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This book is dedicated to Donat el 1 a Barnocchi and Dieter Rddding (t24.8. 1937, t 4 . 6. 1 9 8 4 )
To both of them I owe more than this book  its beginning, its being completed and the best of its contents. I owe them, in particular, their example: i t consists in confronting persons and situations in life and science selflessly and with an open mind, and never abandoning the purpose of recognising what is essential and true and to think and act according1 y.
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PREFACE
The theme of this book is a pair of concepts, already recognised as belonging together by Leibniz, whose mathematical development from Frege to Turing has laid the theoretical foundation of computer science: the concept of formal language as carrier of the precise expression of meaning, facts ,problems, and the concept of algorithm or calculus, that is, formally operating procedure for the solution of precisely described questions and problems. The book gives a unified introduction to the modern theory of these concepts, to the way in which they developed first in mathematical logic and computability theory and later in automata theory, the theory of formal languages and complexity theory. Apart from considering the fundamental themes, and nowadays classical aspects of these areas, the subject matter has been selected to give priority throughout to the new aspects of traditional questions. results and methods which have developed from the needs o r knowledge of computer science and particularly of complexity theory. The aim of this book is twofold: to be a textbook for introductory courses in the abovementioned disciplines as they occur in almost all current curricula of computer science, logic and mathematics, but apart from this, to be a monograph in which further results of new research (to a large extent in textbook form for the first time) are systemetically presented and where the attempt 1s made to make explicit the connections and analogies between a variety of concepts and constructions. A price must be paid by the reader for the knowledge I expect him to acquire when and if the experiment is successful: for the beginner the first lectures of the text will be difficult due to the profusion of concepts, remarks and forward and backward references to currently posed clusters of problems  particularly if he approaches the material by selfstudy unaccompanied by lectures. My advice is to initially skip over those parts which, despite study, are not understood; the connections will spring to mind on second reading. The following remarks on the use of the book might be helpful; I have employed all parts of this book as the basis of introductory o r advanced lectures on the foundations of theoretical computer science, automata theory and formal language, logic, computability and complexitytheory. To enable the reader to recognise the use and interdependence of the various parts I have devised a detailed table of contents and a graph of interdependence. The sections marked with
*
IV
Preface
contain material which is not treated in the basic courses but is suitable to follow them. The arrangesent of p r o p o s i t i o n s as theorem, lemma, remark and exercise mirrors the methodical significance of the various states of affairs from a contemporary point of view. It says nothing about historical or individual achievements to have proved these propositions for the first time. Many a significant proposition becomes a simple example as a result of later progress. I strongly recommend beginners to work out with pencil and paper, at first reeding, all matters of routine or intermediate steps which are not explained in detail and to solve the exercises, or at least, try to solve them. By doing this one not only learns whether one has really understood the preceding subject matter and how to apply i t , but one also acquires a feeling for what is essential in the techniques used. In this endeavour it might help that I have tried to express complicated ideas occurring in proofs without the use of formulas. The reader is advised to use this method of intuitive, but precise substantive thinking which opens the way to a deeper understanding. The r e f e r e n c e s t o l i t e r a t u r e at the end of each section are considered as completions of those references given in the text. I would like to express my heartfelt thanks to the many persons who have helped with the work on this book in the past I name years, by no means all of whom I am able to mention. in particular the following colleagues and collaborators who read the manuscript in whole or in part and who have given me T.Brand, valuable criticisms: K. Ambo6Spies, H. Brlhmik, A. BrUggemann, H. Fleischhack, J. Flum, G.Hensel, H. KleineBUning, U. Ltiwen, L. Mancini, K. May, W. Rtidding, H. Schwichtenberg, D. Spreen, J. StoltefuB, R. Verbeek, S. Wainer. Separately I would like to thank: K. AmbosSpies, whose elaboration of one of my Dortmund logic lectures I have partially used in chapters D/E, and who has given valuable help, particularly to 5BII3; U.Ltiwen for a critical reading of the entire manuscript and the preparation of the symbol and subjectindices; K.May f o r careful corrections and numerous drawings; H.W.RBdding for the intricate control of the bibliography. I would especially like to thank U.Minning, R.KUhn, J. Kossmann, P. Schoppe and K. Gruhlich for the precise transposition of parts of several versions of my manuscript into the typescript for the printer. U.Minning has borne the main burden in this  her engaging and friendly manner has often allowed me to forget this arduous labour. Finally, but not less heartily, I thank Walburga RSdding and many other colleagues who, in the past difficult six
V
Preface
weeks, have given me their spontaneous moral support, thereby decisively helping me to complete this book.
Dortmund, 3. 7. 1985
EGON BURGER.
Note on the second edftfon. At this point I would like to M. Kummer, P. PUppinghaus, express heartfelt thanks to V.Sperschneider: mainly because of their list of errors corrections have been made in the second edition. At this point also I thank in advance all those readers who show me further errors.
Pisa, Spring 1986
EGON BURGER.
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CONTENTS
Graph of dependencies Int roduct ion Terminology and prerequisites
XI v
xv
XVI II
Book One
Chapter A
ELEMENTARY THEORY OF COMPUTATION
1
THE MATHEMATICAL CONCEPT OF ALGORITHM
2
PART I. CHURCH'S THESIS 2 Explication of Concepts. Transition systems, 2 Computation systems, Machines (Syntax and Semant ics of Programs), Turing machines, structured (Turing and registermachine) programs (TO, RO).
§l.
52.
53.
54*
55.
Equivalence theorem. F, 6 F ,
. . . , s3
3
> >
sj
instead of (s,,I
>,
...,
S j ' n c
j
> >)
f
i
or a s S.i, 1 2 , *
A s there
* ,
s j < r  # r 3 ,
>
1
sj.
is only essential advantage in considering such calculi with manyplace rules (read: rules with more t h a n one premiss) by dealing with logical calculi, in the first book we
A1 Church's Thesis
6
consider, in general, premies.
only transitionsystems
with just one
EXAMPLE 1. <Word substitution). A s d  T h u e s y s t e m over an alphabet A is a transitionsystem for the substitution of formally: subwords V, in given words by words W, ; S = A*,
3
1
= {(VVSW, V W i W ; V M A A 3
for 1 E n. The rules +, will mostly be denoted by stating V, and their conclusions W,: their premisses 4 ,
=
V,+W,
or
(V,, W,).
muesystems are semiThue systems which for each rule V, + W, contain its inverse rule W, 4 V, (read: defining equations V, = W,); Thuesystems are reversible. Introduced by Thue in 1914 in connection with grouptheoretical investigations, semiThue systems form, in the intuitive sense, effective procedures for the schematic alteration of sequences of signs, dependent only on the external form of those signs. Stemming from Chomsky, 1956, semiThue systems have entered into the study of formal languages, particularly programming languages. Essential for this was the observation that grammatical analysis of complex concepts could be described by special combinations of simpler concepts via appropriate substitution rules. The decomposition of "sentence" into a subject followed by a predicate. for example, by the rule "sentence
*
subject predicate"
gives the particular case of the derivation of "Luca laughs" by means of the "lexical" rules "subject 4 Luca", "predicate laughs".

We will deal with such applications of semiThue systems in the study of formal languages in Ch.CV.
A normal P o s t c a l c u l u s over EXAMPLE 2. (Post 1943, 1956). alphabet A is a transformationsystem with S = A*, +,= t, denoted mostly by V, + W, or by (V,, W,) for i E n. To apply +, to a word is a transition system, In: I + S is the input function defined Out: S + 0 (the socalled output on the set I of inputs, function is a function defined on S with values in the set 0 of outputs and Stop: S + to, 1 ) is the stopcriterion. The computed (result) relation R e d Kl is Out0 =
g(m
f t R , y + l ) = h t P , y, f t R , y ) )
and
iff V i l y f ( 2 ,y ) r . For example, + arises by p r i m i t i v e recursion form U 1 * NoU",Z*,. from C1,c' and h with h ( x , y , z ) = Xtz, exponentiation from C ' , and h with h ( x , y , 2 ) = x  z . ftrr, y).L
and and
Define the p r i m i t i v e r e c u r s i v e r e g i s t e r o p e r a t o r s PRO (also the socalled loopprograms) as the R O ,but using, instead of the unrestricted (precursive) iteration, the following restricted (primitive recursive) iteration: (slM), where M contains neither a, nor sl. Show that: if f = PR(g,h) and g , h are computable by LOOPprograms, then so also is f. Exercises 13 for TO and exercise 4 for RO show that simultaneous substitution, primitive recursion, and application of the poperator as processes for the definition of new <more complex) functions from given <more simple) functions, give procedure6 for the construction of normalised TMprograms. That, conversely, the functioning of arbitrary (not only normalised) TMprograms can already be described by use only of those definitionschemata and particularly simple basic functions (and thereby inductively, read: solely by means of structured programs), shows the
THEOREM. The Equivalence Theorem The c l a s s F (TM) of t h e Turlng machine computable f u n c t i o n s is equal t o F , , the class of t h e precursive or p a r t i a l recursive f u n c t i o n s , d e f i n e d i n d u c t i v e l y (Kleene 1936) b y 1 ) N, C", c, U", 6 F, ( t h e socalled i n i t i a l f u n c t i o n s ) f o r n , k c N , 1 6 j E n. 2) I f f a r i s e s from g, hi ( 1 < i C n) by simultaneous s u b s t i t u t i o n or from g, h by p r i m i t i v e recursion or from
26
A1 Church's Thesis
g by application of the poperator , and if g,h,,h E F,, then so also is f € F, 3) f E F, only on the basis of 1 ) and 2 ) . (induction principle) The class F,,,, of primitive recursive functions is defined as for F, without the use of the poperators. f is called recursive in case f is total and partial recursive.
S2, PROOF. F, ci F < T M ) follows from exercises 13 on TO and exercise 4 on RO. I Note that the exercises on closure properties of TO'S are true analogously for RO' s. EXERCISE 0. Draw a flowdiagram for a compiler which associates with each precursive definition of a partial f an RO M which works out f (ie so recursive function that M computes f). Hence verify that each f € F,,,, is computed by a LOOPprogram. F o r a precursive normal form representation of all f E. F C T M ) , in part I1 we will effectively (in the intuitive sense) codify machine programs p and machine configurations by numbers and show that because of this codification the elementary machine operations as well as the singlestep function for p g o over into primitive recursive functions, from which  through a single use of the poperator in the description of the stopping state of p for given inputs the result function for p is easily seen to be precursive. Kurt Gadel in 1931 first introduced the effective codification of formal systems by means of numbers in a systematic manner that is now known in this connection as W e l i s a t i o n Through exploitation of the structural similarity between R O ' s and precursive functions we can reduce the Gbdelisation apparatus in the proof of the precursiveness of TMcomputable functions to a minimum by simulating TMprograms through RO's (F(TM) s F (RO)) and by describing the operation of RO's precursively (F(RO) 5 F,). a) In preparation for the proof of these inclusions, as for the briefly mentioned normal form representation of TMcomputable functions, we give some examples of pridtive recursive functions and derive some closure properties of the class of partial (primitive) recursive functions. Because of the (LOOPprogram) computability of the primitive recursive
AI. 2 Equivalence Theorem
27
functions one can consider (primitive) recursive descriptions of definition processes as mathematical reduced forms of (LOOP) program synthesis procedures. The reader who is not interested in the solution of such programming problems may skip over the following six lemmas. with the closure under explicit definition: if for all I? with a term t consisting only of number variables and constants a s well as signs for functions in FF,,,mor F,, then f is primitive or partialrecursive, respectively. More exactly: We
begin
f(27 = t(27
...
v,, v,, be variablesymbols and gi DEFINITION. Let i E m) symbols for r(I)place functions. We define inductively terms over 8 1 , . , g,,, (as names for numbers) by: (1 E
..
and numbers k E N are terms Over 8 1 , gn,. 2). I f t , , , t F n Z i , are terms over g , , . , ,gm, so also are g i < t , , . .. , t , . c : i , ) ( 1 4 i 4 m). Variable6
1).
...
< f E N)
v1
, ..
3).
.
is a term over
t
8 1 , .
. .,g,, only
by virtue of 1 ) and 2).
DEFINITION. f C r t J is called e x p l i c i t l y d e f i n a b l e from exactly when there is a term t over g,,...,g,,, in which at most the variables v , , . . . , v,., say, occur and numbers 1 < f,,. ..,f#E n occur, so that for all x , , , , x,, C N there holds: g,,.. . , g n ,
.
f(x,, In this,
t
VI,
. . . , x,>
.. .
I
= t
[y,,. v,
v,
. . , y,l
*
. . , v,.[ ,
substituting all occurrences of
* *
'
'
denotes the result of
vi
in
t
by
yi,,
DEFINITION. Let D be a process of definition of objects <eg words, functions, predicates) D E K
..
..
.
LEMMA 1. F,,,, def inf t ions.
and
F,
are c l o s e d
under e x p l i c i t
PROOF. B y induction on the construction of terms:
A 1 Church’s Thesis
28
Let Basis. B = x , , . .,xn
.
f(m
and
fCm
= x, for all B or k C N. The f = U n c i
Inductive step. Let ftrn = g . j ( t l ,
..., t
Sj
)
V I ,*
*
*,
C?,
or
= k for a11 f = Cn+k‘.
1
I . . . , ?
Vr
I
for all 2. By inductive hypothesis f , E Frr,,,, defined by f, E F,,, respectively for the functions f , , v , . c x i ” ’ * . ‘ xi, I . f,(m = t , V ’ , Thus f arises by simultaneous substitution from the primitive or partial recursive functions so f E F p r L or m F, respectively.
g,, f , ,
. . . , f.,
or
and I
LEMMA 2. The following f u n c t i o n s are p r i m i t i v e recursive: +, Ax.y*xY, , max, min, Ax. y‘ ( 1 xyl ), sg, sg, E. a ,
PROOF. For addition we give a definition by primitive recursion from primitive recursive functions: x + 0 = U ’ ,’ ( x ) , x + ( y + 1) = h<x,y,x + y ) with h ( a , b, c ) : = N < U J * “ ( a , b, c). Multiplication is defined by a primitive recursion using addition: x.0 = C ’ ,“ ( X I , X < y + l )= h<x,y , x , y > with
h < a ,b, c) : = Us* ’ ( a , b, c)
+ Us, “ ( a , b, c ) .
Analogously,
= h < x, y, xv) with h ( a , b , c ) : = U3*’ ( a , b, c ) * U:’, “ < a ,b, c>. The predecessor function is primitive recursive because
x” = C’, ’ ( x ) ,
XY+’
V( 0) = CCJ. C’, V ( X + l ) = ui’, ’ < x ,V ( X ) ) . It follows that E F,,,, because x A 0 = U ’ *1 ( x ) , x ( y + 1) = h < x , y ,x y) with h ( a , b, c ) = V min(x,y) = x I xyl = max<x y, y x) s g < x ) = 1 A ( 1 A x> x sg(x) = 1 I <max<x,y) m f n < x , y ) ) . c,(x,y) = 1

AI. 2 Equivalence Theorem
29
LEMMA 3.
F, and F,,,, summation, bounded product, minimum and i teration.
are closed under bounded maximum,
bounded bounded
Let g be primitive or partialrecursive. We show that the function f arising from g by the stated processes can be defined by a primitive recursion from functions in F,.,,,,, F,., respectively. In doing this we use, without further mention the closure of F,, F,,,,, under explicit definition. For f t P , y) = 1ic,gt2,1) we have PROOF.
f(2, 0) = 0
ftP,*f)
Similarly for
+.
For
= f t r r , y)
+
fli,,g(2, 1)
gt2, y).
with
f(2, y) = max(g(2, 1); i
1
< y>
and
instead of
0 and
we have
ftn;, 0) = 0 f ( 2 , y t I ) = max(max(g(2, 1); i < y > , g ( 2 , y)). Similarly for min = x+1, s(x) = *l, p(x,y) = < y , x ) for permutation and (for the expression of the “f(x,P, = (x,f < p ) be the left locality of RMinstructions), 2c
. . .
write
46
A1
Church's Thesis
Considering computationuniversal tag systems, the premisses of whose rules all have length 2 , this machine meaning of normal Post calculi can be so improved in a natural manner that the nondeterministic version of the machine type arising is equivalent to Turing machines, whilst the deterministic variant can be proved equivalent to the socalled linearbounded Turing machines, whose available tape for each computation is restricted by linear dependence on the length of the input word (say, by setting leftmost and rightmost boundary symbols) and whose halting and wordproblems are thereby trivially recursive. See Nojima et a1 1973.
EXERCISE (Shepherdson & Sturgis 1963). Define for arbitrary A nword register machines of "configurations" c (consisting of the program M and the current data R) corresponding to the stepwise execution of M are described by a primitive recursive nextconf igurat ionfunction in the sense of: VC,C': nextconfig(€) = €' iff C   + I C ' .
step : = nextconfig, s(z) : = ( z ) , out(z) : = ( z ) ~n ~ r,,
A
:=
and
iter(nextconfig).
Then: f(d) = ( A ( < M , < m > , p y ( ( A ( < &
w
L
(initial configuration) = outo(step),oin(&

< m > Y, ) ) ~ = O ) ) )  ,,., .
' (stop
position)
(out put result)
2)
For the second part of the normal form theorem we define primitive recursive predicate T (the socalled Kleene predicate) with the property that TtH,d,y) exactly when is the Gbdelnumber of a terminating computation of the RO started on the input I so that also f(R) = U ( p y T ( f l 2, y with U ( z ) : = ( z ) * , > , ., , ~I . . Define T by:
a
T y M ))
REMARK. According to the Kleene normal form, each partial recursive function can be defined from primitive recursive functions and at most one application of the poperator (of unrestricted iteration). For recursive functions this happens in the normal case, ie only when (WR) ( 3 ~ ) :g( 2,y) = 0.
The input and output functions "in" and "out" depend on n; we shall suppress this in the notation by assuming that n o misunderstanding will arise because of the context. Furthermore we need only have stated the Kleene normal form can for functions of one variable, as via encoding each f""'
54
A11 Universality
/
Recursion Theorem
f"' where f(d) = f ( < A 9 ) . Kleene normal form the function e : = hk, z. o u t 0 (step),ofn(k, z)
be treated as
Thus, via the
generates a sequence e, : = A z . e ( k , z ) which runs through all (oneplace) partial recursive functions; e" stands for enumeration. . A program (eg on a Turing machine) for the computation of e accordingly represents a the part ial recursive function universal algorithmic procedure: each algorithm P can be programmed in it via the Gadelnumber P, so that by for all arguments I of P, inputting this program and < A 9 this process interprets the description P of P and by stepbystep simulation of the computation of F on the input I yields the output P(R) in the form etP, < m ) . In engineering terms, a computer which computes e is a fixedwired universal machine which by any given program P can be changed into a special computer for solving the problem described in P; the development of a special program is a modif ication of the universal machine into a special machine. The existence of a universal algorithm means epistemologically, that there is a critical degree of mechanical complexity beyond which all further complexity depends only on the storage space available in the course of a computation. It says something about Church's thesis that this critical complexity can be set very low: for one thing, universal we know today of some surprisingly "simple" procedures (see Ch. CIV), and for another, the functions occurring in the proof of the Kleene normal form are particularly "simple"  "elementary" in a sense studied in Ch. CII  and recursive functions arising from this can be classified hierarchically in order of growing complexity according to their storage o r time requirements (see Ch.BII1, CI, 11) I,
The above considerations make it reasonable to consider the enumeration f,>, f , , f,, . . . of all computable functions with a f = A l , x. f , ( x ) as the computable enumeration function explication of the concept of universal programming system (with functional semantics). In Ch. CIV , a s a byproduct of a normal form result for the finite automata handled there, we (that is, for will explicitly give a small universal program the computation of an enumeration function for F&,). Therefore we shall restrict ourselves here to fixing the concept of universality by: DEFINITION. For f: X x Y + Z we define f , : = A y . f ( 1 , y) and f an enumeration function for the class tf,; I Q X. call The enumeration is called primitive recursive or partial f F p r ir,, or f F p, recursive according as respectively. Q
Q
AII. 1 Universal Programs
We shall write [ k l ( A ’ l ( K l e e n e ) i n d e x of e k .
55
instead of
e,(m.
k
is called the
On the basis of Church’s thesis a programming language L with semantics g is universal when there is a computable compiler c of the TMprogramming language into L , ie e, = gCci , for all f . The Kleene enumeration function is for each distinguished by an effectiveness property: computable interpreter f of a programming language L with e possesses a computable semantics A l . fi (read: f Q F,,), (even primitive recursive) compiler (read: an h E Fr,?*,,, which for each fparameter I (program i Q L) constructs an equivalent Kleene index (eprogram) h(f), ie with fi = e,,,,). We stress this property in the g Q F , is called a GCIdalnllloberin~ (of DEFINITION. riplace partial recursive functions) when: (Wf‘”’’’ E F L , ) < 3 h recursive): fl(m = gh, 1,(2) for all
I , 2.
is called the t r a n s l a t i o n f u n c t i o n (of f). For simplification we shall suppress the n. Further: If B is a Gbdelnumbering of F ,, then A 2 , y,z. g ( 2 , < y , z > ) is a Gbdelnumbering of and A 2 , y. gt2, y,y ) is a Gbdelnumbering of F,‘ h
I
F , , ‘ I T 4
’I’
I .
REMARK. The property which characterises a GUdelnumbering is often expressed for arbitrary sequences of parameters as the “subst it ut ion theorem” or ’’ iterat ion theorem“ : (Wf‘
it+n’’
Q
F , , )( 3 h recursive) ( W 2 ,
g,:
fv(R)
= gh, p, (2).
(This version is equivalent to that chosen for the definition, for as f Q F,,so also is A z , L f((z),,.. . , (z)”,, X), so that f .
j))I<m I
NB. The first version going back to Kleene and the fixedpoint version occurring in Rogers 1967 are equivalent in GBdel numberings, not however for arbitrary universal programming systems, see Smith 1979. The recursion theorem also holds for subrecursive programming systems, see Kozen 1980.
A11 Universality
60
/
Recursion Theorem
FIXEDPOINT MEANING. The f ixedpoint version says that each effective program transformation f f F L ,has a "fixed point" k, read: a program k that is carried by f into an As an example of the application of equivalent program f(k). this fixed point meaning (of the diagonalisation method) of the recursion theorem we show that no nontrivial property of partial recursive functions (read: of the input/output relation of programs in arbitrary G6del numberings) is algorithmically decidable:
THEOREM
(Rice 1953) For 0 , F,,, the indexset, n o n  r e c u r s i ve. C
#
PROOF. Let e, c C, then f f F , , for
e, P C.
arbitrary C c FLt, where I n d e x ( C ) : = {I; e , f 0 1s
If
Ax. e,
C
C
were recursive
Thus, by the fixedpoint theorem there would be a fixed point k with e,. = e , , ,* ,. Hence e,. f C i f f e, P C In fact from e,. e, P C. accordingly and so e,. f C.
f
C there follows f(k) = j , and From e, P C there follows f ( k ) = 1 , I
NB. It is essential in Rice's theorem that the index set of C contains f o r each program 1 in it also all programs equivalent to it (ie computing the same function), and thus for a describes an input/output relation of programs; subrecursive analogue see Kozen 1980. EXAMPLE. relations C
Show that 5
F , , x . .
Rice's theorem holds for arbitrary between precursive functions.
. xF,
RECURSION MEANING. The recursion theorem yields a general schema for the definition of computable functions by
AII. 2 Diagonalisation Methods
61
"recursion". The basic idea of recursive definitions is to define f(X) in terms of "previously" defined values for arguments which, with respect to a given ordering of all arguments, precede 2. Simple examples are primitive recursion or the simultaneous primitive recursion reducible to it and courseofvalues recursion. The recursion theorem allows in the computable domain an even wider inference : it allows one to rely on arbitrary values f ( 9 ) , in case this reference back is partial recursive in B and an enumeration index k of f. In terms of functional equations this means that for each g f F,, the recursion theorem guarantees a solution f o f the equation Ckl = A B . g ( k , B ) in k. We also say here that this equation defines the function f = e, implicftly. In the concrete case such a definition means determining a recursive procedure k with action f working s of storage places in which the code k of on the names the recursive procedure to be written will be filed; in the development of this recursive procedure reference can be made to the name of this storage position, in order to anticipate the result of the program on given arguments (later) stored there. A typical application of the recursion meaning consists of the proof of the existence of computable function satisfying A s a simple specifications given by an implicit definition. example we choose an
Implicit definition o f a recursive enumeration function for F r.p ..,. (Wh E F..,,,,,) ( 3 1 ) (V8): h(R) = p r ( i ,< m )
pr
PROOF. We define p r as the solution in k of an equation Ckl = A i , z . g ( k , i , z ) in which, when determining the expression g ( k , i , z) which is partial recursive in k , 1, z. we imagine z as the codification a:
= < 2 , n, i >
P 3 .3: =< 3, n , i >
.
F o r h ( B ) = g ( h , (B),. . . , h , , ( R ) ) let h c B i :' = < 4 , n , g , fi,, . . , fin,>, and for h t B , 02 = h , (B), h ( 2 , y + l ) = 1i2(.F, y,h(B, y ) ) let 6 ~ , , * 1 : = < 5 ,n + l , h , ,h.,>. In the definition of g ( k , i , z ) we follow the definition of F r , v . t , , , :
A 1 1 Universality / Recursion Theorem
62
when ( 1 ) 1 = 1 when ( 1 ) , = 2 when ( 1 ) , = 3
Ik3 Ikl
((I),,, ) > ) when ( 1 ) ,=5, F < P , y+l> z) otherwise.
For each solution k of this equation it can easily be shown by induction on F,,,,,, that , ( V h E FF,L,,,) (Vz): e, (6, z) 4 and that e, possesses the required enumeration property. To show that e, is total, we add, in the case analysis, the condition “ i is the Gddel number of a primitive recursive function”, replace “ otherwise” by “when ( i ) l = 5 h ( 2 ) (i), = 0”
and put
COROLLARY. (NB. e
I
Ckl ( 1 ,z) = 0.
E
pr
is r e c u r s i v e and
p r Z F,,,,,.
F,,Fr,,i.,t)
Implicit definition of an injective trenslation function for en erbitrary W d e l numbering g. ( V f E F k 8(3h ) recursive, Injective)( V i , 2): f ( i , ,W = g,,, i , (23 PROOF. Let g ‘ ” , * l’ be the G6del numbering of F k . < r * ” induced by the given Gddel numbering g c ’  J ” l ’of F,,‘”’, and s”,, the translation function of g ‘ ’  ’ * * in 8‘””’* For f ‘ ” ” ’ E F,. we define implicitly a translation program k which for j < i generates gprograms s ( k , j ) , s ( k , i ) , for f, and f, respectively, which are “syntactically“ different, even though possibly “semantically” equal. We exclude in the implicit definition, the existence of a minimal i with a larger j and equal gprogram s ( k , j ) = s < k , i ) , by fixing a semantic difference on the assumption of the syntactical equality of program translations  from which the semantic equality would follow. i ) for a solution in k of: Set h : = A i . s&,,,(k, 1
AII.2 Diagonalisation Methods 0 when g ( k , i, m
1
=
63
( 3 j < i ) : s = , . , ( k ,j ) = s Z n ( k ,i)
when
(Wj + k ( x , y 4 2 ) (inner square).
C
M I
EXERCISE. Deduce by the Nuremberg funnel principle that there is a 1 ) ( W k ,n ) : [ X c k , a )  Y ( k , a)
(an)
E
nb* mod ( 2 a n

n'
 l)]
Due to this lemma the congruence conditions can be formulated m is congruent by the diophantine requirement that to the number ??y.(an) for a solution mod(2an  n'  1 ) (x,,v) of the Pellequation of a, and that this solution has index k (G1) (G2)
x  ( a 

