Complexity of Proofs and Their Transformations in Axiomatic Theories

Translations of

MATHEMATICALMONOGRAPHS

Volume 128-

a

Complexity of Proofs andTheir Transformationsin Axiomatic TheoriesV. P. Orevkov

American Mathematical Society

Translations of Mathematical Monographs 128

Translations of

MATHEMATICALMONOGRAPHS

Volume 128

Complexity of Proofs andTheir Transformationsin Axiomatic Theories

V. P. Orevkov

,o American Mathematical Societyy Providence, Rhode Island

B. II. OPEBKOB

CJIO)KHOCTb )OKA3ATEJIbCTB H HX HPEOSPA3OBAHHI4B AKCHOMATH3HPOBAHHbIX TEOPHSIX

Translated by Alexander Bochman from an original Russian manuscriptTranslation edited by David Louvish

The translation, editing, and keyboarding of the material for this book was done inthe framework of the joint project between the AMS and Tel-Aviv University, Israel.

1991 Mathematics Subject Classification. Primary 03F20.

Library of Congress Cataloging-in-Publication Data

Orevkov, V. P.[Slozhnost' dokazatel'sty i ikh preobrazovanii v aksiomatizirovannykh teoriiakh. English]Complexity of proofs and their transformations in axiomatic theories/V. P. Orevkov; [trans-

lated by Alexander Bochman from an original Russian manuscript; translation edited by DavidLouvish].

p. cm.-(Translations of mathematical monographs; v. 128)Includes bibliographical references.ISBN 0-8218-4576-4 (acid-free)1. Proof theory. I. Louvish, David. II. Title. III. Series.

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Contents

Introduction 1

CHAPTER I. Upper Bounds on Deduction Elongation in Cut Elimina-tion 9

§ 1. The calculi KGL(2() and IGL(2[) 9

§2. Measures of the complexity of proofs 13

§3. Admissibility of structural rules 17

§4. Cut elimination in KGL(2() and IGL(2[) 20§5. The calculi KH(2() and IH(2() 25

CHAPTER II. Systems of Term Equations with Substitutions 31

§6. Systems of term equations with substitutions. Main lemmas 31

§7. Extension tree of a CTS-system 40§8. Representation of enumerable sets by TS-systems 51

§9. Upper bounds on the height of natural solutions of systems oflinear Diophantine equations 56

§10. Upper bound on the periodicity index of solutions of CTS-systems 63

§11. An algorithm deciding the existence of solutions of restrictedsubstitution width 76

CHAPTER III. Logical Deduction Schemata in Axiomatized Theories 81

§12. Systems of equations in formulas 81

§13. Deduction schemata in axiomatized Hilbert-type theories 94§14. Deducibility of a formula in accordance with a given schema 99§15. Deduction schemata in Gentzen calculi 106§16. Rebuilding of proofs on the level of schemata 114

CHAPTER IV. Bounds for the Complexity of Terms Occurring inProofs 119

§17. Comparison of the lengths of direct and indirect proofs ofexistence theorems in the predicate calculus 119

§18. Complexity version of the existence property of the con-structive predicate calculus 126

vi CONTENTS

CHAPTER V. Proof Strengthening Theorems 131

§19. Proof strengthening theorems in finitely axiomatized theories 131

§20. Proof strengthening theorems in formal arithmetic 134§21. Upper and lower bounds on lengths of deductions in formal

arithmetics 144

References 151

Introduction

The complexity theory of logical deduction is a natural development ofstructural proof theory, which is based on the Cut Elimination Theorem insequent calculi and the related Normal Form Theorem for natural deduc-tions. Cut-free proofs and normal natural deductions have many advan-tages, chief among which is the subformula property. That is why these havebeen successfully applied in several logical and mathematical-logical calculito prove the consistency, decidability, conservativity of extensions, etc. How-ever, these advantages are achieved at the cost of a significant elongation ofdeductions due to cut elimination and reduction to normal form.

Upper and lower bounds for the elongation of proofs in cut eliminationand related transformations of deductions are a central topic of research.Witness, for example, the work of Tseitin [35], Haken [42], and Dougherty[39] on the propositional calculus and Statman [52, 53] on the predicatecalculus.

At the end of the sixties, G. Kreisel proposed the conjecture (see problem34 in [41]) that the rule of infinite induction

A(0) , A(1) , A(2), .. .b'xA(x)

is admissible in formal Peano arithmetic PA if there exist a natural numberc and deductions of the formulas A(0) , A(1) , ... whose lengths are at mostc. Throughout this work, the length of a proof is the number of occurrencesof formulas and sequents in it, that is, the number of steps in the proof.Kreisel revised his conjecture in [46] as follows:

For any formula A(x) and any natural number k, there exists a naturalnumber M such that, for all n > M, the following condition holds: if we canconstruct a deduction of A(n) in PA of length at most k, then, for someN < M, the following formula is deducible in PA

b'x(x - n (mod N) j A(x)).

Parikh [49] proved Kreisel's conjecture for a system of formal arithmeticPA" whose language contains only the constant 0, a one-place function sym-bol ' , equality, and three-place predicates for addition and multiplication.

i

2 INTRODUCTION

No proof for systems of formal arithmetic in languages containing two-placefunction symbols has yet been published. A detailed review of the work todate on Kreisel's conjecture may be found in Krajicek [43]. Yukami [54],Miyatake [47, 48], and Bozhich [3] also deserve mention.

In proof theory one usually supplies applications of rules and axioms withanalyses. An analysis includes the code of the relevant axiom or rule, theindexes of the premises to which the rule is applied, and (in sequent calculi)the indexes of the formulas to be transformed and the formula obtained. Asequence of analyses of axioms and inference rules in an axiomatized theoryis called a deduction schema. A deduction schema can be considered as aneconomical and rather convenient code of a proof. A great many proof trans-formations can be accomplished using only deduction schemata. In finitelyaxiomatized theories, the number of deduction schemata of finite length isfinite -a fact that plays an important role in Parikh's proof of Kreisel's con-jecture for his system of formal arithmetic. Logical deduction schemata arealso used by Krajicek and Pudlak [45], Krajicek [44], and Bozhich [3].

The aim of this dissertation is to develop the tool of logical deductionschemata and use it to establish upper and lower bounds on the complexityof proofs and their transformations in axiomatized theories.

Our main results are as follows.We will establish upper bounds on the elongation of deductions in cut

eliminations, depending on the number of sequents in the deduction but noton the complexity of the formulas occurring in them, valid whether all cutsare eliminated or only those that involve formulas in some prescribed set.

We will prove that the length of a direct deduction of an existence theo-rem in the predicate calculus, provided such a deduction exists, cannot bebounded from above by an elementary function (in Kalmar's sense) of thelength of an indirect deduction of the same theorem.

A complexity version of the existence property of the constructive predi-cate calculus will also be proved.

For formal Peano arithmetic PA, Parikh's arithmetic PA*, and finitelyaxiomatized fragments of these systems, restrictions will be established on thecomplexity of deductions such that, if they are satisfied, the deducibility ofa formula for all natural numbers in some finite set implies the deducibilityof the same formula with a universal quantifier over all sufficiently largenumbers.

We now proceed to a more detailed description of the results.Let Qt be an enumerable set of formulas of the predicate calculus. We

will assume that Qt is given as a finite list of closed formulas (particularapplied axioms) and a finite list of formula schemata. In §1 of Chapter Iwe propose a sequent calculus KGL(2L) for an axiomatized theory with aset of applied axioms Qt, based on the classical predicate calculus, and acalculus I GL(2L) for the same theory but based on the constructive (i.e.,

INTRODUCTION 3

intuitionistic) predicate calculus. Both KGL(2() and IGL(2() contain thecut rule and the cut rule for formulas in 2( (2(-cut). If is a proof inKGL(2() or in IGL(2(), h[am] will denote the number of applications ofrules in the longest branch of and 1[PIJ] the number of different sequentsoccurring in £. Let O be some set of formulas. We will say that O isq+-closed if it is closed with respect to renaming of bound variables andsubstitution of terms for free occurrences of terms. An application of a ruleintroducing a symbol in will be called a O-application if the formula tobe transformed belongs to V. An application of the cut or 2[-cut rule inwill be called a O-application if the cut formula belongs to O. Let h°[]denote the maximal number of O-applications of rules in one branch ofand l°[] the number of different formulas in O obtained in usingthe rules for the introduction of logical symbols. A sequent S will be called2(-pure if no variable occurring free in S is bound in S or in some particularapplied axiom. The main result of Chapter I is the following

THEOREM 4.1. For any q+-closed set of formulas V, any proof inKGL(2() or in IGL(2[) of a 2[-pure sequent S can be rebuilt as a prooflJ' such that:

(1) ' is a proof of the same sequent S in the same calculus as ;

(2) if contains no O-applications of the cut rule;

(3) h[P1J'] <2v[ . h[P1J];

(4) hv[/]! [mil+i

Henceforth,2n = n, 2n = 2[2u].

0 i+1

The bound (4) for the case when V is the set of all formulas was obtainedby Statman [52].

In the last section of Chapter I we will propose an algorithm for rebuild-ing deductions in a sequent version of the predicate calculus as Hilbert-typedeductions; this algorithm yields an almost linear elongation of deductions.Earlier algorithms accomplishing the same were constructed by Gentzen [5]and Kleene [8]. Gentzen's algorithm yields a quadratic elongation, that ofKleene exponential (because Kleene used the Deduction Theorem).

The search for a logical deduction in cut-free calculi using metavariablesfor terms (see, e.g., [7]) involves solving systems of equations in terms. An al-gorithm that recognizes the decidability of such systems is easily constructedon the basis of Julia Robinson's Unification Theorem [32]. The analogoussearch in calculi with cut, in Hilbert-type predicate calculi, or in formal arith-metic involves solving systems of term equations with substitutions. Suchsystems of equations can be regarded as a generalization of systems of wordequations, that is, equations in a free semigroup. Word equations (systemsof word equations are reducible to single equations) are considered in [34].

4 INTRODUCTION

An algorithm that recognizes the decidability of word equations is con-structed in [13]. It can be used to recognize the decidability of systems ofterm equations with substitutions that involve only 0-place and 1-place func-tion symbols. Another such algorithm was published in [40].

In Chapter II it will be shown that there exists no algorithm recognizing thedecidability of systems of term equations with substitutions if the functionsymbols occurring in the equations are of arity more than one. This resultwas published in [28]; other proofs were obtained independently in [45] and[37]. In Chapter II we will also construct an algorithm to recognize whethera system of term equations with substitutions has a solution of a specialform, namely, a solution in each term of which the number of occurrencesof variables for which substitutions are made does not exceed some fixednumber. In particular, this applies to solutions that contain only 0-place and1-place function symbols.

Chapter III is concerned with the solution of the following problems aboutlogical deduction schemata in axiomatized Hilbert-type theories and sequentcalculi.

(1) Given a deduction schema U, construct a proof whose schema isU.

(2) Given a deduction schema U and a formula A, construct a proofof A whose schema is U.

It is clear that solutions to both problems reduce to solving systems of termequations with substitutions. For axiomatized Hilbert-type theories and se-quent calculi in which the sets of applied axioms are given by a list of formulaschemata without explicit occurrences of predicate symbols, the solution ofproblem (1) is reduced in Chapter III to a narrower class of systems: systemsof term equations (without substitutions). Consequently, it is possible toconstruct an algorithm for such theories and calculi that recognizes whetherproblem (1) is decidable for a given deduction schema. The situation withregard to problem (2) is different.

In Chapter III we will prove that the problem of deciding whether formulasare deducible according to a given deduction schema is algorithmically unde-cidable, for a large class of axiomatized Hilbert-type theories (this result waspublished in [29]). This class includes all known Hilbert-type formulationsof the positive, minimal, constructive, and classical predicate calculi. Thealgorithmic undecidability of the problem in the sequent calculus with cutwas proved in [45].

Let k be a natural number and a proof in an axiomatized Hilbert-type theory. We shall say that is k-restricted if, for any term occurringin , the number of occurrences of substituted variables does not exceedk. Any proof that involves only 0-place and 1-place function symbols isobviously 1-restricted. In Chapter III we will construct for any fixed k analgorithm which, given a deduction schema U and a formula A, constructs

INTRODUCTION 5

a k-restricted proof of A with schema U. Upper bounds for the degrees ofthe formulas involved in k-restricted proofs and for the heights of the termsoccurring in such proofs play a major role in the functioning of the algorithm.The height h[T] of a term T is the greatest length of maximal chains ofnested occurrences of terms in T other than variables and constants.

