) word x G x is a word over d (i.e. x is a member of d*"made up entirely of letters from the copy of a).


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[CH.6 , $11

word 0, d x +word y (2) term x

e

+z = y

* x +word

2.

x is a term.

word x +term x. var x + term x.

( 3 ) pseudoatomic x G x is a pseudoatomic formula. re1 z +term x + y

=z

* x +pseudoatomic

pseudoatomic y -+ term x + w

y,

* * x +pseudoatomic w.

=y

0

(4) atomic x G x is an atomic formula (no variables).

re1 z +word x + y

=z

* x +atomic

atomicy+ wordx -+ w = y

( 5 ) pseudoformula x formula.

e

*o*

y,

x -+atomicw.

x is an elementary formal system pseudo-

pseudoatomic x + pseudoformula x, pseudoformula y

+ pseudoformula

z

+x

=y

* 3 *z -+

pseudoformula x.

(6) formula x G x is a formula (no variables). atomic x + formula x, formula y +formula z or

+x = y

* 3 * z +formula x.

(7) proper x e x is a proper pseudoformula, in that none of I?,, . . . , I?, f occurs in the conclusion.

pseudoatomicx + x

=y

* z -+re1 y + y

#,I

I?,+. . .

- . . + y # , , , R k - + y #,,II+properx, pseudoformula x +x

=y

* z + pseudoatomic z +proper

z

+proper x. (8) Sx, y, z , w e x and z are words over A&?* (note), y is a variable, and w is the result of substituting z for all occurrences of y in x.

.dx +var y +Sx, y, z, x, re1 x --+ var y

3

Sx, y, z , x,

UNIVERSAL MACHINES

CH. 6 , §11]

punct x +var y

131

+ Sx, y, z , x,

var x + Sx, x, z, z , var x +var y SXl,

+

x Z,,, y

+ Sx,

y, 2, x,

y, z, WI -+ sx., y, 2 , wz

+ x = x I * x 2 + w = W I * W 2 + sx, y, 2, w. x is a word over d *and w is the (9) w is a partial instance of x result of replacing one or more variables in x by words over d. Sx, y. z , w +word z

-+

w is a partial instance of x,

w I is a partial instance of x +

+

Sw I, y, z , w 2+ word z

wzis a partial instance of x.

(10) w is an instance of pseudoformula x G x is a pseudoformula and w is the result of replacing all variables of x by words over d.

pseudoformula x + w is a partial instance of x +formula w (11) Bx

ex

+w

is an instance of pseudoformula x.

is a base.

properx+y= #*x*#+By, proper x + B y + z

=y*x*#+Bz.

e word x is part of y = u * x * v + p a r t x, y. Cx, y e x is a component of

(12) part x,y

word y

(13)

base y.

pseudoformula x

+ By -+ z =

#*x

* # + part

z, y + Cx, y.

(14) Fx. The axioms for F, unlike those above, accept input and, when given B C d " as input, Fx is derivable G x is the form # A l # A, # . . . # A, # X , where X is derivable in T(%'")in the elementary formal system with axioms A ,,A?,. . . ,A,, using ' 9 as input. (We assume 3, is n,-ary.) I x ~.,. . ,x, + y

-

=I*XI

-

* o * * .

. * o * x , -+ BZ -+ w

R l x l , . . .,x,,,-+Y = RI *XI* o * . . . * o *

=z

x",+ B z + w

* y +Fw, =z

*y+FW,

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B y + Cx, y

+w

is an instance of

pseudoformula x + r u =x

=y

* w -+

Fr,

* y + u = x * y * 3 * z + w = x * z -+Bx -+atomicy

+formula z

+ Fu + Fv + Fw.

(15) Finally, Q is represented by 0 using the following. Fw+w = a * y + B a + y +wordxl +wordxz+

=~ ~ X , * O * X ~ * O * ~ ~ ~ * O * X ,

- - - +wordx,

+ Oa, XI,.

. . ,x,.

This completes the proof if T ( - )is rec(-). Now we list the modifications necessary if T ( - )is w-rec(-). In (1) and (2) no changes. Replace (3) by (3’) pseudo atomic,^ e x is a pseudoatomic formula allowing occurrences of r . re1 z -+term x + y = z * x +pseudoatomic, y, re1 z

-+

y =z

* rr +pseudoatomic,

pseudoatomic, y

-+

term x + w = y

pseudoatomic, y + w

=y

x,

* * x +pseudoatomic, 0

* * T -+ pseudoatomic, 0

w,

y.

Make similar changes in (4) to get (4’) atomic,x. We leave this and other straightforward changes to the reader. Change (5) and (6) by replacing pseudoatomic by pseudoatomic,, and atomic by atomic,, to get axioms for: (5’) pseudoformula, x. (6’) formulaax. (7) Requires more attention. In an w-elementary formal system we don’t want T to occur in axiom conclusions. Now, if x is a pseudoformula,, and z is the conclusion of x, saying z is just plain pseudoatomic says, in particular, that r does not occur in it. Thus we have the following. (7‘) pseudoformula, x +x = y * z +pseudoatomic z -+ proper z 3 proper, x. (8) and (9) require no changes. In (lo), replace pseudoformula and formula by their w-versions to get axioms for: (10’) w is an instance of pseudo formula,^. In (11) replace proper by its w-version to get axioms for: (11’) BJ e x is a base for an w-elementary formal system.

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(12) and (13) require no changes. Before(14) we insert the following x is not a word over d (i.e. x is a member of d*” (13a) word x which includes some letters outside the copy of d).

-

var x +word x,

-

re1 x +word x, punct x +word x,

-

wordx + w

=u

-

* x * u +word

w.

Now, replace (14) by the following. In these axioms, Gx, y, z is intended to represent: x is b * r where b is a base and r is pseudoatomic,, y is a variable, and either (1) z is a word over d and we have a derivation of b * r’ where r’ is the result of replacing y by z in r, or (2) z is not a word over d. (14’) FJ (like Fx, but taking the w-rule into account). I x ,,..., x n + y

~*x~*o*...*o*x.+B,z+w =z*Y+E,w, R l ~ l ,. ..,x,, + y = R 1 * x I * o * * * * * * x,, +B,z + w = z * y ---* F,w, =

0

&XI,.

. .,Xn,+

-

y = Rk

*XI

*O

*

* ‘*

* * Xnk +B,z O

+w = z

* y +F,w,

B,y +Cx, y +w is an instance of pseudoformulad +r = y * w +FJ, ~ = x * y + u = x * y * ~ * ~ + ~ = ~ * ~ + B d +atomic,y +formula,z +F,u

-

-+

F,u

+F,w,

* r +Gx, y, z, =b *r

word z +var y -+ B,b

+pseudoatomic,r +x = b

word z +var y +B,b

3

+Sr,y,z,r‘+x’=

pseudoatomic,r +.x

b*r’-F,x’+Gx,y,z,

B,b +pseudoatomic,r

+x = b

* r +Gx, y, ‘d

+Sr, y, r ,r ’+x = b * r ’ +Fur ’.

And finally (15’) Du is represented by 0 using the following. F,w+w

= a*y+B,a+y

+word x1+word xz+ * .

-

=O*xI*o**~**~*x, +word xm -+ Oa, xl,

...,x,.

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[CH. 6,512

12. Operator indexing

We introduce some terminology, and restate the result of the previous section. Let ?I and 8 be structures, with T(B) an extension of T(V1). By &*,)(n, m ) we mean the collection of all operators of T(?I) of order ( n , m ) . Let q ( n r r i ) be an operator in T(B) of order (n, m + 1). We call %(n"') universal for $,,ul,(n,m ) if: @ E 9r(L,l)(n, m ) if and only if there is some b E W such that ( x l , .. . ,x , ) E @(P)a (b, X I , . . . ,x,,,) E %(P),for all 9" E [d]". We say 9T(\ll)(n, m ) has an indexing in T(B) if some operator in T(B) is universal for 9rr311(n, m). Now suppose, for each n and m, that %("'") is an operator in T(B) which is universal for Y+(",)(n, m).We call the family % = n, m 2 1) an indexing of T(VI) in T(B). is an indexing of T(?I) in T ( 9 ) .By @b""" we mean that Suppose operator (if one exists) in T(?I) of order ( n , m ) such that, for P E [d]",

{%(""')I

We call b an index of @P"".Note that indexes are relative to the particular indexing chosen. In our present terms, we may restate Theorem 11.1 (with Theorem 10.4) as follows:

THEOREM 12.1. Let ?I be a structure for which T(V1) is a theory with equality. Then T(\)I")has an indexing in T(?I*"),an inessential extension of T(?I") via a 1-1 embedding. By Proposition 3.1, T(\)I")is an extension of T(?Q so the operators of T(\)l)are among those of T(91"). Thus we immediately have the following.

COROLLARY 12.2. Under the same hypotheses, T ( % )has an indexing in T(?l*"), an extension, though not necessarily inessential, of T(91). Thus our recursion and o-recursion theories (with equality) have recursion and w-recursion theories that can "talk about" them. We have two main questions: (1) what can be "said", and (2) which theories can "talk about" themselves. We look at these questions in the next two sections.

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135

13. Input and output place-fixing theorems For this section, T(%) is a theory with equality. We have constructed an indexing of T(%“)in T (%* ” ).Informally, T(%*”)can “talk about” T ( 8 ” ) .Now we ask what sort of things can it “say”? In this section we establish two such things, from which, in turn, many others will follow in the course of the next two chapters. Thus these play a key role in our development. The two may be thought of as generalizing what is called the s-m-n theorem of ordinary recursion theory, which is due to Kleene. See Kleene [1952], p. 342. Kleene works with functions, which take n-tuples as input, but which give numbers as output. The s-m-n theorem has to do with manipulating input: changing n-tuples to k-tuples (k < n ) by fixing some of the places. Not much can be done with output: numbers are 1-tuples, and can get no shorter. Here, however, we work with operators, and they can use n-ary relations both as input and as output. Thus both input and output is subject to the kind of “place-fixing” manipulation of the s-m-n theorem. Consequently we have two theorems below, not just one. NOTE. We call a function fpartial recursive if its graph is r.e., though its domain may not be total.

THEOREM 13.1 (Output Place Fixing Theorem). Let @ be a n operator of T(%”)of order (n, q + m). There is a q-ary partial recursive function f, in rec(%*”), such that ( a ] , .. . , a q , b l , . ., b,)E @(P) ( b i , . . . ,b m ) E@ ~ i ~ , ) . . . a q ) ( P ) (for all n-ary P).

REMARKS.The intention in constructing %*” was that rec(%*”) could not only “discuss” rec(%”), but could do it in, essentially, the same way we do. The proof below illustrates this. PROOF.Let E = {A,, A 2 , .. . ,Aj} be axioms for an (o)-elementary formal system for @ in T(%”),in which I represents input and D output. For each a l , .. ., a, ESP”, let E,, . . . . be the (o)elementary formal system with axioms A 1 ,A 2 , .. .,Aj and Da,, . . . ,a¶,xl, . . .,x, + 0x1,. . .,x,. It should be clear that q ,

( a , ,. . .,aq, b l , . . ., hn) E @(S)

e (ai,... , a q , b i , . . . , b , ) E [ E ~ ] ( ~ ) e ., bm)E [(Eal aq)b](P). (b13..

,...,

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[CH.6,014

Now define a partial function f by: for a ] ,. . . ,a, E d'",

f(a,,. . .,a , ) = # A , # A * # +OX1,...,Xrn

* .

. # A , # Dal, . . . , aq,xl, -.. , X,

#.

That is, f ( a l , . . ., a,) is a base, whose components are the axioms of ,Fa,., ( l q . Thus [(E.,, .o,)h](9)= @ ),?: .aq)(B) by the construction of our universal machines in Section 11. We are done once we show f is partial recursive in rec('%*"). This is simple. Let v l , .. . , v,,,be members of var. Now as axioms for fi take those for word (item (1) in Section 11) and the following: word v1-+ . . .+word v, +

-

=D

*

01 *o*.

.* * o *

V,

* 0 * x I* o * . . . * O * X ,

* 3 *~*xI*o*-.'*o*x, + y = # A I # A z # . . . # A , # z + f ( v , , ..., v , ) = Y . This ends the proof.

THEOREM 13.2 (Input Place Fixing Theorem). Let @ be an operator of T(W")of order (q + n, m).There is a q-ary partial recursive function f in rec(a*") such that, for B (d")", @ ( { ( a h .. , a,)} x 9)= @ !.?:;

.,aa)(B).

PROOF.Let E = { A l ,A?,. . . , A , } be axioms for an ( w ) elementary formal system for @ in T('2I'")with J as input and 0 as output. For each a ,,.. ., a, E d ' " , let E a , , . ~ qbe the ( w ) elementary formal system with axioms:

Then

A l , A z , . . , A , and @({(Ul,.

I x l , . . . , x,

. . ,a,)} x 9)= [EJ,]({(Ul,. = [(EL?,.

Jai,.. . , a , , xI, . ..,x,.

+

. . , a , > )x 9)

.&1(9).

Now proceed as in the above proof. 14. Indexing revisited This section is the culmination of a long sequence of sections. We have seen that, if T(%)is a theory with equality, then T(%*'") can "talk about" T(%'").Here we prove some results that imply the nicer fact that T ( a ' " )

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can "talk about" itself. Better still, under certain mild conditions, even T('ZI)can "talk about" itself. Let 011 be an indexing of T ( % )in T(B), as in Section 12. Let 0 be an operator in T('2I) of order (n, q + m ) . By an q-ary output place-fixing function for @ we mean a function f : d q-+ 3 such that, for all 9 C d",

(a1,..., a,,x1, ..., x,,,)E @(9)e (f(a1,..., Uq),XI,..., x,,,)E Ou(""'j(P)

,a (XI,..., Xm)E @;y)u,,(s). We say 011 has the output place-fixing property if, for each operator @ of T ( q ) ,and for each q, there is some q-ary output place-fixing function f for @ which is generated in T ( % ) Moreover, . iff can always be taken to be r.e. in r e c ( 3 ) we say 011 has the strong output place-fixing property. Again, let @ be an operator in T(Y1) of order ( q + n, m ) . By a q-ary input place-fixing function for @ we mean a function f : d q+ 3 such that, for all P C d", ( X I ,..., x,)E

e

@ ( { ( a h . ..,a,)lx

9)

( f ( a 1 , .. . , a,), XI,. . ., x,n)E 011(n."')(9) (XI,.

. . , X",) €

@ j;:;

Llq)(P).

We say 011 has the input place-fixing property if, for each operator @ of T(??l), and for each q, there is some q-ary input place-fixing function f for 4 which is generated in T ( 3 ) .Moreover, i f f can always be taken to be r.e. in rec(B) we say 011 has the strong input place-fixing property. Combining work of Sections 11 and 13, we have:

PROPOSITION 14.1. Let T(Y1) be a theory with equality, and let 011 be the indexing of Section 11. OU is an indexing of T(Y1") in T(41*")which has both the strong input and the strong output place -fixing properties. Now we prove the main result of this section, which will allow us to strengthen the above proposition.

THEOREM 14.2. Suppose T ( % )has an indexing 011 in T ( % ) ,where T ( % )is an inessential extension of T(Y1) via a 1-1 embedding. Then ( 1 ) T(41) has an indexing 'If in itself. ( 2 ) If 011 has the input (output) place-fixing property, so does V. ( 3 ) If rec(B) is an inessential extension of rec(4I) via a 1-1 embedding, then if 011 has the strong input (output) place-fixing property, so does 'If.

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REMARK.If "Ir is an indexing of T ( % )in itself, and if f is an input or output place-fixing function for some operator, f is a total function. PROOF.Say T(B) is an inessential extension of T ( % )via h, where h is a 1-1 embedding. Thus h : T ( 8 ) + T ( % )is a 1-1 embedding, and the relation y E ( h l d ) ( x ) is generated in T(%).By Proposition 4.12.2, ( h rd)"is an operator of T(rU). We note that, €or 8 d", ( h Id),( 8 )= {(yl,. . .,y.) for some (x . . . ,x, ) E 9,y , E h (xI) A . . . A y. E h (x, )}. We will also want an operator like ( h rd);',but which leaves one position alone. We get this from the proof, rather than the statement of Proposition 4.12.2. Thus: define a relation 9 by

I

%(y,y1

e

I . . . ,

y

ymrX,XI,

..., X n t )

= x A yl E ( h ld ) ( x l ) A

*

*.

A

y, E ( h l d ) ( x ,

1.