1)y
= 1,
x  y ( a  n)
s m
x = X(k,a)
mod ( 2 a n

ni

1)
BI. 4 Theorem of Mat 1jasevich
91
(G3) m < 2an  n  1 The second estimate nk < Z e n  n i  1 follows from n" < a: for n = 1 observe that a > 1 (cf G4) and for 1 < n we Thus by the hypothesis n E n'* < a we also have a E n a  1. have
a < (na

+
1)
n(a

n ) = Zan


n
We can give a diophantine condition for following lemma 3:
1.
n"' < a
LEMMA 3. equation.
Growth properties of solutions For a l l a > 1 and n c N
a"' < X ( n , a )
'"I'
= <X(m,a )
I'

+ Y(m, a ) a > +( X < n , a )
 Y(n,a)a>
BI. 4 Theorem of Mat I jasevlch
95
PROOF of the recursion formulas. From the addition formulas there follows: X ( n + 2 , a ) = a a X ( n + l ,a) + ( a  ” l ) Y ( n + l , a ) a s a = X(1, a), and 1 = Y ( 1 , a) X ( ( n + l )  f ,a) = a * X ( n + l ,a)  ( a  ’  1 ) Y ( n + l , a ) .
Thus, X < n + 2 ,a )
Analogously for
+
X ( n , a ) = Z a X ( n + f ,a )
Y ( n + 2 , a)
.I)
Finally
there remains the diophantine description of for 2 < a, x = X(k, a) exactly when the system of diophantine equations and inequalities (Gl)(G7) defined below are solvable for the parameters x , k,a with y , t , v f N and positive b, s, u. x = XI ) ( W k ) : Y ( k , a ) i'l Y ( k . Y ( k , a ) , a ) one we have Y(k,a ) Y(k. Y(k,a ) , a ) and by the other Y ( k . Y ( k , a ) , a) I Y ( 2 k . Y ( k , a ) , a). Hence Y ( k , a ) 'I Y ( n , a ) by the definition of n, and so Y ( k , a ) d Y ( n , a ) a s 1 6 k implies 0 < Y ( k , a). By the Chinese remainder theorem of number theory there is a b > 1 such that b P a mod u and b P 1 mod 4 . y (G3,G6) because 4y and u are relatively prime. by
(Proof: by LEMMA 8 .
( V a > l ,n ) : n even
iff
Y ( n , a) even.
we have that v = Y ( Z k Y ( k ,a),a) is even, and so, because  (az  f ) v  = 1, u is odd: by ud
LEMMA 9. (Wa>1, n ) : Y ( n , a )
and
X ( n , a)
a r e coprime.
u and v are coprime, thus also u and 4y; for a common u and 4y, since u is odd, would divide prime factor of y and therefore also v since yl v by (G5)).
( s , t ) : = ( X ( k , b), Y ( k , b ) ) solve the Pellequation of b, as Hence x = s mod u from required by the second part of (G2). (G6), for from the already established congruence b P a mod u from G6, there follows by lemma 7
s = X ( k , b) Also
k E t mod 4y
=
X ( k , a ) = x mod u.
because by lemma 4 t = Y ( k . b) P k mod ( b  1 )
BI Unsolvable Problems
98
and so also mod 4y as by G 3 bl is a multiple of 4y. The condition k E Y ( k , a ) = y from G7 is finally correct by lemma 3.
PROOF of lemma 9. From tlX(n,a) and tl Y ( n , a ) follows t l 1 in accordance with X ( n , a ) A.  (a'  1 ) Y ( n , a ) . = 1 .
there I
PROOF of T1. We show Y ( n , c)I Y ( n i , c) by induction on when Y ( n , c) I Y ( n l , c) , we also have Y < n , c) I Y ( n ( i + l ) , c) the addition formula Y ( n ( l + l ) , c) = X ( n , c)' Y ( n f , c)
1:
by
+ X ( n i , c)' Y ( n , c).
Suppose, conversely Y ( n , a ) I Y < k ,a ) and nYk. For k = nl+r with O < r < n and 06 f there follows Y ( n , a ) I X ( n f , a ) Y ( r , a ) , because by the addition formula Y ( k , a ) = X ( r , a) Y ( n f , a )
+
X(ni, a) Y(r, a)
and by hypothesiss Y ( n , a ) I Y ( k , a) and Y(n, a)I Ycni, a). As Y(n,a) and X(nf,a) are coprime  each divisor of Y(n,a) also divides Y(ni,a) by case 1, which by lemma 9 is coprime to X ( n i , a )  there follows Y ( n , a)l Y ( r , a ) , contradicting Y ( r , a ) < Y ( n , a ) a s r < n. I
PROOF of lemma 6. From Y ( i , a ) " I Y ( n , a ) T1. Thus, n = i k for some k.
there follows
fln
by I
PROPOSITION ( V f < a ) ( V i , k ) :Y ( f k , a )
1
kX(f,a)"
' Y ( i , a ) mod Y ( i , a )  3
From the proposition there follows Y ( f , a )  l k X ( f , a ) b b  7 Y ( fa, ) .
 for from
c'ld
and
d
5
e mod c' there follows
Y ( i , a ) I k X < i ,a ) h   I . Thus, Y ( f ,a ) I k as Y ( i ,a) and X ( i , a) thereby we have Y ( i , a ) I n.
cI
e

and
are coprime, and
B I . 4 Theorem of Matijasevich
99
The Proposition follows from the unique representation (E) of the elements of a quadratic number domain: from X ( i k , a)
+
Y C I k , a ) a = ( a + a ) '', = < X ( f , a )
+
Y(1, a)a)'.
I
case of PROOF of T2: special k = Y ( 1 , a). (NB: from u f v"w mod v,'
PROOF of le
the F: oposition with there follows v'I u. ) I
Because of the recursion formula
8.
Y ( n , a ) mod 2
Y ( n + 2 , a) = i"aY(n+I, a )  Y ( n , a)
we have: Y ( 2 n , a ) r Y ( 0 , a ) = 0 mod 2 Y ( 2 n + 1 , a) z Y ( 1 , a ) = 1 mod 2.
PROOF
of
formulas:
le
7
by
induction on
X(0, a) = 1 = X(0, b),
X( 1, a ) = a
3
n
I
from the recursion
X l j , n
 X ( j , a) mod X ( n , a)
for all a > l j , n
c
N.
N.
For X(n+(n
?
j),a)
= X(n,a)X(n
_+
+ taA
j , a)
 1 ) Y ( n , a ) Y ( n f j,a) X ( n , a ) Y ( j , a ) ) mod X ( n , a)
5
(ai  1) Y(n,a). t X ( j , a) Y(n,a)
5
t a   1 ) Y ( n , a )  X ( j , a ) mod X ( n , a )
= (X(n,a)

f
as X ( n , a)&  ( a   1 ) Y ( n , a) = 1
l)XCj, a),
z  X C j , a ) mod X ( n , a )
and from this X(2n +
(‘J.1 f
I ) , a)
=
X
1,
all
>
n
0, and all
X ( i , a ) E X C j , a ) mod X ( n , a )
&
0
(Ri),

* *
that
for
all
€,>(R,,)) = 1
(read: Q satisfies f ( m ) . f ( R ) is also called the truthtable condition or the ttcondition and ( R l , . , , R,,) is called the question set (to the "oracle" Q> associated with
.
f(rn
.
EXERCISE. Prove E reduction concept (1)
f
(11)
( 1
(ill)
P E Q
(iv) :
properties
of
the
is reflexive and transitive E
f,,, c
s
(++.
ET
and
Q
recursive
Q and
Q
r.e.
Q P
By defining
iff
implies
implies
P
recursive
P r.e.
(also for m = 1 )
C(P) f , , C C Q )
recursive, t V @ 0 , N):P E.*.+, Q. =.l
by:
P =.r Q i f f P f T Q 6 ,  P and similarly for E . ~ + , P,,,, = 1 , there arise by (1) equivalence classes which we call degrees of unsolvability. By (11) Turing degrees are composed of ttdegrees, these of mdegrees, and these of 1degrees. A degree of unsolvability is called r.e. when i t contains en r.e. element. For the stronger reductions r.e. degrees contain, by (iv), only r.e. predicates, which by (vi) does not hold for the weak reduction concepts (counter example: t t  or Tdegrees of K, see the following exercise); amongst the t t  and Tdegrees there is by (vii), a smallest element, the degree of the recursive sets. EXERCISE. Show that for r,e. A: 1) B
A>,.
follows from
B
f,
A
2) If A is not recursive, there are sets B,C A I+.+ B =*+. C, B e ~ ~ I  Z I , and C E A p  ( I I u I l ) . Consider C < A ) and t < l , x > ;x C A) u t < 2 , x>;x E C(A>).
with (Hint:
b) The determination of the maximal degrees (thus, also of maximum complexity) is fundamental t o the investigation of the structure of degrees of unsolvability as well as for applications of the reduction methods in the following sense:
BII AR / Degrees of Unsolvability
118
DEFINITION. Let 6 be a reduction concept, C a predicate class, and P a predicate. P is called Chard or C  d i f f i c u l t with respect to 4 when for all Q 6 C, Q6P. P is called Gcomplete with reapect to 6 when it is Cdifficult and an element of C. The Chardness of a predicate clearly m a n s that a P is at least as (relative) algorithmic solution for complicated as all the problem in C; Ccompleteness means that with respect to the reduction concept used, P C and each solution of P represents the "essence" of solves all problems in C. EXAMPLE 1
.
KL, is z,complete with reapect to
(Why?)
6,.
Further examples are easily obtained through reduction on the basis of: OBSERVATION. upwards
.
The
reduct ion
concepts
transmit
Chardness
EXAMPLE 2. The following are ,&complete with respect to K, the complement € * W, f 0 ) of the emptiness problem, € x ; f , € W,) and thereby all all special halting problems those problems constructed in Chs. AI, BI, by injective simulations by means of halting, confluence and wordproblems of Turing, register, or modular machines, (semi) Thue systems, Post normal or regularsystems etc. C,:
PROOF of example 2. For an enumeration index k x, y , z the following ars all logically equivalent:
KO,
<X,Y)
K
(X,
Y,z ) , W~,,.,,
v>(z),
S < k ,X, Y)
and all
K
and so are: x 6 K & ( x , z), k t c , , >(Z). &c,. *>(id Thus Ax, y. S < k , x, y ) and Ax. S ( k , x) 1reduce KO to K and K to the emptiness, special halting problems, respectively.
REMARK. &complete to lower classes
c
zn+r.
instead of
W
when
21n.
I
predicates do not belong t o f17 nor < m < n), so that the demonstration
I, &
BII. 3 Concepts of Reduction
12 1
of fin or &completeness locates the complexity of a problem exactly in the arithmetic hierarchy.
For many decision problems and for many numbertheoretic conceptual structures the exact arithmetic complexity could be determined; it is a remarkable phenomenon that up till now only a few "natural" arithmetical undecidable problems, with a simple description in the intuitive sense (or occurring in mathematical practise, independently of the constructs of recursion theory), are known which are not complete in some ar i t hmet ical class. We introduce now further standard exarplea of arithmetical complete sets a s well as some simple methods of determining higher, more complete sets from given ones, which will play a role in the investigation of complexity of decision problems. EXAMPLES of 6 ,
&complete sets are: Infinity problem, ( 1 ; W, infinite) ( 1 ; W, = An Totality problem, Equivalence problem, < ( 1 , j ) ; C 1 1 =C j l I
Correctness problem,
( 1 ; C 11 = f3 (1; = K)
U e ~ r s h i pproblem,
( i ;C 1 1 f 0 , for classes C # 0 recursive functions is flz hard.
w,
for
recursive
flz PROOF. By Markwald's characterisation, for each P is an R f ,& and a k so that P ( m i f f ( g  y ) R ( R , y> if€ We,,, infinite. The iwrinity problem 1s 1reducible t o ii; W, = K) as
f of
there
,,.,
W,,,.,
,,
= K
for some
k
i f f W, with
infinite,
W, x, y c W,, x f x ) S injective. l@ is 1reducible t o the totality problem by means of a such that:
and
< W y ) < 3 z ) : C 1 1< R , y , z ) r
iff iff
k
(Wy):Ckl ( i , R , y ) 4 C S < k , 1, HI
is recursive.
The totality problem is 1reducible t o the remaining problems by means of a solution k for: C S ( k , 1 ) l < x ) = f ( x ) when (Wy(x): y C W,, t otherwise. Clearly, all of the problems, except for the membership I problem, are fl&
B I I A R / Degrees of Unsolvability
122
EXERCISE. (Hooper 1966, Fischer 1970). Show the ,&completeness (with respect to mreduction) of the i m r t a l i t y problem for TMprograms, 10 of i M M a TMprogram h for some configuration C, no end configuration of M is attainable from 0 . Hint: Construct effectively for each f f F, a register f which does not have "immortal" operator W for computing configuration exactly when f is total (the socalled etrong coqutabillty. See Davis 1956) For similar problems eee Herman 1971, Fischer 1969 for a generalisation. sets are: EXAMPLES of 6 . &complete Cofinitenese problem, (1;W, cofinite) Docidability problem, if; W, recursive) il; ( 3 3 c U,): U, infinite ) (1; ( 3 j f U,): U, cofinite ) (1; ( 3 3 f W,): U, total (ie = A D ) ) . PROOF. Clearly, all the examples belong t o &. f = AU. < 3 x ) < V y ) < 3 2 ) R < U , x , y r r )f & are 1reducible to last three for some k such that t the computation of the halting problem instant with use of the oracle qsl only uses corresponding to f xc, (and so from questions t o the oracle which are below t l ) , in case the computation ends in at most s steps. A s each r.e. set W, is looked at at most once, the PROOF. If positive requirement < P a ) puts at most one element in Q
BII AR
140
/ Degrees o f U n e o l v a b i l i t y
) >6 n < x , m ( x , n ( x ,6 ' e c 3 , ( x ) ) ) >= : n ~ ' ( x , @ ' * , ~ , ( x ) ) with m' recursive and monotone in the second argument. To m' there is for each p p r o g r a m i for f a 9'program t < i ) for f, and, because of the surjectivity of t, cp'program t ( j > for f  m'faster in 6'  so t h a t .Iis a p p r o g r a m for f with n ( x , m ( x , 6 , , ( x ) ) ) E m ' ( x , @ ' , , , , , ( X > ) < @'*, , ( x ) 6 n ( x , O , < x ) > AE Then because of the strong monotonicity of n there follows m(x, 6 , < x )) < 6,( x ) AE. I
REMARK. From the speedup theorem it follows that in principle with each replacement of a (slow) universal computing machine by a new one (faster) there remain jobs which the "slower" computer works out faster than the "faster" 6 is estimated one: namely, if the old complexity measure according to the recursive conversion lemma by m in relation to the "faster" complexity measure 6':
B I I I . 1 Speedup Phenomena
O,(x)
151
6
m < x , O ' , ( x ) ) AE
then, f o r f u n c t i o n s f with mspeedup i n O', t o each program i i n t h e new c a l c u l a t o r , t h e r e 16 a n m s p e e d e d up e q u i v a l e n t program j w h i c h o n t h e o l d c a l c u l a t o r almost a l w a y a w o r k s f a s t e r t h a n i o n t h e new o n e b e c a u e e
O,,<x) 4 m ( x , O ' . < x ) ) NB:
i, j c o m p u t e
< y or # , ( x ) a m < x , y ) l t, satisfies the inequalities by definition i f ( V x ) : t,,,(x) &. For x, y,, : = 0, y , + , : = m < x , y . , ) define a strongly arbitrary monotone sequence; as there are at most x defined values # , ( x ) with i < x, one of the x + l intervals l y , , y , , , C with j < x contains no @ , ( X I with i < x, so that O,(x) < y, or O , < x ) 3 y , , , . I
REMARK. The gap theorem has a similar methodological consequence to that of the speedup theorem: for a universal u, each replacement by a new machine u' say, computer with faster computing time or extended instruction code admits in principle, infinitely many, even effectively u and u' presentable recursive runtime bounds inside which solve the same class of problems.

The statement of the gap theorem becomes even more clearer in terms of the socalled complexity classes of all recursive functions which are computable in a fixed, prescribed recursive runt ime: : = tf; f
C,
If
O
r e c u r s i v e B ( 3 i ) : p l = f 8 O, ( x ) 6 t ( x ) AE) is understood from the context we write C, instead of
c,. The gap theorem says that each strongly monotone recursive m generates a complexity class C, which is unaltered by raising the permissible computation expense by the factor m 2 1 ie, such that C, = Ccn,*,.,o,. The following remark C+ is properly extended when, shows that on the other hand as a new lower bound on runtime, not t but a sufficiently t in m large runtime is substituted for
BIII Abstract Complexity of Computation
160
REMARK. By means of a refinement (see Rabin 1960, Blum 1967) which will not be dealt with here it follows from the Rabin Blum construction of arbitrarily complicated functions that there exists a recursive, monotone function m by means of which, for recursive functions with sufficiently strongly growing runtime, larger complexity classes can be effectively generated, namely c c for all recursive cpl with @ , < X ) ~ X . @,
EXERCISE 1. (Hartmanis % Stearns 1965). class generated by a recursive t F : = rfi f total 8 f < x ) = O AE) is r.e. find a stage arbitrary i and a majorant m of all more exactly Hint:
for
@,(x) 6
t(x)
cp,(n)
..,
P ~ , , . ~ , ,.,>(n):=
lo
when @ , < X I E t < x ) % CPl<x) 4 m
Each complexity which includes s
O,(x)
beyond which with x < s,
for all for all
s 4 x x < s
has complexity 0 for j P K and all other programs have positive complexity. Then for the constant function t with the value 0, c, = which is considered up to that moment satisfies however 9, ( x ) > f , ( x ) , then we also ensure that d(x) < 9,(x)  i f this happens infinitely often then by this d is not an AE bound for 9, and replace < i , n > by a new candidate, eg < i , x > . The following effective procedure enumerates stepwise the Ks and elimination sets E:* and recursive candidate sets defines d ( x ) by this heuristic: El:=@
K  l : =
Case E:* = 0. Put
Case E x
f
0. Put d( x)
k
K  := ( P  '  E " )
LJ
E
E3,
( < < k ,, ) x>; k
Ex)
LJ
C < X , x>)
I f 9* ( x ) I d ( x ) AE then 9, ( x ) 4 f , , ( x ) AE for some n; for Cx; (3n): Em) would be infinite  for otherwise < i , n >E K x  K X r  ' there would be by hypothesis a minimal y > x with UJ,(Y) > f , , ( y ) , so that < i , n > P EY and < i , y > € K v  K v I , etc  thus for infinitely many x with E E' by construction d(x) = mintf
(k)
(x); k
f
E 9 E
fn,.(x)
AEI. The latter is true because for each n there is a stage s after which there can occur in E" only k ' s with n < ( k ) 2 . Therefore
d<x) = f , ( x ) 3 f , , ( x )
or
(3n<m): d ( x ) = f , , ( x ) 3 f , , ( x ) .
I
162
BIII Abstract Complexity of Computation
§3. Decomposition theory of universal automta. The Kleene normal form shows that each G6del numbering with recursive input, output, transition and stopfunctions I s analysable into the form o(t),i ; such a pair < t , d consisting of an t and stopcriterion elementarycomputationstep predicate s defines a "universal automaton", that is a computer which is capable of interpreting a universal programming language, if one supplies it with suitable coding and decoding mechanisms  including a compiler and which operates an i on arbitrary input 2 by iteration of arbitrary program t and extracts the computed value when the stop criterion s is sat isf ied. This threephase analysis of operat ion of computation into the production of the initial configuration, iterated application of the transition function until satisfaction of the stopcriterion and final output of the result of the computation characterises equally the abstract formalism of computation (see Ch. A. ) and modern programming We therefore say languages (see German0 & Mazzanti 1 9 8 4 ) . with Buchberger 1974:
DEFINITION. t is the transition function of a universal t is recursive and there exist automaton exactly when i , 0,s such that o(t),i is a G6del numbering. recursive Correspondingly for stop, input and output functions. I is called the runtime of a unlversal automaton exactly when o(t),i is a Gi3del there exist recursive t, s,i, o such that numbering and for all j , R 1 , (2) = pn(s( t i ( j , R) ) ) = 0) 'I
, , ( r ) , + l > + 0
I
The weakness of the general concept of complexity II 'asure manifests itself here in the artificiality of the tran ition function constructed merely as a step counter which does not exclude the strange examples of universal autome say, without infinite cycles or without confluences of different computation paths: EXERCISE : 1. is with ix;
Give a cyclefree universal
(u),(x)T
L (3n#m):
c
maton, that
uri(x)= un8(x).= 0.
2. Give a cycle and confluencefree universal automton , that is without computations which break off or become of different periodic and which have no confluences configurations: (Vx#x'): f o r all s,t: u  ( x ) # u * ( x ' ) Hint. For each computation step encode the total computation which has been simulated up till now.
With similar, simple simulations we get, according to Buchberger & Roider 1978, a general characterisation of those recursive functions which can take on the role of input, output, or stopfunctions in universal automata.
THEOREM Characterisation of input functions of universal The input functions of universal automata are automata exactly those recurs1 ve functions which are injective on a nonempty cylinder.
REMARK. By this characterisation, the input function of a universal automaton must feed in without merging program and datainputs ( i , x )  injectivity  and thereby leave free a
BIII Abstract Complexity of Computation
164
recursive infinite set of storage locations for filing intermediate results  the complements of r. e. cylinders contain infinite recursive subsets. The proof clearly shows this: is a GBdel numbering then for injective PROOF. If o(t),i f,(x) = o(t),i(h(j),x). there is a recursive h with f c F,, Conversely, if for recursive h, Ax, y. i ( h ( x ) ,y ) is injective, then i ( h ( 2 N ) x N ) E R for a coinfinite recursive by means of codification in coinfinite recursive sets set R; R one can simulate arbitrary computations of universal automata according to the following :
LEMMA. For each r e c u r s i v e f f F, R there a r e recursive r e c u r s f ve f ( x ) = o ( t ) , . < x ) f o r a l l x 6 R.
and 0, t
coinfinite ,r wi th
Put g  ' < Z ) = < j ,X > Then for the above R f : = Az. e ( , g 7 ( z ) ) , ,( g   ' ( z ) ) > ) there are recursive such that fcL all j , x
when and
gCJ, x ) : = f ( h < Z j ) ,x ) ; Let gCj, x) = z, t otherwise.
0,t , s
.
o ( t ) , ( f ( h ( Z j ) x, ) ) = f < f ( h ( Z j ) ,x ) ) = e < j , x ) Thus the GBdel numbering e is translatable into o ( t ) , f h j . h ( 2 . f ) and so o < t ) , i is a G6del numbering.
by I
PROOF of lemma. By means of recursive (de)coding functions c , d with c < N ) = C(R) and dc=U7, and a Kleene index k of f one can encode the computation of e,.(x) = o u t ( s t e p ) , i n ( k , x ) in R by input of c(in(k,x)), output o(z) : = o u t ( d ( z ) ) and c ( i n ( k ,z)) # O when z c R t ( z ) := r ( z ) := c(step(d(z))) s(d(z)) otherwise
t
I
rn
THEOREM. Characterisat ion of output functions of universal These a r e e x a c t l y t h o s e r e c u r s f ve s u r j e c t f ve automata. f u n c t i o n s f o r which f n j e c t f v i t y f a i l s i n f i n i t e l y o f t e n . PROOF. If o(t),f is a GBdel numbering and i f o were injective starting from a position n then the GBdel numbering would take no value o ( i ( k , x ) ) for inputs i ( k , x ) > n of nonterminat ing computations for from o ( f ( k , x ) )= o ( t " ' ( f ( 1 , y ) ) ) at a stopping position m there
BIII.3 Decomposition of Universal Automata
165
f0 1 1OW6 because of the inject ivit y assurnpt ion i(k,x) = t ” ’ ( f ( I , y ) ) and s o s ( f ( k , x ) ) = 0 and o < t ) , < ( k , x ) & . Conversely we simulate Kleene’s e = o u t ( s t e p ) , f n , again by recursive c,d with c ( N ) = R e p : = tn; ( 3 d n ) : o ( m ) = o ( n ) ) and dc = CJ’, ’ by: L
z:=c does not accept
f
in
6
ti fl
steps).
In considering asymptotic relations of time and requirement functions we use the f ollowinu abbreviated
tape
NOTATION
For diagonalisation over timeor taperestricted computations, we must be able, in the course of simulations, to compute the counters used inside the given time or space restrictions. We therefore use the following (after Paul 1978)
DEFINITION. A function f is called tape constructable exactly when Aw. bin f. F o r as the length of the result of the computation is bounded by the tape requirement of the computation, bin<s'(l wl ) ) can be computed for arbitrary input word w with tape requirement E O<s'l w l ) as follows: compute f(I w l ) 2*s(lwl)l 8 f < l w l ) , EXERCISE Give, for each recursive function t i meconst ruct able func t ion t '
.
f
a larger
THEOREM Space hierarchy theorem If a tape constructable function S grows more strongly than s (ie s = o < S ) then TAPE. For the diagonalisation we make use of the universal program sketched in SO with the bounds given there on the simulation expense.
w, M tests whether Construction. On the input of pSv with a p f (0,I ) * and a v E (0,1 , s ) . In case yes M computes the binary representation of S(I $ 4 ) and sets tape s(l wl ). (Because of the boundary marks in the space S(I wl ) S the taperequirement of this is tapeconstructability of O<S ) iff w P L(M, contrary restrictions, so that to the hypothesis. I
THEOREM (FUrer 1982) T i m e hierarchy theorem, For a l l k 2 2, I f a function T which is timeconstructable on a ktape Turing machine grows asymptotically faster than t (ie t=o(T)) then TIME, ( t )
c
TIMEk 1 natural number A by A =
H
numbers for distributed we represent an arbitrary
I j e Ihah.fBh
. ..
where 0 C ar,d < B and I(h) : = 10, , 2'+"  1 1 , that is, the coefficients of B" are sums of digits a h . f . We order H as these digits as node labels on a binary tree of height follows:

the root is labelled with
a,.,,,.
 I f a node is labelled ah., then its left son is labelled with ahl .>.+ and its right son with a,.,, , 2 j + , With h is the height of the counting started from the leaves, nodes labelled a,,,.
.
These labelled binary trees are stored in the following list which we call the (binary) tree representation of height H of A: b,,.
. . , bztCH+l>l
with
bCPthl.tpj+l,
: = a,,j.
EXAMPLE H = 2
h = 1, j c c 0 , 1)
h = 0,
jcco, . . . , 3
project ion on line. In this tree representation each odd numbered position gets a b,,,, is the coefficient of digit with position value 1 , ie BC'. We consider each position In the tree representation as a I f each counter is full we have the representation "counter". of the largest number representable by a tree of the given N height H
116
Ir,(*
B" =
ch(
= (B1)
2'+',Br'(B1) 4
(B1i" l  2 b f +  l ) / ( & 2 )
2 the mean length of the propagation of transfers (arising in the leaves during serial subtraction of constant < 1.
1)
in tree representations is bounded by a
PROOF. Let all the counters in a tree representation of b be full, ie a,,,, = B  1 for all h, j. In the course of m subtractions of 1 in the leaves a,,., ( 0 < j S Z h L 1 ) at most Lm/B"J times is a digit carried from a father node of height h to a son of height hI; the worst possible case occurs when a 1 is subtracted m times from the same leaf. In the tree representation of b the distance between the position of a node aP,.$ and its father node is 2': so that the sum of the lengths of the paths of the carries done in b is bounded by
The mean length of such carry paths (propagation of transfers) 1 in leaves of a tree in serial subtraction of I representation with initially full counters is thus < 1 . PROOF of the time hierarchy theorem. For the construction of M as given in the preliminary remarks we use 5 auxiliary procedures
.
a) A program U on a ktape TM with fixed alphabet which is universal for all programs on that machine and which has a linearly restricted simulation time Lin(l j l ) for simulation of a single step of the program j to be simulated. ad a. Such a U is got by suitable alteration of the construction in §O: because of the assumed fixed alphabet no input coding is required; the shift of the combinations of state and workingcell inscriptions during program coding, which is costly in time, is superfluous if one stores on two different tapes the program to be simulated and the combination of state and workingcell inscription to be
188
CI Recursive Complexity Classes
operated on  this is possible because 2 < k; finally, one can omit the initial test for the correct form of input as the simulation will only need to work on correct inputs. construction of tree b) represent at ions of numbers.
representations
for
Badic
c) conversion of tree representation with root inscription 0 into Badic representation. d) & e): procedures for the shift of counters and for the bringing down of digits in tree representations. construction of W : For the initialisation of the simulation of O ( t ( 1 wi ) ) steps of ktape TM programs I on inputs w with w = jSv M inserts the initial state of J and constructs from the Badic form a tree representation of T(Iwl) so that to begin the simulation the simulation workingcell stands on the first tape in its first track in such a way that it is over the middle of the tree representation placed there on the third track. After U has carried out the simulation of any single step of computation of the program to be simulated M decreases its counter (at a leaf under or near the workingcell) by 1 and calls up the carry procedure if necessary. If M tries to carry from the counter at the root when it has fallen to 0 then the simulation is interrupted and, by means of procedures b) and c), a new tree representation with a smaller tree but full counters is constructed and the simulation then continued. M leaves the domain of the tree If the workingcell of representation then the simulation is interrupted and, by means of the procedure d), the tree representation is shifted so that aforementioned workingcell of M stands over the middle of the tree representation. Afterwards the simulation by U is continued.
Before showing the restriction on the time requirement of of this simulation we give further details of the construction of the auxiliary procedures. O(T(I w i ) )
Coding. The first two of the k tapes are laid out with 6 trecks the first of which each time contains the tape of the program to be simulated: the remaining tapes contain the corresponding tapes to be simulated. The remaining tracks of the first tape contain the program (track 2 ) to be simulated, the respective tree representation b,, . , bc of a number p < T ( I w i ) (track 3 ) and an auxiliary counter with a Badic
..
CI. 1 Hierarchy Theorems
189
representation of length 0(10g,I w l ) of a small remainder p' (track 6) determined later. For the execution of the carry procedure there stands in track 4 under each a,,,, of the tree representation lying above it an indication of whether the associated node in the Trace tree is a left (L) or a right (R) son or the root (W). 5 contains a counter for the current height of the carry propagation. With both these details it can be determined on which side (track 4 ) and for how many nodes (depending on the height stored in track 5 ) the carries must be propagated; Tape 1: 1. Contents of the simulated first tape. 2. 3. 4.
5.
6.
Tape 2: 1 2
5.
Coding of the simulated program Tree representat ion of p : b, , left (right) son, root: L. , . W.. R height of the carry propagation counter for the remainder p ' . I
. . . . b, .
Contents of the simulated second tape State, workingcell inscription) of simulated program. for operations (specially copying) on the contents of the corresponding tracks in tape 1
6.
ad b. Construction of tree representations:
The given number z < T(I w l ) is smaller than a Case 1: c Bi : then we count z by carrying out the constant simulation not in tree, but in the existing Badic representation Case 2: c E z. The tree representation of height H with v(H) < B"'i (see above). full counters represents a number H : = LlogmzJ2 so that v(H) < z and the Thus we choose length O( 1 og,) of the Badic representation of p' : = z  vC'?'', which for i= c *S < n ) and an accepting configuration C' is the main question, according to the Roman principle "divide and conquer", into two subquestions < G Bi"'C'' ' & C'*+c ""C')? for some configuration C", for whose successive answers the same segment of tape can be used. By iterated application of this strategy of analysis we "C+
CI. 2
Nondeterministic P r o g r a w
195
obtain the following recureive REACH procedure for the determination of the reachability relation AC, C', 1. C, i'C': Procedure REACHCC,C', 1 ) ; When 1 = 0 8 < C = C' or When
0
If
,
lz*,
3",, = mln{3", y>.
PROOF. By the E2 Kleene f c E,,, f(23 = UAE
a,<x) C g < x ) for s o m e 1 .
in I
ad 2. By the Kleene E2 normal form theorem and the mjorising of En,functions by iteration of an, or by a, each f 6 Enk has, for certain M, 1 the representationi
f<m =
UtR, 2. anr : =
CII. 2 Computationtime Hierarchy f ( i, X, y,
2) : =< f
217

, ( Z , y) , sgc u( i, X, a;,* ( C L  ~  ~ ='(Y) ~ ))) r
Here < > denotes the Schwichtenberg coding, bounded by a function in Enk. Describe f recursion on z from En* functions.
E . . B

8
so that f is by a primitive
EXERCISE. f arises from g, h, t by primitive recursion with substitution at parameter places by definition iff f(2, y, 0) = g(2, y) f ( 2 , y, r+l) = h ( 2 , y, 2, f ( 2 , t < 2 ,y, z ) , a)). , is not closed under E,bounded primitive Show that: 1 ) . E recursion with Substitution at parameter places, 2) F,,,, is closed under primitive recursion with substitution at parameter places. Hint: for 1 ) give a definition by primitive recursion with SUbStitUtiOn at parameter places from functions from the nth Grzegorczyk class of an "enumeration function" of all one place 01valued functions in the nth Grzegorczyk class; say, by recursion on k for f < i , x , y , k ) : =S g ( ~ ( i , X , c Y n r < y );Z) )En. For 2) define the given function by courseofvalues recursion, wherein an appropriate bounding function for the SUb6titUtiOnS which could possibly occur at the parameter places is to be defined. EXERCISE. f
iff , , , yn)
recursion
9=
( y ,,
.
arises from g,h,, for some term t there holds: f(2, 0) =
..
g<m,
. . . ,h,
over
by nfold nested f , h,,...,hp, 2,
and f < 2 , p = t
for 9 # 6 = ( 0 , . , O ) , where each term f < a ,Q) occurring in t Satisfies the condition Q f with the lexicographic wellordering: (v,, , v,) < f ; f recursive) f',, f , , . . . of the one Hint: For an effective enumeration place primit ive recursive functions, there holds f : = A f , x f , < x ) C Mz  M I ; analogously for l < n . This is the hierarchy of multiply recursive functions (see Peter 1951, 1967) which aroae from an exact analysis of the dmfinition processes of the 2fold, but not 1fold, recursive Ackermann function. The Grzegorczyk hierarchy theorem a s well
CII Primitive Recursive Classes
2 18
as the hierarchy theorem for the nth Grzegorczyk class can be carried over to the multiply recursive functions, see Robbin 1968 0, b > 1 l e t (x), b e d e f i n e d by t h e "Goodstein process'':
a ) subtract b:
1
from
x
and
represent
... +
x  1 = c,b" + c n  , b n  ' + b) r e p l a c e t h e b a s i s b in t h e by b t l :
c,b0
x1
with
t o t h e base M c,< b  1 .
r e p r e s e n t a t i o n so o b t a i n e d
CII. 3 Goodstein Sequences
22 1
.. .
(x), = C , ( b t l ) " + c,](b+l)"' + + co) analogously for
NEXPTIME,
N)
6 6
N)
EXPTAPE.
With regard to polynomialtime cornputable mreductions we show characteristic examplee of complete eetr for these claoses; in particular we prove, (as a modification of the completenoes of the halting and the dominoproblems in the domain of the recursively enumerable) the NPcompleteneso of restricted halting and dominoproblems and deduce from it (after Lewis & Papadimitriou 1981 and Savelsbergh & van Emde Boas 1984) the NPcompleteness of the partition and rucksack
226
CIII Polynomialbounded Classes
problem, the clique problem. the problem of Hamiltonian cycles, respectively, circles, the travelling salesman and of binary integer programming. The reader will find further natural examples of complete sets for the above complexity classes in bCIV4 (PTAPEcompleteness of the loop problem), 8CV4 (decision problem for regular expressions), bFIII13 (decision problems for propositional logic, firstand higher order logical expressions).
B 1. NPcomplete problers. For the concept of completeness used here we use the following quantitative refinement of the concept of mreduction which was first investigated by Karp 1972 (see BBI1): A is called p l y n o r i e l  t i m reducible t o MAIN DEFINITION. (abbreviated to A 6 B or A is preducible to B ) iff A 6 m B by means of a reduction function which is computable by a deterministic program on the polynomialtime bounded Turing machine.
B
Thereby the concepts of Chardness and Ccorpletenese from BBII3 carry over t o 6for complexity classes C. In this section, whan not otherwise mentioned, by Ccompleteness we always mean completeness with respect to Gp. In proving Ccompleteness we restrict ourselves in this section in general t o the critical proof of Chardness and leave the proof of membership of C 'to the reader. We will thereby presuppose, without mentioning it. an appropriate coding (for example, in binary) of the problem concerned.
EXERCISE
(i)
a segment
..
. ..
of H.).
PROPOSITION 1: forms a segment ( v , ( p , q ' ) ) of H, must therefore be a numberpoint; for with v = ( q " , p ) , where q" C q and i q " , p ) , Cp, q ) f K, . Then < p r ,. . . , p , > of generality that is a Hamiltonian cycle in G. I
..
of
COROLLARY 2. NPcompleteness travelling salesman
the
problem
of
the
OEi,j(n),k); kC N & 8 Z i < n d p C i D p C i + l+D
(Vi,j):Cdd., = dji 4 N & d p c m J p c l D d kl for some permutation p of CO, , n> }. Informally, for given d,, between cities i , j and length of route distances k, a route of length 6 k must be found which visits each city exactly once. {(td,,; d,,=O
...
PROOF by preduction of the problem of Hamiltonian circles by means of G =
f
K,
i,j
d,,:=
f
1
P), 0)
otherwise.
I
EXERCISE 1. NPcompleteness of the problem of undirected Hamiltonian paths, that is, of all finite undirected graphs G with one path which visits each point exactly once. Hint: preduction of the problem of Hamiltonian circles. For a 9 of the given graph G introduce three new points point 0,I , 9' so that the Hamiltonian circles 9,; , 9 in G correspond to the Hamiltonian paths 0,9 , . , 9 , 1 or their reversals I , 9 ' , , q, 0 in the associated graph G'.
. .,
..
..
EXERCISE 2.NPcompleteness of the problem of directed Hamiltonian paths. Hlnt: preduction of the problem of undirected Hamiltonian paths, so that each undirected Hamiltonian path is also directed and conversely.
CIII. 1 NPcompleteness
239
THEOREM on the N P  c o w l et m e s s of binary integer programming, t h a t i s , t h e s e t of a l l f i n i t e s y s t e m s of equations ~ICjC,a,,ix.j = bi (16iEm) where a , , j , n i E. N and xi a r e v a r i a b l e s , which a r e simultaneously sol vabl e by a sequence ( x , , , . . , x,?) 6 {O, 1 ) '? The same h o l d s f o r a , , E. f0, I ) and b, =. = b,, = 1 .
..
PROOF by preduction of the restricted originconstrained S = ( D ,0, H, V) be a originconatrained domino problem. Let domino game with pieces 0,1 , . , 1, originconstrained domino We define a system of equations which piece 0 and 0 < s. is solvable over CO, 1 ) exactly when the srestricted originS is solvable. For each domino constrained domino game piece k 6 I and arbitrary coordinates ( I ,j ) with 0 6 1,j < s we introduce the variables x c i , with the intended meaning: x , , , = 1 or 0 according to whether f ( i , j ) = k or # k respectively, for some ssolution f of S.
..
The
equation
(1)(4):
system
consists
of
the
following equations
for i , j < s. Read: In position lies exactly one domino piece k E 1 ) . (1)
&,,xki, = 1
(Existence and uniqueness: ( 2 : xcij
+
E k6
~rkz,u*>rw~~#r*Cl+lDj
for
( I ,j )
there
1 I & i < s l & . l < s
(Horizontal neighbourhood condition N when k lies in (1,j ) then no k' with (k,k') P H lies next to position it on the right.) For clarity we have formulated ( 2 ) as an inequality as an abbreviation of the equation x c i j
+
xce,c.>HaXe'Ci+l>j
+ Y&id = 1
with new variables y e . , . A s the y c i i do not occur in any of A ( i , av) = A < i , a)A(6 0 c (x, y> c E to, 1 1 , 1 c {a, b) c = minCa, b)
computes
The following transducer computes the addition of two binary numbers on successive input of their digits, that is, in the sense of ACz,,, a,b.,.
. . a,b,,)
= c,,.
. . c,,,,
= binary sum of
O/
o/o, 111
1/
o/ 1
1 /o
O/
. . a,
a,,
11
o/o, 111
and b,,.
. . b,,
C
IV. 1 Acceptors and Regular Sets
(edges
v, w
245
are abbreviations for two edges
+
and
V
+) W
EXERCISE. Construct a finite acceptor M for controlling the correctness of addition of two binary numbers, that is, M accepts exactly those words 5 ., .s,, over the alphabet of threerowed column vectors
.
with
for which the rsequence and ysequences.
xi,y,, z d
€
represents the addition
(0,1 )
of the x
EXERCISE. Construct a "ticket machine automaton" which sells tickets of values DM 1,2,3,4,5 if coins of values DM 0.50, 1,2 or 5 are inserted in arbitrary order but to the total value of the chosen ticket. EXERCISE. Construct finite, deterministic subword w L(M) iff aaabccc is a recognition a u t o m t a ( eg and error rejection automata (eg subword of w C {a,b, c)*) w € L(h9 iff w contains no subword abc or aabccc , where u f {a,b, c)* ) . EXERCISE. Construct a finite nondeterministic acceptor which accepts the binary representations of numbers which are divisible by 3, 5, 7. EXERCISE. Construct a deterministic and a nondeterministic finite automaton with the least possible number of states which accepts exactly those binary representations which have a 1 in the 5 t h place from the right. In the preceding exercise you have presumably found a non6 states but no deterministic deterministic automaton with automaton with less than Z h states, see Meyer 8 Fischer 1971. It can easily be shown that for each finite automaton (with n states) there can be found an equivalent deterministic automaton (with at most 2" states).
CIV Finite Automata
246
THEOREM (Rabin and Scott 1959). F o r each nondeterministi c finite automaton M there can be constructed an equivalent deterministic finite automaton M', that is with
L(M) = L<M'). PROOF. The idea of the proof is as follows: for arbitrary w follow step by step, simultaneously on all input words w in the state diagram of M the states paths labelled reached and finally test whether one of the paths ends in a distinguished state; a state of the constructed automaton M' therefore consists of a set of possible states of M Formally, M' is given by the definition: Z ' : = , L ( a ) : = (a).
2. I f a , B are regular expressions over A then so also are (a@), (avB), a* regular expressions for LCCap)) : = L ( a ) L ( B ) , L((w/3)) := L when I itself is a distinguished final state. ) .
.
Regular expressions for E(i,J,k) are obtained from the state diagram of M by extending, inductively on k starting i to j when k = 0, the with the direct transitions from i to j , set of permitted intermediate points on paths from and describing the newly adjoined paths from i to j allowing possibly several visits to k by regular expressions:
C IV. 1 Acceptors and Regular Sets
249
EXERCISE. Conversely, develop, for each regular a the state diagram of a nondeterministic finite automaton which accepts L , :
also for as sub,
M a c c e p t s wl w l l . .
.W,~I
quantification initial, end
with word
S2. Algebraic character lsat Ion. After the inductive characterisation of the languages accepted by finite automata by regular expressions, we now deal with an algebraic which characterisation going back to J. Myhill and A. Nerode, plays a role in minimising the number of states of a finite automaton, in certain decision problems and certain related periodicity properties. From the latter there arise proof methods for impossibility propositions about finite automata and also for the observation that by allowing leftward movement the finite 2way automata that arise in this way only accept regular languages.
CIV.2 Algebraic Characterisation DEFINITION. An equivalence relation K r i g h t invariant iff for all v, w, u vKw implies
25 1
over
A'
is called
vuKwu
K is o f f i n i t e Index iff K has a finite number of we denote by K(L> the equivalence classes. For L E A* canonical r i g h t invariant equivalence r e l a t i o n belongfng t o L:
vK, v) = a(&,, w) for initial state z,, of M (Notice that we require this relation of equality only for the initial state as we are only M as an acceptor.) Clearly, K is an interested in equivalence relation and, because of the determinacy of M it is right invariant; K is of finite index because there are only finitely many Mstates dw I". 1 for some w, with the w in K(L<M)). (If i were not equivalence class of reachable in M' from the initial state by any such w, then 1 would be superfluous and M' would not be minimal; because I K' 5 K ( L < M ) ) the definition is unique.)
EXERCISE. Construct two equivalent, minimalstate nondeterministic finite acceptors which are not isomorphic to each other. REMARK. For the practical utilisation of finite automata whether for modelling of hardware constructions or for implementation in test programmes  the number of states influences the expense of the realisation, so that more efficient minimisation algorithms than the one given above have been developed such as Hopcroft's procedure (1971) which determines the equivalence classes of the canonical equivalence relation belonging to L<M) in O(n1og n) steps.
CIV. 2 Algebraic Characterisation
253
Minimisation algorithms yield at the same time a decision procedure for the question of the equivalence of given finite automata by comparison of the constructed minimal variants. This and related questions will be considered further below. The MyhillNerode algebraic characterisation languages also yields a method of forming languages: COROLLARY. regul ar.
The bracket
language
t Cn)”;
of regular nonregular
0 4 n)
I s non
PROOF. If this language were regular then it would be a union of equivalence classes with representatives from L of a right invariant equivalence relation K of finite index. As (” represent only finitely many equivalence the words n < m with (“KP ; because of right classes there is an (“)”K(”>” and because C (“)“I, E L invariance there follows there is the contradiction (”’)“ f L. I
EXERCISE. t w Rv(u);
twq
tq
w E w
f
Prominent examples of nonregular languages are w
f
{a, b)*)
ta, b)*) Ca, b)*
8 V,(w) = V,(w))
where V, ( w ) : = number c of occurrences of in w.
set of all correct bracket expressions over the 2element t ( , )) (the socalled Fftch domain or Dyck alphabet language defined by t q w f t ( , ))* 8 w +F A1 with the semiThue system F consisting of the single rule 0 + A) and the correspondingly defined set over n bracket C,, )i. symbols The above corollary is frequently expressed as the impossibility for finite automata of counting arbitrarily far on account of their fixedwired finite memory. Formulated positively, however, there emerges the periodicity property in the form: For each regular LOOP LEMMA for regular languages. a number p can be given so that all longer language L words x c L can be analysed In the form x s uvw with I uvI E p, v # A and U V ’ ~ W€ L f o r all n.
CIV Finite Automata
254
PROOF. Let p be the number of states of a finite M with initial state i,, which deterministic automaton L. If x = x r . . .x,, C L is longer than p , then in accepts i,, to 6(i,.,,x) there is a the transition sequence from first state repetition
.
i,: = 5(1,,,xI.. x.,) = This loop from 1, to oft en. Theref ore put w : = x, + I . xn,.
..
.
6(i,,, xI.. x,)
=: 1,
for OSj is n o t r e g u l a r . PROOF. If tI6: p prime) were regular, there would be a u + v + w with sufficiently large prime number of the form v f 0 and prime numbers u + vn + w for all n; the latter however does not hold for: u + v < u + 2 v + w t 2 ) + w = ( u + 2v + w) + v ( u + 2 v + w) = < v + l ) ( u + 2v + w). I Application example 2. The set of q u a d r a t i c numbers unary r e p r e s e n t a t i on> i s n o t r e g u l a r .
(in
PROOF If L = < I n ; m = nx, n N) were regular then, b y the loop lemma, PZ would be represented as a sum p2 = u + v + w for certain u, v, w and quadratic numbers u + vn + w for all n; because 1 E v S p we have p'= u + v + w < u + 2v+ W E p = + p < < p + 1)s and hence u + 2v + w cannot be a quadratic number. I
EXERCISE. Show that no finite automaton can be constructed for controlling the correctness of the multiplication of two binary numbers by deducing from the loop lemma that the following set is not regular: the set of all words s,>...s,> over the 3rowed 01 column vectors
for which the r s e q u e n c e represents the multiplication of the x and ysequences. Deduce that no deterministic transducer can compute the binary multiplication of two numbers.
CIV.2 Algebraic Characterisation
255
As the third example of the application of the MyhillNerode construction we show that admitting the left movement operation in finite automata does not extend the class of accepted languages. Formally we define twoway automata M as given by  a deterministic Turing machine programme without print operations which stops on reading the empty tape symbol a,,,  an initial state z,,, a set Z of distinguished final states. The language accepted by M is defined as L(M) : =
tw,
w
c
(a,,.
. . , a,)*,
z,,w
I
wz
for 2 E
z>.
some
THEOREM (Rabin. Scott and Shepherdson 1959). With each f i n i t e twoway automaton t h e r e can b e a s s o c i a t e d an equi val en t f i n i t e nut oma t on. PROOF. A s in the theorem of Myhill and Nerode we define for M an Mindistinguishability the given twoway automaton relation K whose equivalence classes describe the possible M on input words in relation to their behaviours of recognition  read: leaving the input word to the right or to the left: vKw iff for arbitrary Mstates i, j there holds: when started at the left (right resp.) end of the input word v in state i M leaves the word v on the left (right, resp.) in state .I iff this is the case for y that is i vt,., ja,,,v
iff
i +,. j, a , ,w
tstart at the left and stop at the left.)
iff
iwImWJ)
tstart at the left and stop at the right)
iff
wia+,.,ja,,wa
and i*”Vj
and via+,ja,,va
(start at right and stop at the left)
and tstart at the right and stop at the right)
K
is clearly an equivalence relation and rightinvariant;
K
256
CIV Finite Automata
is of finite index because, by the finiteness of M there are v can only finitely many ways in which M on an input word v to the right or to the left. The equivalence leave v E L(M) cover L<M) so that L<M) is classes of the regular. I
13. Decomposition theorems. The MyhillNerode method of description of finite automata uses the wellknown algebraic method of forming quotient structures by an equivalence relation. The question therefore arises of whether there are other representation methods in algebra which could fruitfully be carried over to finite automata. As representative of a large class of examples we deal here with the question of whether and how arbitrary finite automata can be analysed into products of particularly simple automata. For handling such a question of decomposition initial and distinguished final states play no role so that in the following, by finite automata we mean in general reduced (without loss of generality, deterministic) finite automata, which are given by their transition function d:BA+A with The decomposition problem input and statealphabets A and Z asks whether each finite reduced automaton can be analysed (embedded) up to isomorphism into an automaton which is suitably compounded of simple automata. The fundamental concept of embedding used here arises naturally as follows:
DEFINITION. Let Mi be reduced finite automata with input alphabet A. M , is lsomorphlcally embeddable in M2 iff Mi. includes a subautomaton isomorphic to MI M, is a subautomaton of M, iff d,lZrxA = 6 , ; M , is Isomorphic to M , iff for some bijection h:Z,+Z, the diagram
.
commutes, ie for all f
E
Z,
a
Q
A,
h < d l(f, a)) = 6 , 2 ( h ( l )a). , Instead of I'M, is isorn3rphically ernbeddable in &'' one also simulates M, is o m o r p h ic a ll f,' , "MI can be 6imulated says "I% isomorphi c a l l y b y M2".
.
C I V . 3 Decomposition Theorems
257
NB: In t h e c o n c e p t o f i s o m o r p h i c s i m u l a t i o n o f a transducer M by a t r a n s d u c e r M' we r e q u i r e a d d i t i o n a l l y t h e agreement of t h e o u t p u t f u n c t i o n on t h e i s o m o r p h i c a l l y r e l a t e d states: h ( f , a) = A'(h(f), a). NB:. A s a l r e a d y i n e a r l i e r c o n c e p t s of e m b e d d i n g w e n e g l e c t what t h e e m b e d d i n g ( r e a d : s i m u l a t i n g ) s y s t e m y i e l d s o u t s i d e t h e embedding ( r e a d : c o d i n g of t h e s i m u l a t e d s y s t e m ) .
As a n s p e c i a l l y s i m p l e example of a reduced finite automaton we choose t h e s o  c a l l e d f l i p  f l o p F' w h i c h a s s u m e s t h e t w o s t a t e s "up" a n d "down" a n d c a n b e a l t e r e d o n i n p u t o f 8 signal I; to simplify the proof of the product decomposition theorem we e q u i p F' with a further signal 0 t h e i n p u t o f w h i c h d o e s n o t c h a n g e t h e c u r r e n t s t a t e of F'. F' i s t h e r e f o r e d e f i n e d by:
1
0
F i n a l l y w e must d e t e r m i n e t h e c o n c e p t o f decomposition i n t o products. For t h i s w e o r i e n t o u r s e l v e s t o w a r d s t h e m o d e r n hardware t e c h n o l o g y f o r t h e c o n s t r u c t i o n of computers from p a r a l l e l processors; imagine n processors P,  possibly i d e n t i c a l w i t h each o t h e r  w i t h state and i n p u t  a l p h a b e t s Z,,A, w h i c h work c o m p l e t e l y i n d e p e n d e n t l y o f e a c h o t h e r , b u t a l l c o n n e c t e d up t o a c e n t r a l s e n d i n g a n d r e c e i v i n g s t a t i o n F by a n i n f o r m a t i o n e x c h a n g e s y s t e m If P sends a signal a  f r o m i t s own i n p u t a l p h a b e t A  t h e n t h i s w i l l be r e c e i v e d a n d p r o c e s s e d by a l l o f t h e P i w h i c h are n o t d i s c o n n e c t e d a t t h e t i m e ; d u r i n g t h e p r o c e s s i n g t i m e of a s i g n a l by P,, P, is d i s c o n n e c t e d f o r t h e r e c e p t i o n o f f u r t h e r s i g n a l s a n d is c o n n e c t e d a g a i n o n l y a f t e r d e l i v e r j n g t h e r e s u l t of t h e processing. The i n f o r m a t i o n exchange s y s t e m t t h e r e f o r e y i e l d s , f o r each state vector ( i, , . . . , f , > ) o f the processors P I , . , P,, a n d e a c h P s i g n a l a 6 A, a vector ( a , ,. , a,.,) o f e v e n t u a l i n p u t s a , 6 A , f o r P , , which are t h e n p r o c e s s e d by Pi. For t h e f r a m e w o r k w h i c h i n t e r e s t s u s here we combine these methods of construction in the f ollowlng:
I.
..
..
DEFINITION. Let automata, and let I,&
(6,: Z,xA,+Z,) be mappings:
Mi =
be
reduced
I: ( x C Z , ; 1 1 f 6 n ) )
x
A
+
x(A,;
14fGn)
d: ( x C Z , ; 1 6 i 4 n ) )
x
A
+
xCZ,;
16iEn)
finite
CIV Finite Automata
258
M
with the transition function 6 is called the rproduct Mi iff 6 1s defined by the diagram:
automaton of the
(xCZ,;lEi , J = ( j , , ..,.I,,), take from the F' the signals air.. . ,a,, with the diagram of F'transitions ia rjr for 1 E I E n and define i(i,a) : = ( a , , , a,.,) < i arbitrary otherwise). Thus, we have defined an rproduct automaton from F'components with transition function 8 whose restriction to t i ; i E Z) x A is isomorphic to M because btf, a) = < & , ( f ,, a,), . , S F . (I,,, a,?)) = ( j , , , j , ) = 6 ( i , a) where S F . is the transition function of F'. I
...
..
...