In the last section of Chapter III it will be shown that one can construct acut-free proof schema relying only on the schema of the original proof withcut, and not on the actual proof. This result will be used in § 19.

Let A be a formula, x an object variable, and t a term. Denote theresult of substituting t for all free occurrences of x in A by [A]. A proof

of the formula 2xA is called a direct proof of an existence theoremif contains a proof of the formula [A]X for some term t. Otherwise

is called an indirect proof. Not every existence theorem has a directproof in the classical predicate calculus. Thus, for example, the formula2x (P (x) dyP (y)) is deducible in the classical calculus, but for no termt is (P(t) dyP(y)) deducible in this calculus. In the constructive pred-icate calculus, the Cut Elimination Theorem implies that any deduction ofan existence theorem can be rebuilt as a direct proof. This feature of theconstructive calculus is known in the literature as the existence or explicitdefinability property (see, e.g., [6, 33]).

Let 91 be some axiomatized Hilbert-type theory and a proof in 91.The number of occurrences of formulas in is called the length of .

The expression 91 I1 A means that one can construct a proof of A in 91of length at most 1. In what follows KH will denote the Hilbert versionof the classical predicate calculus, and I H that of the constructive predicatecalculus. The following proposition compares lengths of direct and indirectproofs of existence theorems in the predicate calculus.

THEOREM 17.1. There exist a sequence of formulas A1, A2, ... and a con-stant c such that, for any natural number k,

(1) I H Fc(k+1

axAk+1

;

(2) IH F 21 [Ak+1 ]2k k+l

(3) for any term 0, if the formula [Ak+1 ]e is deducible in KH, thenh[e] >_ 2k+1 and the length of any proof of [Ak+1 ]e in IH or KH isgreater than

3 2k - 'where 2k+1 denotes 0 with 2k+ primes.

We will prove in § 17 that the height of a deduction obtained by eliminat-ing all cuts from a given deduction cannot be bounded above by a Kalmarelementary function of the length of the original deduction. This result waspublished in [15]. Statman [51] (a detailed account was published in [52])proved that in the classical predicate calculus with equality the normalization

6 INTRODUCTION

of natural deductions also lengthens deductions, by an amount that cannot bebounded above by a Kalmar elementary function of the length of the originaldeduction.

Let O be a set of formulas and a proof in some axiomatized Hilbert-type theory. Denote the length of by the number of formulasfrom O that occur in by 1H and the number of formulas from Owhich occur in as logical axioms by !° [2uJ] . For any formula A, we leth* [A] denote the greatest height of the terms occurring in A. The followingcomplexity version of the existence property of the constructive predicatecalculus will be proved in §18:

THEOREM I8.1. For any proof of a formula 3xA in IH such that nobound variable of A occurs free in 3xA there is a term T which is free forx in A and

h[T] < h[A]I H FL,N[931 [A]T,

where

M = 221+12 L = 22131E'' [ ]+2 ' 3/E.2[ ]+3 '

and the constant c does not depend on the choice of and A.

Here E.1 is the set of formulas B such that B contains either the quan-tifier 2 or a bound occurrence in the scope of a function symbol; E.2 is theunion of E.1 and the set of formulas containing an occurrence of disjunc-tion.

Let be some axiomatized Hilbert-type theory and µ a set of complexitymeasures of proofs in 3. By a proof strengthening theorem in we meana theorem of the following type:

Suppose that, for any n in some finite set of natural numbers, one canconstruct a proof of a formula in 3 such that the set of numbersi[] satisfies a prescribed system of inequalities. Then there exists k suchthat dx[A]X(k) is deducible in 3.

Here Oand xdenote 0 with n primes and x with k primes,respectively. In other words, proof strengthening theorems in 3 impose re-strictions on the complexity of proofs in 3 such that, when they are satisfied,the deducibility in 3 of an assertion for all natural numbers in some finiteset implies its deducibility for all sufficiently large natural numbers. If theset of measures p consists only of the length of the proof and the system ofinequalities is simply an upper bound for the length of the proof, then theproof strengthening theorem for 3 implies Kreisel's conjecture for q3.

Proof strengthening theorems for finitely axiomatized theories were ob-tained in [16, 45]. Bozhich [3] proved the following proof strengtheningtheorem for the system PA of formal arithmetic and related systems.

INTRODUCTION 7

THEOREM. Let 3 be one of the systems of arithmetic constructed in [3].For any natural numbers 1, h, k, and q, there exists N such that, for anyn > N, the formula is deducible in 3 if A satisfies the followingconditions:

(1) for any natural number m, if 0(`") occurs in A, then m < q ;(2) the number of logical connectives and quantifiers in A is at most k ;(3) the weight of A is at most h ;(4) there is a proof of the formula [A]n) in 3 such that 1H[21] < 1

and the weight of the induction axioms in is at most h.

Here the weight of a formula B is the largest m such that B contains anoccurrence of a term y('") that generates a bound occurrence of the variabley in B. Theorem 18.2 will give a formula for computing N from 1, h, andq

In the last chapter we will establish some strengthening theorems for finitelyaxiomatized theories and formal arithmetic. The main result of § 19 is thefollowing sharper version of Kreisel's conjecture for finitely axiomatized the-ories:

Let be any finite list of closed formulas, m any natural number, n asufficiently large natural number, and A a formula. Suppose that, for all i

(0<i<n),x

F log2(n+m) [A]o(m+i).

3 log2 log2 (n+m)

Then for some k <m + n

F dx[A]X(k).

Here is an axiomatized Hilbert-type theory with set of appliedaxioms based on the calculus KH.

The following sharper version of Kreisel's conjecture for Parikh's arith-metic will be established in §20 (Theorem 20.1) :

For any formula A and sufficiently large n and k, if A contains only0-place and 1-place function symbols and, for all i (0 < i < k),

PA* F l to to n+i [A]o(n+i) ,

then PA* F dx[A]X(n).We have not been able to establish strengthening theorems that use only

length when the language of formal arithmetic contains two-place functionsymbols and have to use other measures for the complexity of proofs. Definethe ji-rank of a formula A as the maximal number of occurrences of avariable x in a term T for all bound occurrences of x in A and all termsT whose occurrence generates a bound occurrence of x. The expression

PA Fk Al,pwill mean that one can construct a proof of A in PA such that

(1) the length of " is at most 1;

8 INTRODUCTION

(2) the maximum of the 'u-ranks of the applied axioms in and of the'u-rank of A is at most k ;

(3) the number of formulas in of 'c-rank greater than k does notexceed p.

Note that PA f-' o A if and only if the 'u-rank of A is at most k and itis possible to construct a k-restricted proof of A in PA of length at most1.

THEOREM 20.3. For any natural numbers m and p, any formula A andsufficiently large n and k, if

PA F .log2log2log2(n+i) , ps

for all i (0 < i < k), then PA I- dx

In the last section of Chapter V we introduce a finitely axiomatized theoryRHA* based on the calculus IH. Parikh's arithmetic and the theory ob-tained from Robinson's intuitionistic arithmetic (see, e.g., [8, 10]) by replac-ing the function symbols + and by three-place predicates are extensionsof RHA*. The main result of §21 is the following

THEOREM 21.1. There is a natural number cl such that, for any consistentaxiomatizable extension 2t of RHA*, there exist a one-parameter formulaA and a natural c2 such that

(1) for any natural number n,*

RHA Fci[log2(n+1)]+c2 [A]0(11)

(2) for no natural k is the formula dx[A]X(k) deducible in 2t.

This theorem was announced in [16] and published in [30]. A counterex-ample to Kreisel's conjecture for a system of arithmetic containing as appliedaxioms all valid term equations not involving the multiplication symbol, wasconstructed in [55].

To end Chapter V, we will establish upper and lower bounds on the lengthsof deductions in RHA* and PA* for the following assertions:

(1) The natural number n is even.(2) The natural number n is composite.

For all even n , the length of a deduction of (1) in RHA* is bounded fromabove by c[log2(n + 2)] ; in PA* there is a constant upper bound. For allcomposite n, the length of a deduction of (2) for both RHA* and PA* ismajorized by c[log2 n]. For infinitely many n, the length of a deduction of

logn(1) or (2) in RHA* is greater than5 tog2 2 og2 n

; for all sufficiently large n, itis greater than 3 Jlog2 n. The length of a deduction of (2) in PA* is greaterthan a Jlog2 n for infinitely many composite n.

CHAPTER I

Upper Bounds on Deduction Elongationin Cut Elimination

The aim of this chapter is to obtain upper bounds on the elongation of de-ductions in cut elimination, depending on the number of sequents appearingin the deduction but not on the complexity of the formulas occurring in thesequents.

We will consider the following systems of logical inference in the predicatecalculus: (1) Hilbert-type system, (2) Gentzen sequential calculus with cut.The Hilbert-type calculus KH for the classical predicate calculus consideredin this chapter coincides with the calculus from [12]. The sequential calculiKGL for the classical predicate calculus and I GL for the constructive (i.e.,intuitionistic) predicate calculus have some methodological and technical ad-vantages, the main advantage being that one can reduce the number of reduc-tions in cut eliminations by merging of some of them in a single reduction.The calculi KGL and I GL were published in [22], where an upper boundwas established for the elongation of deductions due to cut eliminations.

In the last section of the chapter we present an algorithm that translatesdeductions in the sequential calculus into Hilbert-type deductions. This al-gorithm yields an almost linear elongation of deductions. The derivation ofthe bound relies on the idea of balanced binary search trees (see, e.g., [1] or[11]).

§1. The calculi KGL(2L) and IGL(2t)

Let 2t be a recursively enumerable set of formulas of the predicate calcu-lus. The axioms of the calculus KGL(2L) are sequents of the form

F, A, F' -p A, Lx',

where A is an arbitrary formula; F, F', L, 0' arbitrary lists of formulas.The inference rules of KGL(2L) are the following figures:

F->B,LI'->0 AFB 0' -'O1' F->0 AFB 0'

I' ADB

9

10 I. UPPER BOUND ON DEDUCTION ELONGATION

r->A,D,AyB,L' r->B,t ,AyB,t' .

r->o AvB o' v1' r->o AvB o' v2'T,AyB,F ,A->0; T,AVB,r',B->0

Vr AvB r'->L im'

r->A,D,A&B,0'; r->B,D,A&B,0'r-o,A&B,o

r,A&B,r',A_Or,A&B, r' 0 r,A&B,r' 0r,A-iA,O'r-p O, -IA, t'

T- > [A], o , dxA, o'The variable a is free for x in A,

a does not occur freein T, t, VxA, Lx'.T -p L, dxA, t'

r -> [A]x , 0 , 2xA, 0'The term t is free

for x in A. -p2;r->S, 2xA,O

F,-IA, F'->A,0

T, dxA, T', [A]X -LThe term t is free

for x in A.r,dxA,r'->0

r, 2xA, r', [A], -*0The variable a is free "

for x in A,a does not occur free

in T,2xA,F',L L. _

T, axA, r' -La -> ;

[A E QL]-cut' A' cut.

Below, the expression [A]X denotes the result of substituting the term tfor all free occurrences of the variable x in A. Let ICI denote the length ofa list of formulas , i.e., the number of distinct occurrences of formulas asmembers of .

The axioms of the calculus I GL(2t) are just the axioms of KGL(2L) butwith single formulas in the succedents, the rules ->>1 , v -p, & & - 2 ,

-> d -p, 2 -p, and 2t-cut are the same as in KGL(2t). The other rulesare:

BT -Br->ADB 2 r r ->o

->v r->B ->v r-*A; r-Br->AvB 1' r->AvB 2' r->A&B

ICI <1]]T,,A,r'- -1 -p.

0

F -> [A]a

The variable a is free for x in A,a does not occur free in T, dxA.

T - `dx A

§1. THE CALCULI KGL(?i) AND IGL(?t) 11

T -> [A]X

[The term t is free for x in A.] . T->A; T,A-0cut.

The following proposition is obvious:

LEMMA 1.1. If a sequent is deducible in IGL(Q[), its succedent contains atmost one occurrence of a formula.

Let S be an arbitrary sequent. Let us index the occurrences of formulasand of the symbol -> as members of S by integers. The occurrence of ->will be indexed by zero, and the indexes of occurrences will increase fromleft to right; antecedent occurrences will receive negative indexes, succedentoccurrences, positive indexes.

Following [9] and [10], explicit occurrences of a formula in an axiom ofKGL(2() or IGL(2L) will be called principal occurrences of the axiom, and theformula itself, the principal formula. An explicit occurrence of a formula inthe conclusion of an application of a rule will be called a principal occurrence,and the formula itself, the principal formula of the application. Explicitoccurrences of formulas near -> in the premise of a rule will be called sideoccurrences, and the formulas themselves, side formulas. For rules with twopremises, the notions of left and right side occurrences and formulas aredefined similarly.