Then 9 is generated in T ( % )so by Corollary 2.5.2 (l), the following, J,+l, is an operator, of order ( m + 1, m + I), in T (%):for 9 C d"'+l, J,+1(8) = 9?"(8). We note that Jm,+l(8)= {(x, xI, . . . , x,) for some ' ' ' A y m E ( h 1d ) ( x r n ) } . (X, YI,. . . , ym)E 9, YI E ( h &)(XI) Now, we turn to the proof of (1). Let &(,,,(n,m) be the collection of operators of T ( % )of order (n,m).021("."') is an indexing for 9T(:)l,(n, m ) in T(B). By the preceding, 'V'","') is an operator in T(%),where:

I

"Ir'n."') = Jm+l((?l(n.n'))h

(h r d)m.

We claim "Ir'".'")is an indexing for 9T(!11,(n,m ) in T ( % ) . Let @ be an operator in T ( % )of order (n,m).Then @ has indexes in T(B), using the 021 indexing. Let b be one of them, and let a E h ( b ) . We claim, for 9 d " , (XI,. .

. ,x,)

E @(9)e

(a, XI,. . . ,x,) E cv.(n.nr)(8),

and thus a is an index, using 'V("""),for @ in T(\21).Actually, what must be shown is

(b,xi,. . . ,x,) E .("."')(9) e ( a ,X I , . . . ,x,) E v(n.M)(9). This is straightforward, and is left to the reader. Next, (2). Suppose @ is an operator of T(2Q of order (n,q + m ) , f is a q-ary output place-fixing function for @ (relative to the 021 indexing) and f is generated in T('X3).As we did above, by use of Corollary 2.5.2 (l), we may produce an operator K in T('%),of order (4 + 1, q + l), such that, for 9cdqtl,K ( P ) = { ( x ,..., ~ x,,x)lfor some (yl ,..., Y ~ , X ) E ~ 'y, l ~ h ( x l ) A ' ' ' A yq E h(X,)}.

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139

Now, f is generated in T ( % ) ,hence f ' = K ( f h )is generated in T(91). That f ' is a function follows from the fact that h is 1-1. We leave it to the reader to check that f' is a q -ary output place-fixing function for Q, relative to the .Ir indexing. Thus 7'" has the output place-fixing property. The input place-fixing property is treated similarly. Finally, (3) has a proof similar to (2). We note that if rec(8) is an inessential extension of rec(9I) via h, it follows by Corollary 3.5.2 and by Proposition 1.9.1, that T(B) is an inessential extension of T ( % )via h. We leave the rest of the proof to the reader. Now for the corollaries we have been after all along. COROLLARY 14.3. (1) If T ( % )is a theory with equality, then T(9III")has a n indexing in itself having both the input and the output place-fixing properties. (2) If rec(%) is a theory with equality, T(91") has a n indexing in itself having both the strong input and the strong output place-fixing properties. PROOF.We show (1) only. T(91) is a theory with equality, so by Theorem 10.4, T(9I*") is an inessential extension of T(91'")via a 1-1 embedding. By Theorem 11.1, there is an indexing of T(Y1") in T('71''") which, by Theorems 13.1 and 13.2, has the output and the input place-fixing properties. The result follows from Theorem 14.2.

COROLLARY 14.4. Every recursion theory has an extension which is a recursion theory having an indexing with the strong input and strong output place -fixing properties. Similarly for w -recursion theories. PROOF.If rec(YI) is a theory with equality, we are done, by Corollary 14.3 (and Proposition 3.1). If not, let Y1' = (91, # &). Then, trivially, rec(91') is a theory with equality, which is an extension of rec(Y1). Now finish as before. COROLLARY 14.5. (1) If T(91) is a theory with equality and a n effective pairing function (where sd is infinite), then T ( Y ) has a n indexing in itself which has the input and the output place-fixing properties. (2) If rec(91) is a theory with equality and an efective pairing function (where sd is infinite), then T(91)has a n indexing in itself which has the strong input and stong output place -fixing properties. PROOF.We only prove (1). By Corollary 8.5, since T(?I) has an effective

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[CH.6,015

pairing function T(9P") is an inessential extension of T ( f l ) ,via a 1-1 embedding. Since T ( % )is a theory with equality, by Corollary 14.3, T(%'") has an indexing in itself, having the input and the output place-fixing properties. It follows easily that T(91) has an indexing in T(Y1'")having these properties, and we are done, by Theorem 14.2.

14.6. (1) If T(%) is a theory with equality, both T ( % ' )and COROLLARY T(") have indexings in themselves, having the input and the output place -fixing properties. (2) If rec(%) is a theory with equality, both T('2I') and T ( % " )have indexings in themselves having the strong input and the strong output place -fixing properties. PROOF.We only prove (1). By Theorem 6.1, T(B1') has an effective pairing function, and trivially, so does T(9P).Since T ( % )is a theory with equality, so is T ( % ' ) by , Lemma 5.1. It is not hard to show T(%')is also a theory with equality. The result follows by Corollary 14.5. EXAMPLES. (1) Let l?l be one of our "standard" structures, G(N), S(L,,) or G ( a , ,. . . ,a n ) .Then rec(%) is a theory with equality (shown in Ch. 1, §4), and has an effective pairing function (shown in Ch. 4, $10). Hence both the recursion and the o-recursion theories of these structures have indexings having the strong input and the strong output place-fixing properties. (2) Let G(R) be the structure (R; +, x , > ) where R is the set of real numbers. In w-rec(G(R)) there is an effective pairing function (Chapter 4, $10) and it is a theory with equality. Hence w-rec(G(R)) has an indexing having the input and the output place-fixing properties. 15. Relational indexing Suppose T(91) has an indexing in itself. This is an indexing of operators, but it induces various indexings of relations that are of interest. Here we say how, then in the next chapter we investigate the structure that results, before returning to operators in Chapter 8. Let r be a collection of sets and relations on d. r is said to be parametrized if for every n 3 1 there is an n + 1 place relation U" E r such that

9' is an n-ary relation in

a

r

for some i E&, B = (x I ( i , x ) ~u").

CH. 6, $151

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141

I

If r is thus parametrized, we write 3; for {x U"(i,x)} and we call i an index of 3;. Suppose, in addition to being parametrized, r meets the following conditions: For each n, there are functions f :d"+ s4 called basic such that (1) the collection of basic functions is closed under composition, (2) (the graph of) each basic function is a member of r, (3) (iteration property) for each n + k-place relation, 22 E r there is a basic function f : d n+ d such that g(Xl,.

..

3

xn,

y l , . . . yk) 7

e

3 F ( x j ,

.xn)(yl?. . . yk). 9

If this happens, we will say (the indexing of) r has the iteration property.

EXAMPLES. (1) Let T ( % )have an indexing % in itself, which has the output place-fixing property. Let r be the collection of relations generated in T(%) [the r.e. or w-r.e. relations over 9.l depending on whether T ( - ) is rec(-) or w-r.e.(-)]. Then is parametrized and has the iteration property. We may show this as follows. For each n 1, define an n + 1-place relation U" by:

r

U" = Q ( I . n ) ( 0 ) * U" is generated in T(%),hence is in r (Proposition 2.3.1). Further, if 24 is any n-ary relation on d which is generated in T ( % ) ,then for some operator @ E T(9.l)of order (1, n ) , 3 = @(0) (Proposition 2.3.1 again). Now @ has an operator index, say i. That is, x E @(9) e (i, x ) E

%y9).

In particular, xE3 @ xE

@(0) e ( i , x ) %('.")(0) ~

( i , r ) E U".

Thus r is parametrized. Note that the operator indexing and the relation indexing are connected by $28; = @!'."'(0).

Now, take the basic functions simply to be those functions generated in T ( % ) .It is easy to see that the iteration property for is an immediate consequence of the output place-fixing property. (2) Let T ( - )be w-rec, and suppose rec(9.l) is a theory with equality and an effective pairing function. Then, based on Corollary 14.5, we may construct a parametrization for the set of generated relations of w-rec(%),

r

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[CH.6,516

having the iteration property, but in which the basic functions are those which are r.e., rather than o-r.e. The notion of basic function is thus not always as trivial as it was in Example 1. (3) It is a standard result of ordinary recursion theory, rec(G(N)), that the generated (r.e.) relations may be parametrized so that the iteration property holds, where the basic functions are the primitive recursive ones. This is a rather narrow class of functions that has no exact analog for arbitrary structures. Nonetheless, analogs have essentially been constructed for set structures, Jensen and Karp [1971], and for word structures, Asser [1960]. (See also Machtey and Young [1978].) We do not continue this topic here, though. (4) Let 9 be an arbitrary relation on d, and let r be the collection of relations s 9 (in T(i?I)).Again, r is parametrized and has the iteration property. We may show this exactly as in Example 1, but writing 9 wherever 0 was written. Finally, we present a non-example. For all of this chapter, T(-) was either rec(-) or w-rec(-) while in earlier chapters the theory assignment f.0. was generally given equal billing. The reason for omitting it in the present chapter is that no first order theory can have an operator indexing, since we have the following simple diagonal argument. If f.o.(%) had an operator indexing then, as in Example 1 above, the generated relations of f.o.(i?I) would have a parametrization. So there would be a generated (first order definable) relation U' so that

~ € 3e :(i,x)EU'. Consider the set

I

9 = {x U'(x,x)l.

9 would also be first order definable, hence so would 4 (since in f.o.(%) negation is available). Let a be an index for 9, so 4 = 3;. Then a

~

e9 U

e u'(u,u) e

E ~ ;

aE9,

which is a contradiction.

16. Finite codes In recursion theories, every enumeration operator is compacf (Proposition 1.103); loosely speaking this means that, though input may be infinite, it is used in finite chunks. (The same need not be true of w-enumeration

CH. 6, §16]

FINITECODES

143

operators, a fact we will elaborate on in Chapter 8.) Consequently, the finite sets can be expected to play a major role in recursion theories. The problem is, as things stand, we have no mechansism for "telling" an enumeration operator that the finite input we may have supplied thus far is all it is ever going to get. It can't tell the difference between finite, and infinite but incomplete yet. Neither, for that matter, can we where outputs are concerned. But there is the feeling that one ought to be able to deal with finite sets as single, complete objects. For this purpose, finite codes are introduced. The consequences of their existence are developed in Chapter 8; here we consider only the circumstances. , Let 8 be a structure, with d C 93. Suppose, to each finite subset F C I one or more members of 93 have been assigned, so that distinct finite subsets of d never have the same member of 93 assigned to them. Then we say d has a finite coding in 93. We call the members of 93 assigned to F codes for F. We write D, for the finite subset of d with code c (if c is not a code, D, is not defined). Now suppose d has a finite coding in 93. We say d has a positive canonical coding in rec(8) if (1) There is an enumeration operator @ in rec(8) of order (1,l) such that, for each 9 C d, @(9) = {c D, C 9}.

I

(2) The following relation is r.e. in rec(%):

M(x,y )

e

y is a finite code and x E D,.

We say, simply, d has a canonical coding in rec(B) if, in addition to the above we also have (3) The following relation is r.e. in rec(B): N(x,y )

e

y is a finite code and X GD,.

Let ?I* be defined from % as in Section 11. Our first goal in this section is to show d " has a canonical coding in rec(%*"), under the right circumstances. Afterwards, we will improve on the result. Select a member of punct in d *(see Section 11)that we have not used so far, let us denote it by 0.Now, if { a , ,a 2 , .. . , a , } is a finite subset of I " we code it by the word Oa,OazO 0a.O. Note that n element subsets of d " thus have n ! finite codes in d*".

LEMMA16.1. The finite coding defined above is a positive canonical coding of d" in rec(%*"). If also rec(%) is a theory with equality, then it is a canonical coding of d" in rec(%*").

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PROOF.Part 1. Let @ be the enumeration operator of order (1,l) in rec(%*") defined by the following axioms (I represents input, 0 represents output): (item (1) in Section ll),

axioms for word 000, Zx +wordx

+Ow

+ y = w * O * x * O + Oy.

It should be clear that, for P C d , @ ( 9 ) = ( c ID, C P } . Part 2 . Let M ( x , y ) e y is a finite code and x ED,. M is r.e. in rec(%*"), having the following axioms: axioms for word, y = O * x *O-+wordx-+M(x,y),

M(x, u ) + y

=u

* w * 0 +word w 3 M ( x , y),

M ( u , u ) + y = u * x *O-+wordx-,M(x,y).

(3) Now suppose rec(3) is a theory with equality. rec(G(N)) is certainly a theory with equality, so by Proposition 10.3, rec(%*) is a theory with equality, and then by Lemma 5.1, rec(%*") is a theory with equality. Let N(x, y) a y is a finite code and x e D,. N is r.e. in rec(%*"), having the following axioms. axioms for word, axioms for # , N(x, OO), N ( x , u ) + y = u * w *O+word w - + x # w +N(x,y), Now we set to work on bettering this result.

LEMMA16.2. Suppose rec(B) is an inessential extension of rec(%). Then (1) If d has a positive canonical coding in rec(B), d has a positive canonical coding in rec(%). (2) If d has a canonical coding in rec('P), d has a canonical coding in rec('21). PROOF.Say rec(%) is an inessential extension of rec(%) via h, so that h : rec(B)+ rec('U) is an embedding, and the relation y E ( h Id ) ( x ) is r.e. in rec(%). Suppose d has a (positive) canonical coding in rec(B). Using it we define

FINITECODES

CH. 6 , 9161

145

a finite coding of d in d as follows. Let F be a finite subset of d,let b be any finite code for F in 9, let c E h ( b ) ;we call c a finite code for F in d. We now show this coding is (positive) canonical. (1) Suppose the original finite coding of d in 9 is positive canonical; we show the finite coding of d in d defined above is also positive canonical. Let Qfl be the enumeration operator in rec(23) that takes P C d to the collection of 23 codes for finite subsets of 9. By Proposition 4.12.2, ( h Id), is an enumeration operator in rec(8), since the relation y E ( h Id ) ( x ) is r.e. Then the following is also an enumeration operator in rec(Y1): @ = @k(h I &)I.

Now, if P C d , @(P) = hl@,hi’(h Id)l(S)= hl@,(P)

and this is simply the collection of d codes for finite subsets of P. Thus we have a suitable finite-subset-operator. Let Mfl(x, y ) e y is a finite code in 9 for a finite subset of d with x as a member. We are given that Me is r.e. in rec(23). Let MA (x, y ) e y is a finite code in d for a finite subset of d with x as a member. We must show that MA is r.e. in rec(2l). We note that M A

( 4Y )

e

( 3 Y ’ ) [ Y E h(Y’)A M f l k Y ’ l l .

Now the following is r.e. in rec(Y1):

(32)[(Z,Y)EMh,A E ( h t d ) ( x ) ] . This is equivalent to

Now, if z E h ( z ’ )and z E ( h Id ) ( x ) , it follows that x = z’, so the above is equivalent to ( 3 2 ) ( 3 Y ’ ) [ Z E ( h I&)(XI

A

Y E h ( Y ’ )A M ? ( x ,Y’)1

and this in turn is equivalent to ( 3 Y ’ ) [ Y Eh(Y’)AMJ3(X?Y’)I Thus M A is r.e. in rec(3). This completes (1). (2) The N-relation is treated exactly the way the M-relation was treated above. We omit the details.

146


[CH.

6,616

DEFINITION. If d has a canonical coding in rec(8) we simply say rec(%) has a canonical coding. THEOREM16.3. If rec('21) is a theory with equality, rec(%") has a canonical coding. PROOF.The above lemmas and Theorem 10.4. THEOREM16.4. If rec(fl) is an infinite theory with equality and an effective pairing function, then rec('2l) has a canonical coding. PROOF.By the previous theorem, rec(8") has a canonical coding. It follows easily that d has a canonical coding in rec(?P'), and the result follows by Lemma 16.2 and Corollary 8.5. EXAMPLES. By Theorem 16.4, our "standard" recursion theories, rec(G(N)), rec(G(L,)) and rec(G(a,, . . . , a , ) ) all have canonical codings. In fact, for rec(G(L,)) a simple, direct canonical coding can be defined: code each finite subset of L, by itself. For rec(G(N)) there is a well-known canonical coding defined in Rogers [1967], p. 70 (where it is called a canonical indexing). This one is the archetype for all canonical codings. Notice that these are 1-1, onto codings. In general this is too much to hope for, and more than we need.

CHAPTER SEVEN

INDEXED RELATIONAL SYSTEMS

1. Introduction

This is the first of two chapters in which we investigate the consequences of the existence of indexings, as established in the previous chapter. Here we consider just relations; in the next chapter, operators. We proceed axiomatically; Chapter 6 supplies us with many models for our axioms, but it should be kept in mind that other, rather different models will occur later on, most immediately in Section 6. We have chosen the axiom system below because it arises naturally out of our elementary formal system approach, but there are other treatments possible. The most elegant axiomatic development of essentially the material of this chapter is that of Wagner [1969] and Strong [1968]; see also Friedman [1971].