CIV.3 Decomposition Theorems
259
EXERCISE. Consider that the preceding construct ion can also be carried out with the flipflop with only one input signal, namely, 1. EXERCISE. Show that each set of reduced automata from whose elements each finite automaton can be isomorphically simulated as a product automaton, contains at least one automaton with the following flipflop as Cup to isomorphism) subautomat on:
In the proof of the product decomposition theorem the inary coding establishes a “microscopic” analysis of the *transit ions in n parallelworking (but independent 0 1 each other) flipflop transit ions: I
A
j
simulated by
I.d*j.
for
1 E
1
< n
So the product automaton M reflects the struct’ + of M by means of the information exchange function L; connect s the n flipflops up to form a circuit for i.le (product) automaton M which simulates M In fact I comprises part M and so depen’s on M The of the computation work of question arises of whether, in the decomposition theorem, the form of interconnection of the elementary automata in the circuit can also be modularised by fixed prescribed schemata. In the following we give a positive solution, going back to Rt)dding, to this problem for tne example of Mealyautomata.
Clearly, Mealyautomata can be considered in the form:
x,
*.......

*....... ,Yo.*
2
..... *....
y,
Y
with input and outputleads x, and y,, respectively for a, and b* respectively. the input and output of signals This suggests the following mode of connection of Mealyautomata M , , , , , M., into a Mealy automaton M (with possibly partial transition and output functions): each output lead of each M, can be identified with (“joined to“) an input lead of an M,; the remaining input and output leads in M,, . .,M,, that are unused by this are the input and output leads of the the states of M arise from the states of the network 4
.
.
CIV Finite Automata
2 60
components Mi. If M in state I receives a signal a on M i , ,, then M i ,I , changes its state an input lead of according to 6, I , and sends according to A,, an output a,, I , on an output lead of M i , ,; if this is signal connected in M with an input lead of an M,,2 , , then M,,,, reacts on a , . , ) by a change of state according to 6,,p , and a A, ,,,outputsignal a, > , on an output lead of M i ,.?, etc until eventually a signal appears on an output lead of M this is the output signal A ( i , a ) of M and the state of the M I , . ..,M,, reached in M is the successor state 6 ( i , a ) of M on Input of a in state 1; if no signal appears on the output leads of M, then & < I , a ) and A ( 1 , a ) are undefined (read: the network M gets into a loop in input of a in state i. ) I
I
EXAMPLE 1 Let E be a Mealy version of the flipflop defined for the productdecomposition theorem with the state diagram:
3
set/ s
t e s t / t,+ U
t elt ~I
I
d
' '
t,, t,
On input on the t e s t l e a d the internal state of E is tested ("down") and by choice of the t e s t o u t p u t s t,., ("up") or t , communicated to the outside without altering the state of E; input on the s e t l e a d effects a change of state of E and the reporting to the outside on the s e t o u t p u t s. E is thus a combinat ion of flipflop and decision automata.
EXAMPLE 2. Let H be the variant of E in which setting into positions "up" and "down" is realised by separate set inputs s e t , . , and set,, respectively, with associated set outputs s,, and s,,: s e t . , / s,,
set,,/s,, s e t ,,I s.,
CIV.3 Decomposition Theorems contracted to:
test
26 1

.
+ I
1
Y
set, s,, is isomorphically simulated by the following circuit which connects up four copies of the automaton E if one associates u , d of H the state combinations X inductively as the smallest class of automata which includes X and is closed under parallel coupling and feedback connection.
THEOREM. R6dding' 6 normal form Each Mealy automaton can be simulated isomorphlcally by a network of flipflop/decision automata E and the nodeautomaton M = contracted to
PROOF (RUdding 8 Brllggemann 1 9 8 2 ) By the above example it suffices to simulate an arbitrary Mealy automaton M by a H and N. Let M be given by the transition network of function 6, output function A, input signals 0,. ..,f, states 0, . . . , J and output signals 0, . . . , L. For ( i , j ) with i E f and j E J we choose copies H , . , , H g i , , ,H" , as well as ~ ' ( i j,) when d ( i , j ) t, are not used In the simulation and represent no output leads of the total network. Hence, by
connectlne; It with an Nautomaton which itself feedbackconnected we "kill" it. :
EXERCISE 1. Show that the Reddin8 normal form theorem also holds for nondeterministic Mealy automata, If one also adjoins to E (respectively H) and N the indeterminator:
contracted to
CIV F i n i t e A u t o m a t a
266
( H i n t : I n t h e p r e c e d i n g p r o o f , b e t w e e n p h a s e 1 a n d p h a s e 2, let t h e s i m u l a t i o n of t h e s i g n a l run through a network of I which f o r ( i ,j ) chooses o n e of t h e p o s s i b l e t r a n s i t i o n s i n Ef see Brllggemann et a1 1984).
EXERCISE 2. P r o v e t h e R 6 d d i n g n o r m a l f o r m t h e o r e m w i t h t h e b a s i c a u t o m a t a 0. F, N w h i c h are d e f i n e d by: test/ t., R=
test/ t,, contracted to
The f l i p  f l o p F (read E without s e t t i n g of p o s i t i o n s s e p a r a t e f r o m tests. ) tests and c h a n g e s s i m u l t a n e o u s l y its i n t e r n a l state. The p a r t i a l automaton 0 ( O t t m a n n 1978) s w i t c h e s i t s s t a t e o n e a c h t e s t t o "down" and u s e s I t s p o s i t i o n i n p u t t o c h a n g e o n l y f r o m "down" t o "up":
test/ t., Q =
set/ t,, contracted to
(Hint: I n t h e proof of t h e theorem o n l y t h e subautomaton of H without t h e t r a n s i t i o n s UP is u s e d .
set,.,/ s,,,
Simulate
Deduce t h a t basic automata arises from 0
H,,
*up,
down
by a n e t w o r k o f
set, / s..,
H,,
bdown
0, N, F. ) .
t h e normal form theorem also h o l d s w i t h t h e 0, a n d N, w h e r e t h e c o m p l e t i o n 0, of 0 by a d j o i n i n g t h e t r a n s i t i o n UP
set/ t . ,
rdown
EXERCISE 3. Show t h a t each register machine can be E.N and t h e simulated i s o e o r p h i c a l l y by a network of register automaton RC ; t h e l a t t e r h a s i n f i n i t e l y many s t a t e s n E N and t h e state diagram
C I V . 3 Decomposition Theorems
#O
267
=O
Hint: Consider the proEram M as a finite automaton with n register automata  one for each register of M  realised as a network M of N,H,RG, so that for arbitrary states i , j of M, and corresponding states i , j of ihe finite program automaton and arbitrary register contents 2 , g of M (states of the n RG automata) there holds:
+,
(i,II) ( j , p iff 6 when started in state (i,!?) with a signal on a special lead L , corresponding to i, attains the state ( J , p with a signal on L,.
Diagrammatically M h a s the form:

input lead L,, for initial state U
=o realisation of program M
output lead L,. for stop state r
'
*u
RG
add
RG I
The proof of the R6dding normal form gives a good opportunity to take a look at concept of parallel signal processing which has become important in modern computer technology but has not yet been satisfactorily investigated. We will show how, by allowing parallel instead of merely sequential signal processing in networks of automata, the computation time bounds on the simulation in the R6dding normal form theorem can be essentially reduced without essentially complicating the network construction  the types of basic automata, their interconnection into circuits and the magnitude of the simulation program  or the proof of Thereby we get acquainted correctness of the simulation. of the modular basically with a natural extension decomposition and synthesis technique from sequential to parallel processes. In choosing the speedup phenomena for illustrating useful parallel operations on signals we were led by the following idea: the simplifying condition of having to simulate purely sequential Mealy automata M by a network M of parallelM in the working automata, allows us, reasonably, to use
CIV Finite Automata
268
event of sirnulation as a total network only sequentially ; that is, a new input signal for the simulation of the next M reaches R only after the output of computation step of the output signal indicating tho and of the simulation of the previous computation step of M We exclude conflict situations, which can occur between the asynchronous A (ie those that work without a coupled clock components of fi pulse), by ensuring in the construction of the network that each individual basic automaton in is only nused hg ie sequentially" whilst simulating a computation sltep of that at any time it only receives a new input signal if the signal being emitted as a reaction to the previous input signal has left its output lead and this is free of signals. In particular, the deterministic Mealy automata occurring in M are only used in a simulation event in a manner which is well known to us and do not fall into conflict Situations, although in the total network in the course of the sirnulation many signals are allowed to move, and in fact do move simultaneously and independently of each other at different posit ions. It says something for the power of the modular decomposition approach of the R6dding normal form that in modelling asynchronous and parallel networks with a view to faster simulation of sequential automata, besides the sequential deterministic basic components  read: H and N only two further automaton units are needed, which are well known in switching theory and extensively used: 1 ) Signal doubler D
B=
{=:' ::
(one state, one input, two outputs). On input of a signal in v, that signal is emitted by both v, and vz.
(logical "and", reverse of D) On input of a signal in each of w, and wIz a signal is emitted on w
D and G are representatives of a class of automata which, in a given state, on simultaneous input of a block of signals distributed over several input leads, can react by change of state and output of signals on several output leads, and are thereby tailor made for the simultaneous processing of many signals. We fix this variant of the concept of a transducer and its circuit connections in the
CIV. 3 Decomposition Theorems
269
DEFINITION. A (finite) block transducer M is a semiThue i e w j for states i , j , input sets a 5 system with rules A, output sets w 5 B for state, input, outputalphabets Z,A , B respectively. To apply a rule i e w j means that M in state i "reads'* a signal on each of the input leads in a emits a signal on each of the output leads of w and changes state to j . An APAnet work N (asynchronous parallelworking automaton) of a set X of blocktransducers is a directed graph with (copies of) elements from X in the nodes, and edges which always connect one automaton output lead to exactly one input lead. The automata at the nodes of N are called the basic automata or components of Ni the input and output leads of Ncomponents into which, and out of which, respectively, no edges run from the input and outputleads, respectively, of N. EXAMPLE
What does the automaton
s:=
l

I
S do ? f
'
S has states "u" (signals can pass through), "d" (signals u t i > + t i ' > d, d t s ) + Bu. cannot pass through) and transitions What does the following block transducer R do? (see Priese 1983)
1
r3'
R=
R has a single state for 1 f 3.
0
and transitions
Oti,i+f)+{i'+2)0
In sequential computation there is always only one signal in the network, and thereby, according to our usual stepnumber definition for Turing machines, the computation time of the network measures the number of transitions of its basic automata; by computation time of parallel computation in an N one understands the number of "transitions of APA network N', where a "transition of an APA network" is so determined of the network components activated in it that all simultaneously carry out each individual transit ion
270
CIV Finite Automata
(application of one rule out of those of the defining semiThue system which can be applied). (NB: The "clock" introduced by this definition only plays a role in the counting of computation steps. The asynchronous and parallel working network6 to be constructed even work correctly without the clock, that is, if, at arbitrary instants some <not necessarily all) of the activated components of the network make a transition. 1 EXAMPLE In the following tree form of APA network of CD) it requires three network transitions for a signal at the root to appear copied eightfold at the leaves and correspondingly for N, G
in the reverse direction.
EXERCISE. Formalise the above definition of "transition" and "computation time" for APA networks and consider how, following the preceding pattern, for each n, the generalised 4 C, D basic components, with n inputs, respectively, outputs and one output respectively, input, can be realised as (N,G,D) with computation time bounds of APA networks of f+LlOgnl (Llognl is the whole number part of log.,n) The sequential network M in the proof of the Rbdding normal form theorem simulates each individual Mtransition in c.IAl ' I Z I steps for some constant c. (NB: the factor I YI can be neglected in comparison with I A l * I Z I , because in K" at most I A l * I Z l Nautomata are needed for encoding A ( i , j ) . ) Schkitz 1 9 8 2 has been able to improve the simulation time bound to log1 Al * log1 ZI whilst for parallel signal processing this i s reduced to c * (log1 Al + log1 ZI ) : c d
THEOREM (Schatz, Rodding & BrUggemann 1982) Each Mealy automaton can be simulated isomorphically by an APA network of N, H,D, G with bounds on simulation time of and bound on size of simulation c.(loglAl + 10glZI) for some constant c S 17 which is program of c. I A l . I ZI and input and state independent of the given automaton, alphabets A, Z respectively. PROOF. We take over the basic idea of the proof and notation But a new idea for the of the Rijdding normal form theorem.
CIV.3 Decomposition Theorems
271
testing of states or output values as well as for deleting or setting up states is to distribute a pulse simultaneously over all Itcomponents of the row or column concerned by generalised doublers D and to control if necessary the complete execution of such a parallel running operation in a row or The generalised N , G , D used column by generalised gates G. in this have at most I A l I ZI input and output leads and therefore can be run through in 1oglAl + 1 o g l Z I i 1 steps, from which we get the desired bound on computation time of the total network M to be constructed.
Let us recall once more that for the process of simulation we will use M only sequentially, thus does not receive a i for the simulation of a new signal on an input lead corresponding Wtransit ion before the processing of the preceding input signal has come to an end and the associated out put signal has been emitted. We therefore organise the i in state simulation of an Wcomputation step on input of j in two phases: I. Configuration recognition running in parallel: 1. erasing J in K
consisting
2.
intermediate storage of
SCj, i )
in
3.
intermediate storage of recognition control.
A(.j,i)
in
4.
of
four
processes
K' K"
give an endofoperation signal to Processes 14 transition control, which supervises the beginning of coding of the Wtransition.
the the
11. Transition consists of three processes which are separated from each other by controls and follow each other, and which themselves are built out of parallel running subprocesses: 1. again looking for the successor state in K', setting control of the termination of up the successor state in K, the output operation of the previous simulation step and signalling endofoperation to the output prercontrol. 2. looking again for the output signal in K", passing it over to the output unit K"' and signalling endofoperation to the output endcontrol.
3. identification of the output signal, output and endofoperation signal to the output precontrol.
CIV Finite Automata
272
According to this plan we order the "hardware I' of R (without showing Wautomata) by parallel connection of basic automata as follows:
D
... G
Hi,
'
...
G
. . .H',... GLL. . H",. . . GD. . H"',.
4 G
\G
With regard to the details of the proof of the Radding normal form there are five alterations: a). The H i , in K have a fourfold upper test output lead be,, for I=l,2,3,4 for triggering the Ith named subprocess of configuration recognition. In addition .K possesses per row one and per column two doublers D with associated (generalised) gates G for distributing signals to J+1 Hcomonents of the row for initiating the state the i+l Hcomponents of the column recognition and to the concerned for deleting or installing states in this column, respectively; the corresponding gates form the recognition i?") control ("has each recognition signal left the Krow with input leads RC,, and output lead RC,, the delete j again stand at control ("does each Ifcomponent of column IIdll?ll) with input leads DC,, and output lead DC,, the stand at install control ("does each Ifcomponent of column j "u"?") with input leads InC,, and output lead InC,. b). K' is a row vector of H e f each with two upper test output leads btf and b',I for initiating the installation of the successor state and the endofoperation signal to the output precontrol, respectively. In addition K' has a D for distributing the pulse for finding the doubler Itcomponents in K'. successor state t o the J + 1
c) K" and K"' are vectors of Hautomata H", and H " ' , , respectively, each with one doubler D and the associated gate G. H"' has two upper output leads b'" ,'. D serves to distribute signals to the l + l Hcomponents of the vector for G in K" forms the the recognition of the output symbol; output precontrol ("recognit ion of successor state ended and K' free of signals? Successor state installed in h? Output to previous simulation step terminated and K"' free of signals?") with inputs OPC,.,,, ,, OPC,,, ,, OPC, and output OPC; G in K"' forms the output endcontrol ("output value from K" transferred to K"' and N" free of signals?") with inputs OEC, and output OEC.
C I V . 3 Decomposition Theorems
TC,
273
d). the transition control is a gate with four input leads and output lead TC.
e). J is defined by all H,,, for i 6 f : in "u", one signal on each of the leads OPC, for k 6 k' and otherwise signalfree leads and the other Ifcomponents in "d". We now program M by placing feedback connections between the parallel coupled basic automata of the matrices K, , KW I and the transition control. Here we will not indicate feedback connections from or to (generalised) D,G, and Wcomponents explicitly, but merely note in the margin "via (signal) distributor", "via nodes" or nothing at all: in the circuit diagram we have used at critical places the notation:
.. .
4
G,
.. ...
...'t:..
D,
Numbers ( i ) ,(j), ( k ) denote the number of the lead of the indicated connection which stands for a set of f + 1 , J i l , k'+l leads. For arbitrary i , t & f, j 6 J , k 6 I? the following feedback connections are therefore established:
0. i  + a A , i via
D
to the ith row of
K
(which u" ?)
H,,
is in
J,
store
*I
I. Configuration recognition
1. Deletion of
found: b,
fti
I+
in
j
via
D
K
(seek
and
delete
6(j , I > , A ( J , i ) : in parallel! )
for the state
j
at the jth column of
K
which has been (put II
gA
i
4
DC,
2.
eli+
3.
via N
(column j deleted)
Intermediate storage of successor state b, i  d ' , , *
TC,
in
< H i., deleted?)
i
DC,+TC,
H,,
d8l )
,,,,
if 6 ( j,i ) & ,b, i;
via N
in
a(j,i)
otherwise
K':
(store)
(successor state stored in K')
Intermediate storage of output signal
A(j,i)
in
K":
K
.
go t o intermediate
D
storage of n e x t state d(f, j )
+. . D
s gnals
t o statecolumn .I
'1 L,
(1:
H',
.
D
L
:
signals to
I
I
g o t o i n t e r n e d i a t e o u t p u t s t o r a g e of
(k)
A(i,j)
I

U
~
 output  2  c o n t r o l I.
si p a l s t o l  t h row
=1%P z w
U D >
 3 r m o r
;f
G
> m r C O
+or
>w>
2
G
"Il'Cj)
recoanition control
(i):
:
d e l e t e c o n t r o l ( s t a t e column e r a s e d ? )
K
(k) 1.
signal install control (next s t a t e installed?)
...
(out put ensured)
2IL
statecolumn 1
urn
e)Cffl
. ... D
2 ?lete i n
N'
H" ' k
...
G
9 2

r'
*
out put end c o n t r o l
n.+ Ki1
0,
Y D
3
275
CIV.3 Decomposition Theorems
b, ,  u d " , c 3 ,i
e",+ TC,
RC,+
configuration RC,
Ci
t v
I+
11. Transition signal k);
successor
Seek succes6or state in out put ended?:
e
TC+a',
at all
K', H',
state
H,
J,
install {seek u" ? )
K,
which
j:
is
,
has
row of
emit
In
f
E')
K
in
{recognition in Ith has been effected)
1.
D
recognition
K
(Install
via
{store)
trecognit ion in been effected)
3
via N
TC,
otherwise
{output signal stored in
whether
RC,
i4+
N
via
4. Control, closed: b,
i f h < j ,I ) & b,,," ,
,
output
previous is in
H',
11
via
b' ' , 3 d i , e,
{H,
via
InC,+OPC,,,
b'
2,4 f'
{f
N
, ,,
c'
,+ OPC,,,,,
Starting from OPC transfer i t to N"': OPC+a".
,
K
set In
{H,,
"u")
Installed)
installed)
{delete intermediate storage: set H', back to "d")
g',+ OPCo,.+ 2.
D at jth column of
,+ I n C ,
via
n has a path of length > n+l (with at least n + 2 nodes), on which some variable occurs more than once. We choose an Hsubtree H, generated by a node i 2’’ has height at least n+1. Therefore we put p := 2”. Because of the height bound nkl on H, the result vwx is not longer than q : = 2 v 1 ’ 1 .
As with regular lanuuages, the loop lemma yields a method of proof t h a t c e r t a i n lanppages are not c o n t e x t  f r e e :
L: = C a i 7 b i i c i : 0 1 n )
EXAMPLE.
is not contextfree.
PROOF. For n > p ar1b”c’? can have no loop lemma decomposition uvwxy: for otherwise if a ’nb’ = uvwxy neither v nor x could contain occurrences of at least two uv”’wx”’y not all a’s different letters (as otherwise in would occur before all b‘s and these before all c’s); if, however, v and x each consisted of occurrences of just one letter, then the number of occurrences of the third letter in I uvr”wxil’y would be constant with growing m 3
~ 8 ’
EXERCISE .The following languages are not contextfree: t arib17c’1:n 1 m)
Ca’’bicn: n 4 i < m) { anbmc’ndn, 0 4 n,m)
iw {wtq
w
{ a , b, c)*
f
w
f
81
V,(w) = V,,(w) = K ( W ) >
Ca, b > * )
COROLLARY
(Scheinberg 1960) The c l a s s of c o n t e x t  f r e e l a n g u a g e s is c l o s e d u n d e r u n i o n b u t n o t u n d e r i n t e r s e c t i o n or compl emen t a t i o n .
C V . 2 Periodicity Properties
307
PROOF. t a 1 ’ b i i c ” : 0 C n, ml
NB:recall that
EXERCISE. twaRv(w):
A* 
n
w
n
Show f
u
n).
and complementation.
(a’)’h
,
I
and
O C n )
The proof of the loop lemma shows a method for extending a Riven derivation tree H‘ by substituting heightbounded H ’ , at appropriate variable positions. Iteration of trees this process yields an analysis and synthesis algorithm which G constructs arbitrary for each contextfree grammar derivation trees from Gderivation trees restricted in height by the square of the number of variables of G. This alRorithm reveals a Reneralisation of the periodicity property formulated in the l o o p lemma, which characterises the frequency distribution of letters in words of contextfree lanRuaKes and shows that in this respect they are not distinRuished from regular languages; as we shall see later the difference comes in the expressibility of bracket structures by contextfree grammars. (for another sharpening of the loop lemma see Ogden 1968: cf also Ogden 1969, Boasson 1973).
DEFINITION. A s a generalisation of linear sets of numbers + pi; 0 4 i l we define l i n e a r s e t s L(B, P ) of ntuples of natural numbers for finite B (“Basis“), P (“periods“) E N ” by: t b
L(B, P ) : = { b + F < p , ; p ,
€
P,
= { b + F t i , p , ; jCm>;
i&kl; b b
f
B,
B,
E
i,
f
k Nl
€
N) for P = t p , , , .
. . ,p,)
with the usual componentwise vector addition. S e m i l i n e a r s e t s are finite unions of linear sets. For alphabet A = ta,, . , a , > > the (letter) distribution function K A * + N ”  we drop the index A  is defined by with Va ( w): =number of occurrences V ( w): = ( V ( w), . , Va, ( w) )
..
a,
. .
,
of a, in w. V is called the P a r i k h f u n c t i o n and the P a r i k h map of L .
EXAMPLE. Each semilinear reKular language because for P = tp,,, . , pnr) c N,
..
set
is
A =
ta)
..
L ( B , P ) = v ( U { t a b ’ l *t a p ’ > * . .
the Parikh map and B = tb,,,
V(L)
of
a
. . . , b,,),
*
t a p ” ’ )*;
16 n ) )
CV Contextfree Languages
308
N k instead of N). So the regular (analogously for )" and the Fitch domain of all correct bracket language expressions of (,> are equal under V.
THEOREM (Parikh 1966) The 1 anguages is s e m i 1 i n e a r .
Parikh
map
of
contextfree
PROOF. For a semilinear description of the "result" of Gderivation trees, we can take over the construction from the proof of the loop lemma under the assumption that the tree H', (unshaded) substituted in i, with root Z and result vZx, contains the same variables as the tree Hi. attached Q below. For this we check in Gderivation trees the set of variables which occur as denotations of nodes: let L, be the set of variablefree results of a Gderivation tree with root Ax and variableset Q (that is, apart from terminal letters, all and only the variables of Q occur as denotations of nodes). A s there are only finitely many such Q, it suffices to prove the semilinearity of V ( L , > ) . For the following let Q be fixed, n : = I Q I . A simple consideration leads to an appropriate height bound generated for the trees to be substituted: the subtrees Hi i along a path have a smaller set Q, of from the node i lies t o the leaf, so that on variables the nearer the root n + l nodes I with the same inscription Z, each path with because 0
f
Qm+v s
5
Qd+v
5
Qs s
a
.
*
C
91
5
Q
9
at least one pair of nodes i and i ' : = i t 1 generate trees with the same set of variables; the tree arising by excision of Hs. from Hs can then play t h e role of the unshaded H', in the loop lemma. Each path of length > nx subtree contains n + l repetitions of at least one variable, therefore we choose here n1 as the height bound (instead of n in the loop lemma). A basis t r e e is each Gderivation tree with root Ax, variablefree result, variable set Q and height bound n+l, a p e r i o d t r e e (substitution tree) is each Gderivation tree with root Z and result WZX with variablefree words v,x with variable set included in Q and height bound n + l . B, P are by definition the sets of results of the basis and period trees, respectively. We will show that
v
contextfree.
Fitch domain.
EXERCISE. L is said to be accepted by final states by a pushdown automaton i f f L = ty
O w,.,t
vf
for some final state
and accepted by empty stack
L = tq
O w I,., i
f
and word
v)
iff
for some Mstate
1).
Show that each such language L is accepted by a pushdown automaton in the sense of our original definition.
Schlltzenberger 1963) THEOREM (Chorsky 1962. Evey 1963, Characterisat ion of contextfree derivations by automata. Contextfree languages are exactly those languages accepted by pushdown automata.
CV. 3 Machine Characterisat ion
313
NB: Finite automata with two instead of one pushdown store are recursively equivalent to 2register machines. PROOF. After the above example it remains to construct, for an a contextfree grammar M arbitrary pushdown automaton y with L < M ) = L + A
in
A For
t+l
decompose the computation into a first step and the rest of length t which is composed of n parts Bj as given above for the elimination of the topmost pushdown symbols X,: Xiav
+
X,,.
For all
we
.. X , i ' v
j
n:
6
XJiJWj
't_Mij+I
with v = w , . . . ~ , , , i , = i ' , i,,,, = k. By inductive hypothesis there follows < i j , X s , i j + , > + m w , and 60 in fi there holds:
. . +
< i ,X, k> + a < i ' , X I , i,>.
If the first &transition is hypothesis < i ' ,A. k> tm v for i ' v
"t
X i a v *
awl..
. w,.
i ' v then by inductive
k,
thus
+ a < i ' , A.k>
I
av.
If, conversely, t
< i ,X, k>
p)
w
is given, then :
For
the rule used could only have been so that X = w = A, i = k and 1 +,.I i because of the reflexivity of I.
For
t = 1
+
A,
t+l
let
+
< i ,X, k>
.
a < i ' , X I , i , > < iX,,, , i p > .
with av = w be the first step of the given derivation so that by inductive hypothesis: for all j < n, X , , i j w d kM i,,, for & t
< i dX,, . i j + , >19
wj
,
i, = i'
and
i,,,, = k,
and so Xiaw,.
with
.
. . w,,
bn
X,,.
. . X , l ' w , . . . w,, +,.,k
w = a w , . . w,,. In the case of a first step of derivation < i , X , k > + a C n  i . B C C in at least one configuration part has no state symbol 1 or C
6
UtAg+12i; i 4 n) * IJ t Ad' ( ZB) A* 1) ( ZB) AmkA gm* l ' I ; 1 & j
..
.
I
where
X
:=
U ( A J ~ ~ C A ~ , ~(AL’   ”. <M(abc)>)*AnvJ2; J < m 81 abc
C
A=}
Clearly the length of Form u Initial u Final u Transition is bounded by a polynomial in I M I , I I J I , m, n. The error description can be shortened with the help of the *operation and so lemma 1 can be sharpened to:
2. By means of a deterministic TM program with m linear bounded space requirement, for each W and there can be constructed a regular expression R<M, W,m) of linear bounded length so that
LEMMA
L(R(M, W,m)) = A*:

AccCM, W,m)
for the set Acc(M, W,m) of a l l W accepted by Mcomputations with space requirement m PROOF of lemma 2. Final u Transition Final
We define regular error sets by (  binds stronger than u ) :
Initial
u
: = we generate th. left C = ( v i , w i ) i * , 7with and right halves vi, I > I . . vi, , . > I and wirr * * W i ' , . . respectively of Csolution attempts together with a ecord of iCl) those pairs used by gruirunars C , the numbers iCr) a C , with disjoint rules together with the rules ( * ' ?',''>", are new symbols): Ax + L L  + v,L(l> L vi a (i r + w,a(i> R + w,RCi> Ax + R
...
There holds: via
. . v**,,a(i(r)). . . Ci u
C ,
i s
ambiguous ) consisting of all words V,aI'aJWJ. where
v,
= vir 7,. * vie,>, I * = (lCr)>.,,( i C l ) > ,
J =
W,.
CjC 1)
>.
. . (JCs) >
= RvCw,,,,).
. . Rv(w,,
I
,),
we determine the Csolutions by intersection with the language R v f : = twaRv<w); w arbitrary) of all reverse forms.
Then there holds:
CV Contextfree Languages
324
l.
3. The class of contextfree languages is the smallest 0,and closed under class of languages containing homomorphic inverse images, intersection with regular sets and homomorphi c images.
b) Left and rightderivations have several times played a role in our considerations of contextfree derivations , That this was not accidental, but characterises the possibilities of contextfree description, i s shown in the following variant of the bracketlanguage characterisation of contextfree languages going back to Mathews 1967:
THEOREM. Characterisation of contextfree languages by leftrightderivations.. Contextfree languages a r e exactly those languages which can be generated from a Chomsky grammar by restriction to left and rlghtderivation steps.
V I"'
W
means that
W
is derivable in
C
from
V
CV. 5 Chomsky Hierarchy
329
with a derivation in which at each step a outermost variable word is replaced:
Cq A x t
L,,,(G):=
'bR
W & W
C
left or right
L(G))
By paraphrasing the proof of the bracket language characterisation we establish the assertion of the theorem for the case of grammars in which the conclusions of the rules contain, besides possibly variables, exactly one terminal letter and the premisses consist only of (at most two) variables. LEMMA. (Gineburg (L Greibach 1966) I f G c o n s i s t s of r u l e s o f t h e form U+ VaW f o r variable words UVW with I U I < 2, t h e n L(G) is c o n t e x t  f r e e PROOF of theorem (using the lemma): We simulate the G (which without l o s s of LRderivations of an arbitrary generality can be assumed to be in Chomsky normal form) by a G' of the form stated in the lemma. For this we mark the position at which LRderivations are taking place by boundary symbols (new variables) L, R and describe Gderivation steps at these boundaries so that the results of grammatical transformations remain in the interior (between L and R ) and the results of lexical transitions are transferred outside the boundaries; on applying grammatical rules each time an occurrence of a new terminal letter (dummy symbol) e is passed to the outside, so that for all terminal words UV and all variable words W o f G with the homomorphism defined by h ( e ) : = A, h ( h ) : = a for all a # e there holds Ax
k$ U W
iff
(3L7,
For new variables L x v , Ax'+
n: [Ax' G ' ULWRv
&
h ( n = U 81 h < f i = L ' l
l e t G' be defined by the rules:
Rwv
eLAxR
[initial rule for to set up beginning of simulation1 LX+
eLU,
XR+ U R e
for each grammatical G r u l e X + U LX
eLwv,
L,,Y+
eLU
,
YR+ R x v e ,
XRxv+
URe
for each grammatical G r u l e X Y  3 U LX+aL,
XR+Ra
for each lexical Grule X + a L+ e , R+ e C closing rules for ending simulat ion1 is contextfree by the lemma, so also is h ( L ( G ' ) ) As L(G') and hence so also is L L . , W ( G ) because L , , , < G ) = h ( L ( G ' ) ) .
CV Contextfree Languages
330
PROOF of LEMMA. The final letter a, which by an application of a rule LI + VaW occurs in the produced word but in no rule premiss, decompose6 the result of the application of the rule into 2 halves, which in the course of further derivations have no common context. Because of this one can simulate such rule applications  wordforword as in the right linear simulation of contextfree left derivations  by applications of contextfree rules: let G be given by rules U,+ V,b,W, Without loss of generality we with b, c Ca ,,.. .,a,,), i < m ,0 , ; namely by the can consider bl as a bracket expression Cwb,, >,HA. Let G' be defined by all homomorphism G r u l e s with premisses of length 1 as well as, for U, = XY, the rules:
x+ v,, Y+
),
[generate left derivation tree, record rule numberl
w,
[generate right derivation tree, record rule numberl
(Exercise!), so that Then L(G) = L,(sr,, s,, 1 for the signature of r, 1place s, Iplace predicatesymbols with functionand r,, si f N u ( w ) ; here we also write ( p , , , , p , > ) instead of (PI,* , p,., 0,. I 0).
...
..
..
. ..
. .
DI. 1 Formal Languages
339
Inductive definition of the terms of a language 1.
Each individual variable is a term of
S:
S.
.
2. If t , , , . , t,, are terms and f is an rrplace function sign of S then f t , , . * t , , is a term of $. 3. t is only a term according to 1. and 2. (inductive clause) t, t,,, t , , . . . , s , s o , s 1, . . . T(B).
metavariables for terms we use The set of terms of S is denoted by As
Inductive definition of the expressions of
1. If t , , t, expression of S
are terms of , a , ,
.
. ..
...
..
...
ar: I A,AVA, 0 ) are propositional logic expressions. 2. If a, /3 are propositional logic expressions, so also are .a and avg. We call 0 and 1 propositional constants and the P,, n3 0 proposi t f onel vari abl es. 1.
0, 1,
P,
EXERCISE. (Characterisat ion of tautologies by analytical tables, see Beth 1959, S67 ff. ) The basic idea of this a a tautology by systematically method consists of proving leading each attempt t o construct an evaluation satisfying  a to a contradiction; if, on the contrary,  a is satisfiable then the procedure yields an (respectively, all) evaluation<s) satisfying a. Because of the tautologies .m/3 and  . < p v y ) ~  / 3 ~  y , expressions 7B and  < / 3 v y ) , respectively, 1 exactly when this is the case for /3, get the truthvalue and /3,  y . respectively: for expressions /3vy however, there results a treetype branching, as both /3 as we13 a s y can have the truthvalue 1 if /3vy has it. A formalisation yields the following definition: An analytical table for a propositional logic expression a is an ordered binary tree whose points are occurrences of expressions and which are determined recursively as fOllOW6: 1.
is an analytical table for
a
a
with root
a.
If T is an analytical table for a, e an endpoint of and /3 an expression of conjunctive type (ie of the form  ( y 6 ) or y) which occurs on the path from a to e , then
2.
T
7
I 6'
is also an analytical table for /3'
8'
=
I
u with
y
for
J
for /3 e  < y , v y s )
/3 L   y ,
is called a s u c c e s s o r of
with
6
Q
365
DII.2 Propositional Logic Completeness Theorem 3.
If
T and
r6
)
T
is an analytical table for a, e an endpoint of an expression of disjunctive type
(#)
P<J(~,.
= 1
ff by o ( B ) ) : P " ( t I , . . ' , t,)
3
in
A" be
.
We define a canonical algebra < W P ) ( W t ,,
. . . I
t ,
f
in
From ( # ) it follows that for all
t,
,*(A Proof.
iff
*
Then for an arbitrary interpretation XI
. .. AXna)
3 ( P t , . . . t,)
= 1
B
= 1 f
o(B)
..
3 < t , > . 3(t,) (a) = 1 x, x,and so by the substitution lemma, 3(d t , , , t , l ) = 1. By B , 3 and 3* agree, for terms the definition of t l f oCB), on BC t , , . . . , t,l for all subformulas /3 of a (induction on 8 ) so that, to sum up:
...
...
REMARK. Because of the fixed canonical interpretation of terms, the Herbrand models of a are determined by, and therefore often identified with, the set of true ground statements of a which occur in i t , that is the prime formulas Ic C t,, , t,,l with terms t , from the Herbrand XI,.
.
*,Jill
...
universe of a and prime formulas R occurring in a. For the special case of Horn formulas 1 and so p occurs in Y: f. 2. If R is a critical quantifier rule then the variable condition is satisfied: the eigenvariable occurs neitb3r in 3, f, Q ,  because of the variable condition 1 I the original derivation  nor in n, Z  because c the disjointness of the eigenvariables in the or ,ginal derivation. A s we reduce the rank of the mix by 1 the newly pising mix can be eliminated by the subsidiary inductive hypothesis. We obtain a mixfree derivation with the end figure
In order to continue this derivation to a mixfree derivation of fl, Li, P I Zc, Q2 we must distinguish the two following cases: Csse 1.2. 2. 1: p does not occur i n 8 ie 6 = SC. get the desired derivation from ( 1 ) by interchange.
Then we
Cese 1.2.2.2: p occurs i n g Then as remarked earlier, S z p and p is the principal formula of the application of the rule R. We amend ( 1 ) in the following way
when when when
A.,a A,.a
f
ri  r Ti
A..a Z Ti
EIII. 2 Interpolation Theorem When kLk f i ' + A , , interpolant for
then also
h,. f i
I
and
A,,, 9
I
S o let
A,.
that is
T I ' , A,,, A l ) ,
fa,'
kL(a) and notVal, ir, ( 8 ) but S " i n t h e f l n l t e " , ie with there is no Interpolant Val,, ( (uld)A (838)) . deflnes F e x p l i c i t l y I n f l n l t e algebras, but P 2. y cannot be d e f i n e d e x p l i c l t l y I n f i n i t e algebras. 1.
PROOF. (according to Gurevich 1 9 8 4 ) . The assertion is easily obtained from the following proposition on a weakness of first order expressions:
.
1. F i r s t order closed expressions i n LEMMA. (Ehrenfeucht ) $(O; 5,K) cannot d i s t i n g u l s h between f i n l t e the language 2+(quantifier depth + 1 ) l i n e a r orderings with at l e a s t el emen t s. 2. The f i n 1 t e l i n e a r orderings w l t h even cardinal1 t y cannot be character1 sed by a f i r s t order closed expressl on In t h e language S ( K ) . 3. There is no f i r s t order expression IZ i n t h e language $(K) with f r e e variable x whlch is s a t i s f i e d i n a l l
f i n i t e l i n e a r orderlngs e x a c t l y by those elements with an even order number.
PROOF of theorem from the lemma. Let ord, be the axiom of linear ordering with smallest element 0 and successor relation S (cf example 2, §DIl). Let even,. (for a oneplace predicate symbol F ) describe in finite linear orderings the set of elements with even order number, that is, x,,, x,., x4,.. , with x,, = 0 and S X , ~ , , , for all 1: even,. : PO A*.A,(Sxp ( P x H  F y ) . Then val.,,,(ord,
A ( N y 3 y = u ) ) . EXERCISE. The sets I and N of the (only) infinitely satisfiable, respectively, contradictory expressions are recursively inseparable. Hint: Deduce from the assumption that there is a recursive R with I E R & R N = 0 , that then (a; sata) would be recursive (because F = (.a, sat,,,a) and N are r.e,1 Trachtenbrot 1953 showed that also the sets F and N of the finitely satisfiable and the contradictory expressions are recursively inseparable and thereby the Hilbert Entscheidungsproblem even for finite satisfiability is recursively unsolvable. Because val,,,.,a iff notsat,,,.,a, this means that the set of first order expressions which are valid in finite algebras is not recursively axiomatisable, in other words, that there is no complete and correct calculus for the concept of validity in the finite. This statement of impossibility sets a principled boundary on applications of the first order predicate logic, particularly to database theory; we recommend to the reader Gurevich 1984, SFIII2 for a further complexitytheoretic analysis of first order theories of finite algebras. As a consequence of our reduction lemma we now prove,besides the recursive inseparability of F and N , also that of I and F and formulate this as extending the
FI Undecidability and Reduction Classes
448
(Trachtenbrot). The s e t s F, I, N  which a r e r e s p e c t i v e l y t h e s e t o f a l l f o r m u l a s which a r e f i n i t e l y satisfiable, t h e s e t o f f o r m u l a s which a r e s a t i s f i a b l e i n f i n i t e l y and o n l y i n f i n i t e l y , and t h e s e t o f f o r m u l a s which a r e n o t satisfiable  are pairwise recursively inseparable f o r languages o f t h e pure p r e d i c a t e logic w i t h a r b i t r a r i l y many 2  p l a c e p r e d i c a t e s y m b o l s .
THEOREM
PROOF. After the preceding exercise it remains to show the recursive inseparability of N, F and of I, F. We get these by application of our reduction lemma to a reduction of the recursively inseparable 2  R M halt ing problems H,, H,, respectively, H, H2. (see exercise to the lemma on the existence of r.e., recursively inseparable sets in BBII3d), whereby for i = 1, 2: Hs : = C h i , C, = (0,(0,O))I,, (1,(0,0)) & M a 2RM program whose only stop states are 1 , 2 ) H
:=
< M Co
= (0,(0,0))IL halting configuration 8 M a 2RM
program whose only stop states are i , i i a,, be, as in corollary 0 , the expression of the pure Let predicate logic with Skdlem normal form ~ , . ~ , , ~ , , ~ K , , O O ~  K , 0 0 . We show that for arbitrary programs M (1) M E H, iff cona,, (2) M c H.z iff sat,,,,a, The assertion of the theorem follows from these: for if there were a recursive R with F E R and N n R = 0 ( I n R = 0, respectively), then the recursive set .V,Nxv
(axiomatisat ion of successor).
( c iff C ; 6 , < J , ( 0 , O ) ) as well as of The interpretation as C < n , n + l ) ; n € N) yields a model of y',.,. We show indirectly that y',., has no recursive model.
N
Suppose, to the contrary, that A = (0; P I , . .. , p ,  ,fi is a y',.,. Because sat[& V,,A,Z,.,l there is a recursive model of u,, such that 3,J uc,l ( A y Z n ) = 1 for each number 3 in A . Starting with this u,, we can interpretation because sat[$ A,.V,.Nxvl and because of the recursiveness of #  effectively enumerate a subset &I,, = Cub; k 6 N > of a, S O that muk, u,,,) holds for all k 2 0. From the validity of the first two conjunction members of y',., in A" and from the definition of &I,, it follows that
F I . l Theorem of Aanderaa and BBrger
455
.
A", : = (a,,; a; 4 a l l , , ,. . , ff,Io,,) with 6 : = u,, and .%u,) : = u L , , is a recursive model of in contradiction to the above remark
y,, I
EXERCISE 1 . The following variant y",, of y',, is, under the hypotheses of corollary 2, a Krom formula with equality
and without function recursive models:
signs which
is consistent
y",: = V,,,A,,fM A A  V Y A y ( N X V A T ' ~ ) A A,.A,,A=V,.,(NXFy=W) Hint: because of the functionality of N b y " ,,3y',,,
EXERCISE 2. Under 6following formula
models:
and A
without
(Nx~Fw)
the conditions of corollary is satisfiable and without
2 the lTluX,
for a 3place predicate symbol P, where y"',., arises from y',., by replacing all prime formulas K,st by Px,st and Nst by P,st; r is the number of states of M Hint: via P represents all K, and N as well as parametrisation their negations, so that an interpretation of P by a II,or a X,predicate satisfying 6, yields a recursive interpretation of the K, and of N by the negation lemma (CiBII 1) which satisfies y',,. NB: The lower bound for the complexity of the model of S,, is optimal because each consistent first order theory with theorem predicate in As and language in A, has at least I , , see Carstens 1972, one model in the Boolean closure of p42.
EXERCISE 3 . Construct from E,,,,  E,, separable sets E,,.,sets of expressions which have a &+, model but no E, model.
COROLLARY
4. Arithmetical complexity of metalogical For recurs1 ve formula classes F The decision probleas. following problems for Ded(F) : = {a a E F & kPL. a) have
the stated complexity:
n , c omplet e : emptiness problem n,compl et e: totality problem
AC+~, than one configurations at any one time and (using trivially formalisable abbreviations) is defined as the conjunction of: a) In the second argument only W s t a t e s the last argument only 0,1 , 2: A,A,A,A,(CtiPy
f
3
6
t o , . . . , r)
A possible Horn description
* * *X,. V Cx,,=O
11
c
y
to,
E
occur, and in
r
1, 2)).
of this state of affairs reads:
A Sx~xi,,
VX,,
A
): =
+
I e c ",", I
1)
+ Qd(pu,,.
M ,with G6del number
For a RMprogram let:
for
atorial principles lying at the basis of Gentzen's cut elimination process which are formally undecidable in PA), Cichon 1983 ("The theorem of Goodstein mentioned in §CII3 is formally undecidable in PA"), Kowalczyk 1982 (regularity of a certain language over the oneelement alphabet), Simpson 1985 (with further references to the literature). Chaitin 1974 gives an informationtheoretic interpretation of the G6del incompleteness phenomenon: with respect to an informationtheoretic complexity concept for programs, logical theories, random numbers etc , G6del's theorem says that to each theory of information complexity n there can be given a program of information complexity n which does not halt on a particular input  or a binary sequence of complexity greater by a 7l constant  without the latter statement being provable in as an example, it follows from this that the randomness property of numbers can only be proved in T for numbers with En. (See Chaitin 1982 for further information complexity literature). The first order theory of addition (Presburger REMARK 4. arithmetic) and of multiplication (Skdlem arithmetic) is decidable, see Presburger 1929, Skdlem 1930. Decision procedure for these theories, however, according to Fischer & Rabin 1974, need double and triple exponential time, respectively; cf Ferrante & Rackoff 1979 for a more exact discussion. Young 1985 showed how such a result of difficult decidability (in the example: the exponential inseparability of the provable and the refutable statements of Presburger arithmetic) can be deduced from a simple sharpening of the The idea proof method of the Church inseparability theorem. is that only a restricted part of multiplication is necessary
FII. 1 G6del's Incompleteness Theorem
477
for the description of restricted recursive halting problems which can be expressed by short additive formulas by means of the method of Fischer & Rabin 1974. For this interesting interpretat ion of the Gadel incompleteness phenomenon cf €3160 Machtey & Young 1978, 1981. FURTHER LITERATURE (references): Smorynski 1977
PART
III.
RECURSIVE LOWER COMPLEXITY BOUNDS.
In part I we applied the method of logical doocription to expressing halting and dofinition p r o b l o m and in doing eo placed n o reotrlctionr on the length or epaco roquirennt of the described computations. Thoreby w e csrriod over the undecidability of these unreetrictod halting p r o b l e m t o the associated clarsee of formulas. In this part VI) uro the method of logical description of program for the forpalisation of time or spacerestricted halting p r o b l e m in restricted claeses of expressions uheroby those recursivoly solvable decision problem6 turn out t o be complete for the underlying time or spacecomplexity clamsee. OVERVIEW. In $ 1 w e prove the theorem of Cook on tho NP completeness of the propositional logic eatisfiability problem by a propositional logic interpretation of polynomial t imebounded computations of nondeterminist ic Turing machines. As a corollary to our proof method w e get tho polynomial equivalence of two natural complexity m o a ~ r o rof P = NP7 its Boolean functione, by which the Cook problem shown to be equivalent to the quootion of short Horn definitions of Boolean functions. The interest in this derives from the fact that the decision problem of propoeitional logic Horn formulas lire in PI we doviee a corresponding algorithm in order to prove the theorem of Henechen 1c Woo on the completonees of the unit remolution calculus for Horn formula.. Allowing quantif lore over proposition variables yields an extenrion of the deecription method t o polynomial rpacebounded TMcomputations, from which there follows the PTAPE completenere of the decioion problom of quantified propositional logic n , by: arbitrary q = q , ,
...
A N, e,l ( q ) : = q s A N, kl ( q ) : = op(k) ( A N, & , I ( q ) , A N , k,l for logical nodes k with label k,, kZ. The network complexity the minimal number of logical computes it.
EXERCISE. For a set
B
(q))
op+J and
F I I I . l Complexity of Boolean Functions
hence
b(y9
= b(y'.*')
1
Put
.
C
1 C f 6 R,.
d
b,(y,)
conjunct a,,[x / ql
for all
49 1
of
: = b definable as a global 0place predicate by a second order expression of the form VpV,,Vr;.a with first order a. Thus, under the assumption that P # NP there is a global t w r p l a c e predicate which is not computable in polynomial time because, by the Fagin characterisation a global predicate lies in NP i f f i t existent la1 second order expression.
Is definable by an
The following extension of first order definable global predicate6 by the formation of smallest fixed points is motivated by application of the theory of relational data bases and permits a logical characterisation of the class P.
DEFINITION. ; at the lefthand end of the tape and reads the Mletier & here, as in the following, stands for a sequence t,,, , t,.I which is to be thought of as the I Aladic coding of t. (Without loss of generality we can omit the finitely many, first order describable cases of algebras with A = 10) and sequences R over A for which h'&<Xl is true. )
...
It therefore remains to give a intended interpretation 0. For arbitrarily accepts X in time 2 w , fixed in the following, we construct chosen input formula y f VAAV with O(I wl * log1 wl )length and computable w, and which has only monadic in polynomial time from predicate symbols, which simulates the Mcomputat ion defined by w in the sense that:
M , on input of w, has at least computation with 2 c ' w '  1 steps.
satO.. 0.
.
The universal closure of A A a A o is, by the simulation M accepts w. If w were not lemma, contradictory when 4 then the intended interpretation yields a accepted by canonical model for this expression. F A " n Krom it For the reduction to the subclass suffices to carry out the above coding over the twoelement
FIII Lower Complexity Bounds
526
(0, 1 ) . alphabet constant factor.
The length of
y
is thereby raised by a I
REMARK. A small change in the above proof yields additionally, because M ie deterministic, that the Krom and Hornformula arising from R A ff A o by contraposition of its implications represents a deterministic PROLOGprogram : The Kleene Symposium. NorthHolland, 6183.
CONSTABLE, R. L. & BORODIN, A. B.
1972. Subrecursi ve programming languages, part 1: efficiency and program structure. JACM 19, 526568. 1983. Constructive mathematics a s a programming logic I: Some principles of theory. SLNCS 158, 6477.
CONWAY, J. H.
1971. Regular
algebra
Hal 1, London.
and
finite machines.
1979. The relative efficiency systems. JSL 44, 3650
of
Chapman and
propositional
proof
COOK, S. A.
The compl exi ty of theoremprovi ng procedures. STOC 151158. 1983. An overview of computational complexity. Corn. ACM 26, 401408.
197 1.
COOK, S . A.
&
RECKOW, R.
1974. On the lengths of proofs in 6%”’ STOC, 135148. cal cul us. SIGACT News 6 (1974) 1522.
the propositional cf Correction in
COOPER, D. C .
1967. BUhm and Jacopini’s reduction of flow charts. ACM 10, 463+473.
Cow.
CORBIN, J. & BIDOIT, M.
1983. A rehabilitation of Robinson’s unification algorithm. in: R. E. A. MASON (Ed. ) : Information Processing 8 3 . , 909914.
CRAIG, W.
1957, Three uses of the HerbrandGentren relating model theory a n d proof theory. 268.
Theorem
in
problems
of
JSL 22, 250
CUDIA, D. F. 1970. The degree hierarchy of undecidable formal grammars. 2““ STOC, 1021.
538
Bibliography
DAHL, 0.J. , DIJKSTRA, E. W. & HOARE, C.A. R. 1972. Structured programming. Academic Press. DAHLHAUS, E., ISRAELI, A. & MAKOWSKY, J.A. 1984. On the existence of polynomial time algorithms for interpolation problems in propositional logic. Technlon, Israel Inst. of Technology, Halfa TR 320. van DALEN, D. 1980. Logic and Structure. Amsterdam 1980. DALEY, R. P. 1978. On the simplicity of busy beaver nets. ZMLG 24,207224. 1980. The busy beaver method. in: The Kleene Symposium., NorthHolland, 333345. 1980. Busy beaver sets and degrees of unsolvability. JSL 46, 460 sqq. DAVIS, M. 1956. A note on universal Turing machines. In: Automata Studies., Princeton, 172175. 1957. The definition of universal Turing machine. in: Proc. AMS 8, 11251126. 1958. Computability and unsol vabili ty. McGraw Hill. 1971. An explicit diophantine definition of the exponential function. Comm. on pure and appl. math. XXIV, 137145. 1977. Unsolvable problems. I n : J. BARWISE <Ed. ) : Handbook of Mathematical Logic., NorthHolland, C.3, 567594. 1982. Why Gddel didn't have Church's Thesis. Inf. and Contr. 54, 324. DAVIS, M., MATIJASEVICH, Yu. & ROBINSON, J. 1976. Hilbert's tenth problem. Diophantine equations: Proc. of positive aspects of a negative solution. the symp. on the Hllbert problems, D e Kalb, Illinois, May 1974. Amer. Math. Soc., 323378. DAVIS, M. & PUTNAM, H. 1960. A computing procedure for quantification theory. JACM 7, 201215. DAVIS, M., PUTNAM, H. & ROBINSON, J. 1961. The decision problem for exponential diophantine equations. Ann. Math. 71, 425436. DAVIS, M. D. 8 WEYUKER, E. J. 1983. Computability, complexity and languages. Academic press. DENENBERG, L. A. 1984. Computational complexity of logical problems: Aiken Comp. formulas, dependencies and circuits. Lab., Harvard Unlv., TR2084.
Bibliography
539
DENENBERG, L.A. % LEWIS 1984. 1984. The complexity of the satisfiability problem for Krom formulas. TCS 30, 319341. DENNISJONES, E.C. % WAINER, S . S . 1984. Subrecursive hierarchies via direct limits. in: Computation and Proof Theory, SLNM 1104, 117128. DEUTSCH, M. 1975. Zur Darstellung koaufzdhlbarer Prddikete bei Verwendung eines einzigen unbeschrankten Quant ors. ZMLG 21, 443454. 1975. Zur Theorie der specktralen Darstellung von Prddihaten durch AusdrLIcke der Prddikatenlogik 1 . Stufe. AMLG 17, 916. 1982. Zur Kompl exi tItsmessung primi t I vrekursi ver Funktionen Uber Quotientermengen. ZMLG 28. 345363. DIJKSTRA, E. W. 1968. A constructive approach to the problem of program correctness. BIT 8, 174186. 1976. A discipline of programming. Prentice Hall. D ~ P K E ,w. 1977. Klassen primi t i vrekursiver Funkt i onen auf Termmengen. Di pl omarbe1 t . Uni versi t 11t MUnst er, Inst . f. math. Logik u. Grundlagenforschung. DREBEN, B. % GOLDFARB, H.D. 1979. The decision problem: Solvable cases of quant i fi cat i onel f ormulas. Addi sonWesley. DURNEV, V. G. 1973. Pozitivnaja teorija svobodnoj polugruppy (Positive theory of free semigroups>. Doklady Akad. Nauk SSSR, Mat. 211, 772774. DWORK, C., KANELLAKIS, P. % MITCHELL, J. 1984. On the sequential nature of unification. J. of Logic Programming 1, 3550. DYSON, V. H., JONES, J. P. 8 SHEPHERDSON, J. C. 1982. Some diophantine forms of Gtldel's theorem. AMLG 22, 5 160.
EARLY, J.
1970. An efficient contextfree parsing algorithm. ACM 13, 94102.
Comm.
EBBINGHAUS, H.D., FLUM, J. % THOMAS, W. 1980. Ein f Uhrung in die mathematische Logik. Wissenschaftliche Buchgesell6chaft Darmstadt <English edition Springer Verlag, 1984. ) 1982. Undecidability of some domino connectability problems. ZML 28, 331336.
Bibliography
540
EDMONDS, J. 1965. Paths, trees and flowers. 467.
Caned. J. Math. 17, 449
EHRENFEUCHT, A., KARKUMKI, J. & ROZENBERG, D. 1982. The (generalised) Post correspondence problem wi th lists consisting of two words is decidable. TCS 21, 119144.
EHRENFEUCHT, A. & ROZENBERG, G. 1981. On the (generalised) Post correspondence problem with lists of length 2. SLNCS 115, 408416. EILENBERG, S. 1974. Automata, Languages and Machines. vol A, vol B us1 n g Mat f jasevf ch ' s "masking". in: L. PRIESE <Ed.) : Report on the l V L GTIworkshop. Paderborn, 105 116. 1984. Unf versal Turing machines CUTM) and JonesSLNCS 177, 248253. Matijasevf chmasking. 1983.
HATCHER, W. S . % HODGSON,B. R. 1981.
Compl ex1 ty bounds on proofs.
HEIDLER. K.
1970. Domf nospf el e
und problems auf AVA.
df e
JSL 46, 255258.
Reduk t f on
des
En t .chef dungs
Diplomarbeit, Universitktt Freiburg
i . Brag. pp VI+67. HEIDLER, K. , HERMES, H. MAHN, F. K. 1977. Rekursf ve Funktfonen. Bibliogr. Inst. hannheim. HE INERMANN, W.
196 1 .
HELM, J. 1971.
Un t ersuchungen Uber d i e Rekursf onszahl en rekursf ver Funktfonen. Dissertation, Univ. MUnster, Inst. f. math. Logik u. Grundlagenforschung.
%
YOUNG, P.
On size vs. efficiency for programs admitting speedups. JSL 36, 2117.
HELM, J. , MEYER. A. 1973.
%
YOUNG, P.
On orders o f translation and enumeratfons. Pacific J.
Math. 46, 185195. v. HENKE, F. W., INDERMARK, K., ROSE, G. 8 WEIHRAUCH, K. 1975. On prfmi tlve recursf ve wordfunctions. Computing 15, 2 17234.
HENKIN, L.
The completeness of the firstorder functional calculus. JSL 14, 159166. 1950. Completeness In the theory of types. JSL 15, 8191. 1949.
HENNIE, F.C. 1965.
Onetape,
offline
Turing
machine
Information and Control 8 , 553578. HENNIE, F.C. % STEARNS, R. E. 1966.
Two t ape
simul at i on
JACM 13. 533546.
of
mu1 t f tape
computations.
Turf ng
machines.
Bibliography
548
HENSCHEN, L. h WOS, L. 1974. Unit refutations and Horn sets. JACM 21, 590605. HERBRAND, J. 1932. Sur la noncontradiction de l'arithm6tique. J . fur die reine u. angew. Math. 166, 18. HERMAN, G.T. 1969. A new hierarchy of elementary functions. Proc. AMS
20, 557562. 1971. Strong computability and variants of the uniform halting problem. ZMLG, 115131. 1971. The equivalence of various hierarchies of elementary functions. ZMLG 17, 115131.
H E R E S , H. 1971. A simplified proof for the unsolvability of the Logic Colloquium decision problem in the case Y 3 Y . ' 69, NorthHolland. 1976. EinfUhrung in die mathematische Logik. Stuttgart. HILBERT, D.
BERNAYS, P.
1939, 1970. Grundlagen der Mathematik I, II.
HILL, R.
1974. Ludi resolution and its completeness. Univ. of Edinburgh.
Berlin. DLC Memo 78,
HOARE, C. A. R. 1969. An axiomatic basis for computer programming. Corn. ACM 12, 576580, 583. HOARE, C.A. R. & WIRTH, N. 1973. An axiomatic definition of the programming language Pascal. Acta Inf. 2, 335355. HOOPER, P. K. 1966. The undecidability of the Turing machine immortality problem. JSL 31, 219234. HOPCROFT, J. E. 1971. An n log n algorithm for minimizing the states in a in: Z.KOHAVI (Ed. ) : The Theory of finite automaton. Machines and Computation. .Academic Press, pp189196. HOPCROFT, J. E. & ULLMAN, J. D. 1979. An introduction to automata theory, languages and computation. AddisonWesley, Reading, Mass. HORN, A.
1951. On
sentences which are true of algebras. JSL 16, 1421.
direct
unions of
HOSKEN, W. H. 1972. Some Post canonical systems in one letter. 509515.
BIT 12,
HOTZ, G . & ESTENFELD, K. 1981. Formale Sprachen. Eine aut omat entheoret i sche EinfUhrung. Bibliographisches Inst. , Mannheim.
Bibliography
5 49
HUNT 111, H. B. 1973. On the time and tape complexity of languages, I . 5tp8 STOC, 1019. 1979. Observations on the complexity of regular expression problems. JCSS 19, 222236. HUNT 111, H. B. & ROSENKRANTZ, D. J. 1974. Computatlonal parallels between the regular and contextfree languages. 6") STOC, 6473. HUNT 111, H.B. , ROSENKRANTZ, D.J. & SZYMANSKI, T.G. 1976. The covering problem for linear contextfree grammars. TCS 3 6 1382. the equivalence, containment, and covering 1976. On problems for the regular and contextfree 1 anguages. JCSS 12, 222268. HUNT 111, H.B. & SZYMANSKI, T.G. 1975. On the complexity of grammar and related problems. 8*" STOC, 5465. HUWIG, H. & CLAUS, V. 1977. Das Xquivalenzproblem fur spezielle Klassen von LOOPprogrammen. SLNCS 48, 7382. IMMERMAN, N. 1979. Length of predicate calculus formulas as a new complexity measure. 2 0 ' L r 0FOCS, 337347. 1982. Upper and lower bounds for first order expressibili t y . JCSS 25, 7698. 1982. Relational queries computable in polynomial time. 14"' STOC, 147152. 1983. Languages which capture complexity classes. 15%" STOC, 347354. ITAI, A. & MAKOWSKY, J. A. 1983. Unification as a complexity measure for l o g i c programming. TechnionIsrael Inst. of Technology TR 301.
JEDRZEJOWICZ, J. 1979. Inverse derivabili ty and confluence problems of deterministic combinatorial systems. Proc. 2v's* Int. Conf. Fundamentals of Computation Theory, FCT ' 79. AkademieVerlag, Berlin. 1979a Oneone degrees of combinatorial systems decision Bull. Acad. Pol. des Sciences, SOr. sci. problems. mathem. XXVII, 819821. JERVELL,H. R. 1983. Gentzen games. Preprint Univ. of Troms6, Inst. of Math. and Physical Science, Math. Reports.
550
Bibliography
J. P. 1975. Diophantine representat i on of the Fibonacci numbers. The F i b o n a c c i Q u a r t e r l y 13,8488. 1978. Three universal representations of recursively enumerable sets. J S L 43, 335351. 1982. Universal diophantine equations. J S L 47, 549571.
JONES,
Yu. V. new representation for the symmetric binomial coefficient and its applications. A l g e b r a a n d l o g i c . 1984. Register machine proof of the theorem on exponential diophantine representation of recursively enumerable sets. J S L 49, 818829.
JONES,
J.P. 1981. A
& MATIJASEVICH,
J . P . , SATO, D . , WADA, H. 8 WIENS, D. 1976. Diophantine representation of the set numbers.. Am. Math. M o n t h l y 83, 449464.
JONES,
of
prime
N. D. 1966. A survey of formal language theory. U n i v . of W e s t e r n O n t a r i o , Comp. S c i . D e p t . Techn. R e p . 3. 1968. Classes of automata and transitive closure. I n f o r m a t i o n a n d C o n t r o l 13, 207229. 1975. Spacebounded reducibility among combinator1a1 problems. J C S S 1 1 , 6885.
JONES,
N. D. & LAASER, W. T. 1977. Complete problems for deterministic polynomial time. T C S 3, 105117.
JONES,
N. D., L I E N , Y.E. & LAASER, W. T . 1976. New problems complete for nondeterministic log space. Math. S y s t . T h e o r y 10, 117.
JONES,
JONES,
N. D. 8 SELMAN, A. L. 1974. Turing machines and the formulas. J S L 39, 139150.
spectra
of
firstorder
N . D . , SESTOFT, P. & SILINDERGAARD, H. 1985. An experiment in partial evaluation: The generation of a compiler generator. i n Science of Computer
JONES,
Programming.
A. S. 1962. Improved
KAHR,
reductions of the Entscheidungsproblem to subclasses of AEA formulas. P r o c . S y m p . o n Math.
T h e o r y of A u t o m a t a , New Y o r k , 5770.
Brooklyn Polytechnic
Institute,
A. S. ,MOORE, E. F. & WANG. H. 1962. Entscheidungsproblem reduced to the AEA case. N a t . Acad. S c i . U S A 48,367377.
KAHR,
KALMAR,
Proc.
L. 1933. Uber die Erfllllbarkei t derjenigen ZB’hlausdrUcke, we1 che in der Normal form zwei benechbart e A1 1zei chen enthal ten. Math. Ann. 108, 466484.
Bibliography KARP,
R. M. Reducibility
1972.
KARP,
55 1
among combinatorial problems. in: R . E. MILLER & W. THATCHER ( E d s . ) : Complexity of Computer Computations. P l e n u m P r e s s , 85104.
R. M.
1969.
& MILLER,
KANELLAKIS, 1984.
R.
Parallel program schemata. P.C.,
DWORK, C.
147195.
J.C.
On the sequential nature of unification.
~
Programming KETONEN, J . 1981.
J C S S 3,
& MITCHELL,
1
& SOLOVAY,
J.
Logic
(1).
R.
Rapidly growing Ramsey functions.
Ann.
o f Math.
2673 14.
KIRBY,L. 1982.
& PARIS,
J.
Accessible independence results for Peano ari thme i c. Bull.
London Math.
SOC.
14,
285293.
KLEENE, S . C . 1936.
General Math.
1938. 1943.
recursive
Ann.
112,
functions
of
natural
numbers.
727742.
On notation for ordinal numbers. JSL 3, 150155. Recursive predicates and quantifiers. T r a n s A M S 53, 4173.
Representation of events in nerve nets and finite i n Rand R e s . Memo. R M 704, see a l s o : automata. Automata studies. P r i n c e t o n U n l v . P r e s s , 342. 1952. Introduction to metamathematics. A m s t e r d a m . of an effectively generated class of 1958. Extension functions by enumeration. C o l l o q . M a t h . 6, 6778. pp 1979. Origins of recursive function theory. 2 0 t t ' FOCS,
1951.
371382.
The theory of recursive functions approaching its centennial. B u l l . ( N e w S e r i e s ) AMS 5, 4361. Ann. Hist. 1981a Origins of recursive function theory. 1981.
Comp.
KLEINE
BUNING.
1975.
3,
5267.
H.
Netzwerke endlicher Automaten fur Potentialautomaten.
Diplomarbeit. Inst. fUr math. G r u n d l a g e n f o r s c h u n g , Univ. MUnster. 1977.
1982. 1983.
Decision systems.
problems
Uni v .
in
M U n s t er.
generalised
vector
addition
Ann. SOC. M a t h . Polonae, Series F u n d a r n e n t a I n f o r m a t i c a e 111. 4, 497512. Note on the E,*E?* Problem. ZMLG 28, 277284.
Klassen definiert durch syntaktische Rekursion.
29, 1984.
u.
#ber Probleme be1 homogener Parket tierung von ZxZ durch Mealyautomaten bei normierter Verwendung. Dissertation. Inst. fUr math. Logik u.
Grundl agenf orschung, 1980.
Logik
IV:
ZMLG
169175.
Complex1 ty of 1oopprobl ems in normed net works. in E. BURGER. G. HASENJAEGER, D. RUDDING ( E d s . ) : Logic and
;
Bibliography
552
Machines: Decision Problems and Complexity. 171,
SLNCS
254269.
KLEINE BUNING,H. % OTTMANN, Th. 1977. K1 eine uni verse1le mehrdimensional e Turingmaschlnen. EIK 13, 179201. KLEINE BUNING,H. 8 PRIESE, L. 1980. Uni versa1 asynchronous iterative arrays of Mealyautomata. Acte Inf. 13, 269285. KNUTH, D. E. 1965. On the translation of languages from left to right. Information and Control 8, 607639. 1974. Structured programming with g o to statements. Computing Surveys 6, 261301. KOEBER,P. & OTTMANN, Th. 1974. Simulation endlicher Automaten durch Ketten aus einfachen Bausteinautomaten. EIK 10. KOSARAJU, S. R. 1964. Decidability of reachability in vector addition systems. STOC, 267281. KOSTYRKO, V. F. 1964. Klass svedeniya V 3 ” V . Algebra y Logika, 4565. KOWALCZYK. W. 1982. A sufficient condition for the consistency of P=NP with Peano arithmetic. Annales SOC. Math. Pol. Ser. IV, Fundamenta Informaticee V. 2. 233245. KOWALSKI, R. A. 1974. Predicate 1ogic as a programmlng 1anguage. IFIP Proceedings (Stockholm), 569574. 1979. Logic for problem solving. New York. 1983. Logic programming. Inf. Proc. 8 3 (Ed. R. H. A. MASON), 133 145.
KOZEN, D. C. 1976. On parallelism in Turing machines. 17L’i FOCS, 8997. FOCS 18, 1977. Lower bounds f o r natural proof systems. 254266. 1980.
Indexings of subrecursive classes.
TCS 11, 277301.
1980a Complexity of Boolean algebras. TCS 10,
KRAL, J.
221247.
A modification of a substitution theorem and some necessary and sufficient conditions for sets to be contextfree. Math. Systems Theory 4, 129139. KREIDER, D.L. & RITCHIE, R. W. 1966. A universal twoway automaton. AMLG 9, 4358. KRATKO. M. J. 1966. Post canonical systems and finite automata. (Russ. ) Probl. Kibern. 17, 4165. KREISEL, G. 1953. Note on arithmetic models for consistent formulae of 1970.
Bi b l i ograph y
553
the predicate calculus II. Proc. XIL” Int. Cong. Philos., Vol. 14, 3949. Appl. Math. and Stats. Labs., 1961. Techn. Report No. 3, Stanford Univ.
KREISEL, G.
KRIVINE, J.L.
&
Amsterdam.
Elements of Mathematical Logic.
1967.
KREISEL, G . , SCHOENFIELD, J. & HA0 WANG. 1960. Number theoret 1c concepts and orderings. AMLG, 4263. KROM, M. R. A
1966.
property
NDJFL VII,
of
sentences
that
define
we11
quasiorder.
349352.
The decision problem f o r a class of firstorder formulas in which all disjunctions are binary. ZMLG
1967.
13.
1520.
Some interpolation theorems for firstorder formulas in which all disjunctions are binary. Logique et
1968.
Analyse
43,
403412.
The decision problem for formulas in prenex conjunctive normal form with binary disjunctions.
‘970.
JSL
35,
210216.
KUMMER, E. E.
uber die ErgB’nzungssB’tze zu Reciproci tdtsgesetzen. Crelle 44,
1852.
KURODA.
recursi ve
S.
1964.
Y.
Classes
of
lanpages
Informatlor, and Control
and 7,
linear
den
allgemeinen
93146.
bounded
automata.
207223.
LADNER, R. E. 1973. 1975.
Folynomial time reducibility. STOC 5, 122129. On the structure of polynomial time reducibility.
JACM 1979.
22,
155171.
Complexity theory with emphasis on the complexity of 1 ogi cal theories. in: F . R.DRAKE & S . S . WAINER (Eds. ) : Recursion theory: its generalisations and applications. London Math. SOC. Lecture Notes Ser. 45,
286319.
LADNER, R. E. , LYNCH, N. A. & SELMAN. A . L. 1975. A comparison of polynomial time reducibilities. 1,
LANDWEBER, H.H. 1970.
1977.
&
ROBERTSON, E.L.
Recursive properties of abstract complexi ty classes.
STOC, LEONG, B.
TCS
103123.
&
3136.
SEIFERAS, J.
New
STOC
realtime 239240.
simulations
of
mu1 tihead
tape units.
Bibliography
554
LEVIN, L. A. 1973. Universa1 sequential search problems. Problerny Peredacki Inforrnatcii 9, 1151 16. (Engl. trans in Problems of Inforrnation Transmission 9, 2 6 5  2 6 6 ) . LEWIS, F. D. considerations in computational 1970. Unsol vabili ty complex1ty. STOC, 2230. 1971. Classes of recursive functions and their index sets. ZMLG 17, 291294. 1971a Enumerabili ty and invariance of complexity classes. JCSS 5, 286303. LEWIS, H. R. 1978. Complexity of solvable cases of the decision problem for the predicate calculus. FOCS 19, 3547. 1978. Description of restricted automata by firstorder formulas. Math. Systems Theory 9, 97 104. 1979. Unsolvable classes of quantificational formulas. Add1 sonWe6 l e y . 1980. Complexity results for classes of quantificational formulas. JCSS 21, 317353. LEWIS, H. R. & PAPADIMITRIOU, C. H. 1981. Elements of the theory of computation. PrenticeHal 1. 1982. Symmetric spacebounded computation. TCS 19, 161187.
LEWIS, H. R. 1981.
& DENENBERG, L. hard problem for NTIhE('n">. Harvard Univ. TR238 1.
A
Aiken
Cornp.
Lab.,
LEWIS 11, P. M., ROSENKRANTZ, D. J. 8 STEARNS, R. E. 1976. Compiler design theory. AddisonWesley, Reading. LINDSTR~M, P . 1969. On extensions of elementary logic. Theoria 35, 111. LISCHKE, G . 1974. Eine Charakterisierung der rekursiven Funktionen mi t end1i chen Ni veaumengen (KFunkt i onen >. ZMLG 20, 465472.
FluBblldma6e  E i n Versuch zur Definition natUrlicher Komplizierhei tsmase. EIK 11, 423436. 1975a Uber die ErfUllung gewisser Erhal tungssdtze durch Kompliziertheitsma6e. ZMLG 21, 159166. 1976/ 7 NatUrliche KompliziertheitsmaBe und Erhaltungssdtze I, rr. ZMLG 2 2 / 2 3 , 413418, 193200. 1981. Two types of properties for complexity measures. Inf. Proc. Letters 12, 123126.
1975.
LIVCHAK, A. B. 1982. The relational model for systems of automatic testing. Automatic Documentation and Math. Linguistics 4, 1719 (cited by Gurevich 1 9 8 4 ) .
Bib1 iography 1983.
LI XIANG 1983.
555
The relational model for process control. Automatic Documentation and Math. Linguistics 4, 2729 (cited by Gurevich 1 9 8 4 ) . Effective immune sets, program index sets and generalisations and effectively simple sets in: C. applications of the recursion theorem. T. CHONG, & M. J. WICKS (Eds. ) : Southeast Asian Conference on Logic. NorthHolland, 97106.
LLOYD, J.W. 1984. Foundations of L o g i c Programming. Springer. L6B, M. H. 1955. Solution of a problem of Henkin. JSL 10, 115118. L6B, M. H. 1970.
WAINER, S. S. Hierarchies of numbertheoretic functions. 3951,
AMLG 13.
97113.
LOVASZ, L. & GACS, P. 1977. Some remarks
on
generalised
spec ra.
ZMLG
23,
547554.
LOVELAND, D. W. 1978. Automated theorem proving: a logica Holland, Amst erdam.
L~WENHEIM,L. 1915.
lfber MUglichkei ten im RelativkalkUl.
basf s.
North
Math. Ann. 76.
447470.
LUCKHARDT, H. 1984. Obere Komplexi tkitsschranken fur TAUTEntscheidungen. in: G.WECHSUNG (Ed. ) : Frege Conference 1984., 331337.
LUDEWIG, J . , SCHULT, U. L WANKMULLER, F. 1983. Chasfng the busy beaver  notes and observations on a competition to find the 5state busy beaver. Bericht. Nr. 159, Abtlg. Informatik, Universitkit Dortmund. LYNCH, J. F. 1982. Complexity classes and theories of finf te models. Math. Systems Theory 15, 127144. 1982. On sets of relations definable by addition. JSL 47, 659668.
MACHTEY, M. 1975. On the density of honest subrecursfve classes. 10,
JCSS
183199.
MACHTEY, M., WINKLMANN, K. & YOUNG, P. 1978. Simple Gadel numberings, isomorphisms and programming properties. SIAM J. Comp. 7, 3960.
Bibliography
556
MACHTEY, M. & YOUNG, P. 1978. In introduction to the general theory of algorithms. NorthHolland, Amsterdam. 1981. Remarks on recursion versus diagonalisation and exponentially difficult problems. JCSS 22, 442453. MAIER, W., MENZEL, W. & SPERSCHNEIDER, V. 1982. Embedding properties of total recursive functions. ZMLG 28, 565574. MAKOWSKY, 3 . A. 1985. Why Horn formulas matter in computer science: initial structures and generic examples. Proc. CAAP 85, SLNCS. MALCEW, A. I. 1958. On homomorphisms of finite groups. Ivan0 Gosudarstvenni Pedag. Inst. Uchenye Z a p . 18, 4960. 1974. Algorithmen und rekursive Funktionen. Vieweg, Br aunsc hweig. MANIN, Y. I. 1977. A course in mathematical logic. Springer, Berlin. MANNA, Z. 1974. Mathematical Theory of Computation. McGrawHill, New York. MANNA, 2 . & WALDINGER, R. 1981. Deductive synthesis of the unification algorithm. in: Science of Computer Programming 1. MARCENKOV, S . S . 1969. The elimination of recursion schemata in a Grzegorczyk class. Mat. Zametki 5, 561568 (Russ. ) MARKOV, A. A. 1947. The impossibility of certain algorithms in the theory of associative systems. Dokl. Akad. Nauk SSSR 55, 587590
(RUSS.
)
On the representation of recursive functions. Izv. AN SSSR, serija matem. 13, 5, pp 417424 (Russ.) (s. MALCEW 1974, S6. 4. ) 1951. Theory of Algorithms. AMS Transl. ( 2 ) 1 5 < 1 9 6 0 ) ,
1949.
114.
Theory of Algorithms. Israel Prog. for Scientific Transl. , Jerusalem. MARKWALD, W. 1955. Zur Eigenscheft primi ti vrekursiver Funktionen, unendlich viele Werte anzunehmen. Fund. Math. XLII, 1954.
166 167.
MARQUES, I. 1975. On degrees of unsolvabili ty properties. JSL, 529540.
and
complexity
Bibliography
557
MARTELLI. A. 81 MONTARI, U. 1982. An efficient unification algorithm. ACM Trans. on Programming Languages and Systems 4. 258282. MARTINLUF,P. 198 2. Const r uc t i ve ma themat ics and comput er pr ogrammi"8. Proc. 6 L t i Int. Congr. Logic, Methodology and Philosophy of Science, Hannover, 153175. MASLOV, S . Ju. 1964. An inverse method for establishing deducibilities in the classical predicate calculus. Dokl. Akad. Nauk. SSSR 159, 1720. 1970. On E.Post's ''Tag'' problem. AMS Transl. ( 2 ) 97, 114 ( = Trudy Mat. Inst. Steklov 72, 1964, 5 7  6 8 ) . MATHEWS, G. 1967. Twoway languages. Inf. and Control 10, 111119. MATIJASEVICH, J. V. 1967. Some examples of undecidable associative calculi. Soviet Math. Doklady 8, 555557. 1967. Simple examples of unsolvable canonical calculi. Proc. Steklov Inst. Math. 93, 61110. dicphantine. Soviet Math. 1970. Enumerable sets are Doklady 11, 354357. 1976. A new proof of the theorem on exponential Diophantine representation of enumerable sets. Zap. Nauk. Sem., Leningrad. Otdel. Matem. Inst., V. A. Steklova Akad. Nauk SSSR 60, 7589 (Engl. trans J. Soviet Math. 14 ( 1980)
1475 1 4 8 6 ) .
MAURER, H. 1969. A direct proof of the inherent ambiguity of a simple contextfree language. JACM 16, 256260. MAYER, 0. 1978. Syntaxanalyse. Bibl. Inst., Mannheim. MAYR. E. L MEYER, A. 1981. The complexity of the finite containment problem for Petri nets. JACM 28, 561576. McALOON, K. 1982. Petri nets and large finite sets. AMS Summer Res. Inst. on Recursion Theory, Cornell Univ. Ithace. McCREIGHT, E. 8 MEYER, A. 1969. Classes of computable functions defined by bounds on computa t i on. STOC 7988. McCULLOCH, W. S. 8 PITTS, W. 1943. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophysics 5, 115133. McKINSEY, J. C.C. 1943. The decision problem for some classes of sentences without quantifiers. JSL 8, 6176.
Bi bl iography
558
McNAUGHTON, R. 1967. Parenthesis grammars. JACM 14, 490500. McROBBIE, M. A. 1982. Interpolation theorems: A bibliography. Dept. Philosophy, Univ. of Melbourne (preprint) . Mc W H I RTER, J . 1971. Substitution expressions. JCSS 5, 629637. MENZEL, W. & SPERSCHNEIDER, V. 1982. Universal automata with uniform bounds on simulation time. Information and Control 52, 1935. MEYER, A. R. 1972. Program size in restricted programmlng languages. Information and Control 21, 382394. MEYER, A. R. & FISCHER, M. J. 1971. Economy of description by automata, grammars and formal systems. 1 2 t t ' SWAT, 188191. MEYER, A. R. & FISCHER, P. 1972. Computational speedup by effective operators. JSL 37,
5568.
MEYER, A. R. & RITCHIE, D. M. 1967. Computational complexity and Program Structure. IBM Res. RE1817, I B M Watson Res. Center, Yorktown Heights, New York, p p 1 5 . 1967. The complexity of l o o p programs. in: Proc. of the 22'' d Na t i onal Conference. Thompson Book Co., Washington D. C . 465469. 1972. A classification of the recursive functions. ZMLG 18,
7182.
MEYER, A. R. & STOCKMEYER, L. J . 1972. The equivalence problem for regular expressions with squaring requires exponential space. 13"" SWAT, 125129.
Word problems requiring exponential time. STOC 19. MILNE, R. & STRACHEY, Ch. 1976. A theory of programming language semantics. Chapman 1973.
& Hall.
MINSKY, M. L. 1961. Recursive unsolvability of Fost's problem of 'tag' and other topics in the theory of Turing machines. Ann. of Math. 74, 437455. 1967. Computation, Finite and Infinite Machines. Englewood Cliffs, N. J . MOLL,R. 1973. Complexity Classes of Recursive Functions. Ph.D. Thesis, MIT (MAC TR110)
Bibliography
559
MOLL, R. & MEYER, A. R. 1974. Honest bounds for complexi ty classes of recurs1ve functions. JSL 39, 127138. MONIEN, B. 1977. The LBAproblem and the deterministic tape complexity of twoway, one counter languages over a oneletter alphabet. Acta Inf. 8, 371382. MOORE, E. F. 1956. Gedanken experiments on sequential machines. Automata Studies, Princeton Univ. Press 129153. MOSTOWSKI, A. 1947. On definable sets of positive integers. Fund. Math. 34, 81112 1953. On a system of axioms which has no recursively enumerable model. Fund. Math. 40, 5661. 1955. Examples of sets definable by means of two and three quantifiers. Fund. Math. XLII, 259270. 1955a Contributions to the theory of definable sets and functions. Fund. Math. XLII, 271275. 1955b A formula with no recursively enumerable model. Fund. Math. 43, 125140. 1956. Concerning a problem of H.Scholz. ZMLG 2, 210214. MUCHNICK, S.S. 1976. The vect orized 441480.
Grzegorczyk
hierarchy.
ZMLG
22,
M L L E R , H. 1970. vber die mi t Stackautomaten berechenbaren Funktionen. AMLG 13, 6073. 1974. Klassifizierungen der primi ti vrehurslven Funktionen. Dissertation, Univ. MUnster, Inst. f. Math. Logik u. Grundlagenforschung, pp 51. MUNDICI, D. 1982. Complexity of Craig's interpolation. Ann. SOC. Math. Poloniae, series IV: Fundamenta Informat icae, V. 34. 1983. A lower bound for the complexity of Craig's Interpolants in sentential logic. AMLG 23, 2736. in: G.LOLLI, 1984. NP and Craig's interpolation theorem G. LONGO, A. MARCJA (Eds. ) : Logic Colloquium '82. NorthHolland, Amsterdam, 345358. 1984a Tautologies with a unique Craig interpolant, uniform vs. nonuniform complexity. AFAL 27, 265273. MYHILL, J. 1957. Finite automata and the representation of events. WADD TR57624, 112137, Wright Patterson AFB, Ohio.
Bibliography
560
NEPOMNIASCHY, V. A. 1971. Conditions for the algorithmic completeness of the in Proc. IFIP Congr., system of operations. Ljubljana. NERODE, A. 1958. Linear aut omat on transformat i ons. Proc. AMS 9, 541544.
NEWMANrn M. H.A. 1942. On theories with a combinatorial definition equivalence. Annals of Math. 43, 223243 OBERSCHELP, W. 1958. Varianten von Turingmaschinen. AMLG 4, 5362. ODIFREDDI, P. Bull. AMS (New Ser.) 198 1. Strong reduci bi 1i ties. 3786.
of
4,
0'DONNELL, M. 1979. A programming language theorem which is independent of Peano arithmetic. ll*p'STOC.
OETTINGER, A. G. 1961. Automatic syntactic analysis and the pushdown store. Proc. Symp. in Appl. Math. 12, AMS. OGDEN, W. G. 1968. A helpful result in proving inherent ambiguity. Math. Systems Theory 2, 191194. 1969. Intercalation theorems f o r stack languages. 1t STOC,
3142.
OTTMANN, Th. 1975. Mi t regularen Grundbegriffen definierbare Pradikate. Computing 14, 2 13223. 1975a Einfache universe11e mehrdimensional e Turingmaschinen. Habll It at lonsschri f t , Univ. Karlsruhe. 1978. Eine einfache universelle Menge endlicher Automaten. ZMLG 24, 5581. PACHONOV, 5,V. 1972. Some properties of the graphs of functions of the Grregorczyk hierarchy. Zap. naucnych seminarov Lenigradskojo otdelenlja mat. inst. AN SSSR, 2, 2123 (Russ. ) . PAGER, D. 1969. On the problem of finding minimal programs f o r tables. Inform. Contr. 14, 550554. 1970. On the efficiency of algorithms. JACM 17, 708714. 1970a The categorization of Tag systems in terms of decidability. J. London Math. SOC. (2), 473480.
Bibliography
56 1
PANSIOT, J. J. 1981. A note on Post's correspondence problem. Inf. Proc. Letters 12, p 233. PARIKH, R. J. 1966. On contextfree languages. JACM 4, 570581. PARIS, J. 1978. Some independence results for Feano arithmetic. JSL 43,
PARIS, J. 1977.
725731.
HARRINGTON, L. mathematical incompleteness in Peano arithmetic. in: J. BARWISE (Ed. ) : Handbook of mathematical logic., &
A
11331142.
PARSONS, Ch. 1968. Hierarchies of Drimiti ve recursive functions. 14.
ZMLG
357376.
PATERSON, M. S . & WEGMAN, M. N. 1978. Linear unification. JCSS 16, 58167. PAUL, W. J. 1977. On time hierarchies. STOC, 218 222. 1978. Kompl ex1 tCitstheorie. Teubner, St ut tgart, PAULL, M. & UNGER, S. 1968. Structural equivalence of contextfree SCSS 2, 427463.
grammars.
PENNER, V. 1973. PushDownStoreberechenbare Funkt ionen. SLNCS 2. PETER, R. 1951. Rekursive Funktionen. Verlag Akaderniai Kiad6, Budapest. 1967. Recursi ve Functions. Academic Press, New York. PETERSON, J. P. 1977. Petri nets. Comp. Surveys 9, 223252. PETERSSON, K. LPM Memo 21, 1982. A programming system for type theory. Dept. of CS, Chalmes Univ. of Techn., Gdteborg. PETRI, C. A. 1962. Kommunikation mit Automaten. Universitkit Bonn. PLAISTED, D. A. 1984. Complete problems in the firstorder predicate calculus. JCSS 29, a35. POST, E. L. 1936. Finite combinatory processes