Note that applications of the 2(-cut and cut rules do not involve principaloccurrences and formulas and that in any application of the cut rule the leftand right side formulas are the same.

In what follows we will assume that the set of applied axioms 2( is a unionof sets 2(1, 2' ... , 2(r ,where for each i (1 < i < r) either

(a) 2Lr consists of one closed formula (such a formula will be called aparticular applied axiom), or

(b) 2L; is a countable set of formulas closed with respect to renaming ofbound variables and substitution of terms for free variables (it is assumed,of course, that terms are free for the variables).

Proofs in KGL(Qi) and IGL(2L) will be represented in the form of planarrooted trees with analyses of applications of rules and axioms. An analysisof an axiom will contain the word "axiom" together with the indexes of theprincipal occurrences. An analysis of an application of a rule will include,besides the name of the rule, the index of the principal occurrence wheneverthe rule has a principal formula; an analysis of an 2(-cut will include theindex i (1 < i < r) such that the side formula of the application belongs to

Let be a proof in KGL(2() or IGL(2[) and U an occurrence of asequent in . The number of occurrences of sequents below U in willbe called the level of U in " . Let U1 (UZ) be an occurrence of a sequent

S1 (S2) in , Vl (V2) be an occurrence of a formula Al (A2) in S1 (S2)as a member of the sequent. Suppose that U1 and U2 belong to the samebranch of and that U2 does not lie above U1 on this branch. We willdefine the relation "the occurrence V2 is a successor of the occurrence Vl "by induction on the difference between the levels of U1 and U2 in asfollows:

If U1 coincides with U2, then V2 is a successor of Vl if and only if Vlcoincides with V2 .

Suppose that U2 lies strictly below U1 in . Let U* be an occurrenceof a sequent S* in and L an application of some inference rule insuch that U2 is the conclusion of L and U* a premise of L on the samebranch of as U1. If no occurrence of a formula in S* is a successor ofV1 , then no occurrence of a formula in S2 is a successor of J/ .

Let V * be an occurrence of a formula in S* which is a successor of j,If V * is a side occurrence of L, then the principal occurrence of L will bea successor of V1 .

Suppose that V * is not a side occurrence of L. Let n be the index of Vin S* and m be the index of a side occurrence of L in S* . If n <0 < mor m <0 < n, the occurrence in S2 of the formula with index n will be asuccessor of V1 . If 0 < m < n, the occurrence with index n - 1 will be asuccessor of V1 . If n < m < 0, the occurrence with index n + 1 will be asuccessor of V.

It is clear that for any occurrence of a sequent below U1 in , the oc-currence Vl has either exactly one successor or no successors at all.

Let A and B be two formulas. We will say that A is transformable intoB if B can be obtained from A by simultaneous substitution of terms forfree variables (where it is assumed, of course, that the terms are free for thevariables).

LEMMA 1.2. For any formula B1, if V2 is a successor of V, then for anyoccurrence of B1 in Al one can construct a formula which occurs in A2 andis transformable into B1.

This assertion can be proved by induction on the differences between thelevels of Ul and U2 in .

LEMMA 1.3. For any occurrence U of a sequent S in and any occur-rence V of aformula in S as a member of the sequent, one of the followingconditions holds:

(a) there is an application L of the cut or 2(-cut rule in such that theconclusion of L lies below U in and the side occurrence of L isa successor of V ;

(b) there is an occurrence of a formula in the last sequent of which isa successor of V.

§2. MEASURES OF THE COMPLEXITY OF PROOFS 13

This assertion is proved by induction on the level of U in . Lemmas1.2 and 1.3 can be considered as yet another formulation of the subformulaproperty (see, e.g., [8] and [10]) of Gentzen sequential systems.

Let L be an arbitrary one-premise (two-premise) rule of the calculusKGL(2i). We can write L as

L,

where II + fl _ + II' = 1, and the lists , fl and ' , fl' coincide,respectively, with the side formulas of L. If L is a rule introducing alogical symbol in the succedent, then the principal occurrence of L resultsfrom the occurrence of a formula in 0 . If L is a rule introducing a logicalsymbol in the antecedent, then the principal occurrence of L results fromthe occurrence of a formula in T.

Let and ci) be lists of formulas. Define an operation o over lists offormulas as follows:

{ci), if the list is empty,

otherwise.It is easy to prove the following assertion.

LEMMA 1.4. For any lists of formulas , fl, and :

(1) if is empty, then o II a II o 1 a II ;(2) if is not empty, then o II a o a ;

(3) o [IIoE] a [ion] oE.Using the operation o, one can write any rule of the calculus I GL(2t) as

follows:(f,4flOL)L.

In what follows, if 2t is empty, the calculi KGL(2L) and I GL(2t) will bedenoted by KGL and I GL , respectively.

§2. Measures of the complexity of proofs

Let be one of the calculi KGL(2L) or IGL(2(), a proof in 3.Let denote the height of , that is, the number of different appli-cations of rules in the longest branch of , l[] the number of differentsequents occurring in , and 1[2/J] the number of different formulas thatoccur as principal formulas of applications of rules in . Any proof whichis included in will be called a subproof of .

LEMMA 2.1. If contains identical subproofs above identical sequents,then

h[am] < l[2/J]- 1 < 2h[]+' -2; (1)

l[217]2h[J] - 1, (2)

l[]l[]-1. (3)


PROOF. Suppose that satisfies the conditions of the lemma. Inequali-ties (1) and (2) can be proved by induction on h[am]. To prove (3) we rewrite

as a linear deduction deleting all repetitions of sequents. Now, onNthe one hand, l[am] does not exceed the number of applications of rules in

* . On the other hand, this number is at most l [?iY] - 1.An occurrence U of a term in a formula or a sequent S will be called

free if all occurrences of variables generated by U are free in S. Let A andB be formulas. We will say that A is q+-transformable into B if B can beobtained from A by renaming bound variables and subsequent substitutionsof terms for free variables (it is assumed, of course, that these terms are freefor the variables). We will say that A is a q+-type of B if

(1) A is qttransformable into B ;(2) A does not contain 0-place function symbols;(3) any object variable has at most one free occurrence in A ;(4) if a term t occurs free in A, then t is a variable.

Suppose that A is a q+-type of B and C a formula qttransformableinto B. Then it is easy to show that A is qttransformable into C. Itfollows that q+-types of the same formula coincide up to renaming of freeand bound variables. Given any formula, we can obviously construct its qttype. We will say that a formula A is qt equivalent to a formula B if it ispossible to construct a formula which is a gttype of both A and B. It iseasy to prove the following assertion:

LEMMA 2.2. For any formulas A, B, and C the following statements hold:(a) A is q+-equivalent to A ;(b) if A is qt equivalent to B, then B is q+-equivalent to A ;(c) if A is q+-equivalent to B and B is q+-equivalent to C, then A is

qt equivalent to C ;(d) if A is q+-transformable into B, then A is qt equivalent to B ;(e) if a term t is free for x in A, then A is qt equivalent to [A];(f) if an object variable y is free for x in A, then QxA is q+-equivalent

to Qy[A]y , where Q a , V.

Let be a proof in 3. We will say that a formula participates in if itis either the principal formula of some axiom in , or the side or principalformula of some rule application in T. We will say that a formula A q+-participates in if A is q+-equivalent to a formula participating in .

We will say that A evenly qtparticipates in if there exist an applicationL of a rule of introduction of a symbol in the antecedent in , and anapplication L' a rule of introduction of the same symbol in the succedent,such that the principal formulas of both L and L' are qtequivalent to A.

We will say that a proof is qt embeddable in a proof ' of the samecalculus if:

(1) whenever a formula participates in , it gtpartlclpates in J';

§2. MEASURES OF THE COMPLEXITY OF PROOFS 15

(2) for any application L of a rule of introduction of some symbol in, there is an application of the same rule in whose principal

formula is q+-equivalent to the principal formula of L ;(3) for any application L of the 2t-cut rule in there is an application

of the 2t-cut rule in such that its side formula is q+-equivalentto the side formula of L.

The following assertion is easily proved, using Lemma 2.2:

LEMMA 2.3. For any proofs , , and " in q3:(a) is q+-embeddable in ;

(b) any subproof of 221 is q+-embeddable in ;

(c) if is q+-embeddable in and is q+-embeddable in " ,

then is q+-embeddable in " ;(d) if ' is q+-embeddable in " , ' and " have the same end-

sequent and " is a subproof of , then the result of replacing "by in is q+-embeddable in .

Let V be a set of formulas and A a formula. We define the V-degree ofA by induction on the construction of A :

0, if A V ;

1, if A E 0 and A is an elementary formula;

B h h f° f] + 1, i as t ormdeg A E 0 and A ev[deg [A] =-iB, VxB , or 2xB ;

max(deg° [B1 ] , deg° [B2]) + 1, if A E 0 and A has the form(B1 B2), (B1 V B2), or (B1&B2).

The V-degree of A relative to a proof in 3 is defined by inductionon the construction of A, as follows:

0, if A v;if A does not qt-participate in ;if A E V, A q+-participates in , and A does

ii lti t+ n ;e even ypanot q co-par[A ] =deg , , +1 + M1, if A E V, A evenly q -participates in and

has the form -B, VxB, or xB;1 + M2, if A E V, A evenly q+-participates in and

has the form (B1 B2), (B1vB2), or (B1 &B2).

Here M1 = deg° [B , ] and M2 = max ( deg° [B1

, ], deg° [B2 , s ])The following assertions can be proved by induction on the construction

of formulas.

LEMMA 2.4. For any formula A and any proof in 3,

deg° [A , ] < deg° [A ] .

LEMMA 2.5. For any formula A and any proofs and ' in q3, ifis q+-embeddable in ' , then

deg° [A , ] < deg° [A ,

A set of formulas V will be called qt closed if, for any formulas A andB, if A E V and A is q+-equivalent to B, then B E V. The followingassertion is easily proved by induction on the construction of formulas.

LEMMA 2.6. For any proof in q3 and any formulas A and B, if theset V is q+-closed and A is qt equivalent to B, then

deg° [A] = deg° [B ] , deg° [A , T] = deg° [B , T ] .

If is a proof in 3, then l[] (resp., l ° will denote the numberof different qtequivalence classes of formulas that belong to V and areprincipal formulas of applications in of the rules

1 , 2 , V 1 ,

V2, -+ - , -+ &, V -p, or - (respectively, the rules -* , V -p, && -p, - V, or -p). The number of different formulas that belongto V and are principal formulas of rule applications in will be denotedNby l ° [s ] . Clearly,

max{ll [ T] ,TV[]} < l °[r]

LEMMA 2.7. If the set V is q+-closed, then

1,

deg° [A , T] + 1.

PROOF. Suppose that V is q+-closed. Let L[] denote the number ofdifferent qtequivalence classes of formulas A such that A E V, deg°[A]k, and A is the principal formula of an application in of one of the rules

1' 2, V 1, V2 , -p - , - &, V -p, or -p It follows fromLemma 2.6 that

L° [PlJ] <

Using Lemma 2.6, one can easily prove by induction on deg° [A ] that

deg° [A , T] < Lgd o[A]WlJ] + 1.

(3)

Inequality (1) follows from this inequality and from (3). The proof of in-equality (2) is similar. O

A rule application L in will be called a O-application if one of theside formulas of L belongs to O. Let U be a branch in that passesthrough L. If L is aone-premise rule, we will say that the side formulaof L corresponds to U. If L is atwo-premise rule and U passes throughits left (right) premise, we will say that the left (right) side formula of Lcorresponds to U. An application L will be called a O-application in abranch U if U passes through L and a side formula of L corresponding

§3. ADMISSIBILITY OF STRUCTURAL RULES 17

to U belongs to V. The maximal number of V-applications of rules ina single branch of will be denoted by h°[JJ]. If V is the set of allformulas of the predicate calculus, we will omit the symbol V in deg° [A] ,N Ndeg° [A , ] , l i /V[]

. The following assertion is obvious.

LEMMA 2.8. For any formula A and any proof in 3,

deg° [A ] deg[A ] , deg° [A , ] deg[A, ],

h [PJ] ,1V[] < l1 [Y] < l [ T ].

Here deg°[ , lJ' ] denotes the maximal V-degree relative to a proof 'of the side formulas of V-applications of the cut rule in " . The followingassertion is obvious.