DEFINITIONS. Let d be a non-empty set. By an indexed relational system on d ;we mean a triple %& = (3,U, S)where: (1) Ce is a collection of relations on d called generated (in %), which contains 0, singletons, =&,and is closed under transpositions, projections, n , U , where defined, and X . (2) U is a function that assigns, to each integer n 2 0, an n 1-place universal relation, U",on d such that (a) U" E Ce. (b) For each n-place R E Ce there is an i E d , called an index of R, such that

+

R(x)

e

U"(i,x).

I

[we write R ; for {x E d" u"(i, x)}.] (3) The members of 9 are functions from d kto d (for k called basic, such that

= 1,2,.

. .),

(a) S is closed under composition. (b) 9C Ce (that is, the graph of each basic function is a generated relation). 147

148


[CH. 7. 82

(c) (Iteration property). For each n + k-place generated relation Q there is a basic function f : d " + d such that

XI,. . .

7

x n , )'I,.

..

9

G R,k,x,. . x n ) ( Y l ? .

yk)

..

9

Yk).

EXAMPLES (See Ch. 6, §15). (1) If rec(\U) is a theory with equality, and an effective pairing function, then the r.e. relations give us an indexed relational system, taking the basic functions to be the total functions with r.e. graphs (recursive functions). (2) Again, if rec(3) is a theory with equality and an effective pairing function, and if B is a relation on d, then the relations C B give an indexed relational system, still taking the basic functions to be the recursive functions. This example includes the previous one as the special case B =O. (3) Similarly, the relations < B in w-rec(%) give an indexed relational system, where we can still take the basic functions to be the recursive ones, provided rec('U) is a theory with equality and an effective pairing function. (4) In ordinary recursion theory, rec(G(N)), the r.e. relations give us an indexed relational system, taking the basic functions to be the primitive recursive ones. Similarly for hyperarithmetic theory, w-rec(G(N)). For the rest of this chapter, CBd system on d .

= (3,U, 9) is some fixed

indexed relational

2. Immediate consequences

THEOREM 2.1. There is a set B complement, B, is not.

.d such that 9 is generated in %* but its

PROOF.Let U ' be universal for the I-place relations, i.e. sets. Let 9 = { x U ' ( x , x ) } . B is generated in %,* since 9 ' = U ' n (=.&). Now suppose 9were also generated. Then it would have an index, say i. Thus, for all x E d ,

1

B(x) a But then, setting x

B(i)

U'(i,x). =

i,

G U ' ( i , i ) G B(i)

which is a contradiction. Hence

is not generated.

REMARK. This diagonal argument compares nicely with the one at the end

CH.7, $21

IMMEDIATECONSEQUENCES

14Y

of §15 in Ch. 6. In both cases we essentially proved one can’t have an indexing and closure under complementation. In Ch. 6, §I5 we had closure under complementation, hence there could be no indexing. Above we are given an indexing, hence there is no closure under complementation. The argument is due originally to Cantor, in essentially the following form. Suppose there were a 1-1 correspondence 0 between a set S and its power set, B(S). “Index” each member 92 of B ( S ) by that member of S which corresponds to it under 8. This gives us an indexing of B(S) in S. But B(S) is closed under complementation, so a contradiction results. Hence 8 does not exist. We make two further observations concerning this proof. First, we made no use of the basic functions or the iteration property; the existence of an indexing is sufficient. Second, by comparing this result with Proposition 2.6.2, it follows immediately that an indexed relational system can not be finite. The following theorem shows that, if we had a second system, V, of universal relations, they could be “mimicked” by the U-system.

THEOREM 2.2. U is the given system of universal relations of %&. W e write R : for the n-place generated relation having U-index i. Let V be another indexing system for $%!,*,satisfying condition (2) of the definition of indexed relational system. W e write S : for the n -place generated relation having V-index i. For each n, there is a basic function f such that S: = R;(x). PROOF. V“ is generated in %,d. Also the iteration property holds for the U-indexing, so there is a basic function f such that V “ ( Xy i, , . . . , y n )

RY(X)(yi,.. . , Y”),

and this, rewritten, says

s:(yl, . . . , yn )

R;w(Yl2...3Yn).

DEFINITION. Keeping the notation of the above theorem we say two indexings U and V are equivalent if, for each n, there are basic functions f and g such that R:= S;,,, and S : = R ; ( x , .

COROLLARY 2.3. A n y two indexings which have the iteration property (satisfy condition (3)) are equivalent. From now on, we use only the “given” indexing, U. The collection of generated relations has certain closure properties,

f l

,

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[CH. 7, $3

U , etc. In fact, we have an effective closure under these operations, in the sense that, for example, given indexes for two relations, we can calculate an index for their intersection. We show this in the following theorem; there are similar effective closure results for U , etc., which we leave to the reader.

THEOREM 2.4. For each n there is a basic function f such that

+ 2-place relation

PROOF.Define an n Q ( u , u, x)

e

U"(u,x)

A

Q by

U"(u,x).

Q is generated, so by the iteration property there is a basic function f such that Now,

Q(w 0, X ) xER:nR;:

Rh.")(x)-

e e e e

R:(x)AR;:(x)

U " ( a ,x)

A

U "( b ,x)

Q(a,b,x) R;(o.bdx).

REIVLARKS. In the above proof we asserted Q was generated. This will be our practice, omitting the details of verification. Just this once, however, we show Q is, in fact, generated, as an example. =,* is generated, and we have closure under projection, hence d is generated. U" is generated, and we have closure under X , hence d X U" is generated. We have closure under transposition, hence 9 = {(x, y, x ) (y, x, z ) E d x U " }is generated. We have closure under intersection, hence 9 n (dX U " )is generated and this, in fact, is Q.

I

3. Fixed point theorems

a,,

We prove, for two so called fixed point theorems. The first, in ordinary recursion theory, is essentially due to Kleene and is an immediate consequence of his Second Recursion Theorem, see Kleene [1938]. The

FIXEDPOINTTHEOREMS

CH. 7, 931

151

second, again in ordinary recursion theory, is due to Myhill, see Myhill [1955, 1955Al. In fact, standard proofs from ordinary recursion theory carry over to 9dwith no difficulty. We begin by following the development in Smullyan [1961], in which Myhill's theorem is proved first, and Kleene's theorem is derived from that. We follow this with a short, direct proof of the Kleene result. We do not present the Kleene Second Recursion Theorem itself here, because we are emphasizing relations, while that concerns itself directly with functions. We note, however, that a functional indexing could be introduced, and then the Second Recursion Theorem is an immediate consequence of Lemma 3.1 below. There is a proof of this in Smullyan [1961], p. 92, for ordinary recursion theory, which works quite well in %d too.

-

LEMMA3.1 (Smullyan). Let M ( z , x, y) be a generated n Then there is a basic function f such that R;,Ydx)

+ 2-place relation.

M(Y7 x,f(Y)).

PROOF.Consider the generated relation P defined by P ( 2 , x ) G R:"(x,z) G U""(z,x,z).

Using the iteration property, there is a basic function d such that

R:&)

e

P(i, x).

R:(,](x)

a

Rf+'(x, i )

That is, Next, consider the generated relation

(gW)[d(y)=w

AM(z,X,w)]

M(z,x,d(Y)).

Again, by the iteration property, there is a basic function cp such that RZ(x, Y)

M ( i , x, d(y 1).

(**)

Finally, set f = dcp. Since the basic functions are closed under composition, f is basic, and

RYdx)

e

Rk(Y0)

cp(Y))

R::;l@? @

M(Y7

e

M(Y7

x 7

&(Y))

f (Y 1).

(*)I (by (**)I

(by

152


[CH. 7,83

COROLLARY 3.2 (Myhill fixed point theorem). For each n there is a basic f such that

R;+'(x,f(Y)).

R;&)

PROOF.Set M = U"" in the lemma.

COROLLARY 3.3. For any n + 1-place generated relation S there is an i such that R:(x) a S(x,i). PROOF.Let a be some index for S, use the previous corollary, and set i =f(a). COROLLARY 3.4 (Kleene fixed point theorem). For any function t : d + d, whose graph is generated in 92, there is an i such that

RXx)

RY(,dX).

PROOF.Define S by

s(4 Y )

'3

(gz)[t(y= ) Z

Ry(y)(x)

A

U"(Z,X ) ] .

Then S is generated, so by the previous corollary, for some i,

R:(x)

a

S(x, i ) G Ry(,](x).

REMARKS. The above proof of the Kleene fixed point theorem is rather roundabout, proceeding, as it does, via the Myhill fixed point theorem. So we include a more direct proof of this important result. It will be noted that the basic ideas of the proof are the same.

SECOND PROOFOF COROLLARY 3.4. Consider the n P(u,x)

e

U"+'(U,X, u)

e

+ 1-place relation

RC"(x, u ) .

P is generated, so by the iteration property, there is a basic function g such that

Ri(")(X)G P ( u , x )

e

R:"(x, u ) .

(*)

g is generated; so is t, hence so is tg. Let u be an index for the generated

relation

V " ( t g ( y ) , x )e R : m ( x ) .

CH. 7, 141

RICE'STHEOREM

153

(**I

We conclude with two simple, and not very important, illustrations of the uses of these theorems. There are others, of greater significance, in the next section.

EXAMPLE 1. There is an i such that Ri'= {i}. (The method of proof below clearly can be modified to produce whole families of related examples.) Consider the equality relation on d,which for the present we will denote by E. Thus E(x,y)

e

x

By definition, E is generated in %, so by the iteration property, there is a basic function t such that R :(x)(y)# G E(x, y). That is, R

= {x}.

Now use the Kleene fixed point theorem.

EWLE 2. In Theorem 2.2 we saw that if there were a second system, V, of universal relations, they could be "mimicked" by those of the U system. Here we show the curious fact that, under the same hypotheses and notation of Theorem 2.2, there must be some n-ary generated relation to which U and V assign the same index. Indeed, very simply, by Theorem 2.2, there must exist a basic function f, such that S : = I?;(=). Now use the Kleene fixed point theorem. 4. Rice's Theorem

In ordinary recursion theory there is a result, due to Rice, see Rice [1953] which embodies many separate undecidability results. Rice's Theorem carries over directly to $Bd as does its proof (though the one we

INDEXEDRELATIONAL SYSTEMS

154

[CH. 7, 84

present is not Rice’s original). After proving it, we illustrate its use, then prove a related result which we will use often, later on. DEFINITION. Let C be a set of indexes for n-ary generated relations of S. C is called closed if i E C and R ; = R ;

+j€C

C is called bi-generated if both C and

are generated.

THEOREM 4.1 (Rice’s Theorem). Let C be a set of indexes for n-ary generated relations on d.If C is closed and 6i-generated, then either C is empty or C contains every index.

PROOF.Let a. and bo be indexes of n-ary generated relations, and suppose (1) C is closed,

c, (3) boE c, (2)

a0

E

(4) C is bi-generated. We derive a contradiction.

Define a function t : s4. + d by (see Fig. 1) r(x) =

bo if x E C,

[

a.

The graph of

t

if x$Z C.

is generated in

$Bd,since it is

( C X (60)) U (C X {GI). Now, by Corollary 3.4, for some i, R l ( x ) R:,i,(x). But this says i and t(i) index the same n-ary relation. Since C is complete, i and t(i) are both in C, or both in but the definition of t makes this impossible.

c,

EXAMPLES.(1) Let C be the set of all indexes for infinite generated subsets of d. C is complete, so C is not bi-generated.

CH. 7, 841

RICE'STHEOREM

155

(2) Let C be the set of all indexes for generated subsets of d containing c (c fixed). C is complete, so C is not bi-generated. (3) Let R be a fixed, generated relation on d, and let C be the set of all indexes for R. C is complete, so C is not bi-generated. (4) An example rather intimately connected with generalized recursion theory. It is trivial that, considered as production systems, rec((N; )) = rec(G(N)), and it is equally clear that rec((N;0))#rec(G(N)). But in general, can one tell, for an arbitrary relation 99 on the natural numbers, whether or not rec((N; 3)) is the same as ordinary recursion theory? Of course, there is no chance of rec((N; being ordinary recursion theory unless 3 is r.e. in the ordinary sense. So we may narrow the question to: in ordinary recursion theory, for fixed n, is the set of indexes i such that

+

a))

rec((N; R y))

= rec(G(N))

a recursive set?

Well, the set is complete, by definition, and neither empty nor all indexes, by the examples above. Hence by Rice's Theorem, it is not recursive. REMARK. The above proof used Corollary 3.4, which itself was proved using Corollary 3.3. It is possible to base a proof of Rice's Theorem directly on Corollary 3.3. We sketch this, because it is similar to, though slightly simpler than the proof of the next theorem.

SECOND PROOF OF THEOREM 4.1. Make assumptions (1)-(4) as in the above proof. Then define an n + 1-place relation S by S(x, y ) G [R",(x)A Y f

c]v [Ri,(x)

A

y E C].

S is generated, so by Corollary 3.3, there is an i such that

Rl(x)

S(x, i ) .

Now, if i E C,then S(x, i ) e R io(x), so R '(x) e R io(x). Thus i and bo index the same relation, but i E C while bogC. There is a similar contradiction if i E

c.

THEOREM 4.2. Let C be a closed set of indexes for n-ary generated relations on d.If C is generated, then a. E C and R :o C R & bo E C.

+

PROOF.Suppose (1) C is closed, (2) C is generated, (3) RZ, C Ria, (4) a. E C, ( 5 ) bog C.


156

[CH. 7, $5

We derive a contradiction.

C

Fig. 2.

Define an n

+ 1-place relation

S by (see Fig. 2):

"

[R2,,(x)l [ R W )A Y

S(4Y )

E

CI.

S is generated, so by Corollary 3.3, there is an i such that

R:(x)

e

S(x, i ) ,

Case 1 : ist' C. Then by definition of S,

R:(x)

e

S(x, i )

e

R2,,(x),

so i and an index the same relation. But a,,E C while i$Z C, which is impossible. Case 2: i E C. Then, again, using the definition of S together with R :, C R :,,

R:(x)

S(x, i )

e

RiJx),

so i and bo index the same relation, which again is impossible.

5. Creative sets

Myhill [ 1955AI showed, in ordinary recursion theory, that any two creative sets (definition below) are recursively isomorphic. This is too much to manage in Pitd without assuming more structure, but a portion of Myhill's work does go through in the present generality. In this section we show that, in CRd, any two creative sets are many-one reducible to each other.

DEFINITION. C C d is creative if C is generated and there is a function cp : d -+ d, whose graph is generated, such that if R ! n C = 0 then cp(i)E R : U C. REMARKS. In the above definition we assumed cp was a total function. In

CH. 7. $51

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157

ordinary recursion theory the definition is generally given using a partial function, as follows. C C d is creative if C is generated and there is a partial function cp whose graph is generated, such that if R ! n C = 0, then i is in the domain of cp, and cp(i)E R : U C. In ordinary recursion theory these two definitions are equivalent. More generally, they are equivalent in every indexed relational system in which every two-place generated relation can be “cut down” to a generated function with the same domain. See Rogers [1967], p. 71. This is probably not the case for all indexed relational systems, however, and the exact status of the two definitions of creative is not known in full generality. We assume, throughout this section, that cp is total. We showed in Section 2 that there was a set which was generated but whose complement was not. Creative sets have this feature in an “effective” way. Say C is creative. Then C is generated, by definition. Suppose we try to see if C is generated, say we suspect = R ! . Then R ! n C = 0, so c p ( i ) @ R : U C ; that is, cp(i)E but c p ( i ) 6 f R : , so we have “calculated” differ. an instance, c p ( i ) , where R : and

c c

c

5.1. There is a creative set. PROPOSITION PROOF.The same set which was used in proving Theorem 2.1 works here. Let 8 = {x U ‘ ( x x)} , = {x R : ( x ) } .8 is generated. Let cp be the identity function, cp(x) = x. The graph of cp is generated since it is simply =.d. Now suppose R! fl 8 = 0.Then in particular, i E R : n 9. If i 6 f R I, by definition of 8, ifZ 8, and conversely. Hence c p ( i ) = i E R : U 8.

I

1

DEFINITION. Let 8,9? C d. 9 is many-one reducible to 9,written Ps,,,~?, if there is a function f : d + d,with generated graph, such that x E 9 f ( x ) E 3. THEOREM 5.2. Suppose A creative.