formulation 1 .
JSL 1,
103 105.
Formal reduction of the general combinatorial decision problem. Arner. J. Math. 65, 197215. 1946. A variant of a recursively unsolvable problem. Bull. AMS 52. 264268.
1943.
B i bl i ogr aph y
562
1947.
Recursive 12,
u n s o l v a b i l i t y o f a problem o f Thue.
JSL
111.
Absol ut e l y unsol vabl e problems and relat i vely propositions. An account of an undecidabl e in: M. DAVIS: T h e Undecidable. Raven anticipation. Press, New York, 340433. P RATHER, R. E. con veni en t crypt omorphi c v e r s i on of r e c u r s i ve 1975. A f u n c t i o n t h e o r y . Inform. and Control 27, 178. 1977. S t r u c t u r e d Turing machines. Inform. and Control 35, 1965.
159 17 1.
PR AWI TZ, D. 1965.
Na t ural Deduct i on.
St oc k hol m.
PRESBURGER, M. 1929. uber d i e V o l l s t 8 n d i g k e i t e i n e s gewissen Arithmetik ganzer Zahlen, i n welcher d i e e i n r i g e Operation h e r v o r t r i t t . C. R. I. Math. des Pays Slaves, Warechau, 192301,
Systems der Addition a l s Congres des 395.
PRIDA, J . F. 1974.
E l problema d e d e c i s i d n del c.4lculo de predicados. Centro de c&lculo de la universidad cornplutense, Madrid.
PRIESE. L. von Markov’schen und Post ’ schen 1971. Normal formen Algori t hmen. Inst. f. math. Logik U. Grundlagenforschung der Univ. MUnster i. W., pp 124. und e i n f a c h e uni verse1 1 e 21976. Reversi bl e Automat en dimensional e ThueSyst eme. ZMLG 22, 353384. 1976. On a s i m p l e combinatorial s t r u c t u r e s u f f i c i e n t f o r sublying nontrivial self reproduction. J. Cybernetics 6, 101137. 1978. A n o t e on asynchronous c e l l u l a r automata. JCSS 17. 1979. uber e i n 2dimensionales ThueSystem m i t zwei Regeln und unentscheidbarem Wortproblem. ZMLG 25, 179192. 1979. Asynchrone, modulare Netze: P e t r i  N e t z e , normierte Habi 1 it at lonsschri f t , Uni v. Netze, APANetze. Dort mund. 1979. Towards a p r e c i s e c h a r a c t e r i s a t i o n o f t h e complex1 t y SIAM o f uni versal and nonuni versal Turing machines. J. Comp. 8, 508523 1983. Automata and concurrency. TCS 25, 221265. PROSKURIN, A. V. 1979. The p o s i t i v e r u d i m e n t a r i n e s s o f t h e graphs o f t h e Zap. naucnych Ackermann and Grzegorczykfunc t i ons. seminarov 88, 186191 (cf Nauka, Leningrad) (Russ. ) PUTNAM, H. 1960. An unsolvable problem i n number theory. JSL 15, 220232.
Bibliography
563
RABIN, M. 0. 1958. On recursively enumerable and arithmetic models of set theory. JSL 23, 408416. 1960. Degree of difficulty of computing a function and a TR 2, Hebrew partial ordering of recursive sets. Univ., Jerusalem. 1967. Mathematical theory of automata. Proc. Symp. in Applied Math. 19, AMS, 153175. RABIN, M. 0. 6 SCOTT, D. 1959. Finite automata and their decision problems. IBM J. Res. 3, 115125. RADO, T. 1962. On noncomputable functions. Bell System Technical J. 41, 977884. RASIOWA, H. & SIKORSKI, R. 1963. Metamathematics of Ma themat ics. Warsaw. REEDY, A. & SAVITCH, W. J. 1975. The Turing degree of the inherent ambiguity problem for contextfree languages. TCS 1, 7791. REINGOLD, E. M. & STOCKS, A. J. 1972. Simple proofs of lower bounds for polynomial Complexity of Computer Operat ions, eval uat ion. Plenum Press, New York  London, 2130. REYNOLDS, J. C. 1969. Transformational systems and the algebraic structure of atomic formulas. in: B. MELTZER, D. MICHIE (Eds. ) Machine Intelligence, vol. 5 American Elsevier 135 15 1.
RICE, H. G. 1953. Classes of recursively enumerable sets and their decision problems. Trans. AMS 89, 2559. 1956. On completely recurs1 vely enumerable classes and their key arrays. JSL 21, 304341. RICHTER, M. M. 1978. LogikkalkUle. Teubner.
R ITCHIE, R. W.
Trans. 1963. Classes of predictably computable functions. AMS 106, 139173. JSL 30, 3501965. A rudimentary definition of addition. 354. 1965. C1asses of recursi v e f unc t i ons based on Ackermann' s function. Pac. J. Math. 15. iO271044.
RITCHIE, D. M. structure and computational 1968. Program Doctoral Dissertation. Harvard Univ.
complex1ty.
564
Bibliography
RITCHIE, D. 81 THOMPSON,K. . The UNIX timesharing system. The Bell System Technical J. 57, 19051930. ROBBIN, J. 1965. Subrecursi ve hierarchies. Doctoral Thesis, Princeton Univ. ROBERTSON, E. L. 1971. Complexity classes of partial recursive functions. STOC, 258266. ROB INSON, J . 1950. General recursive functions. Proc. AMS 1, 703718. 1965. A machineoriented logic based on the resolution principle. JACM 12, 2341. ROBINSON, J. A. 1965. Automatic deduction with hyperresolution. Int. J. of Comp. Math. 1, 227234. ROBINSON, R. M. 1947. Prim1 tive recursive functions. Bull. AMS 53, 925942. 1971. 1972.
Undecidability and nonperiodicity for tilings of the plane. Inventlones Math. 12, 177209. Some representations of diophantine sets. JSL 37, 572578.
ROBSON, J . M. 1979. A new proof of the NPcompleteness of satisfiability. Proc. 2’”’ Australian C o m p . Scl. Conf., 6269. RODDING, D. 1962. Darst el 1ungen der (im KALMARCSILLIG’schen Sinne) elementaren Funktionen. AMLG 7, 139158. 1964. uber die Eliminierbarkeit von Defini tionsschemata in ZMLG 10, der Theorie der rekursiven Funktionen. 3 15330. 1964a uber Darst el 1ungen der el emen t aren Funk t I onen I I . AMLG 9, 3648. AMLG 13, 1966. Anzahlquantoren in der Kleene Hierarchie. 6165. rekursiver Funktionen. in: M. H. L6B 1968. Klassen
1969.
(Ed. ) : Proc. of the Summer School in Logic, Leeds 1967. SLNM 70, 159222. EinfUhrung in die Theorie der berechenbaren Funktionen. Vorlesungskrlpt, Inst. f. math. Logik u.
Grundlagenf orschung der Unlv. MUnster. Reduktionstypen der Prddikatenlogik. Vorlesungsf. math. Logik U. nachschrift, Inst. S. der Unlv. MUnster. Grundlagenforschung Zentralblatt f. Math., 207, Nr. 02034. in: L. PRIESE (Ed. ) : 1983. Networks of finite automata. Report on the l W i GTIworkshop, UniversitBtGH Paderborn, Fachber. Math. Informatlk, 184197.
1970.
Bibliography
565
19838 Modular
decomposition of automata : Foundations
M. KARPINSKY (Ed. 1 Theory., SLNCS 158,
1984.
194412.
Some logical problems connected decomposi tion. Computation and Proof 1104,
(Survey). in : of Computation with modular Theory. SLNM
363388.
R6DDING, D. & BRUGGEMANN, A. 1982. BasissB’tze in H , K (und I) fur (indeterminierte) MealyAutomaten. Arbeitspapier Nr. 1. Inst. f . math. (s. Logik u. Grundlagenforschung der Univ. MUnster. BrUggernann et a1 1 9 8 4 ) . RGDDING, D. & SCHWICHTENBERG, H. 1972. Bemerkungen zum Spektralproblem. ZMLG 18, 112. ROGERS, H. 1958. Gtldel numberings of partial recursive functions. 23, 1959.
JSL
331341.
Computing degrees of unsolvability.
Math. Ann. 138,
135 140. 1967.
Theory of recursive functions and computability. McGrawHill, New York.
effective
ROSE, H. E. 1984. Subrecursion: Functions and hierarchies. Oxford Univ. Press. ROSSER, B. 1936. Extensions of some theorems of Gddel and Church. JSL 1,
8791.
RUSSELL, B. 1908. Mathematical logic as based on the theory of types. Amer. J. Math. 30, 222262. SAVAGE, J. E. 1976. The complexity of computing. WileyInterscience. SAVELBURGH,M. & van EMDE BOAS, P. 1984. Bounded tiling, an alternative to satisfiabilf ty? in: G.WECHSUNG (Ed. ) : Frege Conference 1984., 354363.
SAVITCH, W. J. 1970. Relationships between nondet ermini s t i c and deterministic tape complexities. JCSS 4, 177192. SAZONOV, V. 1980. Polynomial computability and recursivity in domains. EIK 16, 319323.
finite
SCARPELLINI, B. 1984. Complete second order spectra. ZMLG 30, 509524. 1984a Lower bounds on the lengths of second order formulas. Manuskript, Math. Inst. der Univ. Base1 see APAL 29 ( 1 9 8 5 ) 2958.
566
Bibliography 1984b Complexity
of subcases Trans. AMS 284, 203218.
of
Presburger
arithmetic.
SCHXTZ, R. 1982. Basissatz fur MealyAutomaten bei schneller, sequent iell er Simula t ion. Arbeltspapier Nr. 3, U. Inst It ut f. mat hemat ische Loglk Grundlagenforschung der Univ. MUnster. ( s. BrUggemann et a1 1984). 1985. Komplexi tat von sequentiellen und lokal sequentiellen Netzwerken. Diplomarbeit, Inst. f. math. Loglk u. Grundlagenforschung der Unlv. MUnster 1.W. SCHXTZ, R., R6DDING, D. 81 BRUGGEMANN, A. 1982. Basissatz in K,H, V, W fifr determinierte MealyAutomat en (para11el e Signal verarbei t ung>. Arbeltspapier Nr. 2, Inst. f. math. Logik u. Grundlagenf orschung der Univ. MUnster. ( s . Brllggemann et a1 1984). SCHEINBERG,S. 1960. Note on the Boolean properties o f contextfree languages. Information and Control 3, 372375. SCHINZEL, B. 1984. Vorlesung Uber formale Sprachen und Automatentheorie. RhelnlschWest f . Techn. Hochschule Aachen. 1984a Kompl exi tdtstheorie. Vorlesungausarbeitung, RheinlschWest f, Techn. Hochschule Aachen. SCHIRMEISTER. R. Funkt i onen und Funkt i onenklassen. 1975. Gutartige Diplomarbelt, Univ. Freiburg I.Brsg., pp 74. SCHMIDT, D. on separating nondeterministic and 1984. Limi tations Unlverslt kit deterministic complex1ty classes. Karlsruhe, Fakultkit fur Informatlk, Interner Berlcht 1 /84.
SCHNORR,
c.P.
1973. Does
computational speedup concern programming?. ICALP. 1974. Rekursive Funktionen und ihre Komplexi tdt. TeubnerVerlag, St ut tgart. 1974. Optimal enumerations and optimal Gddel numberings. Math. Systems Theory 8, 182191. the Turing machine 1976. The network complexity and complexity of finite functions. Acta Informatlca 7, 95107. 1977. The network complexity and
the breadth of Boolean functions. In: R. 0. GANDY, J. M. E. HYLAND (Eds. ) : Logic Colloquium ' 76, 491504.
SCHOLZ, H. 1952. Ein ungelCistes Problem in JSL 17, p 160.
der
symbolischen LOgik.
Bibliography SCHOLZ, H. & HASENJAEGER, G. 1961. Grundzffge der mathematischen LOgik.
567
Springer.
scH~NFELD,w.
JSL der Re1ationen. Publlkat Ion,
1979. An undecidabili ty resul t for relation algebras. 44, 111115. Refinement In : Gleichungen in
A1gebra der bindren Habi 1 it at ionsschrl f t , S5, Mlnerve MUnchen 198 1. SCH~NHAGE,A. 1980. Storage modification machines. SIAM J.
Comput.
9,
490508.
SCHULTEWNTI NG, J. 1976. Interpolation formulae for predicates and terms which carry their o m history. AMLG 17, 159170. SCHUTTE, K. 1934. Untersuchungen zum Entscheidungsproblem der mat hema t i schen L ogik. Mathematische Annalen 109, 572603. 1960. Beweistheorie. Springer Verlag. 1977. froof theory. Springer Verlag.
SCHUTZENBERGER, M. P. 1963. On contextfree languages and pushdown automata. Information and Control 6, 246264. SCHWICHTENBERG, H. 1967. Eine Klassifikation der elementaren Funktionen. Unpublished manuscript, s. Rdddlng 1968. 1969. Eine K1assi fika t i on der mehr fachreku r s i ven Funkt ionen. Dissert at Ion, Inst it ut f. mat hemat lsche Loglk u. Grundlagenforschung der UnIversltHt. MUns t er. 1969. Rekursionszahl en und die GrzegorczykHierarchic. AMLG 12, 8597. 197 1. Elne Klassi fikat ion der a,,rekursiven Funktionen. ZMLG 17, 6174. 1977. Proof theory: some applications of cutelimination. in : J. BARWISE (Ed. ) : Handbook of matheinatical logic., 867895.
SCOTT, D.
1967. Some
definitional suggestions for automata theory. JCSS 1 , 187212. SCOTT, D. & STRACHEY, C. 1967. Towards a mathematical semantics for programming 1 anguages. in: J.FOX (Ed. ) : froc. Symp. on Inst. of Computers and Automata., 1946, Pol. Brooklyn Press. SEBELIK, J. & STEPANEK, P. 1982. Horn clause programs for recursive functions. in : K. L.CLARK, TiiRNLUND, S.A. (Eds. ) : Logic Programming. Academic Press, 325341.
Bibliography
568
SEIFERAS, J. J . , FISCHER, M. J. & MEYER, A. R. 1973. Refinements of nondeterministic hierarchies. SWAT 4, 130137.
time
and
space
SELMAN, A . 1979. Pselective sets, tally languages, and the behaviour of polynomial time reducibilities on NP. Math. Systems Theory 13, 5565. SHAPIRO, E. Y. 1982. Alternation and the computational complex1ty of logic Proc. first int. logic programming conf., programs. 154163. see J. of Logic Programming 1, 1984, 1933. SHAY, M. & YOUNG, P. 1978. Characterising the orders charged by program translators. Pac. J. Math. 76, 485490. SHELAH, 1977.
S.
Decidability of a portion of the predicate calculus. Israel J. Math. 28, 3244.
SHEPHERDSON, J . 1959. The reduction of twoway automata to oneway automata. IBM J . RES. 3, 198200. 1964. Machine configuration and word problems of given degree of insolvability. in Proceedings of the 1964 International Congress for logic, Methodology and Philosophy of Science held in Jerusalem, Aug 26 Sept 2, 1964. NorthHolland, Amsterdam. SHEPHERDSON, J. & STURGIS, H. E. 1963. Computability of recursive functions.
JACM 10,
217
255.
SHOENFIELD, 3. R. 1967. Mathematical Logic. Reading 1871. Degrees of unsol vabili ty. NorthHolland, Amsterdam. SIMPSON, S.G. 1985. Nichtbeweisbarkelt von gewissen kombinatorischen Eigenschaf t en end11cher Blume. AMLG 25, 1 1 1. SK0LEM, Th. 1930. Ober einige Satzfunktionen in der Arithmetik. Norske Vielenskaps Akademi, Oslo, nr. 7. 1944. Remarks on recursive functions and relations. Det Kongelige Norske Videnskabers Selskab, Fordhandlinger 17,
8992.
SLOBODSKOI, A. M. 1981. Unsolvability of the theory Algeb,a and Logic 20, 139156.
of
finite
groups.
SMITH, C.H. 1979. Applications of classical recursion theory to in: F.R.DRAKE & S.S.WAINER computer science. (Eds.) : Recursion Theory: Its Generalisations and
Bibliography
Applications. Cambridge.
569
London Math.
SOC. Lecture Note6 Ser.
45,
SMITH, J
.
1983.
The identification of propositions and types in MartinLUf's type theory: a programming example. SLNCS 158, 445456.
SMORYNSKI, C . 1977. The incompleteness theorem. in : J. BARWISE (Ed. ) : Handbook of mathemat ical logic. SMULLYAN, R. 1961. Theory of formal systems. Ann. of Math. Studies, Princeton Univ. Press, Princeton NJ. 1968. Firstorder logic. Springer. SOARE, R.
Computational complex1ty, speedable and levelable sets. JSL, 545563. 1985. Recursively enumerable sets and degrees: a study of computable functions and computably generated sets. Springer. SPECKER, E. & STRASSEN, V. 1976. KomplexitSlt von Entscheidungsproblemen. SLNCS 43. 1977.
SPREEN. H. D. 1983. On the equivalence problem in automata theory. in: Coll. on algebra, combinatorics and logic in computer science (Sept. 1216, GyUr. Hungary). s . Schriften zur Informatik u. Angew. Math. 89, RWTH Aachen. 1984. Rekursionstheori e auf Tei1mengen part i el 1er Funktionen. Habi 11tat ionsschrif t , RWTH Aachen. STANLEY, R. J. 1965. Fin1 t e st ate representat i ons of languages. Quarterly Prog. Rep. no. 76, Res. Lab. Elect., Cambridge, Mass.
context free MIT
276279,
STATMAN, R. 1977. Herbrand's theorem and Gentren's notion of a direct proof. in: J. BARWISE (Ed. ) : Handbook of mathematical logic, 897912. 1977a Complexity of derivations from quantifierfree Horn formulae, mechanicel introduction of expli ci t defini tions, and refinement of completeness theorems. in: R. GANDY h M. HYLAND (Eds. ) : Proc. Logic Colloquium ' 76. NorthHolland, Amsterdam, 505518. STEARNS, R. E. 8 HUNT 111, H. B. the equi valence and containment problems 1981. On unambiguous regular express1ons. grammars automata. 22 FOCS, 7481.
for and
STILLWELL, 3. 1982. The word problem and the isomorphism problem groups. Bull. (New Series) AMS 6, 3356.
for
570
B i b 1i o g r a p h y
STOB, M. 1982. Index sets and 24 1248.
degrees of unsolvability.
JSL
47,
STOCKMEYER, L. J . 1974. The complexity
of decision problems in automata theory and logic. P h . D . t h e s i s , Report TR133, MIT,
P r o j e c t MAC, C a m b r i d g e Mass. 1977. The polynomial time hierarchy. TCS 13, 122. 1979. Classifying the computational complex1ty of problems. I B M Research R e p o r t . RC 7606 (#32926). STOCKMEYER, L. J . & MEYER, A. R. 1973. Word problems requiring exponential time. 19
5*"
STOC,
STOY, J . E. 1977. Denotational semantics:
The Scot tStrachey approach to programming language theory. MIT P r e s s .
STRAIGHT, D . W. 1979. Domino fsets.
ZMLG 25, 235249.
SURANY I , J . 1959. Reduk tionstheorie
des En t scheidungsprobl em PrddikatenkalkU1 der ersten Stufe. B u d a p e s t .
TAIT, W. W. 1968. Normal derivabili ty J.BARWISE (Ed. ) : The
infini tary languages.
in classical syntax and Springer,
logic. semantics
204236.
TANIGUCHI. K. & K A S A M I . T. 1970. Reduction of contextfree grammars. C o n t r o l 17, 92108.
im
in: if
Information and
TARNLUND, S.  A .
1977. Horn clause computabili ty.
B I T 17, 215226.
TARSKI, A . 1936. Der Wahrhei tsbegriff in der formalisierten Sprachen. S t u d i a P h i l o s o p h i c a 1, 261405. 1954, Contributions to the theory of models. I,II. N e d e r l . Akad. W e t e n s c h . P r o c . Ser. a 57, = I n d a g . Math. 16, 572588. THOMPSON, D. 8. 1972. Subrecursi veness:
Machineindependent notions of computability in restricted time and storage. M a t h .
S y s t e m s T h e o r y 6, 315. THUE, A. 191 4. Probleme
Uber VerSlnderungen von Zeichenreihen nach gegebenen Regeln. S k r . V i d e n s k . S e l s k . I , 10, p p 34.
Bibliography
57 1
THURAISINGHAM, M. B. 1979. System functions and their decision problems. Ph.D. Thesis, University of Wales. TOURLAKIS, G. J. 1984. Computability. Reston Pub. Co. TRACHTENBROT, B. 1950. Impossibility of an algorithm for the decision problem in finite classes. Dokl. Akad. Nauk SSSR 70, 569572. (Engl. trans. in: AMS Transl. Ser. 2, vol. 23 (1963) 15.0)
TRACHTENBROT, B . A. 1953. 0 recursivno otdelimosti. 953955.
Dokl.
Akad.
SSSR
1967. Complex1ty of algori thins and computations.
notes, 1974. )
Novosibirsk
Univ.
(cited
in
Moll
8
88,
Course Meyer
TSICHRITZIS, D. 1970. The equivalence problem of simple programs. JACM 17, 4. 1971. A note
on comparison of subrecursive hierarchies. Inf. Proc. Letters 1, 4244. TURING, A. M. 1937. On computable numbers with an application to the Entscheidungsproblem Proc. London Math. SOC. (2) 42. 230265. A correction ibid. 43 (1937) 544546.
USPENSKIJ, V. A. 8 SEMENOV, A. L. 1981. What are the gains of the theory of algorithms? in: A. P. ERSHOV, D. E. KNUTH (Eds. ) : Algorithms in Modern Mathematics and Computer Science. SLNCS 122. VALIANT, L. G. 1975. General contextfree recognition in less than cubic time. JCSS 10, 308315. VARDI, M. Y. 1982. Complexity of relational query languages. STOC 14, 137 146.
VERBEEK, R. 1978. Prim1ti vrekursive GrregorczykHierarchien. Dissertation. Univ. Bonn. Bericht Nr. 22 des Inst. f. Informatik, 1 1 137. VOLGER, H. 1982. A new hierarchy of elementary recursive decision problems. Methods of Operat ions Research 45, 509519. 1983. Turing machines with linear a1 ternation, theories of bounded concat enation and the decision probl em of firstorder theories. TCS 23, 333337.
Bibliography
572
1985.
On theories which admit Manuscript, Univ. TUbingen.
initial
structures.
WAGNER, K. & WECHSUNGG, G. 1986. Computational Complexity. Reidel, Dordrecht. WAINER, S . 1972. Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy. JSL 37, 281292. 1982. The " 5 1 owgrowing'' approach to hierarchies. Proc. AMS Summer Institute on Recursion Theory, Cornell Univ., Ithaca NY., 487502, (Eds. ) A. NERODE, R. A, SHORE: Recursion Theory. Proc. Symp. Pure Math. 42,
1985.
)
WANG, H.
A variant of Turing's theory of computing machines. JACM 4, 6392. 1961. Proving theorems by pattern recognition  II. Bell System Techn. J. 40, 141. 1963. Tag systems and lag systems. Math. Annalen 152, 1959.
6574. 1981.
WATANABE, 1963.
Popular lectures on mathematical logic. Van Nostrand Reinhold (New York). Science Press (Be1jing). S.
Periodicity of Post's normal process of Tag. Proc. of the symp. on math. theory of automata. New York, 8399.
WEGENER, I. 1987. The complexity of Boolean (Stuttgart) h Wiley (New York) WEIRAUCH, K. 1985. Computabi1 i ty.
1971. 1973. 1974. 1975. 1975a 1984.
Teubner
SpringerVer lag.
WIRSING, M. 1979. Small universal Post systems. WIRTH, N.
functions.
ZMLG
25,
559564.
Program development by stepwise refinement. Corn. ACM 14, 221227. Systematic Programming: An Introduction. Prent iceHall, Englewood Cliffs, N. J. On the composition of well structured programs. Comput ing Surveys 6, 247259. Syst emat ische Programmieren. Teubner. Algori thmen und Datastrukturen. Teubner. Compilerbau. TeubnerVerlag.
YATES, C. E. M. 1966. On the degrees i f 309328.
index
sets.
Trans.
AMS
121,
Bibliography
573
YNTEMA, M. 1971. Cap expressions for contextfree Information and Control 18, 311318.
languages.
YOKOMORI. T. & JOSHI, K. 1983. Semilinearity, Parikhboundedness and tree adjunct languages. Inf. Proc. Lett. 17, 137143. YOUNG, P. 1971. Speedups by
enumerated.
changing the order in which sets are
Math. p352 (Corrigenda).
Systems 5,
148156. ibid.,
1973,
1971a A note on "axioms" for computational complexity and computation of finite functions. Information and Control 19, 377386. 1971b A note on the union theorem and gaps in complexity classes. Math. Syst. theory 5. 1973. Easy constructions in complexity theory: gap and speedup theorems. Proc. AMS 37, 555563. provably equivalent programs. 1977. Optimization among S A C M 24, 693700. 1983. Some structural properties of polynomial reducibilities and sets in NP. STOC 15, 392401. theorems, exponential difficulty and 1984. Gddel
undecidability of ari thmetic theories: an exposition. (Eds. ) A. NERODE, R. A, SHORE: Recursion Theory. Proc.
Symp. Pure Math. 42, 1985. )
ZLOOF, M. M. 1977. Querybyexample: a database language. 16, 324343..
IBM Syst. J.
INDEX A a n d e r a a , theorem of  and Borger 445,450,488 365
absorption
accepted  l a n g u a g 175,191,243 e  o v e r end s t a t e s 312  o v e r empty pushdown s t o r e 312 acceptor, f i n i t e
243
Ackermann  class  function
522 197 12 12
address next a 1g e b r a Herbrand canonical  over N
389 389 380 3
a l g o r i t hm a l m o s t e v e r y w h e r e , AE
71
a 1phabe t
XIX
a 1t e r n a t or
338
ambiguity inherent 
322 325
a n a 1y s i s  tree  procedure
303 8
analytical table
364
answer s u b s t i t u t I o n
398
antecedent
401
arithmetical hierarchy
108
A s s e r ' s complement
problem
182,502
automaton 167,243 confluencefree 163 cyclic universal 163 finite 243 243 Mealy product 258 pushdown 312 reduced f i n i t e 256 register 267 sub256 two way255 universal 162.168 axiom nonlogical