LEMMA 2.9. For any proofs and ' in 3, if is q+-embeddable iniY' and deg° [ T , T'] = 0, then there are no V-applications of the cut rule

in .

LEMMA 2.10. If V is q+-closed, is a proof that is q+-embeddable ina proof ' , and the principal formula of some application L of a rule in

belongs to V and evenly q+-participates in , then the V-degree of theprincipal formula of L relative to ' is greater than the V-degree of a sideformula of L relative to T'.

PROOF. Let V, , ' , and L satisfy the assumptions of the lemma.Then the principal formula of L evenly qtparticipates in jY' . Suppose thatthe principal formula of L has the form -B, dxB, or xB. By Lemma2.2, B is qtequivalent to a side formula of L. It follows from Lemma2.6 that deg° [B , '] is equal to the V-degree of the side formula of Lrelative to T'. This implies the desired conclusion. Similar arguments yieldthe conclusion when the principal formula of L has the form (B1 i B2),(B1vB2),or (B1&B2). 0

§3. Admissibility of structural rules

Let be a proof in 3, where 13 is one of the calculi KGL(2[) orIGL(2[). The variable a in an application L of the rules -> d and 3 - in

will be called the proper variable of L. The term t in an application L'of the rules d - and -> 3 in will be called the proper tern of L'.

If S is a sequent, [S] ([] ) will denote the sequent (figure) obtained byreplacing all free occurrences of the variable x by the term t in all formulasof S (in all sequents of ). The following assertion can be proved byinduction on h[T].

LEMMA 3.1. If no variable of t is bound in any sequent of and neitherx nor the variables of t are proper variables of applications of the rules -* dand - in them [] is a proof in 3 of the sequent [S], where S isthe end-sequent of '.


Let a be a list of object variables. The following assertion can be provedby induction on h[lJ], using Lemma 3.1.

LEMMA 3.2. The deduction can be transformed by renaming the propervariables of applications of the rules -> d and 3 -> into a deduction in 3with the same end-sequerct, in which the proper variables of applications of therules - d and 3 -> are not included in a and are not bound.

LEMMA 3.3. If the variables in a do not appear as bound variables inparticular applied axioms and in the last sequent of T, then can betransformed by renaming bound variables into a deduction in q3 with thesame end-sequent, in which the variables of a are not bound.

This lemma is also proved by induction on h [ T ] .

We will say that is a pure variable proof if no bound variable fromoccurs free in the sequents of .

LEMMA 3.4. If the bound variables of do not occur free in the endsequent of ?iY, then can be transformed by renaming free variables into apure variable deduction in 3 with the same end-sequent.

PROOF. First, using Lemma 3.2, transform into a deduction in whichthe bound variables are not proper variables of applications of the rules -+ dand -p; then, using Lemma 3.1, transform each subproof ' of intoa pure variable proof by induction on h['].

Renaming of bound variables of a sequent and substitution of terms forfree occurrences of variables will be called q+-transformations of the sequent.

The following transformations will be called structural transformations ofsequen ts:

(1) addition of formulas in the antecedent or succedent;(2) permutation of formulas in the antecedent or succedent;(3) contraction of repetitions in the antecedent or succedent.

We will say that a proof ' in 3 is obtained from by structuraltransformations and q+-transformations if:

(1) and ' coincide as planar rooted trees;(2) each sequent in J' can be obtained from the similarly placed se-

quent in by structural transformations and q+-transformations;

(3) the analyses of similarly placed rule applications in and ' maydiffer only in the indexes of their principal formulas;

(4) the principal formulas of similarly placed rule applications and ax-ioms in and ' are q+-equivalent;

(5) the side formulas of similarly placed applications of the cut and 2t-cut rules in and T' are q+-equivalent.

The following assertion is proved by induction on h[], using Lemma2.6.

§3. ADMISSIBILITY OF STRUCTURAL RULES 19

LEMMA 3.5. For any q+-closed set of formulas O and any proofs , ' ,

and in 3, if ' can be obtained from by structural transformationsand q+-transformations, then

(1) is q+-embeddable in iY' and ' is q+-embeddable in ,

(2) hV[T]=hV[],(3) V[,]7V[9TF](4) degV[, *] = degV[, *]

A one-premise rule L will be called strongly admissible in 3 if its conclu-sion can be obtained from any deduction in 3 of its premise by structuraltransformations and q+-transformations.

Consider the following structural rules:(1) Thinning

in the succedent

T- ,A,z"(2) Contraction (of repetitions)

in the succedent

T- ,A,z',A,e".T - e, A, e" '

(3) Permutation

in the antecedent

r, A,r'-4e'

in the antecedent

r, A, F', A, F" -z

in the succedent in the antecedent

T- ,B,A,O" r,B,A,r'-0'(4) Substitution for an object variable

ryeno variable of the termt is bound in T, it, or

the particular applied axioms[r - 0]X

LEMMA 3.6. Thinning in the succedent and in the antecedent, contractionin the antecedent, permutation in the antecedent, and substitution for an ob-ject variable are strongly admissible in both KGL(2() and IGL(2(). Thin-ning, contraction, and permutation in the succedertt are strongly admissiblein KGL(2(). Thinning in the succedent is admissible in IGL(2() in the casewhen 0, 0'I= 0.

PROOF. Let be a deduction of the premise of one of these rules. Therequired deduction is constructed by induction on h[?iY],using Lemma 3.2for thinning and Lemmas 3.1, 3.2, and 3.3 for substitution. 0


§4. Cut elimination in KGL(2() and IGL(2()

The theorem stating that O-applications of the cut rule can be eliminatedwill be proved simultaneously for KGL(2() and IGL(2(). Let 3 be oneof these calculi. The expression r will denote the list IT, E if q3 isKGL(2(),and the list IT o E if q3 is IGL(2[).

LEMMA 4.1. For any q+-closed sets of formulas O and CJ and any proofsand in 3, if is q+-embeddable in is a pure variable proof,

and ends with a O-application of the cut rule, them one can construct a proof' in 3 such that

(1) and ' have the same end-sequents;(2) is q+-embeddable in(3) h[iY'] h[] + h[iY2];(4) ' is a pure variable proof;(5) deg max{deg 1},

where £1 and 2 are respectively the subproofs of' the left and theright premise of the last application of the rule in and A is theside formula of this application.

PROOF. Let V and C be sets and and deductions that satisfy theassumptions of the lemma. The required deduction ' will be constructedby induction on h[1] + h[2]. We consider three cases.

1. There exists i (i = 1, 2) such that is an axiom. We have twosubcases:

1.1. The indexes of the principal occurrences of differ from 3- 2i, thatis, from the index of the corresponding side occurrence of the last applicationof the rule in . In this case the end-sequent of is an axiom and theindexes of its principal occurrences are determined by the indexes of theprincipal occurrences of the axiom and by i.

1.2. One of the indexes of a principal occurrence of coincides withthe index of the corresponding side occurrence of the last application of therule in . In this case the end-sequent of either coincides with theend-sequent of the subproof or is obtained from it by contraction. Inthe latter case we use Lemma 3.6 to rebuild as the required deduction(see Lemma 3.5).

In the remaining cases, in order to reduce the number of rebuildings ofdeductions, we will replace by the figure

r,1I \ 2q3

L, (1)

where the list , 11 coincides with the list A. If is empty, L is an

§4. CUT ELIMINATION IN KGL(21) AND I GL(21) 21

application of the cut rule with side formula A and the figure (1) coincideswith . If U is empty, L differs from the cut rule only by a permutationof the premises, coincides with 2' and 2 with . We will assumethat 1 has the form

X1' 1 n®A , 1f,,2-H2® H ®A X1,21

L', (2)

2. There exists i (i = 1, 2) such that ends with an applicationof a rule that either has no principal formula, or the index of its principaloccurrence is not 3 - 2i. Suppose that coincides with ; we considertwo subcases.

2.1. is a proof in KGL(2t). Here we will also consider the casewhen is a proof in I GL(2t) and one of the lists U or II1 is empty. Inthe latter case i = 2 or ends with an application of one of the rules

V & 1' & - , V -p, -p, or 2t-cut. We rebuild thefigure (1) as

1, 1 ' 2 L, 1,2 ' 2 L\T,2-4II2®0

LIB

r-4 A(3)

where the analysis of the application of L' may differ from the analysis ofthe application of L' in (2) only in the index of the principal occurrence,and the deductions

, , 2' and are obtained in accordancewith Lemma 3.6 by structural transformations and qttransformations from

, 2' and 2' respectively. The principal formulas of theapplications of L' in (2) and (3) coincide. The index of the principal oc-currence of L' in (3) is determined by that of the principal occurrence ofL' in (2), by the code of the rule L' and by i. Using Lemma 1.4, one eas-ily shows that the succedents of the end-sequents of 2' and22 coincide with the succedents of the end-sequents of

1 , , 2'and 2' respectively. If L' is an application of a quantifier rule, then therestrictions on the proper term or proper variable of L' in (3) are satisfied,since is a pure variable proof. The only exception is when i = 1 andL' is an application of the rule - in IGL(2t). In that case the propervariable of L' may occur free in i\. This collision can be avoided by usingLemma 3.2.

Thus, figure (3) is, up to a permutation of the premises in L, a proofof the sequent r - i\. While is being rebuilt as (3), the pure variableproperty may well be violated, but it can be restored by using Lemma 3.4. To


obtain the required proof ' , it remains to apply the inductive hypothesisto the premises of L' in (3).

2.2. is a proof in I GL (2t) , and neither U nor U 1 is empty. Inthis subcase i = 1 and cannot end with an application of the rule-4 V 1, - V2 , - &, - V, or -+ 2, since the index of the principal occur-rence of the last rule application in is not 1. Therefore, ends withan application of one of the rules , -, or cut, and the listsand U2 are empty. We transform (1) as follows:

1,2 2L\T,2-4Irye L'

where the analysis of the application of L' coincides with the analysis of theapplication of L' in (2), 2 is obtained from 2 by structural transforma-tions and q+-transformations (Lemma 3.6). It is clear that the new figure isa proof of the sequent r - O in I GL(2t) . If L' is a two-premise rule, itremains to apply the inductive hypothesis to the right premise of L'.

3. For each i (i = 1, 2), l ends with an application of a rule in whichthe index of the principal occurrence is 3 - 2i. We will assume that L' in(2) is an application of the rule V -, - &, -, - V, or -.In that case 2 ends with an application of a one-premise rule and has theform

2,1 r, U, u1 - u2 ® J L\

L".

Let j = 1 if L" is an application of the rule -1

, - V1

, & 1'- - , V -, or - 2 and j = 2 if L" is an application of the ruleV2, or & 2 . It is clear that

unless L" is an application of a quantifier rule. Otherwise

a 1[]t a 2 ' [U]'

where a is the proper variable of L' and t the proper term of L". Sincethe restriction on a is satisfied in L' in (2), the sequent

a

flL (4)

coincides with the sequent

r, , [fl ]i ® [-j ®O .

§4. CUT ELIMINATION IN KGL(21) AND IGL(21) 23

The substitution in (4) is admissible since is a pure variable proof. Weconsider two subcases.

3.1. is a proof in KGL(2t). Here we will also consider the case whenis a proof in I GL(2t) and L" is an application of the rule 1 or .

In the latter case the lists U and u2 are empty. We rebuild (1) as

rye '

(s)

where L* is, up to a permutation of the premises, an application of the cutrule and 2, 1' and 2 are obtained from

1 , 2, 1' 1 , j

and 2' respectively, by structural transformations and qttransformations(Lemma 3.6). Using Lemma 1.4, one easily shows that the succedents of theend-sequents of 2, 1, and 2 coincide with the succedents ofthe end-sequents of X2,1 , and , respectively. We now applythe inductive hypothesis to the premises of L* in (5). By Lemma 2.10 andthe inductive hypothesis, the result is the required proof.

3.2. is a proof in I GL(2t) , L" an application of the rule - 1 ,

V 1 , V2, & 1 , & V -p, or -+ El. Let l = 1 if L" is anapplication of & - 1 , & or V -p, and l = 2 if L" is an applicationof -+ 1, -+ V 1, - V2 , or -+ 2. In this case the list X3_1 is empty, isempty if l = 1, and U is empty if l = 2. We rebuild (1) as

P11; P13-1, s cut

rye cut,

where k = min(3 - 1, j) and s = min(l, j), and apply the inductive hy-pothesis to the right premise of the last cut.

Since all cases have been considered, the proof of Lemma 4.1 is complete.