S,,,B, A

is creative, and B is generated. Then B is

PROOF.A s,,,B. Let f : d + d be a generated function such that x E A G f(x)EB. Also, define Q by Q(i,x)

( ~ Y ) [ R ! ( Y )=f(x)1 AY G R!Cf(xN.

(*I


158

[CH. 7, 85

Q is generated, so by the iteration property there is a basic function t such that R : ( , ] ( x e ) Q ( i , x ) , hence

(**I

xER:,,, G f ( x ) E R : .

Also A is creative so there is some generated function g : d +d such that R : n A =0 3 g ( i ) E R ! U A . We need a function to play a similar role for B. We claim it is cp = fgt. First, cp is a function from d to d with generated graph, since this is true of f , g and t. Now, suppose R f n B = 0. Then by (*) and (**), R:,,, f l A = 0. Then gt(i)E R i(z)U A, and by (*) and (**) again, fgt(i)E R:U B. This concludes the proof. LEMMA5.3. Let A be a generated set, and let g : d -+ d be a function with generated graph. There is a basic function f such that, for all i,

i EA

+ Rj(i)={gf(i)},

i6ZA

+ Rf,i,=O.

PROOF.Let M ( z ,x, y ) e z E A A x = g ( y ) . Then M is generated, so by Lemma 3.1 there is a basic function f such that

R ) ( i , ( x )e M(i, x, f ( i ) )

e

i EA

A

x = gf(i).

THEOREM 5.4. Suppose B is creative and A is generated. Then A S m B .

PROOF.B is creative, so there is a generated function g : d + d such that, if R f n B = 0 then g ( i )$2 R t U B. It follows trivially that if R f = 0 then g ( i ) 6 B, if R = { a }and a 6Z B

(*)

then g ( i )# a.

(**I

Now A is generated, so by Lemma 5.3,there is a basic function f such that

iEA

+ R)(,)= {gf(i)},

i6Z A

+ R)(i , =0.

We claim x E A @ g f ( x )E B,which says that A S m B . Suppose i E A. Then Rj(i,= {gf(i)}.It follows by (**) that g f ( i )E B. Suppose i6Z A. Then R j ( i )= 0. It follows by (*) that gf(i)$2B. This concludes the proof. THEOREM5.5. Let B be generated. B is creative if and only if every generated set is many-one reducible to B.

C H . 7, $61

KLEENE-MOSTOWSKI SYSTEMS

ISY

PROOF.Let B be generated. If B is creative and A is generated, A s m B by Theorem 5.4. Next, suppose every generated set is many-one reducible to B.By Proposition 5.1, there is a creative set, call it A. A is generated, so A S m B .Then B is creative by Theorem 5.2.

COROLLARY 5.6. Let A and B be creative. Then A S v z Band B s,,,A. PROOF.B is creative, A is generated, so A s m B . Similarly B j. (The cases i = j and i < j are similar). By the induction hypothesis

8' u (d'-J x 9') E ZI" (9?.d), and it is not hard to see that

9 u9 = M ( P ' U ( d '

J

X.2'))

and hence is in Zn+l(9?d). Thus Z.+l(%!d) is closed under U . We show 2 n + l ( % isd closed ) under n . Say P,22 E 2n+,(9?d), where both are k-place. Then 8 = M ( 8 ' ) and 22 = N ( 9 ' )where P',9' E L ', ($BE,) and M and N are strings of projection operators. Again, since ZI,, ($BE,) is closed under transpositions, we can choose P' and 9'so that: in M we are projecting off the front, so its terms are of the form P?; in N we are projecting off the back, so its terms are of the form P:. Now, say 8' is i-place and 22' is j-place. By the induction hypothesis,

(8'x d J - k n )(d' x 2 ' )E n,ptd).

CH. 7, 961


161

Let N' be like N but with each PZ: replaced by P:::::. Then

MN'((B'x d J - k n)(ai-'x 9')) is in and it is not hard to see that this is 9 n 9. Thus is closed under n . That Ztl(9d) is closed under transpositions follows easily from the corresponding closure of I7"(!Bd); we omit details. Finally, we show the various closure properties of Z7n+l(9d). Suppose B,9 E IL,+l(9d) where B is i-place and 9 is j-place. Then 9, 3 E Z n + l ( 9 d )By . what was just shown,

(9x d')u (d'x 3)E Z"+l(%d). This is B X 9,so the complement, 8 X 9,is in ZIn+l(%d). Thus IZn+l(9d) is closed under x . Closure of LIn+l(9d) under n and U follows similarly using DeMorgan's laws. And closure under transpositions is simple since complementation and transposition commute. This ends the proof. PROPOSITION 6.2. (1) n,,(9&) C Z(n+l(9d), (2) 2"(%) fl"t1(9d), ( 3 ) Z"( 9 d ) Z " + l ( 9 d ) , (4) n n ( 9 d ) nn+I(%d).

c c c

PROOF.(1) Let 8 E IIn (!Bd). By the previous Lemma, d X B E n. (9$), so 9,which is a projection of d x 8, is in Z n + l ( 9 d ) . (2) Let B E Z n ( 9 d ) . Then 9E By (l), 9E Z + i ( 9 dso) 8 E n"tI(9d). (3) By induction on n. If n = O,Zn(2dd)= n. (9L) and the result follows by (1). Then Suppose inclusion (3) is true at n. And say $3' E Zn+l(%d). B = M ( 8 ' ) where B'E nn(Bd), and M is a string of projections. Then 9'' E Zn('iBd),so by the induction hypothesis, 8 ' E 2 n + l ( 9Then d ) . 8' E Iln+l(9d) so B = M ( 8 ' )E Zn+2(9d). This completes the induction. (4) Suppose B E ('&). Then 9E so by (3), 9E Z n + l ( 9 d ) , hence B E Un+,(Bd).

n,

Before proceeding further, we must impose some extra conditions on Bd.We use ordinary recursion theory as a guide, though there are other indexed relational systems which meet our conditions as well. We return briefly to this point in the next section.

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[CH. 7, $6

In Ch. 4, §lo we gave an example of an effective pairing function for ordinary recursion theory. It has the nice feature that it is onto, i.e. every member codes a pair. Also, since ordinary recursion theory is a theory with equality, and the effective pairing function is total, it is recursive (Lemma 2.6.7). DEFINITION. We say 9'td has a strong pairing function if there is a 1-1 onto function J : d x d -+ d such that both the graph of J, and its complement, are generated in 9L. PROPOSITION 6.3. Suppose 9.dhas a strong pairing function. If 8 E S n + l ( 9 . is d )k-place, then there is a k + 1-place relation 8' E U,,($Be,.> such that 8 = P : + ' ( S ' ) .[That is, iterated projections can be collapsed to single ones.] PROOF.Suppose 8 E ,Zn+l(%d). Then 9' = M ( 9 ) for some 9 E L',($Be,) and some string M of projections. We suppose M has two terms; more than two can be treated by iterating the procedure below. Also we may suppose the projections are off the front, since L'n(%d)is closed under transpositions. So we have

8 = P:+'P:+2(9). Now, 22 E L',, ($Bd), so

S(w,z )

e

g

E Z,, (9L). Define a relation S by

( 3 x ) ( 3 y ) [ J ( x y7 ) = w

A

a x , Y,

211

where J is a strong pairing function. Since both J and its complement are and hence by Proposition 6.2(3), J E Zn( % d ) . generated, J E Z[,($Bd), Then, by the closure properties of Lemma 6.1, S E -%(%I), and hence E IIn ($Rd). We claim 9' = P ; " ( s ) , that is, P : + ' ( s )= P:+'P:+'(9). Now z E P:+'(s)e

e

(3W)S(W,2 )

( 3 w ) - ( 3 x ) ( 3 y ) [ J ( x , y ) =w . % X , Y J ) l ( 3 w ) ( v x ) ( ~ Y ) [ J ( x , Yw ) =3 9 2 x , y , z ) I

and since J is a 1-1, onto function, this is equivalent to

( 3 x ) ( 3 y ) 9 ( x y, , z )

e

2

E P:+'P:+'(9).

In ordinary recursion theory there is a result, due to Kleene, that says the

CH.7, §6]


163

r.e. relations are the projections of the recursive relations (indeed, stronger versions are true, but this is all we need). DEFINITION. We call 3&a Kleene-Mostowski system if (1) %& has a strong pairing function and (2) the generated relations of %,4 are exactly the members of Zl(%,4). THEOREM 6.4. Let 3.& be a Kleene-Mostowski system. Then for each n > 0. 2,(%.d) is itself an indexed relational system, under a suitable choice of universal relations, and using the same notion of basic function as in 3.&. PROOF.0, singletons, and =& are generated in 3.4, which is a Kleene-Mostowski system, hence all are in XI(%&). Then by Proposition 6.2(3), all are in X,,(3&) for n > 0. By Lemma 6.1, 2"(9L)is closed under transpositions, projections, n, U and X . Next there is the matter of universal relations. Suppose Xn(3&) has appropriate universal relations (by hypothesis, X,(%&) does). Say Uk+IE 2"(3&) is universal for the k + 1-place relations of Xn(9&). Then Uk+l E n. (%), so P:"( ,'+I) E Zn+l(3.d). We claim it is universal for the k -place relations of Xn+l(3d). Let 9 E X n + 1 ( 3 d )be k-place. By Proposition 6.3, there is a k + 1-place such that 9 = P ! + ' ( 9 ' ) . Further, E 2" (3.4). relation 9'E Iln(3&) Since U k + 'is universal in 2"(3&), for some i, x

EF

(i,x

) U~k + '

or, in a handier form, Then, so

or

(u,u)Ee F

( i , u , u ) E uk+'.

-

(u,u ) E 8' G (i, u, u ) E U k + ' ,

-

u E P f + ' ( 9 ' ) e (i, U ) EP;+'(Uk+') uE

9 e (i,u ) E P;+*(F).

Thus we have universal relations in Zn+l(%d). That the iteration property holds in Xn(%&), n > 0 , is proved similarly. We leave this to the reader. COROLLARY 6.5. Let 3& be a Kleene-Mostowski system. Then the classification produces a hierarchy, in that, for n > 0,

Zn,IZ.


164

[CH.

7, 87

PROOF.(1) Is by Proposition 6.2. (2) Since Z,, ($&) is an indexed relational system, by Theorem 2.1 there is a relation 8 E 2" (9L)with P G 2" (%), i.e. SG nn(%L). (3) Let 8 be the relation of (2). 8 E Zn (%), SO 9 E 17. (9L),but

@ G Zn(%d).

(4) If 2" = 2"+,(%)n n,+,($&), then 2" (%) would be closed under complementation, but item (3) says it is not. (5) Similar to (4).

(a)

7. The Rice-Shapiro Theorem

In Section 4 we proved Rice's Theorem, which characterized the closed, bi-generated sets of indexes in 5%. In this section, under some additional assumptions about 9 L , we characterize the closed generated sets of indexes. The result, in ordinary recursion theory, is a conjecture in Rice [1953], which was established by a number of people, Shapiro among others. (See Rice [1956].) In ordinary recursion theory the Rice-Shapiro characterization is in terms of the finire sets. It turns out that, in order to prove the Rice-Shapiro Theorem only certain special properties of the finite sets are needed. What we do here is postulate that there is a collection 9 of sets having these properties with respect to $Rd,and we state our Rice-Shapiro characterization in terms of 9. Whether the members of 9 are actually finite or not is of no importance. In fact, after establishing our results, we present an important class of examples in which some of the members of 9 are infinite. This is the first instance in this book where we consider a generalization of finiteness. We will return to this important concept in the next two chapters. For the rest of this section we make the following additional assumptions about the indexed relational system %.d: (1) There is a collection 9of sets and relations on d,called 9-finite. We write 9"for the collection of n-ary members of 9.

CH. 7, 871

THERICE-SHAPIRO THEOREM

165

(2) The 9-finite relations are closed under sections. That is, if D E E d is fixed, then {x (x, c ) E D } is 9-finite. (3) T o each member of 9", one or more members of d have been assigned (called 9 "-finite codes) so that different members of 9 " do not share a code. We write 0 : for the member of 9"with code x . (4) The following relations are generated:

1

9"+', and if c

y is a 9"-finite code and x E D;, y is a 9n-finitecode and x$Z DY. ( 5 ) (9-finite approximation condition). If 9 C d" is generated, then there is a chain Ce of 9-finite n-ary relations such that (a) {x 10: E %} is generated. (b) Ce = 9. (c) If 8 is any proper initial segment of ~ e , U 8 is 9-finite.

u

PROPOSITION 7.1. Under the assumptions of this section, every finite relation on d is also 9-finite. PROOF.Since singletons are generated, and the generated relations are closed under U and x , each finite relation 9 on d is generated. Then there is a chain Ce of 9-finite relations with Ce = 9.But since 9 is finite, and % is a chain, it follows that 9 E Ce, hence 9 is 9-finite.

u

Now we start out on the Rice-Shapiro Theorem. Recall, a set C of indexes for n-ary generated relations is called closed if i E C and R 7 = R ; imply j E C. LEMMA7.2. Under the general assumptions of this section. Suppose C is a closed set of indexes for n-ary generated relations, and also that C itself is generated. If a . E C then there is some b E C such that

RZ is 9-finite

and

R: C RZ,,.

PROOF.Let a. E C. We produce an appropriate b. Choose two distinct members of d,let us designate them 0 and 1. Now, RI,and C are generated, hence so is the n + 1-place relation

T = ( RZt, X (0)) U ( C x (0) x . . x {0}x {l}). n-1

I

Then there is a chain Ce of 9-finite n + 1-ary relations such that { x D :+' E

166

u%

% } is generated, % is 9-finite. Let S(X1,.

e

[CH. 7 , 6 7

INDEXED RELATIONAL SYSTEMS =

T, and the union of any proper initial segment of

. ., X", y )

(3C)[C

E{ X

I D:"

E %}A

(XI,

.. ., X n , o ) E D:"

A

(Y,o,. . . ,o, 1) $Z ,:+I].

If follows, from our assumptions, that S is generated. Then by Corollary 3.3, there is an index b such that R : ( x ~ ,.., .x " )

e

S ( X ~ ,... , x , , b).

We claim this is the b we were looking for. First we show R : C R:,,.Well, suppose x E R :. Then S(x, b ) , and so for some c, (~,O)ED:" where c E { x I D : + ' € % } .Then (x,O)E % = T, and so x E R i0. Next we show b E C. Well, suppose b $Z C. Then (b,0,. . . ,0 , l ) $Z T, and so (b,O,. . .,O,l)$ZD:+'for every D : + ' E %. It follows that

u

S ( X 1 , ..., X , , b )

( 3 C ) [ C E { XI D : " E % } A ( X i

e

( x , ,..., x , , O ) E

u% = T

,..., X , , O ) E D : " ]

(xI,...,xn)ERz(,.

Thus x E R : e x E R z,,. But a,,E C and C is complete, hence b E C after all. Finally we show R : is 9-finite. Since b E C, then (b,0, . . . ,O, 1)E T, and so 8 = { D E % (b,0,. . . ,O, 1)$Z D } is a proper initial segment of %. Thus 8 is 9-finite. But also,

I

u

S ( x l,..., x,, b )

e ( 3 c ) [ cE { x ID:"€

"}A

(xl ,..., x,,O)ED:"]

since the other clause of S, (b,0,. . . ,O, 1) D:" is equivalent to D:+'E %. Thus

u

R : ( x I ,.. . ,x,,)

e

S ( x l , .. . , x , , b )

e

(xl,. . . ,x,,O)E

u 8.

Since 8 is 9-finite, and the 9-finite relations are closed under sections, R : is 9-finite. LEMMA7.3. For each n, (1) There is a basic function f such that 0 : = RY(o,. (2) The set of Bn-finite codes is generated. PROOF.Let 9 ( a , x l , .. . ,x , ) e a is a 9"-finite code and ( x , , . . . ,x , ) € Dz. 9 is generated. By the iteration property, there is a basic f such that

CH. 7, $71


9 ( a , x l , .. . , x n )

167

e R ; ( a d x l , . ., x n )

and this is (1). The set in (2) is a projection of 9, and hence is generated.

THEOREM 7.4. Under the general assumptions of this section. Let C be a set of indexes for n-ary generated relations. If C is closed, generated, and non-empty, there is a generated set B whose members are gn-finite codes, such that

c € C G (3a€B)[D:CR:]. PROOF.By Lemma 7.3 there is a basic function f such that D2= R;(,,. Define a set B by x EB

x is a 9”-finite code and

f(x) E C.