4,298,353 453
axiomatised, f i n i t e l y
453
B a s i s t h e o r e m , E,, 
211
BernaysScho nf i nke 1 class 462,517 Beth, d e f i n a b i l i t y thm. of 
437
binary integer programming
239
b i s e c t i o n problem
233
block transducer
269
Boolean function bound,
116,343,487
least upper
35
boundedness p r o b l e m
71
bounded s u m / p r o d u c t
197
Calculus expression Hilbert predicate propositional  of s e q u e n t s
353 353 350 354 353 401
I BDEX
575
canonical  algebra  term domain
389 389
Cobham, thesis of  and Ednnnds
224
canonically satisfiable 389
cofinite, complement is finite
Cantor, thm of Schroder127 BernsteinMyhill
cofiniteness problem
122
coincidence lemmn
347
combination lemma
147
compact
119
cardinality  statement  formula
112 350
cases, definition by categorical
30 38 1
compiler
categorical1 y describable
38 1
chain deduction
358
choice sequence
193
Chomsky  grammar 298  normal form 299,302  normal form thm 299 Church, theorem of  and Turing ChurchRosser property Church's thesis
443 8
48
circuits of Mealy automata
259
clause fact goal ground halting Horn Krom program 
396 396 396 396 39 1 444 396
clique  problem
186 186
closed
compactness theorem 365,379
27,303,342
16
complement problem of Asser comp1ete NP 
182,502
95,225,244,469 226
completely defined completeness  theorem  lemma
244
355,372,410 470
complex1ty  class  measure abstract  theory general 
159 196 144 145
computab1e  with space  in time t
505 174 174
s
computation  tree  system  universal
4 193 11 38
computat i on time  in parallel computation  closed En hierarchy thm.
269 212 213
computationuniversal
38
576
INDEX
concatenation
248
conclusion
4
configuration end initial stop 
4 5 13 13
confluence problem
7
coincidence lemma
347
XVI I I ,339
conjunction
conjunctive normal form 349 348 7
cons1stency  predicate 474 propositional logic 343 consistent
343,361 268 268 364