LEMMA 4.2. For any qt closed sets of formulas V and C and any proofsP1 and P1* in q3, if P1 is q+-embeddable in P1*, is a pure variable proof,and deg° [P1 , P1*] > 0, then we can construct a proof P1' in 3 such that:

(1) P1 and ' have the same end-sequents;(2) is q+-embeddable in P* ;(3) deg°[ T , P1*] < degV[P1 , *]

(4)

(5) hv[P'] < 2h(6) P1' is a pure variable proof.

PROOF. The lemma will be proved by induction on h[P1]. The basis ofthe induction is obvious. In the inductive step the inductive hypothesis is first

applied to the premises of the last rule application in ; then, if necessary,we use Lemmas 3.3, 3.4, and 4.1. o

In what follows we set

20 = n, 2n = 2[2r ].t+ 1

LEMMA 4.3. For any natural n, m, and k the following (in)equalitieshold:

2(2k) - 2n 2n < 2n+ 1

2n 2nm k+m ' m m ' m m+1'

2m 1 <2m + m <2n,

1+2n <2, 1,+1, n m m

2n+ 1 + 2n < 2n+ 1 2 2n < 2n 2n+ 1 2n < 2n+ 1m m+1 - m+ 1 m- m+1' m+1 m+2 - m+2

m2n+1

m+1'i=0

This lemma is proved by induction on m.A sequent S will be called Qt pure if no variable occurring free in S is

bound in S or in some particular applied axiom.

LEMMA 4.4. For any q+-closed sets of formulas V and 1, any proofin KGL(Qt) or IGL(Qt) of an Qt pure sequent S containing a V-applicationof the cut rule can be rebuilt as a proof g'' such that

(1) ' is a proof of the same sequent S in the same calculus as(2) ' contains no V-applications of the cut rule;

() L

deg [ , ]L J

4by . <

deg° [ , ]

(5) g'' is q+-embeddable in .

PROOF. Using Lemmas 3.3 and 3.4, we rebuild as a pure variable proofof S. Obviously,

Hence the proofs , satisfy all the conditions of Lemma 4.2. Iteratingthe application of that lemma, we obtain the required proof (in view ofLemma 2.9). Lemma 4.3 is used to obtain inequalities (3) and (4).

THEOREM 4.1. For any q+-closed set of formulas V. any proof inKGL(Qt) or in IGL(2() of a Q(-pure sequent S can be rebuilt as a proof

' such that(1) ' is a proof of the same sequent S in the same calculus as(2) gy' contains no D-applications of the cut rule;

(3) h[7'] 2V

§5. THE CALCULI KH(2t) AND IH(2t) 25

(4) h° g"' <2 < 21[x]-1.- 10 +1 - l[am]

This theorem follows from Lemmas 2.1, 2.7, 2.8, 4.3, and 4.4.

§5. The calculi KH(Qt) and IH(Qt)

We now present a Hilbert-type system of logical inference in axiomatictheories in the form of the calculi KH(2() and IH(2[), where 2(, the setof applied axioms of the theory under consideration, satisfies the conditionsgiven in § 1. The logical axioms of the calculus are all the instances of thefollowing axiom schemata:

(la) (AD(B3A)),(ib) ((A3B)D((A3(B3C))3(A3C))),(2a) A 3(B D(A & B)),(2b) ((A & B) 3A) ,

(2c) ((A & B) 3B) ,

(3a) (A3(AVB)),(3b) (B3(AvB)),(3c) ((ADC)3((B3C)3((AvB)3C))),(4a) ((A3B)3((A3-iB)3--iA)),(4b) (-i-iA3A),(5a) (Vx(C 3 A) 3 (C b'xA)) [x is not free in C],(5b) (VxA 3 [the term t is free for x in A],(6a) (Vx(A C) j (xA 3 C)) [x is not free in C],(6b) ([A] xA) [the term t is free for x in A].The applied axioms of KH(2t) are the formulas in 2(. The rules of infer-

ence of KH(2t) are modus ponens

A; AFB

and the rule of generalizationB

m.p.

,xA Gen.

The calculus IH(Qt) differs from KH(2() only in that the axiom schema(4b) is replaced by

(-iA3(AB)). (4b')In what follows, if 2[ is empty, KH(2t) and IH(2I) will be denoted by

KH and IH, respectively. We will show below that KH and IH areequivalent to KGL and IGL, respectively. Proofs in KH(2[) and IH(2I)will be written as linear sequences of formulas with analyses of axioms andapplications of inference rules. The analysis of a logical axiom will indicatethe schema of which the axiom is an instance; the analysis of an applied axiomwill specify the index of the subset in the preassigned finite partition of 2t.Finally, the analysis of an application of a rule will point out the code of the


rule and the indexes of the premises. It is assumed that the occurrences offormulas in a proof are indexed from left to right, and that the indexes of thepremises of an application of a rule are smaller than that of the conclusion.

Let tR be one of the calculi KH(2[) or IH(2i) and a proof in R.The number of occurrences of formulas in will be called the length ofand denoted by 1H[2] . In what follows the expression

R

will mean that one can construct a proof of A in R of length at most 1.Let D be a set of formulas. Let IH denote the number of different

formulas of O that occur in ,and 1[92J] (l,[]; 1V[]) the numberof different formulas occurring in that belong to O and are logical axioms(resp., right premises of applications of modus ponens; obtained by the rule ofgeneralization). When O is the set of all formulas of the predicate calculus,we will omit the symbol O in the expressions l[],, m and1V[]

. It is obvious that

LEMMA 5.1. If contains no repetitions of formulas, then m is equalto the number of formulas that occur in and are obtained by applicationsof modus ponens with right premises in V.

This lemma is proved by induction onWe let KGL(2t)+ denote the calculus obtained by adding the following

variants of thinning in the antecedent and succedent to KGL(2()

r-4oA,T-,0' T-+O,A.

IGL(2[)+ will denote the calculus obtained by adding only the above vari-ant of the rule of thinning in the antecedent to IGL(2[).

Let p be one of the calculi KGL(2[) or IGL(Qt) and ' a proof inLet h[r'] denote the number of different applications of rules other thanthinning in the longest branch of ' , and the number of differentsequents occurring in '.

LEMMA 5.2. Any proof gy in KH(2[) (in IH(2I)) of a formula A can betransformed into a proof ' in KGL(2()+ (in IGL(2()+) of the sequent - Asuch that

2 IH[] + 7, 12. 1H[]'PROOF. We will construct the required proof by inverse induction on

1H[]' writing it as a linear sequence of formulas. If ends with a logicalaxiom, we append to the previously constructed deduction a proof without

cut of the sequent --> A. If ends with an applied axiom, we add theinference

A -- AA

2t-cut.

If ends with an application of the rule of generalization, we add theinference

-9 V .

Finally, if ends with an application of modus ponens, we add the deduc-tion with the following tree form:

-' A thinning°. K cutB A -.> B--> A ,

--> Bcut,

where K denotes the expression-AB

thinning°

thinning ;

The symbol ° attached to the applications of thinning in the succedent oroccurrences of B means that if is a proof in I H(2t) , the correspondingrule applications and occurrences of formulas must be deleted. Thus, thelemma is proved.

A set of formulas V will be called qt essential if the following conditionshold:

(1) V is qtclosed;(2) for any formulas A and B, if B is included in A and B E V, then

AEV.LEMMA 5.3. For any q+-essential set of formulas V, any proof of a

formula A in KH(Qt) (in IH(2[)) can be transformed into a proof ` ofthe sequent -9 A in KGL(Qf) (in IGL(2i)) such that:

(1) the side formulas of cuts in ` are q+-equivalent to the formulas of

(2) the side formulas of Q(-cuts in * are q+-equivalent to specific axiomsin ,

(3)[*]<3V[]

(4)[g7*] 21H[] + 7,

(5) <2[]+9+l[] °< 2l,[]+9.PROOF. After deleting repetitions of formulas in , we use Lemma 5.2 to

first construct a proof ' of the sequent --' A in the calculus with the rule ofthinning. Then, rearranging ' in tree form, we use Lemma 3.5 to eliminatethe applications of thinning. Denote the resulting proof of the sequent -> Aby . It is clear that satisfies conditions (1) and (2). Inequalities (3)and (4) follow from Lemma 3.5 and inequality (5) from Lemma 5.1. o

If I' is a list of formulas, we let Ti (I'2) denote the list of formulasoccurring in I' in evenly (oddly) indexed positions. Obviously,

Ir' I + ICI = iri , Ir2i ir2i + 1.

We define formulas {t}& and {I'} by induction on TI. If r is empty,

{l'}(P3P),where p is a fixed closed formula. If II = 1,

{r}& a {r} = r.

If II > 1,

{} ({Ti}&{T2}) {} ({'} V {T}).In what follows we will use the following notation:

{T - L} ({} {} IT -9 L + ILI.

LEMMA 5.4. There are natural numbers co and cl such that for any listsof formulas t, E and II, if III < 1, then

IH(2t) I- {r, ,}&IH(2O {II} {I', II, }v'IH(Qt) ({T}&{fl}) 3 {f, 11}&'IH(2I) I-co[1os2(Irl+i)]+i {II, r} ({} v {}),

IH(Qt) {I'} {I', II} ,

IH(2O {II, r}& {r}&.

The proof proceeds by induction on II', II,I

and I1.Let q3 be one of the calculi KGL(QO or IGL(2[), a proof in The

maximal value of If --. 0I for all sequents I' -- O in will be denoted byII

LEMMA 5.5. For any proof of a sequent S in

The proof is by induction on the construction of .

LEMMA 5.6.. There is a natural number co such that any proof inKGL(2()+ (in IGL(2t)+) of a sequent S can be transformed into a proof

' in KH(Qt) (in IH(Q()) of the formula {S} which satisfies the inequality

co ' 1 [lOg2 II].

PROOF. Let us rewrite as a linear sequence of sequents

SI [,] ,

where SI[ ] is S. Consider the following sequence of formulas:

{S}, {2}' ... ' {5l[Pir]}(1)

In order to convert this sequence into the required deduction of {S},we introduce additional formulas in (1) before each formula {S} (1 < i <l [ciJ]) . For any i, the choice of these additional formulas depends on theanalysis of the axiom or inference rule attached to S, in . The number ofadded formulas will not exceed c[log2 II] , where c is some natural number.We consider two cases.

1. is a proof in KGL(2)+. This case has two subcases:1.1. Si is either an axiom or is obtained from preceding sequents by a

rule of KGL(2t). This application of an axiom or an inference rule can bepresented as a figure

'1 1' ' 2' ''' 1' k k' ' 2'

(2)where k = 0, 1, 2;

I l , " 2 I < 2; the list1

(E2) is empty if S, is notan axiom and is not obtained by a rule introducing a logical symbol in theantecedent (succedent)-otherwise '~

1(E2) coincides with the principal for-

mula; the sequent F', E1

, F --> coincides with S, . Denote thelist F', '~

1, I" by r and the list 0' , 2' 0" by O . We introduce the fol-

lowing notation:

s max(IS I , Ir, 1--- nl , of , ... , Ir k - nk I)It is obvious that

IrI+1<sl, Iol+1<s,,There are two subcases.1.1.1. Si is either an axiom or is obtained by a rule other than -' V and

-. In that case we can find c2 such that

IH(2.)Hc2

({---> II1} ... IIk} 2})...). (3)

Hence, by Lemma 5.4, it follows that for suitable c3

KH(2t) Hc3[log2s;] ({T, 1 --> U1, 0} 3 ... 3 ({1', k -''Tk, O} {S,}) ... ).

We now insert the resulting proof in (1) before {S,}. To construct thelist of additional formulas we need only apply modus ponens k times and,if necessary, contract repetitions of formulas.

1.1.2. Si is obtained by the rule V or -'. Let a denote the propervariable of the application of the rule. Then we can find c4 such that

IH(2t)F-

(Va{1 -- H1 } (4)

Hence, by Lemma 5.4, it follows that for suitable c5

KH(2t) Hc5[log2 s,] (Va{T, 1 --9 U1 , O} 3 {S}).

We now insert the resulting proof in (1) before {S,}. To complete the listof additional formulas, we add b'a{I', El -- II1, 0} and apply the rules ofgeneralization and modus ponens.

1.2. S, is obtained by thinning. We write the corresponding applicationof the rule as a figure

r - o

E 1 , 2I = 1 and the sequent 0 , coincides with S,. NowLemma 54 implies that for suitable c6

IH(2t) Hc6[log2 IS;I]({r - 0} {S,}).

We now insert the resulting proof in (1) before {S,} and apply modusponens.