Then B is generated, and we claim it is the desired set. Suppose c E C. By Lemma 7.2 there is some b E C such that R i is 9-finite and R C R :. Since R z is 9 -finite, there is some Qn-finitecode a such that RE = D:. But 0:: = R;,,, so RE = R;(a).C is closed and b E C, so f ( a ) E C, and hence a E B. Conversely, suppose, for some a E B, 0:: C R:. Well, D ; = R ; ( o l ,so R;,,, C R:, and f ( a ) E C since a E B. Then c E C by Theorem 4.2. APPLICATION. Let C be the set of all indexes for finite (in the usual sense) subsets of d. Under the assumptions of this section, C is not generated. For, suppose it were; then there would exist a generated set B such that c E C e ( 3 a E B ) [ D : C R:].Now, C is not empty, so B is not empty. But, for any a E B, 0 :C d,and d is generated, hence indexes for d must be in C, and since YL is indexed it follows easily that d must be infinite. A little later in this section we present another, rather curious application. EXAMPLES. We give some indexed relational systems which meet the assumptions of this section. More will be presented in Chapter 9. I. The r.e. relations of ordinary recursion theory provide an indexed relational system such that, taking 9-finite to mean finite, all the assumptions of this section are met. This, of course, is the original setting for the Rice-Shapiro Theorem. We sketch why this is an example. As remarked in Ch. 1, §5, in ordinary recursion theory our non-deterministic elementary formal systems can be replaced by deterministic versions, and these deterministic versions can be

168


[CH. 7,#7

“discussed” within ordinary recursion theory, much as in Chapter 6. Then, given a deterministic means of generating a set 9,the function f, defined as follows, is itself a recursive function:

I

f ( n ) = a finite code for {x E 9 x is generated in < n steps by our deterministic procedure for generating 9}. Now, f has the following properties. (1) The range of f is a set of finite codes, and is r.e. (2) If n < k then DrCn,Df(k), so the range of f is a chain. (3) U(range of f ) = 9. Thus we have the finite approximation condition for r.e. sets. In a similar fashion, relations can .be handled. A finite coding for finite relations can be introduced by using an effective pairing function to “collapse” them to finite sets. We omit details. 11. Let 8 be a structure such that the following four conditions are met: (1) w-rec(8) is a theory with equality (2) Every “given” relation of ‘u is w-recursive in w-rec(8). ( 3 ) w-rec(8) has an effective pairing function. [Then by Proposition 6.8.3, w-rec(8) contains an w-r.e. copy of the natural numbers; but we need the following stronger assumption.] (4) In w-rec(’u) there is an w-recursive copy of (N, G ) . If these four conditions are met, then the w-r.e. relations of w-rec(8) provide an indexed relational system such that, taking 9-finite to mean w-recursioe, all the assumptions of this section are met. In all examples of this sort, some sets which are actually infinite are 9-finite. In particular, the domain, .d,is 9-finite. Specific examples of this sort are w-rec(G(N)), that is, hyperarithemetic theory, and w-rec(G(R)) where G(R) is the structure of the reals: (R; +, X , > ). Before going through the rather technical verification of our claims in Example 11, we pause to present an application. In ordinary recursion theory, every infinite recursively enumerable set contains an infinite, recursive subset. The analog of this for hyperarithmetic theory is: every infinite IIl set contains an infinite hyperarithmetic subset. Now, this is true, but the hyperarithmetic sets are the analogs not only of the recursive sets, but also of the finite ones. If we consider them that way, the result becomes a consequence of the Rice-Shapiro Theorem, as we now show. LEMMA7.5. In hyperarithmetic theory, let o-r.e. (that is, Ill).

(e = { a

1 R h is infinite}. Then % is


CH.7, 971

169

PROOF.Se has o-elementary formal system axioms: x E R f, +x > y + Gy, a,

GV,a+aE%. THEOREM 7.6. In hyperarithmetic theory, every infinite Ilt set contains an infinite hyperarithmetic subset. REMARK.Hyperarithmetic arithmetic = w-recursive.

theory

= w-rec(G(N));

Il! = w-r.e.; hyper-

PROOF.By the remarks above, the finiteness assumptions of the present section apply to hyperarithmetic theory, taking 9-finite to mean hyperarithmetic. Let Se = { a R: is infinite}. By the lemma, Se is generated, and it is trivially closed and non-empty. Then by Theorem 7.4, there is a generated set B whose members are 9I-finite codes, such that

1

cE%

e

(3aEB)[D:cR:].

Now, let Rb be some infinite n: set. Then c E %, so for some a E B, DAC R : . DA is hyperarithmetic. Further, it is easy to see that if DA were finite, then % would contain some indexes for finite sets (in particular, indexes for Dt). Hence DL is infinite and we are done. We now return to our discussion of the general class of examples under I1 above. What follows is a sketch of the verification of our claims, that these are, in fact, examples. We rely heavily on results in Moschovakis [1974], and what we will say will probably not be intelligible without an understanding of that book. However, nothing in later chapters here depends on this verification, it is enough to take our word for it that the examples described above really are examples. We should also mention that it is likely that the four conditions listed above can be weakened somewhat. See the remarks in the last section of Barwise [1975A], which is a review of the Moschovakis book, and in Chapter 6 of Barwise [1975]. We have not followed u p on this, however. We rely, throughout the rest of this section, on Propositions 1.13.6 and 1.13.9, and so we use the terms w-r.e. and inductive interchangeably; similarly for w -recursive and hyperelementary. We also remark that, since we are supposing w-rec(%) is a theory with equality, total functions are w-recursive iff they are w-r.e. (Lemma 2.6.7). In particular, this applies to a pairing function. Thus, if w-rec(%) has an effective (w-r.e.) pairing function, it has an w-recursive pairing function.

170


[CH. 7.17

The definition of an acceptable structure is given in Moschovakis [1974], p. 22. If w-rec(%) meets the four conditions above, it follows as in exercise 1.7 on p. 22 of Moschovakis [1974], that there is an acceptable structure '3' having the same w-r.e. relations. But then, using Theorem 3.5.1, the identity map is an embedding of w-rec(%) to w-rec(%'), and thus w rec('3) = w-rec(%') (considered as production systems). So from now on, we simply assume % itself is acceptable. Since w-rec(%) has an effective pairing function, the w-r.e. relations provide us with an indexed relational system (Ch. 6, 815). We need to verify that the w -recursive relations satisfy our 9-finiteness assumptions. For this, we simply cite results from Moschovakis [1974]. From now on, by a 9-finite relation we mean an w-recursive relation of w-rec('3). That these are closed under sections is straightforward. The notation and results assumed in this paragraph are from Theorem 5D.4, Moschovakis [1974], p. 75. By a 9"-finite code we mean a member of the set I",and if a E I",by 0:: we mean Hz. Then every n-ary w-recursive ( = 9-finite) relation has a 9"-finite code. The relation: a is a 9"-finite code and x E D ; , is w-r.e. since it is: a E I" A (a,x ) E H", and both I" and H" are inductive. Similarly for the relation: a is a 9"-finite code and xfZ D:, since it is: a E I" A (a,x ) E i H", and fin is coinductive. Finally we must check that the 9-finite approximation condition holds. The terminology and results in this paragraph are from Moschovakis [1974], pp. 38-40. Let 9 be an n-ary w-r.e. relation. By the Prewellordering Theorem (Theorem 3A.3, p. 40), 9 admits an inductive norm, u.Let % be the collection of resolvents of P relative to u.Each member of % is hyperelementary, that is, 9-finite. % is a chain; = 9; and if 8 is a proper initial segement of %, 8 is 9-finite. It remains to be verified that {x D:E %} is a-r.e., that is, {x D : is a resolvent of P relative to a},For this we give the following w-elementary formal system, in which it is represented by 0.

u

I

u%

I

axioms €or P, axioms for J, and for

i.fo.

[Let Ax, y , a mean a is a 9"-finite code and x E 0:: U resolvent y , where resolvent y = {x E CP u ( x ) a ( y ) ) . ]

I

Py +J, (x, y ) + a is a 9"-finite code +Ax, y , a, P y + a is a 9"-finite code and xfZD:-Ax,y,a,

A V, y, a +D ::C resolvent y .

CH. 7, §7]

171


[Let Bx, y, a mean a is a 9"-finite code and x E D: U resolvent y.]

9 y + a is a 9"-finite code and x E D : - + B x , y , a , P y + a is a finite code + i j ( x ,y)-, Bx, y, a,

BV, y, a +resolvent y

D:,

0 :C resolvent y +resolvent

y C 0:- resolvent y

9 y +resolvent y = D:+ Oa.

Thus all the 9-finiteness assumptions are verified.

=D:,

CHAPTER EIGHT

INDEXED PRODUCTION SYSTEMS

1. Introduction

In Chapter 6 we showed many recursion and w-recursion theories have operator indexings that have the input and the output place-fixing properties. This is the second of two chapters in which we investigate the consequences. In Chapter 7 we concentrated on relations, now we look at operators themselves. There is a summary presentation of much of this material, from a computer point of view, in Fitting [1979]. Also in Chapter 6 we showed that in many recursion theories there was a “usable” coding of the finite sets. Here we consider the implications of that, one being that for such recursion theories, an analog of the customary definition of enumeration operator holds. (Rogers [1967], p. 147.) It turns out that the compactness of enumeration operators plays a key role in this. We then proceed to find something that can play the role, for w enumeration operators that the finite sets do for enumeration operators. We find that, no matter what choice is made, non-compactness is inescapable. It is in trying to deal with this non-compactness phenomena in our axiomatic setting that partial production systems are forced on us. In the following chapter we study some particular production systems of great interest, and which are closely related to those which develop in the present chapter. 2. Indexed production systems

We define the primary objects of study in this chapter, give examples, and establish connections with the previous chapter.

DEFINITION. Let A be a production system. Let %‘“.k’ be some operator in A of order (n, k + 1). We say %‘”.*’ is universal for the operators in A of order (n, k ) if, for each operator @ E A of order (n, k), there is at least one a E d (called an index for @) such that, for all 9’E [d]”, 172

CH. 8, 831

THEOUTPUT PLACE-FIXING PROPERTY

173

x E @(9) e ( a , x ) E %(".k)(9).

REMARK. The following is an equivalent formulation which some may find "neater". Recall, S: is the a-section operator from Ch. 2, 43. Now, let ~ h k ) be an operator of order (n, k +1) in A. Then, for each a € d , S : +' 021 ( n . k ) is an operator in A of order ( n , k ) . We call %'""' universal if every operator of order (n, k ) is of this form. DEFINITION. By an operator indexing for A we mean a family of operators in A, OU = { % ( n , k ) n, k 2 1) where, for each n, k, % ( n , k ) is universal for the operators in A of order (n, k ) . We say A is an indexed production system if there is some operator indexing, OU, for A. We often write @ F k )for the operator in A of order (n, k ) and index a. That is, @ p k=) S ! + ' % c n . k ) .

1

EXAMPLES. By Corollary 6.14.5 every infinite recursion or w-recursion theory with equality and an effective pairing function is an indexed production system. Also see Corollary 6.14.4. Additional examples will be given in the next chapter. We begin discussing connections with the notion of indexed relational systems, as defined in Ch. 7, $1. This discussion concludes in the next section. Let A be a production system, and let 9 E Consider the collection of relations S 9. It is easy to verify that this collection satisfies part 1 of the definition of indexed relational systems. We note that if 9 = 0, then by Proposition 2.3.1, the relations involved are exactly the generated relations of A. Now suppose further that A has an operator indexing $2. For each k, define a k + 1-place relation U k ,by

[a]".

Uk = q p . k ) ( $ p ) .

It is quite easy to see that these relations satisfy ( 2 ) of the definition of indexed relational systems. In fact, the resulting relational indexes are connected with the operator indexes by R : = @?"(9).

3. The output place-fixing property

In Chapter 6 we showed that many recursion and w-recursion theories had indexings which had the output and the input place-fixing properties. In this section we investigate the consequences of the output place-fixing

174

[CH. 8, $3

INDEXED PRODUCIION SYSTEMS

property, then we go on to the input place-fixing property in the next section.

DEFINITION. Let A be an indexed production system with operator indexing %. Let $23 be a collection of (total) functions f : d n+= d for n = 1,2,3,. . . , which we call basic, meeting the conditions (1) 93 is closed under composition, (2) (the graph of) each member of $23 is generated in A. We say the triple (A, a,$23) has the output place-fixing property if, for each operator @ E A, of order (n,q + m ) say, there is a basic function f : d q+ d such that, for all 8 E [d]",

h , .. .)ff,,Xl,. . . , x , ) E

@ ( 8e )

{XI,..

. ,X,)E

@E:'.)

.a,).

Generally, we will say A has the output place-fixing property, rather than (A, %, a), leaving % and 93 to be understood by context.

EXAMPLES. By Corollary 6.14.5, every infinite recursion or

w -recursion

theory with equality and an effective pairing function has an indexing such that the output place-fixing property holds, taking basic to mean recursive, or w-recursive respectively. But in these examples, the notion of basic function is trivial in that it is as broad as possible. It is interesting generally, to take a smaller collection for the basic functions. In fact, Corollary 6.14.5 shows that if rec('?I) is a theory with equality and an effective pairing function, then w-rec(\u) is an indexed production system having the output place-fixing property, even if the basic functions are restricted to the recursive functions, rather than including all w-recursive ones. Further, in both ordinary recursion theory and in hyperarithmetic theory one can narrow things down to the primitive recursive functions as basic. See Kleene [1952] for a definition. Also, the class of rudimentary functions, from Smullyan [1961] will do. This example can be generalized, using the word structures we introduced in Chapter 6, but we do not do so here. At any rate, there are many indexed production systems in which the output place-fixing property holds, and in which the basic functions are a rather narrow, interesting subclass of the collection of generated functions. More examples will occur in the next chapter.

REMARKS. Let (A,%,$23) be an indexed production system having the output place-fixing property. And let 8 E Then the collection of relations < 9 is an indexed relational system, taking basic in the relational sense, to mean member of 93. We checked part of this in the last section,

[a]".

CH. 8, §3]

THEOUTPUT PLACE-FIXING PROPERTY

175

and now merely note that the iteration property is an immediate consequence of the output place-fixing property. Now we turn to consequences. We have assumed that the operators of A are closed under f l , U and x . We begin by showing that under the circumstances of this section, the closure is effective. We postpone a treatment of composition till the next section.

THEOREM 3.1. Let A = ( A , %, 93) be an indexed production system having the output place -fixing property. For each (n, k ) and ( n ,j ) there are basic functions f , g and h such that (1) @ F k ) n @ F k= ) @F:L), (2) @ F k ) u @jcn.k) = @(n.k) g(a.617 ( 3 ) @ F k ) x @b"." = @y&{).

PROOF.(1) Let S be the operator of order (n, k

s=

+ 2 ) given by

~ ~ , ~ ? ~ k ~ I % ( ~ . k ) n ~ k + I % ( " , k )

(Here A k " and T!.12 are place-adding and transposition operators as defined in Chapter 1. % ( n , k ) is, of course, universal.) Tracing through the definitions, it is not hard to see that, for any 9 E [a]", x E (@Fk)n @ P k ) ) ( 9 ) .

(a, b, x ) E S ( 9 )

Now, by the output place-fixing property, there is a basic function f such that

(a, b, x ) E S ( P )

e

xE

@$:2,(9)

and thus - @ ha k )

0.b) -

@;Tk)

n @?".

+ 2 ) defined by

(2) Now let S be the operator of order (n, k s = T;,;?Ak + I % ( n , k ) u A k + l q ( n . k ) Then, for P E [.PI",

(a, b, x ) E S(P)

xE

(@Fk)U @Pk))(9')

and the result follows as in (1). (3) Let L be the logical operator of order (k

L (9, rl, * . .

7

rk,

s, tI, * *

9

t,)-* (4, s, r l , .

+ j + 2, k + j + 2) such that

-

3

rk,

tl3 *.

.

3

ti).

[CH.8 , § 3

INDEXEDPRODUCTIONSYSTEMS

176

Let S be the operator of order (n, k

s=

L (q ( n .k)

+j

+2) defined by

1.

Q1'n.~)

It follows that, for 9 E [a]", and any k-tuple x and j-tuple y,

(a, b, x,y) E S ( 9 )

e e

x E @b"."(S)and y E @b"."(S) (x, y ) E (@Y x @P))(9).

Now finish as before. Next we establish an analog, for operators, of the Kleene fixed point theorem, Theorem 7.3.4. Its proof is essentially the second proof of Theorem 7.3.4, transferred to operators. THEOREM3.2. Let A be an indexed production system having the output place-fixing property. Let t : d +d be a (total) function whose graph is generated in A. Then for each order, ( n , k ) , there is an i such that @(n.k)

= Q2cn.k) l(1)

.