contextfree  expression  grammar  language
299 299 299 299
continuous
35
contradictory
343
control structure
11
correspondence problem Post's 
146
Cook, thm of 
48 1
73 73
Craig, interpolation theorem of 
436
creative
129
cri tical
403
cut
 elimination thm 412  rule 318,366 41,163
cylinder 127  criterion of Young 128 D a v i s , thm of  Putnam & Robinson decidability  problem
82 15 122
decidable 15,69,453.505  in time t 174  with space s 174 decision problem propositional logic Hilbert's  procedure decompositi on thm product 
336 344 442 7 256,258 258
Dedekind, theorem of 
conversion lefor general complexity measures
correctness  problem
355,404
cyclefree
consequence  concept  set
constant individual propositional
 theorem
deduction admissible  system  thm, Dedthm
385
354 5,353 360 367,369
70,121
definability theorem of Beth
437
I BDEX
577
definition by cases
30
77 183
row square 
degree representation theorem
132
Dyck language
Dekker, theorem of  % Yates
135
Effective
Dekker's process
134
Ehrenfeucht lemma
439
e 1genvar iable  condition
403 403
denotational semantics derivable
4,353
derivation left LK tree normal  order right  system  tree
4
305 4 03 424 425 305 4
303
determ1nate Horn formula
517
deterministi c locally non
5 5 15
diagonalisation methods difficult diophantine  equation  predicate dis j unction  s ymbo1
34
58 118
299 3
elementary  honest  expression  function  jump hierarchy
219
embedding
256
emptiness  lemma  problem enumerable recursively
69 69,103

enumerabi 1ity recursive 
15 69
enumeration  theorem  procedure
106 55,109 8
enumeration function Kleene 
XVIII 339
equat 1on diophantine exponential diophantine Pel1 
349
distr 1bu ted numbers
185
75,77 domino problem column 77 diagonal 77 restricted originconstrained  229 NEXPTIMEcomplete  192 PTAPEcomplete 192
equ iva1ent  grammars  programs equ iva1ence  problem  theorem extended 
339 197
124 70.118
82 82
disjunct ive normal form
212
54
82 82 89
298 140
XVI I I ,339 71,121 25,404 48
578
INDEX
essentially undecidable 453
flipflop
evaluation
343
flow diagram
existential  expression  quantifier
339 338
explicitly definable
27
expression  calculus contextfree elementaryexistential  induction nth order prenex propositional logicregular simple weak nth order 
339 353 310 339 339 341 386 349 364 248 339 386
EXPTAPE
225
EXPT IME
225
extension
453
Factor analysis problem
233
feedback connection
262
final state
243
f initary
354
finite  acceptor  automaton  transducer
243 243 243
finitely axiomatised
453
finiteness theorem
379
Fitch domain
299
fixed point smallest  theorem
35,60,507 35,507 35,66
257 12
formula cardinality Horn Krom mix principal prime program secondary side 
339 350 390 444 412 403 339 445 403 403
Friedberg, thm of & Muchnik
140
Friedmann, thm of  h Gurevich
457
function arbitrarily 155 complicated Ackermann's 197 Biber 70 boolean 116 Cantor's (de)coding 105 chain31 characteristic  XIX,15 coding 30,105 computed 14 constant 21 continuous 35 decoding 30,105 elementary 197 enumeration 55 initial 25 input  11.14,51, 162,163 iteration of a XIX 31,221 length monotone 34 recursive 25 nfold recursive 173 output 11,13,52 262,164,195 Parikh 307 partial XVIII partial recursive 25 primitive recursive 25 production 129 21 projection RadoBiber 70
INDEX
579
recursive 26 reduction 72,226 stepcounting  119,145 successor 21  symbol 338 tape constructable 183 test 12 time constructable 183 5,52,162 transition 166,243 translation 55
structure of type  of type  of type

0 2 3
322 298 299 298
grammatical rule
298
XVI I
graph graph lemma relativised 
106 115
Furer's theorem
185
Greibach normal form
301
G a p theorem
159
generabi 1i ty
15
ground  statement  clause
396 396
Grzegorczyk  hierarchy thm.  class  similar
198 196 219
generalisation
339
generation procedure Gentzen Hauptsatz of  cut elimination theorem Godelisation
8
424 412
26,51,66
Godel numbering 55,84 i somorphism theorem for s 63 Gode1 KalmarSchutte 462,517 class Godel's thm Goodste i n  process  sequence thm. of , Kirby grade
104,110,384
&
Paris
Halting problem general initial restricted restricted initial special Ham1 1tonian  circle  cycle  path hard
7,70 71 70 182 182 70 237 238 238
118,181
Hauptsatz of Gentzen
424
220 220
Henschen, thm. of & wos
483
220
Herbrand  algebra  universe
389 389
Herbrand model smallest 
390 391
413
297,298 grammar ambiguous 322 contextdependent  331 contextfree 299 lengthening 330 regular 298
Herbrand's thm. hierarchy thm.
391,427
109
580
INDEX
Encomputation time 213 Grzegorczyk 198

variable
338
Hilbert calculus
350
infinity  statement  problem
Hintikka set
365
initial state
homomorphism
249
1 nput
honest, elementary

HORN
390
 signal
Horn formula determinate
390 517
Horn clause procedural interpretation of
391 396
12,243
 alphabet function
169

110 121
243 11,14,51 162,163 243
inseparabil ity thm.
471
inseparable recursively
135
instance matrix


391 391
Horn complexity
486
hyperresolution
486
Idempotence
365
Immerman, thm of thm of  & Vardi
514 510
interpolant
428
immune
132
interpolation thm.  of Craig
428 436
interpretation
344
implication
XVI I I,339
implicitly defined
61,437
inclusion lemma
124
1 ncomp1ete
469
incompleteness thm. first Godel second Godel 
337 469,473 469 473
index finite Kleene  set
55 251 55 60
individual  constant  domain
338 344
instruction function  number operation test 
interpreter intersection  lemma  problem
12 12 12 12 12
15 125 232
i somorph i c
127,256,380 recursively 127 ally embeddable 256 ally simulatable 256
isomorphism theorem  for Godel number i ngs iteration  complexity  depth  number
381 63 248 196 196,197 169
I PDEX
581
 of substitution of a in L 310  of a function XIX result of a 10 star 248  theorem 55
J unctor
338
219 120
K a h r , thm of , Moore & Wang
463
Kleene's  enumeration thm. 55  normal form thm. 55  thm . 52,55,198,248  Tpredicate 53 55
KleeneMostowski hierarchy
108
Kreisel's theorem
456
KROM
444
Krom  formula  clause notat i on
444,486 444
XIX
298.338 Language accepted . 175,191,243 contextf ree 299 Dyck 299 formal 1st order 337 f o r m a l 2nd order 382  generated by grammar 298 linear bounded 330 regular 248  of type theory 385
leaf
305
length  of a variable  of a word
xrx
length ordering
518
338
letter
jump elementary hierarchy  operator
Kleene index
left derivation
xx
level set
129
Lewis, theorem of
518
lexical rules
298
LFP expressions, determi nat i on of
509
liar paradox
473
lifting lemma
395
limit existence assert ion
110
1 1 mi ted pr i mi t i ve
recursion
197
linear set semi 
307 307
LINTAPE
333
11teral
339
Lob theorem of  condition
474 474
LOGTAPE
227
logic program
396
loop  complexity  lemma  problem  program
196 305 293 196
LOOPn
197
lower sequent
403
582
INDEX
low set Machine oracle register Turing Wang 
138 13
3. 15
22 16,17 44

Markov algorithm commutative 
38 38
Markov thm. of Post & 
38
Matijasevich, thm. of
82
matrix  instance
349 349
measure of program size
operator bounded  in normal case
Myhill theorem of 130 thm of CantorSchroderBernstein 127 thm of  & Nerode 251 Negation  lemma
network APA complexity logical transitions of APA
261 269 488 488 269 402
new, new variable
metavariable
338
Newman's principle
317
midsequent
424
minimalisation problem
252
Minsky's theorem mix
 formula
37 412 412
mode 1 Herbrand 
348 389
modified nadic represention
200
module transformation system
7
modus ponens
354
monotone increasing/ decreasing
507
105 196
121
&
XVI I I,339
nesting depth
membership problem
Meyer, thm. of Stockmeyer
22 30 53
9
NEXPT IME
225
NL INTAPE
333
nodes, logical
488
normal  Post calculus  system
6,43 43
normal case, application of operator in 53 normal form theorem Chomsky Greibach Kleene Skelem NP
_complete
Nuremberg funnel pr i nc 1pl e
299 301 55 387 226 226 10
IBDEX
583
Occurrence, negative/ positive  of predicate symbol 509 operati on elementary
13 13

operational semantics
34
oracle  machine
115 115
ordering complete partial
35
order number  of magnitude  of a type  of a variable
425 175 386 386
ori ginconstrained domino problem 75 restricted 229 output alphabet  function
243 11,13,52,162 164,195 195
 word
225
P
paradox of Richard
58
parallel connection
26 1
Pari kh  function  mapping theorem of
307 307 307

partial function
XVIII
partial program evaluation part it ion problem path

net of Rodding  problem of Rodding
56 23 1
xx
68 68
Peano arithmetic
475
permu tat i on of variables
110
Petri nets
39
Plaisted 522 thm. of theorem of Denenberg 8 Lewis 525 positive occurrence of a predicate
509
Post

canonical system 43  correspondence 73 problem  normal calculus 43 problem of 138 thm of 38,43,47,115
power
248
predicate XIX,338 arithmetical 110 diophantine a2 exponentin1 diophantine a2 506 globally empty global _ 505 Kleene T53  logic of 2nd order 382  symbol 338 prefix 349  class 462,517  problem 462  signature problem 462 premiss
4
prenex  n o r m 1 form
349 349
Presburger arithmetic
476
Priese, thm. of 
283
prime formula
339
prime number coding
30
584
INDEX
primitive recursion 25 limited 197  with substitution at parameter places 217
quantifier depth
490
principal formula
403
quotient, right 
250
priority method
138
procedure  call  declaration  namg
396 396 396
Rabin thm. of  % Scott thm. of  ,Scott % Shepherdson
196
product automaton
257
production function
129
productive
129
Droaram  formula 445,458,478 logic 396 25 loop PROLOG 396 selfreproducing 65 universal 51
A

Quantifier logic, propositional logic 493
255
RabinBlum construction 155 rank rightlleft

375,413 number 413
reachabi1it y  problem  set
7 7
reachable
5
recognition procedure
8
PROLOG program
396
pr oposi ti ona1  constant  variable
364 364
recursion 31 courseofvalues  depth 196 nfold nested 217 nfold unnested 218 nth  class 196  theorem 59.65,66,67
propositionallogic  expression  quantifier logic
364 493
recursion, pr i mi t i ve limited simultaneous 
provable
469
PTAPE
225
pumping lemma
158
pure
402 376
.

predicate logic badic representation
pushdown  automaton  store
223 311 312
25 197 31
recursive 26,29, 114  closure 114  enumerability 68,81  in C 114  i nseparabi 1 ity 135  isomorphism 127 25 partial 25 primitive 25,26  solvability 69  unsolvability 69 recursivit y
68
I BDEX
585
reducible
72 72 226 226 116 116 116 72
In
P
pol ynomlaltttruthtable Turing Ireduction  function  lemma strong weak 
93 72,226 445 93 93
 of Godel
110
resolution  calculus  theorem hyper SLD unit 
366 366 366.394 486 396 446

resolvant predicate logic SLD
366 311 396

resolve
366
446 449
reversal  complexity
249 146
refutable
469
refutation set
453
reversible  closure
register  automaton ~ ~ ~ x i ma contents l  store
266 146 23
register machine parallel  program
23 42
register operator nprimitive recursive
23 23 25
reduction class conservative
.
regular  expression  grammar  language

298 248 298 248
relat ion computable computable result transformation transition 
11 11 4 4
relative compu tabi 1 ity
114
representability
471
representable
470
representation theorem  for re. predicates 104
5 8
Rice theorem of thm of  & Shapiro
60 119
right  der1 vat i on  invariant  quotient
305 251 251
Ro dding
normal form thm of path problem of 
root of a tree rucksack problem

263 78
xx 232
rule 4,353 conclusion of a 4 grammatical 298 left linear 298 left regular 36,298 lexical 298 premiss of a 4 right linear 298 right regular36,298 tautology rule 363 transformation rule 4 run time  complexity
144,162 196
586
I BDEX
Satisfiability lemma
361
satisfiable 343,346 canonically 389  over A 346 finitely 346 non346 propositional logic 343 Savitch, theorem of

194
Skolem  arithmetic  normal form theorem of 
SLD tree resolvant
Smntheorem solvable recursively 
secondary formula
319
self embedding
311
space requirement logarithmic 
self reference
473 34 34 34
398 396

Schatz, theorem of , Rodding & Briiggemann 270
semantics denotational operational 
476 387 378
56
15,230 69
174,191 227
space hierarchy thm space consumption
184
174,191
spectral problem
498
spectra hierarchy thm
499 498 502 498
semantical concept of consequence
348
semilinear set
307
spectrum general ised  problem
semiThue system
6
speedup theorem
148
sequent  calcu 1u s end finite  tree lower mid upper 
401 401 404 403 403 242 403
square domino  problem , game
230
side formula
403
signature
338
simple  set
132 132,138
simplicity concept
135
simu 1atable isomorphically
256
s i mu 1ate
256
stack standard semantics standard stop criterion star  covering problem  iteration
52 384
14
234 235 248
state 12,243 a 1phabet 243 diagram 4,243 direct successor 4 end 5,194 initial 9,12,194  transition 246 12 stop storage 4
INDEX
587
successor 
4.12
step counter function
119
Stockmeyer, theorem of
493
stop criterion standard
11,52 162,165 14

stop state, accepting
175
stora e requirement total 
146 146
structure gramnmr
322
subautomaton
256
subformula property
423
substitution  basis  closure  lemma simultaneous  thm
211 197 352 21 55

succedent
401
success set
379
synthesis process
a
T a g system
44
tape  bounded  Compression lemma
constructable  reduction lemma 
174 177 193 183 176 193
tautology  rule
343 363
TCexpress1on determ1nat ion
513
term  induction
27,339 341
term domain canonical  of closed expression
389
terminal symbol
236
test number
146
theory complete consistent decidable essentially undecidable finitely axiomatisable incomplete number 
453 469 453 453
thesis of Church of Cobham
389 389
453
453 469 469 50
dmonds 225
thread
404
Thue system 2dimensional 
6,38 283
tiling problem
75,229
time  bounded  compression lemma constructable  consumption  hierarchy theorem  requirement 
total, total function totality problem
174 178 193 183 174 185 191
XIX
70,121
Trachtenbrot, thm of
448
track technique
176
transformation system transducer block 
4 243 269
588
finite 
INDEX
243
transition  function
5,52,162 166,243 state 246  of an APAnetwork 269 pure 246 system 4
U n a r y representation
17
undecidability theorem
72
undec1dable formally essentially 
69 469 453
unifiable
393
unification theorem
393
unifier most general 
393 393
uni on theorem
161
unit resolution
446
truth  concept 113,141 logical 346 propositional logic 343 table 343 table conditions 117
universal  automaton  closure computation  program  val i di ty
162 370 37 54 346
ttcondition
117
universally valid
346
tupleformat i on technique
176
unsolvability degree
117
transitive hull of a global predicate tree computation tree representation of natural numbers
513
xx
192 186
Turing machine 16,18 alternating 154 ktape 19.173 khead 19 nondeterministic  191 174 offline 174 online 16,276 2dimensional Turing reducible Turi ng tape  operator
116 17 20
twoway automaton
255
type
385
type theory, language of
385
unsolvable recursively

69 69

upper sequent
403
Valid 343,346 propositional logic 343 logically 346 universally 346 var 1able e 1gen individual input length of a metanth order output propositional 
298,338 403 338 486 338 338 386 486 364
var 1able wi se d 1st1nct
342
INDEX
589
vector addftion system 40 Reneralised 79,80 ordered 40 W a n g mchine word empty length of a
44

XIX XIX XIX
word problem general special 
73
workingcell
17
worstcase complexity
7 7
174
LIST
OF
205 199 71,146 298 88 197 339 XIX 318 344 39 1 183 342 36 1 36% 486,488 N 21 159 c , , CV, c <m, c <s.m 7 C ( X ) (complement of x ) X V I I cona 346 Con, Con(S) 3 4% Const ( t > 342