2. is a proof in I GL(2t)+ . In that case the list of additional formulasis constructed along the same lines as in case 1. The differences are thatinstead of figure (2) we consider the folowing figure:

r r' o

where 1 I < 1 and II < 1, and that instead of (3) and (4) the followingassertions are used:

IH(Qt) F- ({ - } ({ "k ° { -4 L})...),

IH(2() FC4 (Va{1 - II1 0 0} j {- O})

Since all cases have been considered, the proof of Lemma 5.6 is complete.

THEOREM 5.1. A formula A is deducible in KH(2I) (in IH(2()) if andonly if the sequent -> A is deducible in KGL(2() (in IGL(2() ).

The theorem follows from Lemmas 5.3 and 5.6.

CHAPTER II

Systems of Term Equations with Substitutions

In this chapter, given a system IT of term equations with substitutions,we will construct an extension tree for IT by adding new term equalities andintroducing new unknowns. A fairly simple algorithm, deciding whether thesystems assigned to the leaves of the tree are solvable, will be described. Theextension tree of IT will be constructed in such a way that IT has a solutionif and only if some system assigned to a leaf of the tree has a solution. Theextension tree can be used both to prove the undecidability of IT and to findsolutions of IT with small complexity characteristics.

On the one hand, in this chapter we will prove that there is no algorithmto decide whether a system of term equations with substitutions is solvable ifthe terms contain n-place function symbols with n> 1. On the other hand,we will construct an algorithm to decide whether a system has a solutionin which the number of occurrences of substituted variables does not exceedsome prescribed number. For such solutions we will also obtain upper boundson the lengths of periodicities of the terms and on their heights.

The main results of Chapter II were published in [28].

§6. Systems of term equations with substitutions.Main lemmas

A system of equations and constraints in terms with substitutions (CTS-system) with unknowns (metavariables for terms) t1, ... , t1, , function sym-bols f1, ... , fm and object variables x1, ... , xp consists of a list of equal-ities

T1 = 1,..., TN= N'a list of equalities

(1)

S1=U'R1,...,SM=UMRM; (2)yt 1 M

and a list of expressions

gl Vl , ... Sk k' (3)

1Here T ... T ® ... ® S ... S U ... UM R ... R1' ' N' 1' ' N' 1' ' M' ' ' ' 1' ' M'V1, ... , VK are terms constructed from unknowns t 1 , ... , to and variables

31

32 II. SYSTEMS OF TERM EQUATIONS WITH SUBSTITUTIONS

x1, ... , xp using function symbols f1, ... , f,,n (this list may also contain0-place symbols); variables y1, ... , yM belong to the list x1, ... , xp ; sym-bols g1, ..., gK belong to the list f1, ... , f,, x1, ... , xp ; N > 1, M>0, K>0.

Let U be a term that does not contain the unknowns t 1 , ... , t,,. Theexpression UyR denotes the result of substitution of a term R for all occur-rences of a variable y in U.

Let g be a function symbol or an object variable. We will say that theexpression g U is true if g does not occur in U.

A solution of a CTS-system (1)-(3) is a list of terms 0, ... , Bn con-taining no unknowns, such that, after the unknowns t 1 , ... , t, have beenreplaced by 01, ... , 0,,, respectively, in equalities (1)-(2) and in expressions(3) and the substitutions in (2) have been performed, equalities (1)-(2) aretrue graphic equalities and (3) true expressions.

If list (3) is empty, the CTS-system (1)-(3) will be called a TS system;if list (2) is empty, a CT system; and if both (2) and (3) are empty, a Tsystem, that is, a system of term equations. The terms R1, ... , RM of (1)-(3) will be called substituting terms, and the unknowns occurring in them, thesubstituting unknowns. List (1) will be called the list of term equalities, list(2) the list of equalities with substitutions, and list (3) the list of constraints.

Let P be a term, y an object variable. We define the height h[P] ofP by induction on the construction of P : (1) if P is an unknown, objectvariable, or 0-place function symbol, then h[P] = 0; (2) if P has the formg (P1 , ... , PI) , where g is an 1-place function symbol and 1> 1 , then

h[P] = 1 + max (h[PI])1<i<1

The number of occurrences of y in P will be called the y-length of P,denoted by l y[P] . Let a be a solution of a CTS-system (1)-(3). The max-imal length of terms occurring in a will be called the height of a, denotedby h [Q] . By the substitution width of a solution we mean the maximum y.-length of the results of replacing unknowns in U` by their values in ci forall i (1 < i < M). By the height of a system (1)-(3) we mean the maximumheight of a term T1 , ... , TN, ®1 , ... , e, S1 , ... , SM, U', ... , UM,R1,...,RM.

Let a be a variable in the list x1, ... , xp . By a CTS system with unknowns

t1, ... , t,, and a-unknowns v1, ... , vq we mean any CTS-system (1)-(3)with unknowns t 1 , ... , t,,, v 1 , ... , v9 such that, for all i (1 < i < M)and j (1 < j <N), the following conditions hold:

(1) U` is one of the a-unknowns v1, ... , v9 ;(2) yi coincides with a ;(3) the a-unknowns and a itself do not occur in Si or in R.;(4) if an a-unknown or a occurs in one of the terms T or 4 , then T

§6. MAIN LEMMAS 33

and contain no 0-place function symbols, unknowns t 1 , ..., toor object variables other than a and the a-unknowns.

Let f be a CTS-system with unknowns t1, ... , to and a-unknowns v1,... , vq , defined by lists of equalities (1)-(2) and a list of constraints (3),and let a be a solution. By the principal a-unknowns of r we mean thea-unknowns that occur in the right-hand sides of equalities in (2). We leth[f] denote the height of f and lar[Q] the largest a-length of the values of

the principal a-unknowns in a. It is obvious that lr[Q] is the substitutionawidth of ci.

By f-terms we mean: (1) the unknowns t1, ... , to ; (2) the a-unknownsv', ... , v q

; (3) the object variable a ; (4) the subterms of T1, ... , TN,®1 , ... , ®N , S1, ... , SM, R1, ... , R, V1 , ... , VK . By substituting (-terms we mean the substituting terms of f and a ; by f-atoms we meanall f-terms and expressions of the form waR , where w is an a-variableand R a substituting f-term; the principal f-atoms will be the principal a-unknowns and f-atoms that occur on the left or right of equalities in (1)-(2).Let A be a f-atom. We let Aa denote the term obtained by replacing allunknowns and a-unknowns in A by their values in a and performing thesubstitution if A has the form waR .

Let A 1 and A2 be f-atoms. We write A 1 =r A2, if A 1 coincides withA2 or if f includes one of the equalities A 1 = A2 or A2 = A 1 . We writeA 1 <r A2, if A2 is a f-term and A 1 a proper subterm of A2. We let rdenote the relation which is the closure of =r with respect to the followingrules:

Al =r A2 Al r A2 R r a w r aA 1 r A2 '

A 1 Dr A2 ; A2 r AAl Dr A3

W r g(w1

W r g(w 1, . .

A2 r Al , waR ^'r w , waR R ,

Q!)NPj _ r Qj

, w1); waR ^-'r g(Q1 ,

a r Q

.,wJ,...,w1) u ^'r g(u1, ... , u3

waR ^-'r ua Q

...,Q3.,...,Qi)

... , u1) ; wa R ua Q

where A1, A2, A3 are arbitrary f-atoms; g (w 1 , ... , w'), g (u 1 , ... , ul) ,

g (P1 , ... , P,), g (Q 1 , ... , QI) are f-terms; g is any l-place function sym-bol from the list f1, ... , fm; 1 < j < l ; w, w1, ... , wl , u, u1 , ... , ulare arbitrary a-unknowns of f; R and Q are substituting r-terms.

Let <r denote the closure of <r with respect to the following rules:

A 1 <r A2 A 1 r A2 ; A2 r A3 u r wA 1 r A2 ' A 1 r A3 uaR r waR '

A 1 "r A2 ; A2 r A3 A 1 r A2 ; A2 r A3Al r A3 A 1 r A3

where u, w are a-unknowns and R a substituting F-term.The following assertion is obvious.

LEMMA 6.1. The following conditions hold for any solution a of a CTSsystem F and any F-atoms A and B :

(1) if A .r B, then Aa coincides with Ba ;(2) if A r B, then Aa is a proper subterm of Ba.

For F-atoms A and B, A -r B means that either A r B or A r B.A F-atom A will be called a basic F-atom in F if there is a principal F-atomB such that A r B .

Let g belong to the list x1, ... , xp , Jj, ... , f,. We will say that a F-atom A begins with g if we can construct a F-term P beginning with gsuch that one of the following conditions holds:

(a) A P ;

(b) g is not a and there are an a-unknown v and a substituting F-termR such that A r vaR and v r P.

A CTS-system F is called admissible if none of the following conditionsholds:

(1) there is a F-atom A such that A -r A ;(2) there are F-atoms A and B and symbols g and h from the list

x1, ... , xp , f1, ... , f,n such that g is not h, A begins with g, Bbegins with h, and A r B ;

(3) there are a term Q, a F-atom A, and a symbol g from the listx1, ... , xp , f1, ... , f,,n such that gEQ belongs to the list of con-straints, A begins with g, and A r Q.

Lemma 6.1 implies the following assertion.

LEMMA 6.2. If a CTS-system F has a solution, then it is admissible.

LEMMA 6.3. For any F-atoms A 1 and A2, if A 1 r A2, then(1) if there is a natural number i, 1 <i < 2, such that Al is a F-term in

which a or an a-unknown occurs, then one of the following conditionshold:

(a) A3_, coincides with A.,(b) A3_1 is a F-term in which there occur no 0-place function symbols,

unknowns t 1, ... , t,, or object variables other than a and the a-unknowns,

(c) A3_1 has the form vaa, where v is an a-unknown;(2) if there is a natural number i, 1 <i <2, such that Al has the form

vaR , where v is an a-unknown and R is a substituting term of F,then one of the following conditions holds:

(a) A3_, is a F-term that does not contain a-unknowns or a,(b) there are an a-unknown w and a substituting term R* of F such

that A3_1 coincides with waR* ;

§6. MAIN LEMMAS 35

(3) there is no i, 1 < i < 2, such that Al is a r-term containing anunknown, a 0 -place function symbol, or a variable other than the a-unknowns and a, and A3_1 has the form vaa, where v is an a-unknown.

This lemma is proved by induction on the length of a deduction of A 1 rA2 in accordance with the rules for r .

Let A and B be r-atoms, w', w2, ... , w r (r > 0) a list of a-unknowns.We will say that w', w2, ... , w r connects A with B in r if there aresubstituting r-terms P1, Q1, P2, Q2, ... , Pr , Qr such that

1 1 2 rA r wa P1 , wa Q1 r wa P2 , ... , wa Qr r B.

The empty list connects A with B in r if A r B. The followingassertion is obtained from Lemma 6.1.

LEMMA 6.4. Let a be a solution of a CTS system F, w 1, ... , wr a listof a-unknowns, and A and B r-atoms. If w1, ... , wr connects A withB in r and the terms ... , w do not contain the variable a, then Aacoincides with Ba.

We will say that an a-unknown v is free in r if it is not true that a r V.The expression A r B will mean that it is possible to construct a list ofa-unknowns, with all members free in F, which connects A with B in F.It is clear that A r B implies A tir B. We will say that a r-atom A isdistinguished in r if A is a r-term that does not include a-unknowns, ora principal r-atom of the form waR , where R is a substituting term ofr and a r w. A r-term A will be called an a-term of r if A has theform g (w 1, ... , w,), where g is an 1-place function symbol, l > 1, andw

1, ... , wl are pairwise distinct a-unknowns.

A CTS-system r with a-unknowns will be called correct if the followingconditions hold:

(1) r is admissible;(2) for any basic r-atom A there is a r-atom B such that B is distin-

guished in r and A r B ;(3) for any a-unknown v we can construct a principal a-unknown w

such that v r w ;(4) for any principal a-unknown w one of the following conditions

holds: (i) w r a ; (ii) there is no r-atom A such that w -<r A ;(5) for any basic r-atom A and any 1-place function symbol g, l > 1,

if A begins with g, then we can construct a r-atom B such that Bis distinguished in F, B begins with g, B is not an unknown, andB can be connected with A in r by a list of a-unknowns beginningwith g ;

(6) for any a-unknown v and r-atom A, if v r A, then A is ana-term of F, or A r a, or A is v or va a ;

(7) a-terms of r that have common a-unknowns coincide.Let A and B be r-atoms and i a natural number. We will say that B is

an i-argument of A if we can construct a r-term g(P1, ... , PI) such that1 < i < 1 and one of the following conditions holds:

(a) A tir g(P1, ... , PI) and B ^r p';(b) terms P', ... , PI are a-unknowns and there are an a-unknown v

and a substituting r-term R such that A r vaR , v r g(P1,... ,PI ),and B tir PaR .