PROOF.Using the output place-fixing property, there is a basic function d such that k) ( y , x ) E P ~ + 2 E ~ , ~ z ~ c n ~ k +El @) ( (nd. (S Y )d 9 ) .

Tracing things out, this means

e

( y , x ) E @:".k+"(S)

x E @ ( nd.(ky) ) ( 9 )

or, more briefly, using the section operator, for all y, s;+l@(n.k+l) Y

= @(n.k)

d(y).

Next, since d is a generated function, as is t, then the composition, td : d .--, d is generated. Let H be a constant operator, of order (n,2) and value (the graph of) td. Then there is an index e such that @(n.k+I) = p ; + 2 p ; + 3 p ; ? ( x~~ h k ) ) . e Again, tracing things out, this says that for all y, S;+l@Fk+l) = @ .)::

Finally, let i @(n.k)

= d(e). Then

= @Ti;; = s : + I @ ? k + l >

=@hk) -@hk) ld(e1-

((8)

.

COROLLARY 3.3 (Rice's Theorem for operators). Let A be an indexed production system having the output place -fixing property. Let C be a closed

CH. 8, 541

THEINPUT PLACE-FIXING PROPERTY

177

set of indexes for operators of A of order ( n , k ) . If both Cand c a r e generated in A then either C is empty or C = d. PROOF.Similar to the proof of Theorem 7.4.1, but using the above instead of Theorem 7.3.4. 4. The input place-fixing property

We continue the work of the previous section, adding the input place-fixing property as a new assumption. DEFINITION. Let A be an indexed production system with operator indexing %, and let 93 be a collection of basic functions (each is total, generated in A, and 93 is closed under composition). We say the triple (A, %, 93) has the input place-fixing property if, €or each operator @ E A, of order ( q + n, m ) say, there is a basic function f : d q+ d such that, for all P? E [dl",

y E @ ( { ( a 1.,.., a,)) x

9)

y E @Fc,!...aq)(9).

EXAMPLES. Every example given in the previous section for the output place-fixing property is also an example in which the present assumption holds. Now we discuss consequences. We begin with some more results on "effective" combinations of operators. LEMMA4.1. Let A = (A,%, 93) be an indexed production system having the output place-fixing property. Let I? be an operator in A of order (n, k ) . For each q there is a basic function h :d 4d such that @Lk.q)!€f = cD(".~) h(x)* PROOF.For 9 E 2

E

[a]",

CDLk.q)?P

@ (x,

2 )E

%(""'?P(P).

Now use the output place-fixing property on %(k.q)!P.

REMARK.This lemma, and the input place-fixing property, imply the output place-fixing property. THEOREM 4.2. Let A

= (A,%,

93) be an indexed production system having

INDEXEDPRODUCTION SYSTEMS

178

[CH.

8, $4

both the input and the output place -fixing properties. Then for each n there is a basic function f : d ”+d s u c h that f ( a l , . . . ,a,) is an index for a constant operator of order ( 1 , n ) and value {(al,. . . , a ” ) } .[Thus “constants” are effective.] PROOF.Using the input place-fixing property on the operator P::!, there is a basic function g : d ”+ d such that, for 9 E [ d ] ’ ,

P : = : ( { ( u , ,... an>}x 9)= @fi::....,an)(9)Further, if

B# 0,

P z : ( { ( a l , . . ’a,)}x , P)={(a,,.. . , a , > } . Now, let c be some member of d.Let ‘P be a constant operator of order ( 1 , l ) and value {c}.Then, for all 9 E [d]’, ‘ P ( 9 )is never empty. So by the above, for all 9 E [d]’, @fi:,).....an)*(9)

= {(a17 * .

. 7

an>}.

Now by Lemma 4.1, there is a basic function h such that @ ( I d ‘P = @ ( I . hg(a1,. “) . .. a n ) . g ( a l , .. ..a,,)

Basic functions are closed under composition, so f = hg is the desired funtion. The universal operators we have been discussing sort things out by “tagging” the output. We now produce universal operators that “tag” input instead. At first thought this should mean we want an operator ‘V(n.k) such that, for each a, and each 9 E [d]“,

,(“.*’({a}x 9)= @h“.”(9). But further thought shows that this is not going to work when B is empty, since then {a}x 9 = { b }X $3’ = 0, and the “tag” is lost. So what we do instead is replace 9 by ((0)x 9)U {(1,1,.. . , l)}.This is never empty, and the original members of 9 can be recovered from it by using So””.

PROPOSITION 4.3. Let A be an indexed production system with operator indexing (22 (the place-fixing properties are not assumed). Choose two distinct members, 0 , l E d. For 9 E [ d ] ”by, 9’ we mean

CH. 8, 841

THEINPUTPLACE-FIXING PROPERTY

Then for each order (n, k ) there is an operator Vn.') of order ( n such that, for 9 E [a]",

179

+ 2, k )

V(".k'({a} x 9') = @h"."(9). PROOF.Let L be the logical operator of order ( n + 2 , l ) such that L : (xl, x2,. . . ,X , + ~ ) + ( X ~ ) . Then set y(n.k) =p;+Ip;+2~;,;2[~ x %(n.k)s;+lp;+Z].

We leave to the reader the task of checking that this has the right properties. Finally, we are in a position to treat the "effectiveness" of composition of operators, the only method of combining them that we have not dealt with so far. THEOREM 4.4. Let A be an indexed production system having both the input and the output place-fixing properties. For each n, k and q, there is a basic function f : d2+d such that @Lk,q)@?k) =

@P:L).

PROOF.First, let @ be defined by @

= qy(k.q)y(n.k)

where Y"(n,k)is as in the above proposition, and %'k.q' is the standard universal operator. @ is of order ( n + 2, q + 1). For 9 E [d]", let 9'be defined as above, namely

-

9'= ((0)x 9)u {(1,1,.. .,1)}. fl+l

Then, simple checking shows

(b, x ) E @ ( { a }x 9')

x E @~k~q'@b".k)(9).

(1)

By the input place-fixing property, there is a basic function g : d +d such that, for 2 EISe]"+l, z E @ ( { a }x 2 )

z E

e

@$2;,q+1)(22)

( g ( a ) ,2 ) E % ( " + ' , 4 + l ) (9).

(2)

Also, by the output place-fixing property, there is a basic function h : d2-+dsuch that, for 2 E [a]"+',

INDEXEDPRODUCHON SYSTEMS

180

[CH. 8,§4

-

Now, let A be the constant operator of order (n, 1) and value (0);let B be the constant operator of order (n, n + 1) and value

{(1,1, * * . ,I)}; n+l

I" is the identity operator, as usual. Set

V=(A xI")UB. Then, for 8 E [a]", q(8)= P r and thus @;k.q)@(n.k) a

=

@p(i(!$,)b)

q-

The result now follows, using the closure of basic functions under composition, and Lemma 4.1. We remarked in Sections 2 and 3 that, if A is a production system having the output place-fixing property, then, for each 9 E [a]", the collection of relations s 8 is an indexed relational system, using, for k = 1,2,. . . u

k

= qy(".k)(gP)

as universal relations. We will write R :9 for the k-ary relation S 8 having index a under this indexing. Briefly this means R :9 = @jb"."(S). If 8 = 0, the relations involved are just the generated relations of A, and we will generally write R for R :O. With this understood, we have the following.

:

COROLLARY 4.5. Let A be an indexed production system having the input and the output place-fixing properties. Let 8 E [a]'. For each (k,q ) there is a basic function f such that @b"."( R k9) = R $,,a).

PROOF.This simply asserts @bk~"@6"."(8) = @ ( &,)(8) nq) for some basic f , and thus is a special case of the theorem above.

POINTWISE GENERATED FUNCTIONS, AGAIN

CH. 8, 55’1

181

COROLLARY 4.6. Under the assumptions of Corollary 4.5. For each operator @ of order (k,q)there is a basic function g such that @(Rk”)= R’$).

5. Pointwise generated functions, again In Ch. 2, 05 we introduced two plausible notions of a function f being “computable” in a production system A. We repeat the definitions for convenience.

DEFINITION. Let f be a partial function from d ” to d * . ( 1 ) f is generated in A if the graph of f is generated in A. (2) f is generated pointwise in A if there is an operator @ E A of order (n, k ) such that

[;,.,,

@(b)) =

Edomf, u$Z dom f .

Corollary 2.5.5 says that every generated function is also pointwise generated. Now we show that, under the indexing assumptions of the present chapter, the converse also holds. PROPOSITION 5.1. Let A be an indexed production system having the input place-fixing property, and let @ be an operator in A of order (q + k, n). For a fixed 9 E [&Ik, define a q + n-place relation $32 by

B ( x , , . . . ,xq, y 1 , . Then 92

. . ,y.) e ( Y l ,

* * *,

Y ” )E @({(XI,

. f

x,)) x 9).

9.

Since f is basic, it is generated. Let F be a constant operator of order (k,q + 1) and value the graph of f . Then it is easy to see that the above is further equivalent to

INDEXEDPRODUC~ION SYSTEMS

182

[CH.

8, $5

COROLLARY 5.2. Let A be an indexed production system having the input place-fixing property, and let @ be an operator in A of order (4, n). Define a q + n-place relation 9 by

PROOF.Define an operator @' of order (q c E d be fixed. Then certainly Y E @({XI)

Y E @'({XI

+ 1, n ) by

@'= @P:z:. Let

x {CH-

Then by the above proposition, 9 G {c}.But { c }is generated (Axiom 3 of production systems) so { c } S 0 (Proposition 2.3.1). Thus 3 S 0 , so 3 is generated. COROLLARY 5.3. Again, suppose A is an indexed production system having the input place-fixing property. (1) 9is generated in A if and only if there is an operator @ E A such that xE

9 =, @({xl)={OI,

xe3

+ @({x})=0.

(2) For S # 0, allowable in A, 9 G 9 if and only if there is an operator @ E A such that x€9

x e 9

+ @(Sx{r})={O}, + @(Sx { x } ) = 0 .

PROOF.We show (1); (2) is similar. If 9 is generated, the existence of a suitable operator Q> is by Corollary 2.5.3. Now suppose we have an operator @ in A whose output is either (0) or 0. Define a relation 9' by 9 ' ( x , y ) e y E @({x}). [ y can be only 0, of course.] By the above corollary, 9'is generated in A hence, using the section operator and transpositions, so is 9 = {x 3'(x, 0)). It is easy to see 9 has the desired properties.

I

CH. 8, 561

9-FINITENESS

183

COROLLARY 5.4. A is an indexed production system having the input f is generated place-fixing property. Let f be a partial function from d"to dk. in A if and only i f f is generated pointwise in A. PROOF.Half is by Corollary 2.5.5. Conversely, suppose f is generated pointwise in A. Let @ E A be an operator of order ( n , k ) such that

@(bl)=

[;(41

tJEdomf?

ujZ dom f.

By Corollary 5.2, the following relation is generated in A. %(X?Y)

But y E @({x}) in A.

Y E @({XI>.

y = f (x) so $92 is the graph of f, hence f is generated

6. 9-finiteness

Finite sets are important for recursion theories since enumeration operators are compact. And we saw, in Ch. 7, 97, that for certain w-recursion theories, there were sets that played a role somewhat analogous to that which the finite sets play for recursion theories. In this section we introduce a kind of generalized finiteness notion into the machinery of production systems. The consequences will occupy us for the rest of this chapter. We note that our present assumptions are not the same as in Ch. 7, 97. DEFINITION. Let A be a production system, and let 9 be a collection of generated subsets of d,which we will call 9-finite. Suppose, to each member D of 9, one or more members of d have been assigned, called 9-codes for D, so that distinct members of 9 never have the same member of d assigned to them. Then we say we have a 9-finite coding in d.We write D, for the member of 9 with 9-code c. (If c is not a 9-code, D, is not defined.) Suppose now that a 9-finite coding exists. We say it is a positive canonical coding in A if

184


[CH.8,§6

(1) There is an operator F E A of order (1,l) such that, for 9 E [ d ] ’ ,

F ( 9 ) = { c ID,

c 9).

(2) The following relation is generated in A :

y is a 9-finite code and x f 0,. We say we have a canonical coding if, in addition to above items, the following relation is generated in A : y is a 9-finite code and x fZ D,.

EMLES. I. By Theorem 6.16.4,in any recursion theory with equality and an effective pairing function, there is a canonical coding for the 9-finite sets, where the 9-finite sets are simply the finite subsets of the domain. In particular, this is the case for ordinary recursion theory. In fact, Rogers [1967], p. 70, gives a finite coding under which each finite set has exactly one code, and every number is a code. In general, this is too much to hope for, however. IIA. In Ch. 7, 07, we gave some conditions on 8 that would ensure w-rec(8) met the special assumptions of that section, taking 9-finite to mean w-recursive. In fact, the same four conditions also ensure that we have a canonical coding, taking 97-finite = w-recursive. Recall, the four conditions were : (1) w-rec(8) is a theory with equality. (2) Every “given” relation of 8 is w-recursive in o-rec(8). (3) w-rec(8) has an effective pairing function. (4) In w-rec(8) there is a w-recursive copy of (N, 6 ) . In fact, in Ch. 7, 07, it was shown that if the four conditions hold then both y is a 9-finite code and x ED,,

y is a 9-finite code and xfZ D,, are w-r.e. All that we need now is the existence of an w-enumeration operator F such that F ( 9 ) = {x D, C 9}. But, in fact, y E F ( 9 ) @ 0, C 9 @ D,U 9 is the entire domain. So we are interested in the x E ByU 9.With this said, it should be clear that the relation Nx, y desired operator F is [EL] where E consists of:

1

CH.8, $61

9-FINITENESS

185

axioms for the relations y is a 9-finite code and x E D,, y is a 9-finite code and

and

xG! D,,

y is a 9-finite code and x E D, +y is a 9-finite code, y is a 9-finite code and xG! D,

y is a 9-finite code,

.+

y is a 9-finite code +I x .+Nx, y, y is a 9-finite code and x

D, .+Nx, y,

NV, y +. Oy.

This family of examples is extremely important, and in one way or another, will occupy us for much of this chapter and the next. It includes hyperarithmetic theory, w-rec(G(N)), as well as w-rec(G(R)), where G(W) is the structure of the reals; (W;+ , x , > ). IIB. If we want a canonical coding, but do not care about the special assumptions of Ch. 7,§7, we can manage with just the first three of the four conditions above for w-rec(%). We argue for this using material from Barwise [1975], in particular, the notion of HYPE,an extremely fundamental topic which must, otherwise, be ignored in this book. Since w-rec(%) is assumed to be a theory with equality, and in it, each “given” relation of % is w-recursive, then w-r.e. = inductive over 8 and w-recursive = hyperelementary over %. By Barwise [1975], Theorem 5.1, p. 230, (1) w-r.e. on % = Z on HYPE. (2) w-recursive on % = member of HYPv[. (3) HYPE is projectible into 8 . The definition of projectible is on p. 168, Definition 5.1, of Barwise [1975]. We use the notation from there. Let IZ be a notation system for HYPa, projecting into 8 . Recall, 9-finite is to mean w-recursive. Now suppose 59 is 9-finite. Then by (1) we have 9 E HYPa. And then n(9) d.Let the members of n(9)be 9-finite codes for 9. By Lemma 5.2, Barwise [1975], p. 169, Dn is HYPa-r.e., that is, Z on HYPE. Then by (2) Dn is w-r.e on %. Then x is a 9-finite code Consider I. the relation is w-r.e. over ‘?

186


x is a 9-finite code and y E 0,.

We claim this is w-r.e. over ’%.It is equivalent to X

EDn

A

( g % ! ) [ yE %! A 1x1”

=%!I.

This is 2 on HYPlrr,using Lemma 5.2, p. 169, of Barwise [1975], hence it is w-r.e. over 9l by (1). Similarly, the relation x is a 9-finite code A y

D,

is equivalent to x E Dn

h

( g % ! ) [ Y e%! A

I X In

=

91

which is 2 on HYPa, hence is w-r.e. over ’%. is Finally, the existence of an operator F, such that F ( 9 ) = {x D, G 9}, established as in example IIA.

1

7. Rogers’ form for operators

In ordinary recursion theory, enumeration operators generally are defined quite differently than we did in Chapter 1. The definition from Rogers [1967], p. 147 (for the special case of operators of order (1,l)) amounts to this:

CJ is an enumeration operator iff there is an r.e. relation

% ( y , x ) such that x E @ ( B )

%(Y,

e

for some y,D, C 9 and

X).

We will show, in this section, as a consequence of our axiomatic assumptions, that the operators of order (1,l) that satisfy Roger’s condition are precisely the operators that are monotone and compact relative to our generalized notion of finiteness. In the next section we introduce some assumptions about effective pairing functions, and extend the Rogers’ style characterization to operators of arbitrary order. We note that, since in recursion theories all operators are compact, it follows from the results below that our definition of enumeration operators agrees with the customary one for ordinary recursion theory. We begin with a few elementary results about 9-finiteness. A positiue canonical coding is enough for the work of this section.

ROGERSFORMFOR OPERATORS

CH. 8, 971

187

PROPOSITION 7.1. Let A be a production system with a notion of 9-finiteness for which there is a positive canonical coding. Then (1) There is an operator G E A of order (1,l) such that

I

G ( 8 )= u{D, y is a 9-finite code and y E 8}.

In particular

G({Y1) =

[

D,

if y is a 9-finite code,

0

otherwise.

If, further, A is an indexed production system for which the input place-fixing property holds, then (2) The relation 2 ( x , y ) e D, C D,, is generated in A. (3) The relation Y ( x , y) e D, = D,, is generated in A.

PROOF.(1) Let %(y, x ) a y is a 9-finite code and x E D,. % is generated in A. Now use Theorem 2.5.1. (2) Let F be the operator of order (1,l) such that F ( 9 ) = { x D, C 9}, and let G be the operator from (1). Then, D, C 0, e x E ( F G ) ( { y } ) Now . use Corollary 5.2. (3) Is immediate from (2).

1

DEFINITION. Let A be a production system with a notion of 9-finiteness. implies We call an operator @ E A of order (1,l) 9-compact if x E @(8) x E @(9) for some 9-finite set 9C 9. Recall, @ is monotone if 9 C 2 implies @(8) C @(9). Now let us return to the Rogers' definition, given above. The defining clause was for some y,D, C 8 and %(y,x).

x E @(8)

1

Now, using the operator F, having the property F ( 8 ) = {x D, C 9}, this can be rewritten: x E @(8)e for some y E F ( 9 ) , % ( y , x ) .

Further, using the notation of Ch. 2, §5, namely

I

%"(8) = { u for some w E 8,~ ( wu ),} ,


188

[CH.8,§7

this may be rewritten

xE@(P)e xE3"F(P) or, simply @ = 3 ° F . PROPOSITION 7.2. Let A be a production system with a notion of 9-finiteness for which there is a positive canonical coding. Let 9 be a generated, two place relation on d.Define @ of order (1,l) by @ = 3'"F. Then @ is an operator in A, which is monotone, and 9-compact.

PROOF.3"is an operator in A by Corollary 2.5.2, hence so is @. That @ is monotone and 9-compact is straightforward. DEFINITION. We say an operator @ of order (1,l) can be put in Rogers' form if @ = 3 ° F for some generated binary relation 3. REMARKS. In the next section we will extend this definition to handle operators of arbitrary orders.

The above proposition says that Rogers' form always gives monotone, 9-compact operators. The fallowing says when it gives all of them. THEOREM7.3. Let A be an indexed production system having the input place-fixing property, and with u notion of 9-finiteness for which there is a positive canonical coding. The operators of order (1,l) in A which can be put in Rogers' form are precisely the monotone, 9-compact operators. PROOF. In one direction the result is Proposition 7.2. Now suppose @ E A is of order (1, l), and is monotone and 9-compact. Define a relation W by S(Y,X)

x E(@G)({Y))

where G is from Proposition 7.1. By Corollary 5.2, %! is generated in A. We claim @ = 3 ° F . Let P E [d]'. (1) Suppose x E 3"F(P).Then, for some y , Y EF(P) and and hence

%(Y,X)

CH. 8, 581

E F F E ~ I PAIRING VE

0, C 9

189

and W Y , ~ ) ,

Q C 9 and x E @ ( G ( { Y I)), D y G 9 and x E @ ( D , ) , but @ is monotone, so x E @(9). Thus R " F ( 9 ) C @(9). (2) Suppose x E @(9). Since @ is 9-compact, for some y , 0,C 9 and x E @ ( D , ) , 0, C 9 and

Y E F ( 9 ) and

x E (@G)({yI), WY,X),

so x E a " F ( 9 ) . Thus @(9) C %"F(S). This concludes the proof. 8. mective pairing

We have been discussing the consequences of 9-finiteness for sets. The simplest way to extend our work to relations is to introduce a pairing function and use it to collapse relations to sets. After all, we do have effective pairing functions in all the recursion and w -recursion theories that are models for our assumptions up to now. Also, introducing a pairing function provides an easy way to extend Rogers' form to arbitrary operator orders, not just to those of order (1,l). DEFINITION. Let A be a production system. We say A has an effective pairing function J if J is a 1-1 function, J : d x d +d which is generated in A. If J is an effective pairing function, we define

PROPOSITION 8.1. If J is an effective pairing function in A, then J " ( x l , .. . ,x , ) = J " ( y l , .. . , y.) implies x I = y l and * * and x. = y.. Further, each J " is a generated function in A.

190


[CH. 8, $8

PROOF.J is 1-1; and generated functions are closed under composition. DEFINITION. Let A be a production system with effective pairing function J. For each n we define operators [J"]and [J"]-'of orders (n, 1) and (1, n ) respectively by

I

[J"](9) = { ~ " ( x.~ ..,,x,) ( x l , .. .,x,)

I

E

91 for 9 E [a].,

[J"]-'(P) = {(xl,. . . ,x.) J ~ ( x ., ., . ,x.) E 91 for 9 E [&]I. PROPOSITION 8.2. Let A be a production system with effective pairing function J. (1) [ J " ] and [J"]-'are operators in A. (2) [ J " ] and [J"]-'are monotone. (3) [J"1(9 u~~=lJ"l(~)u[J"l(~), [ J " ] - ' ( 9u 9) = [J"]-'(9) u [J"]-'(9). (4)[J"]-"J"] = I " . PROOF.[J"]and [J"]-'are operators in A using Corollary 2.5.2. The other assertions are straightforward. The introduction of a pairing function gives us a simple means of extending the notion of 97-finiteness to relations. Let us consider some examples. Suppose rec(%) is a recursion theory with a pairing function J. For 9 C d", 9and [J"]( 9 )are of the same cardinality, hence if either is finite, both are; and for recursion theories we have been taking $@-finiteto mean finite. Suppose w-rec(%) is an w-recursion theory with equality and an effective pairing function J. We claim that for 9 E [ d ] " 9,is w-recursive iff [J"](9) is w-recursive. This is significant since, in such situations we have been taking the 9-finite sets to be the w-recursive ones. We now turn to verifying our claim. We first observe that range J" is w-recursive. This follows from the equivalences

( 3 y l , .. . ,Y n ) [ J " ( Y l ,.. . ,Y n ) = x ] ,

x ErangeJ"

xSfrangeJ"

e

(Vy,,..., y . ) ( 3 z ) [ J " ( y,..., l y.)=z ~ z i c x ]

and both of these use only machinery available in w-rec(3). We omit details. Next, suppose 9 is w-recursive in w-rec(3). Then 9 is w-r.e., hence so is [J"](9) since [J"]is an operator in w-rec(%), which must take generated

CH. 8, 581

EFFECTIVE PAIRING

191

relations to generated relations. Likewise, @ is w-r.e., hence so is [ J " ] ( @ ) . But,

[ J " ](9) = [J"](9)u range J" and hence is w-r.e. Thus [ J " ] ( 9 )is w-recursive. Conversely, suppose [ J " ] ( 9 )is w-recursive. Then it is w-r.e., hence so is 9 = [ J " ] - ' [ J " ] ( 9 ) ,since [J"]-' is an operator in o-rec(2f). But also [ J " ] ( 9 )is w-r.e., hence so is

[ J " ] - ' ( [ J " ]( 9 )= ) [ J " ] - ' ( [ J " ](@)Urange) = [ J " ] - ' [ J " ](@)u

range J") = 4.

Thus 9 is w-recursive. All this suggests the following as a reasonable addition to our machinery.

DEFINITION. Let A be a production system with an effective pairing function and a notion of 9-finiteness for which there is a 9-finite coding. Let 9 be an allowable n-ary relation. We call 9 a 9-finite relation if [ J " ] ( 9 )is a 9-finite set. By a 9-finite code for 9 we mean any 9-finite code for the set [ J " ] ( P ) .We write 0 ;for the 9-finite n-ary relation with 9-finite code y. Then D = 0,. This also allows an extension of 9-compactness. We say an operator Q, € A of order (n, m) is 9-compact if x E @(9) implies x E Q,(D;)for some D ; C 9. PROPOSITION 8.3. Let A be a production system with an effective pairing function J a n d a notion of 9-finiteness for which there is a 9-finite coding. (1) [J"I(W) = DY, (2) rJ"l-'(DY) = D,", ( 3 ) for 9 E [ d ] "0, , C [ J " ] ( 9 )e D ; C 9, (4) D : c Dy D, 0,. PROOF.(1) is by definition. (2) follows from (1) by Proposition 8.2 (2). ( 3 ) follows from (1) and ( 2 ) using monotonicity of [ J " ]and [ J " ] - ' ,as does (4). Now it is easy to generalize the work of the previous section.

DEFINITION. Let Q, be an operator of order (n,m). We say Q, has been written in Rogers' form if there is a generated m + 1-place relation 52 such that Q, = 9?"F[J"].


192

[CH. 8, $9

THEOREM 8.4. Let A be an indexed production system having the input place-fixing property, an effective pairing function, and a notion of 9finiteness, for which there is a positive canonical coding. The operators of A that can be written in Rogers' form are precisely the monotone, 9-compact operators. PROOF.If @ can be written in Rogers' form, it is simple to show it is a 9-compact, monotone operator. Conversely, suppose @ is of order (n, m ) , is 9-compact and monotone. Let %(y, x i , . . .,x,)

(xi,.

. . ,x,)

E (@[J"]-'G)({y}).

By Corollary 5.2, 92 is a generated relation. We claim @ = % " F [ J " ] . Suppose x E %"F[J"](P), for some 9 E [ d ] "Then . for some y , y E F [ J " ] ( 9 )and % ( y , x ) . Since y E F [ J " ] ( B ) , D, C [ J " ] ( 9 )so by Proposition 8.3, D ; C 9.Also since 9 ( y , x),

x E @ [ J " ] - ' G ( { y } ) or x E @ [ J " ] - ' ( D , ) , so by Proposition 8.3 again, x E @(D;).Since @ is monotone, x E a("). Conversely, suppose x E @(9). Since @ is 9-compact7 x E @ ( D ; )for some D y C 9 . Now, DyCP, so D, and hence y E F [ J " ] ( S ) . Also, x E @ ( D ; )so x E @[J"]-'(D,), so x E @[J"]-'G((y}), and thus % ( y , x). It follows that x E % " F [ J " ](9).

c[J"](9)

9. Non-compact operators

We have seen, in the two preceding sections, that monotone, 9-compact operators play a special role. As the chapter goes on, we will see that such operators play a deep and important role indeed. It is not being facetious to ask: are these notions trivial in the sense that, in the production systems that most concern us, all operators are monotone and 9-compact. In every recursion or o-recursion theory, all operators are monotone. In every recursion theory, all operators are 9--compact, taking %finite to mean just plain finite. What about 9-compactness in w-recursion theories? In this section we show that, in the most important of all o-recursion theories, namely hyperarithmetic theory or w -rec(G(N)), there must be operators that are not 9--compact, under any reasonable meaning of 9-finite. Much of the rest of this chapter will be devoted to dealing with the existence of these non-compact operators.

CH. 8, $101

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193

All the following takes place in w-rec(G(N)). Let @ = [EL]where E consists of the following axioms: axioms for < Zy + x < y GV-.

(see Ch. 1, §4),

+ Gx,

05.

It should be clear that, if P is an infinite set of numbers, @(P) = {5}, but if B is finite, @(9) = 0. Thus Q, is not compact, in the ordinary sense. We show @ is not 9-compact for any reasonable notion of 9-finiteness.

PROPOSITION 9.1. Let S , , SzrS,, . . . be a sequence of infinite sets of numbers. There is a set T which is infinite, but for no n do we have S, C T. PROOF.Construct T as follows. Let n l be the smallest member of SI, put n, 1 in T. Next, let n, be the smallest member of Sz bigger than n ,+ 1 (which exists since S , in infinite), put n Z + 1 in T. And so on. Clearly T is infinite. But we can not have S, C T since n, E S by construction, n,6Z T. Now, suppose a notion of 9-finiteness has been specified for w rec(G(N)), for which there is a 9-finite coding. We show that the w-enumeration operator @ is not 9-compact. Since there is a 9-finite coding, there can be only countably many 9-finite sets. The example above shows that if 9-finite means finite, @ is a non-compact operator, so now suppose 9-finite includes some sets which are actually infinite. Let S1,S,, S 3 , .. . be a listing of all the infinite, 9-finite sets (with repetitions if necessary to make the sequence infinite). By the proposition, there is an infinite set T which has none of the S, as subset. Now, using the w -enumeration operator Q, constructed above,

+

@ ( T )= ( 5 ) since T is infinite. But, if D is 9-finite, and D T, D must be actually finite, since no S, is a subset of T. But then @ ( D ) = 0 .It follows that @ is not 9-compact. 10. All the %finiteness assumptions together

We have been investigating the consequences of certain 9-finiteness assumptions for indexed production systems A. But (assuming the output place-fixing property) the generated relations of A constitute an indexed relational system, and our %-finiteness assumptions for relations in the

INDEXEDPRODUCTIONSYSTEMS

194

[CH. 8, $10

previous chapter were quite different than the assumptions we have been considering here. Now we put all of this together, and find that, under the combined 9-finiteness assumptions, there is a nice connection between non-compact and compact operators. We will make use of this connection in the next section. We note that, by our remarks in Ch. 7, 97, and in Section 6 in this chapter, our present work applies to many w-recursion theories, to hyperarithmetic theory in particular. We begin with a few results about indexed relational systems. Much of the work in this section is based on Myhill and Shepherdson [1955]. DEFINITION. In an indexed relational system 5L,we call a function g n - m -extensional if

R : = R ; implies RRm(,)=RZh). LEMMA10.1. Let B.* be an indexed relational system. If g is an n - m extensional generated function, then g is monotone in the sense that

R Z C R ; implies

R,"(,,CRzb1.

PROOF.Suppose R Z C R ;, and c E R K;.) ; we show c E R z b ) . Define a set C by C = {x c E R z x ) } . (1) C is generated, since x E C ( 3 y ) [ g ( x )= y A U m ( yc, ) ] . (2) Thought of as a set of n-ary indexes, C is closed. For, suppose x E C and R : = R ; ; we show y E C. Well, since R : = RY and g is extensional, R S x )= R Z , , . Since x E C, c E RK;,,, hence c E RK;,, and so y E C. Now, a E C since c E R Z n ) ,and R : C RE, so by Theorem 7.4.2, b E C which means c E R z b ) .

I

LEMMA10.2. Let 9'td be an indexed relational system meeting the 9finiteness assumptions of Ch. 7, 97. If g is an n-m-extensional generated function then g is compact in the sense that, if c E R Z B )then there is some 9-finite relation R R with c E R z b ) .

I

PROOF.Say c E RK;,,. Again let C = {x c E R Z x ) } As . in Lemma 10.1, C is generated and closed (as a set of n-ary indexes). And a E C. The result follows by Lemma 7.7.2. In Ch. 7, 97, when we discussed 9-finiteness, we took n-ary %finite relations as primitive, not just 9-finite sets. It seemed simplest to avoid dealing with pairing functions at that time. But now we have introduced

CH.

ALLTHE %FINITENESS ASSUMFTIONS TOGETHER

8, $101

195

them, so relations can be collapsed to sets. The following deals with this issue. LEMMA10.3. Let A be a production system with an effective pairing function J and a notion of 9-finiteness for which there is a 9-coding (not necessarily positive or canonical). If any of the following holds for n = 1, it also holds for arbitrary n. (1) The relation

y is a 9-finite code and x E DJ:

is generated. ( 2 ) The relation y is a 9-finite code and x!iZ DY

is generated. (3) If B C d" is generated, then there is a chain V of 9-finite n-ary relations such that (a) {x D:E V} is generated, (b) = 9, (c) if 8 is any proper initial segment of V, U%is 9-finite.

1

u%

PROOF.Left to the reader.