It is clear that if B is an i-argument of A, then B - A. If B1 andB2 are r-atoms, each being an i-argument of A, and r is an admissibleCTS-system, then obviously B1 B2.

LEMMA 6.5. For any r-atoms A and B, if I' is a correct CTS systemand A -< B, then we can construct a natural number i, an 1-place functionsymbol g, 1 < i < 1, and a F-atom C, such that B begins with g, C isan i-argument of B, and A , C.

This lemma is proved, using Lemma 6.3, by induction on the length ofa deduction of the condition A - 1, j > 1, A bean r-atom A, and v and w be a-unknowns. If r is a correct CTS-systemand A is both an i-argument of v and a j-argument of w, then one of thefollowing conditions holds:

(a) A ^r a ;(b) i = j and v is w.

We will say that an a-unknown v is minimal in F, if v is free in F andthere is no r-atom A such that A -< v. Suppose that an a-unknown vis free in r and is not minimal in F. Lemma 6.5 implies that there is an1-place function symbol g, 1 > 1, such that v begins with g. It followsfrom Lemmas 6.3, 6.5, and 6.6 that, for any i, there is a uniquely determineda-unknown which is an i-argument of v ; in what follows we will denote itby Argl[v] .

The following assertion is easily proved using Lemma 6.5.

LEMMA 6.7. Let A and B be r-atoms and w1, ... , wr (r> 0) be a-unknowns that are free in F. If none of the a-unknowns w1, ... , wr isminimal in F, the list w1, ... , wr connects A with B in I', and I' is acorrect CTS system, then there is an 1-place function symbol g, 1 > 1, suchthat w 1, ... , wr , A and B begin with g, and for all i (1 < i < 1), the listArg1 [w 1 ] , ... , Argl [wr] connects the i-argument of A with the i-argumentof B in F.

§6. MAIN LEMMAS 37

LEMMA 6.8. For any r-atoms A 1 and A2, if r is a correct CTS systemand A 1 r A2, then the following conditions hold:

(1) If there is a natural number i, 1 < i < 2, such that Al has the formvaR1, where Rl is a substituting term of F, A3_, is a r-term that does notcontain the a-unknowns or a, then one of the following conditions holds:

(1.1) vaRl is a principal r-atom;(1.2) v` _r a ;(1.3) there are a natural number j, an a-unknown w, and a r-term T

such that v` is the j-argument of w, waRi r T, and T (as aterm) begins with a function symbol;

(1.4) there are a natural number j, a substituting term Q of r, and a-unknowns w 1, w2, and w 3 such that v' is the f -argument of w 1,

2 3 1w as the f-argument of w , wa Ri r wa Q , and w 2 r a

(2) If Al and A2 have the forms vaR1 and vaR2 , respectively, whereR1 and R2 are substituting terms of r, then one of the following conditionsholds:

(2.1) va R 1 coincides with vaR2 ;(2.2) there is a r-term T that does not contain a-unknowns and a such

that vaR 1 r T and vaR2 r T ;

(2.3) there are a natural number j and a-unknowns w 1 and w 2 suchthat v` is the j-argument of w1, v2 is the j-argument of w2, andwa R 1 _r wa R2 .

(3) If one of the I'-atoms Al and AZ is not basic, then one of the followingconditions holds:

(3.1) Al coincides with AZ;(3.2) there are an a-unknown v and a substituting term R of t such that

v r a and either Al or AZ is vaR, while the other is either R orhas the form wQ R , where w r a.

PROOF. Let us replace (2.2) in Lemma 6.8 by the following condition:(2.2') Both vaR1 and vaR2 satisfy one of the conditions (1.1), (1.2),

(1.3), or (1.4).The assertion thus obtained can be proved, using Lemma 6.3, by induction

on the length of a deduction of the condition A1 r AZ in accordance withthe rules for r . To complete the proof of Lemma 6.8 we need only pointout that (2.2) follows from (2.2' ).

By the multiplicity of a principal a-unknown v in I' we mean the numberof principal t-atoms of the form vaR , where R is a substituting term ofF. The maximal multiplicity of a principal a-unknown in I' will be calledthe multiplicity of I' .

A CTS-system I' with a-unknowns will be called normal if the a-unknowns


and a do not occur in the list of term equalities and all the a-unknowns areprincipal. Obviously, an admissible normal CTS-system is correct.

LEMMA E).I. For any CTS-system t with unknowns t1, ... , to we can con-struct anormal CTS-system E with unknowns ti , ..., to and a-unknownsv , ... , vq (q > 0) such that

(1) the multiplicity of E is two;(2) the number of equalities with substitutions in t is q ;(3) any solution a of t can be transformed, by adding the values of the a-

unknowns, into a solution a' of E such that lQ[Q'] is the substitutionwidth of v ;

(4) any solution a' of E can be transformed, by deleting the values of thea-unknowns, into a solution a of t such that l[ a'] is the substitutionwidth of a.

PROOF. Let t be given by lists of equalities (1)-(2) and a list of constraints(3). We add a new object variable a to the list of object variables of tand introduce a-variables v1, ... , Define the system E by the list ofequalities (1), the list of equalities

1 1S1 =vaR1, U1=vQy1,

,M MMSM=va RM, U =vQ yM,and the list of constraints (3) enlarged by adding the constraints

1 My1 Ev ,... , yME21

It is obvious that is the required normal system. D

Let r be a correct CTS-system with a-unknowns. A list vl , it , v2, i2,... , ir_1 ' yr , where v1, v2, ... , yr are a-unknowns of r, will be called achain in r if r = 1 or if r > 1 and v2 is an i 1-argument of v1, v3 isan i2-argument of v2, ... , yr is an ir_ 1-argument of vr_ 1. The number rwill be called the length of the chain, v 1 the beginning and yr the end ofthe chain. It is obvious that the beginning of a chain coincides with its endif and only if it is of length 1. A chain in r will be called free if all thea-unknowns occurring in it are free in F. We let str [v ] denote the maximallength of chains in r with beginning v, and f[r] the maximal length of freechains in F. A chain v1, i1, v2, l2 , lr ' vr+ 1 ' ' lr+s-1 ' yr+sin r will be called the union of the chains v1, i 1 , v2, i2, ... , ir_ 1 , yr and

vr' lr' yr+1 ' lr+s-1 ' yr+sLEMMA 6.10. For any a-unknown v and chains SZ

1and SZ2 , if r is a

correct CTS system, the length of SZ1 is at most that of SZ2 , and v is the end ofboth SZ1 and SZ2 , then either v ^rr a or there is a chain SZ3 whose beginningcoincides with that of SZ2 , the end of SZ3 coincides with the beginning of SZ1,and SZ2 is the union of SZ3 and SZ1 .

§6. MAIN LEMMAS 39

This lemma is proved by induction on the length of SZ 1 , using Lemmas6.5 and 6.6.

Let a[F] denote the number of a-unknowns w such that w r a. Let vbe an a-unknown. Let YJ2r[v] denote the set of a-unknowns w such thatw r v and w is minimal in F, andr the set of a-unknowns that areminimal in F. We now define ar[v ] by induction on str[v ] :

0, if v is minimal in I',1, if v r a,ar[v] _

1

> ar[wl] , if v ^r g(w1, ... , w1).i=1

LEMMA 6.11. For any solution a of a correct CTS system r and any a-variable v we have

la[V]=44V]+ (4)WE9J1r[V]

PROOF. We will prove this lemma by induction on str[v ] . Suppose firstthat str[v ] = 1. If v is minimal in F, then ar[v ] = 0, 9Y1r[v ] includesonly v, and hence (4) holds. If v r a, then la [v

Q] =ar[v ] = 1. Nowlet str[v ] > 0. Then, by Lemma 6.5, v r g (w 1 , ... , w1) . Note thatStr[v ] > max1 <i<1 Str[wi ] . Hence

1 1 1

la[v]a => la[w ] = 2 la[wn]+> ar[wJ] = ar[v]+ la[ws].i=1 i=1 WEW1r[w;] i=1 WE9J1r[V]

Here we have used the fact that, by Lemma 6.10, the setsr [w 1 ] , ... ,r[w!] are pairwise disjoint and their union is r[v] . D

Denote the set of principal a-unknowns of a CTS-system F by r , andby IflI the cardinality ofr . Let a be a solution of F. We introduce thefollowing notation:

wE9Jtr

LEMMA E.12. For any solution a of a correct CTS-system F,

vE9r

PROOF. Let a be a solution of F. Using Lemmas 6.5 and 6.10, one easilyproves that

a[I'] _ ar[v],vE9r

vE`nr wEfitr[v]

Lemma 6.12 then follows from this and from Lemma 6.11.

§7. Extension tree of aCTS-system

Let F and E be CTS-systems with a-unknowns. We will say that E isan a-extension of F, if the following conditions hold:

(1) F and E have the same lists of unknowns;(2) all the a-unknowns of I' are a-unknowns of E ;(3) I' and E have the same lists of equalities with substitutions and lists

of constraints;(4) the list of term equalities of E is either the same as that of I' or

is obtained from the latter by adding term equalities containing a-unknowns or a.

LEMMA 7.1. If E is an a-extension of F, then for all I'-atoms A and Bwe have

(1) if A r B , then A B ;(2) if A <r B, then A -E B.

This lemma is proved by induction on the length of a deduction of theconditions A r B and A -r B in accordance with the rules for r andr

Let r be a correct CTS-system with a-unknowns, SZ a sublist of the listof function symbols of F'. We will say that an a-unknown v is a-hangingin r relative to SZ if v is minimal in r and there are an l -place functionsymbol g, l > 1 , and a r-atom A such that g occurs in SZ , A begins withg, and v r A. We will say that v is a-hanging in r if v is a-hangingin r relative to the list of all function symbols of F'. We will say that ana-unknown v is /3-hanging in r if v is minimal in r and at least one ofthe following conditions holds:

(1) there are r-atoms A and B such that A and B begin with differentsymbols, v r A and v r B ;

(2) there are a r-term Q, r-atoms A and B, and a function symbolor object variable g such that A begins with g, A tir v, v tir B,B r Q, and g Q belongs to the list of constraints of F.

We will say that v is /3-hanging in r relative to SZ if v is both /3-hangingin r and a-hanging in r relative to SZ . Let y [F] denote the number ofr-atoms of the form waR distinguished in F, and ic[I'] the multiplicity ofF.

Suppose that a CTS-system r does not have a-unknowns f3-hanging inr relative to SZ . Let v 1, ... , yr (r > 0) be a list without repetitions of allthe a-unknowns that are a-hanging in r relative to SZ , and let A1, ... , Arbe r-atoms such that for all i (1 1, and gl occurs in S2 . We introduce new

§7. EXTENSION TREE OF A CTS-SYSTEM 41

1 1 rrpairuvise distinct a-unknowns w

1 , ... , wl , ... , w 1 , ... , w1, and add the

rfollowing equalities to the list of term equalities of r :

1 1 r rv1=g1(w1,...,wl), ..., vr=gr(w1,...,wl) (1)I r

Denote the resulting a-extension of r by irr . We will say that irr isobtained by prolongation of all a-unknowns that are a-hanging in r relativeto SZ . Henceforth, if SZ is the list of all function symbols of F, we will omitthe subscript SZ in the expression irr . If a is a solution of irr , we leto denote the restriction of a to the unknowns and a-unknowns of F.

LEMMA 7.2. For any correct CTS-system F, if I' does not have any a-unknowns /3-hanging in t relative to S2, then

(1) nI' is a correct CTS-system;(2) an a-unknown of icI' is minimal in icI' if and only if it is either

minimal but not an a-hanging a-unknown of I' relative to S2, or itis not an a-unknown of F;

(3) fl[irF] <_ fl[F] + 1, y[r'],a a solution of icI' , then Qr is a solution of t and ar[ar] -an a-unknown w of nI' is a-hanging in it relative to S2, then

there is an a-unknown v such that v is not an a-unknown of t andv nr w ;n

(6) if an a-unknown v of I' is free in F, then v is free in ni' .

PROOF. Let t be a correct CTS-system without a-unknowns /3-hangingrelative to S2. Let E be the CTS-system obtained by prolongation of allthe a-unknowns a-hanging in I' relative to S2 . Let vl , ... , yr be a listwithout repetitions of all the a-unknowns a-hanging in t relative to S2and w i , ... , w, ... , wi , ... , w a list without repetitions of all the newa-unknowns of E . Let (1) be the list of new term equalities of E .