Now, finally, our main result. THEOREM 10.4. Let A be an indexed production system having both the input and the output place -fixing properties, and an effective pairing function. Suppose also that A has a notion of 9-finiteness meeting the following conditions. ( 1 ) There is a canonical coding of the 9-finite sets. ( 2 ) The collection of 9-finite relations is closed under sections. (3) If B is a generated set then there is a chain V of 9-finite sets such that (a) {x ID, E V} is generated, (b) U V =B, (c) U%is 9-finite, for any proper initial segment % of V. If these hypotheses are met, then if g is an n-m-extensional generated function, there is a monotone, 9-compact operator @ in A of order (n, m ) such that @ ( R:) = R for every generated relation R z. PROOF.The generated sets and relations of A constitute an indexed

196


[CH. 8, $11

relational system, and by Lemma 10.3, all the finiteness assumptions of Ch. 7, §7 hold for it; and consequently Lemmas 10.2 and 10.1 may be applied. Let F be the operator such that F ( 9 ) = { x D, 9}. Let f be a generated function such that D: = Rj,,) (see Lemma 7.7.3). Define a relation P by

1 c

W , Y l

ym)

' . . . I

@

(Yl,.-.,ym)ERz(r,.

P is generated, so the following is an operator in A : @ = P"F[J]".

We leave it to the reader to verify that @ is monotone and 9-compact. We show @(I?:)= I?:(,). (1) Suppose u E @(I?:). That is, u E P"F[J]"(R:). Then for some w E F [ J " ] ( R &P ( w , u ) . Now w E F [ J " ] ( R : )implies D, c [ J " ] ( R :im) plies D ::C R :. Also P( w, u ) so u E R ; ( w ) . Let z = f (w).By definition of f , D z = R : , hence we have u E and R : C R:. Then by Lemma 10.1, 0 E R,",,). ( 2 ) Suppose u E R,",,). Then by Lemma 10.2, there is some 9-finite RE R: with u E R;(b).Since R ; is 9-finite it has a 9-finite code, say c. Thus D : = R : . Then D : C R : , so D c c [ J n ] ( R : ) and hence c E F [ J " ] ( R : ) .Now by definition of f we have D : = Rj,,), hence R i = R;( l (Vx E K ) ( ~ Y ) Q ( ~ , Y ) .

Since Q satisfies weak replacement, there is a 9-finite set H such that and

(Vx E W ( 3 y E H ) Q ( x , Y )

(VY E H ) ( g x f W Q ( x ,Y). Note that, by the second of these, if y E H then, in particular, y E F [ J " ]( 8 )and hence 0 ;C 8. Now, let K ' = u,,,DJ:.By hypothesis, K' is 9-finite. Also, by our previous remarks, K' C 9. Finally, K C @ ( K ' )since: suppose x E K ; then for some y E H, Q(x, y ) . That is, y E F [ J " ]( 8 )and R ( y , x), But then 0 ;C 8. Now if y E H, in fact DJ:C K ' , so y E F [ J " ]( K ' ) ,and R ( y ,x). Then x E R " F [ J " ]( K ' )= @ ( K ' ) . Now we continue by looking at the consequences of every 9-compact operator being 9-metacompact, which appears in the conclusion of Theorem 14.5. First, a lemma.

LEMMA 14.6. Let A be an indexed production system having the input piace-fixing property, a notion of 9-finiteness, and an effective pairing function. Suppose also that the 9-finite relations are closed under projections. Let @ be a monotone, 9-compact operator in A of order ( q + k, n ) . Then there is a monotone, 9-compact T E A of order ( k ,q + n ) such that, for

9 E [&Ik,

( Y I , . . . , y.) E @({(XI,. .

e

(XI,. . . ,xq, y l , .

. 1

Xq))

x 9)

. . , y n ) E W8).

PROOF,By the input place-fixing property for A, there is a basic function f : d q+& such that y E @({x} x 8 ) e y E @F$(8).

Since f is basic, hence generated, there is a constant operator, 1, E A of order ( k , q + 1) and value f . Now, set q+n+1

q+n+2

T = Pq+l Pq+l

q+n+Z Eq+l.q+2(zf

x

wk.n)).

CH. 8, 5141

COMPACT CORES

215

and from this the monotonicity of W follows easily from that of @. We show W is 9-compact. Suppose (x, y ) E W ( 9 ' ) . Then y E @({x} x 9) and since @ is 9-compact, y E @ ( K )for some 9-finite K C {x} X 9'.Then by Lemma 13.1, K C G X H where G, H are 9-finite, G C { x } and H C 9. If K # 0 then G # 0 which implies G = {x} and hence K C { x } X H. If K = 0, trivially K C {x} X H. So, in any case, K C {x} X H. Since' @ is monotone, @ ( K )C @({x} x H ) , so y E @({x} X H ) . Then (x, y) E T(H) where H is 9-finite and H C 8. Hence W is 9-compact.

THEOREM 14.7. Let A be a full production system and suppose, in addition : (1) the union of two 9-finite relations is 9-finite, (2) the collection of 9-finite relations is closed under projections, (3) singletons are g-finite, (4) 0 is 9-finite, ( 5 ) every monotone, 9-compact operator is 9-metacompact. Let A " consist of just the monotone, 9-compact operators of A. Then A o is also a full production system, with the same relations generated as in A (using the same notions of basic and 9-finite as in A ) . PROOF.Since singletons are 9-finite, then, as observed earlier, 9metacompactness implies 9-compactness. Hence, in A, for monotone operators, 9-compact = 9-metacornpact. Constant operators are trivially monotone. Also, if @ is constant it is 9-compact since, if x E @(8) then x E @(0) and 0 C 8 and is 9-finite. Thus A " contains all the constant operators of A. It follows that the same relations are generated in A and in A". In particular, 0, =.d and singletons are generated in A". Projection operators are 9-compact: say x E P ? ( 9 ) ;then for some y , ( y , x ) E 9; but { ( y , x)}, being a singleton, is 9-finite, x E P:({(y,x)}) and { ( y , x)} 9.They are trivially monotone. Hence they are in A". Similarly transpositions and place-adding operators are in A". Suppose @, W E A". Since they are in A, @ T E A. @T is easily checked to be monotone and 9-metacompact, using that these are true of @ and 'I' separately. Hence @7y. E A". (It is here that we use 9-compact = 9met acompact .) Suppose @, T E A " . Trivially @ f l T is monotone; we show it is 9-compact. Suppose x E (@ f ul)(9). l Then x E @(8) and x E T(9).


216

[CH.8,414

Since both are 9--compact, we have x E @ ( K ' )and x E W ( K " )for some 9-finite K', K " C 9'. But K = K' U K" is 9-finite, and by monotonicity, x E @ ( K )and x E ?P(K).So x E (@ r l ly)(K) where K C P. Thus @ n ly is 9--compact, hence in A". That A" is closed under X is a similar argument, while closure under U is easier. Thus A " is a production system, with the same relations generated as A. Next we show A " is indexed. We produce a universal operator for the collection of operators in A" of order (n,m). Since A is an indexed production system, its generated relations constitute an indexed relational system. Let U'"" be a generated m +2place relation which is universal for the generated m + 1-place relations of A. Define a relation S by S = Ty''( Urn+'). Now, define an operator %!L('"."' of order (n,m + 1) by

S"F[J" 1.

~ ( n . m > =

By Theorem 8.4, %!L("."') is a 9-compact, monotone operator of A, hence it is in A". We show it is universal. Let @ E A " be of order (n,m). Then @ E A, is monotone and 9compact. By Theorem 8.4, there is an rn + 1-place relation R, generated in A, such that @ = R " F [ J " ] .Let i be a relational index for R, so R ( y ,x ) e Um+'(i, y, x ) e S ( y , i, x). Then, for allowable 9, x E @(P)e x E R " F [ J " ] ( P )

e

( 3 Y ) [ R ( Y , X ) A Y EF[J"I(9')I ( 3 Y ) [ S ( Y ? i , X ) A Y EF[J"I(P)I

(i, x ) E S " F [ J " ](9)

e

( i , x ) E .(""'(P).

Thus 021'""' is universal, and A " is indexed. Next we show the output place-fixing property holds in A", using the same notion of basic as in A. Let @ E A" be of order (k,q n). Then @ is monotone and 9-compact, so there is a generated q + n + 1-place relation R such that @ = R " F [ J k ] . Now, the iteration property holds for the generated relations of A, so there is a basic function f : d q+ d such that

+

R(y,x i , . . . ,xq, y i , . . ., y n )

~ " + ' C ~ ( X I.,..,

xq),

Y , Y I , . . . ,y n ) .

It then follows easily that, using the indexing of operators defined above,

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217

Thus we have the output place-fixing property in A". Next we turn to the input place-fixing property. Let @ E A" be of order ( 4 + k, n). By Lemma 14.6, there is an operator P E A of order (k,4 + n } which is monotone and 9-compact, hence in A", such that Y E @({x)x 9),3

(X,Y)E

WP).

By the output place-fixing property applied to 9 in A", there is a basic f :d q-+ d such that (X,Y)E

9(W

(f(X),Y>E

.ck.n)(p)

and this immediately gives us the input place-fixing property for @. Finally we turn to the 9-finiteness conditions. A" has an effective pairing function since A does, and the two have the same generated relations. Similarly, the relation: y is a 9-finite code and x E D,, being generated in A, is generated in A". And the operator F E A such that F ( 9 ) = {x D, C 9'1 is easily seen to be monotone and GB-compact, hence it is in A". Thus A'' is a full production system. This ends the proof.

1

Finally, we introduce some terminology, then we combine the effects of several of the preceding theorems.

DEFINITION. Let A be a production system with a notion of 9-finiteness and an effective pairing function. Let 9 be the collection of semihyperregular relations of A. By the compact core of A we mean (A /9)", using the notation of Theorems 14.1 and 14.7. THEOREM 14.8. Let A be a full production system, and suppose: ( 1 ) every generated relation satisfies weak replacement, (2) the 9-finite union of 9-finite relations is 9-finite, (3) the 9-finite relations are closed under projections, (4) every finite relation is also 9-finite. Then, the compact core of A is also a full production system with the same relations generated as in A, in which allowable = semi-hyperregular, and in which every operator is monotone and 9-metacompact. In addition, if the 9-finite coding of A is canonical (not just positive canonical) the same is true of the compact core. And if the 9-finite approximability property holds in A, it also holds in the compact core.

218

INDEXED PRODUOIONSYSTEMS

[CH.8,815

PROOF.The hypotheses here allow the application of Theorems 14.4, 14.5 and 14.7, which give most of the conclusion. The other 9-finiteness conditions mentioned only refer to generated and to $@-finiteness,neither of which changes in passing from A to its compact core. This observation yields the rest of the conclusion.

EXAMPLE. w-rec(Q(N)) = hyperarithmetic theory, satisfies the hypotheses of Theorem 14.8, hence its compact core is a full production system. This is useful since then we have Rogers’ form available. But, using Theorem 13.5, there are sets which are not semi-hyperregular in w-rec(G(N)), hence which are not allowable inputs in the compact core. Thus it is a partial production system. 15. The least fixed point theorem, again

In Section 11 we proved a least fixed point theorem in an axiomatic setting. Now we do it again. The present proof involves different more restrictive hypotheses, but is, correspondingly, more elegant. The material is based on work of Scott and Park, see Scott [1976]. The proof below only applies to operators of order (1,l). At the end of the section we discuss how to reduce the (n, n ) case to the (1,l) case, but the assumption of an onto pairing function seems to be essential. Suppose A is a production system with a notion of 9-finiteness for which there is a positive canonical coding. Then, as in Section 7, certain operators @ of order (1,l) can be expressed in Rogers’ form, @ = %’IF, where 3 is a generated, binary relation. If we also have an effective pairing function J, then every generated binary relation can be written in the form [ J 2 ] - ’ ( 9 where ) 9 is a generated set. For, suppose 3 E [d]’is generated. Let 9 = [J‘] (3).Then $3’ is also generated, is a set, and [J’]-’($3’)= [ J ’ ] - ‘ [ J z ] ( 3 = ) 3 by Proposition 8.2. DEFINITION. Let A be a production system, with a notion of 9-finiteness for which there is a positive canonical coding, and an effective pairing function J. For each generated set 9, by @p we mean the operator in A of order (1,l) given by @p = ([J’]-’(Q))”F. PROPOSITION 15.1 (Existence of a diagonal operator). Let A be a production system with a notion of 9-finiteness for which there is a positive canonical coding, and with an effective pairing function J.

CH. 8, 9151

THELEASTFIXEDPOINT THEOREM, AGAIN

219

There is an operator F € A of order (1,l) such that, for generated 9 E [.PI’, F(9)= @9(9). PROOF.Let

F = P:(T7,2A’Fn [Jz]-’). Then, if 9 E [.PI1is generated,

x E F(9)

e x E P:(T:.2A’Fn [ f 2 ] - ’ ) ( 9 ) e for some y, ( y , x ) E T : . z A ’ F ( 9and ) (j

( y , x )E[J2]-’(9)

for some y , (x, y ) E A ’ F ( 9 ) and ( y , x ) € [.J2]-’(9)

e for some y , y E F ( 9 ) and e x E([J2]-’(9))”(F(9)) e X€@P(9).

( y , x ) E [Jz]p1(9)

REMARKS. Let A be a production system with a notion of 9-finiteness for which there is positive canonical coding, and with an effective pairing function J. Let @ be an operator of order (1,l) and suppose the composition @F turns out to be @9 for some generated set 9 (here ?P is the diagonal operator of Proposition 15.1). Then @ has a generated fixed point of a rather interesting form, namely GP(9). This, being an operator applied to a generated set, is generated. And it is a fixed point since @(QP(9)) = @?P(P) (general behavior of = @?($P)

F)

(since @?P = @).

This construction of a generated fixed point for @ is based on an argument in Combinatory Logic. Its application to recursion theory is due to Scott. See Scott [1976]. Now, one reason this particular fixed point for @ is of interest is that Park has shown that under reasonable assumptions it will be the least fixed point of @. The following argument is essentially his. See Scott [ 19761, pp. 569-570 and p. 577. (Park, himself, never published this argument.)

THEOREM 15.2 (Least fixed point theorem). Let A be an indexed production system having the input place -fixing property, with a notion of 9-finiteness for which there is a positive canonical coding, and with an effective pairing

220


[CH. 8, Pl5

function, J. Suppose also that every operator in A is monotone and 9compact (see Theorem 14.8). Suppose, in addition, there is a well-founded, transitive relation < on d (it need not be generated) meeting the two conditions: (1) x < J ( x , Y >, ( 2 ) if x E 0, then x < y. Let @ E A be of order (1,l). Then @ has a generated least fixed point. PROOF.@ € A is order (1,l). We produce a generated least fixed point for @. Let W be the diagonal operator of Proposition 15.1. P E A , hence @W E A. Since we are supposing every operator in A is monotone and 9-compact, this applies to @P.Then by Theorem 7.3, @!P can be put in Rogers’ form, hence by our remarks at the beginning of this section, @W = @p for some generated set 8. By the remarks preceding this theorem, @p (9) is a generated fixed point for @. What we now show is that it is least among the allowable fixed points. Let $34 be any allowable fixed point for @. We must show @, (8)G % or, equivalently,

c

W(8) %. Suppose we couId show

(*I

c8

(**I

D Y

3 P(DY)5%.

This would suffice to establish (*), for, if x E P(8)then, since T is 9--compact, x E P(D,) for some D, 9. Then by (**), P(D,) 93,hence x E 9. Thus we show (**). We suppose (**) is not true. Let k be a minimal member of d,in the < ordering, such that (**) fails, that is,

c

Dk

c9

but not

c 9.

q(Dk)

Choose some yo such that yo E W ( D k )but y o fZ 93.Since yo E P(Dk)and ( D k ) or

W is a diagonal operator, then yo E yo

E ([J*]-’(Dk)))’F(Dk).

That is, there is some D, C Dk with J(t, yo) E Dk. Now, by our assumptions on
REMARK. The proof is much like that of Theorem 3.5.1, but now Zreplacement must be brought in, to ensure that If-finite derivations arise. We leave details to the dedicated reader.

11. Combining augmented If-strucures

The material in Chapter 4, does not apply directly in the present setting, since we must now work with augmented structures. However, only minor adjustments are needed. We begin in this section by defining U and X to take usable partial orderings into account. Recall, in Chapter 4, U and X were defined for domains in Section 2, certain maps i and j were defined in Section 4, and finally, U and x were defined for structures, in Section 5. We make use of all this, below.

246

ADMISSIBLE SETRECURSION THEORIES

[CH. 9, 812

DEFINITION. Let F be a family of admissible sets, and let (B,

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