Let Al and AZ be E-atoms. The following proposition can be proved,using Lemma 6.3, by induction on the length of a deduction of the conditionAl AZ in accordance with the rules for E .

SUBLEMMA I. If Ai AZ , then we can find substituting terms R and Qof t and natural numbers i, j, p, s, and q, where 1 < i < 2, 1 < j < r,1 < p < r, 1 < s < 1 , 1 < q < lp,such that one of the following conditionsholds:

(a) Al is g( w,, ... , wl) and A3_, is either v or vjas ;J

(b) Al is ws aR , A3_1 is a r-atom and the s-argument of v . aR in F ;

(c) A. is A3-i is wqPaQ g is g' s= q, and v.1aR ^- vpaQ'c s P .1 r

(d) Al is wsaR, A3_, is wpaQ, and there is a F-atom B which is both9

the s-argument of vjaR and the q-argument of vp aQ in r ;

(e) Al is w, A3_, is wsaa(f) Ac . and A3-i are r-atoms and Ar . A3-ir(g) A, coincides with A3_,.

The following proposition can be proved by induction on the length of adeduction of Al -< A2.

SUBLEMMA II. If Al -< A2, one of the following conditions holds:

(a) there are r-atoms B1 and B2 such that B1 -< B2, Al B1, andA2 B2 ;

(b) there are a F-atom B, a substituting r-term Q, and natural numbersj and s , 1 < j < r , 1 < s < l . , such that A 1 ws a Q , B A2,VjaQ

Sublemmas I and II imply the following propositions.

SusLEivtMA III. For any t-atoms A and B, if A E B, then A r B,and if A -E B, then A r B.

SUBLEMMA IV. If a >.-atom A begins with a symbol f, then one of thefollowing conditions holds:

(a) there is a t-atom B that begins with f in t and B A ;

(b) there are a substituting I'-term Q and a natural number j, 1 < j <r, such that f coincides with g and A vjaQ .

SUBLEMMA V. For any >.-atom A, if A -< A, then there is a F-atom Bsuch that B A and B -< B.

Since r is admissible and has no a-unknowns f3-hanging relative to SZ ,

Sublemmas III, IV, V imply that is admissible. The following propositioncan be proved using Lemma 6.8.

SUBLEMMA VI. For any natural number j, 1 < j < r, there exists a F'-atom A such that A3 is a basic r-atom in r', A begins with g3 in F, andV3 tir A.

Sublemmas II and III imply the following proposition.

SUBLEMMA VII. For any basic i-atom A, one of the following conditionsholds:

(a) there is a basic r-atom B such that B A ;

(b) there are a substituting F-term Q and natural numbers j and s,1 < j < r, 1 < s < such that Vja Q is a basic r-atom andAwsaQ.

Sublemma III and Lemma 7.1 imply the following proposition.

SUBLEMMA VIII. For any i-atom A, A is distinguished in if and onlyif A is a distinguished r-atom in F.

The correctness of follows from the correctness of F, the admissibilityof E , Lemmas 6.3, 6.7, 7.1, and Sublemmas I, II, III, IV, VI, VII, VIII. Theother conditions of Lemma 7.2 follow from Lemma 7.1, and Sublemmas III,IV and VIII. This completes the proof of Lemma 7.2. D

Let r be a correct CTS-system and v an a-unknown of F. Let rvdenote the a-extension of r obtained by adding the equality v = a to thelist of term equalities. We will also say that rv is obtained by cutting of v.We will say that an a-unknown v is admissible in r if r' is admissible.

LEMMA 7.3. For any correct CTS system r and a-unknown v, if v isadmissible and minimal in r, then

(1) rv is a correct CTS-system;(2) an a-unknown is minimal in rv if and only if it is minimal in r and

distinct from v ;(3) fl[rv] fl[F], y[F] v[r] + K[r](/J[r] - 1), a[F ] = a[r] + 1;(4) if a is a solution of rv , then it is a solution of I' and ar[a] =

are[a] + 1;

PROOF. Let r be a correct CTS-system, v an a-unknown that is admis-sible and minimal in F. Let be the CTS-system obtained by cutting offv. Using Lemmas 6.3 and 7.1 and reasoning by induction on the length ofa deduction of A B in accordance with the rules for and -< we canprove

SUBLEMMA IX. For any r-atoms A and B, if A B and either A is ar-term containing a-unknowns or a, or A has the form waa, where w isan a-unknown, then

(1) B is a r-term containing a-unknowns or a, or B has the form uaa,where u is an a-unknown;

(2) A r B or a r B and A r v, or v r B and A r a.Sublemma IX implies the following proposition.

SUBLEMMA X. For any F-atom A, if A is a basic i-atom, then one of thefollowing conditions holds:

(a) A is a r-term that does not contain a-unknowns or a ;(b) A has the form waR , where w is an a-unknown and R a substitut-

ing term of r ;(c) A is a basic r-atom;(d) A r a ;(e) A r v .

The following proposition can be proved using Lemmas 6.3, 6.5, 6.6, and6.8.


SuBLEMMw XI. For any natural number i, I'-atom A, a-unknown w,and substituting term R of I', if E is a correct CTS-system, A is a basicE-atom and waR is the i-argument of A in E, then one of the followingconditions holds:

(a) w E a ;(b) waR is a principal I'-atom;(c) there are a natural number j and an a-unknown u such that w is

the j-argument of u and uaR is a basic E-atom.

The correctness of E follows from the correctness of F, the admissibilityof E, Lemmas 6.3, 7.1, and Sublemmas IX and X. The other conditions ofLemma 7.3 follow from Lemmas 6.3, 6.5, 6.10, 7.1, and Sublemmas IX andXI. This completes the proof of Lemma 7.3.

Let I' be a correct CTS-system and S2 a sublist of the list of functionsymbols of F. We will construct trees r and £39 of a-extensions of F,with the a-extensions of I' assigned to the vertices of r and £39 andthe a-unknowns or the letter n to the edges. Let E(O) denote the CTS-system assigned to a vertex O, la(O) the number of a-unknowns of-hangingin E(O) , l(0) the number of a-unknowns /3-hanging in E(O) , l(0) thenumber of a-unknowns a-hanging in E(O) relative to S2, and l(0) thenumber of a-unknowns /3-hanging in E(O) relative to S2. The tree r(9) will be constructed in such a way that the following conditions aresatisfied:

(1) I' is assigned to the root of r(2) for any vertex O, if 1a(0) = 0 (resp., l(0) = l(0) = 0),

then no edges begin at O ;(3) the same a-unknown cannot be assigned to two different edges that

begin at the same vertex;(4) for any vertex O, if l(O) > 0 (resp., l(0) > 0), then all admis-

sible a-unknowns that are /3-hanging (resp., /3-hanging relative tofl, and they alone, are assigned to the edges that begin at O ;

(5) for any vertex O, if l(O) = 0 and l a(0) > 0 (resp., l(0) = 0and l(0) > 0), then ic is assigned to one edge that begins at O,and all admissible a-unknowns that are of-hanging (resp., of-hangingrelative to S2 ), and they alone, are assigned to the edges that beginat O ;

(6) if an a-unknown v is assigned to an edge connecting O1 with OZ ,then v is minimal and admissible in E(OM) ,and E(OZ) is obtainedfrom (0) by cutting off v ;

(7) if ic is assigned to an edge connecting Oi with OZ, then 0

(resp., l(0) = 0) and E(OZ) is obtained from E(OM) by prolonga-

tion of all a-unknowns that are a-hanging (resp., a-hanging relativeto c.

The trees r and £39 may obviously be infinite. Let a be a solutionof I' and O a vertex of r or £39. We will say that O agrees with a ifthere is a solution E(O) whose restriction to the unknowns and a-unknownsof I' is a.

LEMMA 7.4. For any correct CTS system I', the tree r (fir) satisfiesthe following conditions:

(1) for any vertex O, E(O) is a correct CTS-system;(2) for any solution a of I', there is a finite branch, beginning at the

root, that passes only through vertices that agree with a and ends ata vertex O such that l a(0) = 0 (resp., l(0) = 1( 0) = 0).

PROOF. Condition (1) follows from Lemmas 7.2 and 7.3. We prove con-dition (2). Let k be a natural number such that the function symbols of I'include no 1-place symbols with 1 > k. Let O1 be a vertex of r or rand Ql a solution of E(OM) . We introduce the following notation:

h°[a ; °] k + 1)h[va']vE9JtE(o )

Assume that l( O) + l Q(0) > 0 (resp., l (Ol) + l(0) > 0). Then wecan construct an offspring 02 of 01 and a solution Q2 of the CTS-systemE(OZ) such that Ql either coincides with or can be prolonged to QZ . Lemmas7.2 and 7.3 imply that

h°[a2; 02] < h°[a1 ; Ol].

Thus, the finite branch required by condition (2) can be constructed byinduction on h ° . o

Let F be a correct CTS-system, A and B F-atoms. The expressionA- B will mean that there is a r-atom A l such that A l r B andAl r A. The following assertion can be proved using Lemmas 6.3 and 6.8.

LEMMA 7.5. For any F-atom A and F-term T that does not contain a-unknowns or a, if F is a correct CTS-system, T r A and A is a basicF-atom, then T occurs as a subterm on the left or right of term equalities, onthe left of equalities with substitutions or in substituting terms.

Let T and Q be terms and S2 a list of function symbols. We define acomplexity measure h[T] of T by induction on the height of T.

(1) If T is an object variable, an unknown, or a 0-place function symbol,then h[T] = 0.

(2) If T begins with a function symbol not belonging to S2 , then h[T]=0.

(3) If T is g (T1 , ... , TI) , where T1, ... , TI are terms, 1> 1, and gis an l-place function symbol from S2 , then

h[T] = 1 + max1<i<I

We also define a complexity measure OQ, of T by induction on its height.

(1) If T is an object variable, an unknown, a 0-place function symbol,or Q, then a [ T ] = 0.

(2) If T begins with a function symbol not belonging to S2 , then a [T]=0.

(3) If Q does not occur in T, then aQ , [ T ] = 0.(4) If T is g (T1 , ... , T,), where T1, ... , TI are terms, l > 1, g is

an 1-place function symbol from S2 , and there exists i, 1 0, then

°a [T] = 1 +max1<j<I

Let a be a collection of terms. Denote by co [a] the number of subtermsoccurring in terms of a that begin with function symbols from S2 of positivearity; h[ a] ] (aQ , [a]) will denote the maximal value of h[T] (aQ , [ T ] )

over all subterms T occurring in the terms of a.Let F be a correct CTS-system with unknowns t1, ... , t . Denote by

y[F] the number of F-atoms distinguished in F that begin with functionsymbols from S2 and have the form waR . Also denote by R°[F] the unionof the set of left- and right-hand sides of term equalities of F without a-unknowns and a, the set of left-hand sides of equalities with substitutionsof F, and the set of substituting terms of F. We introduce the followingnotation:

so[F] he[r]n

aQ, [r] aQ, [R°[r]] , he[r] a,. , [F],r

i=1

he[r] = he[r] + he[r].

If S2 is the list of all functional symbols of the CTS-system or axiomaticsystem under consideration, we omit S2 in the expressions coy , a , , h,y, h, and h.

Let T1 , ... , TI be terms, b1 , ... , bl a list of pairwise disjoint objectvariables which do not occur in F-terms, a a list of terms. The expressionb b T1 TI will denote the result of simultaneous replacement in a of alloccurrences of b1 by T1 , ... , all occurrences of bl by TI . The following

equalities can be proved by induction on the construction of T.

hsi[Tb ...b Tl ... TI] =max ,max{ab , [T] + (2)l Ee,

aX si[Tb ...b Tl ... 7] = max ox° [T] max {a° [T] + a° [T.]} (3)' 1 ' tE0 ne b,SZ x,SZ t

a

where E is the set of all i, 1 0; D2 is the set of all i, 1 0; x is an unknown, a 0-place function symbol, or an objectvariable distinct from b1, ... , bl .

By a basis of a correct CTS-system F we mean any collection

{A1, ... , A; 1,..., b,}, (4)

where k > 0, b1, ... , bk are pairwise distinct object variables other thanunknowns, a-unknowns, or object variables of F ; Al , ... , Ak are F-atomssatisfying the following conditions.

(1) None of the F-atoms A1, ... , Ak begins with an object variable ora function symbol.

(2) There are no natural numbers i, j, 1 < i < k, 1 <j < k, suchthat Al r A.

(3) For any F-atom B, either B begins with an object variable or a0-place function symbol, or there is a natural number i, 1 < i < k,such that A. r B.

It is obvious that we can construct a basis for any correct CTS-system. Fixa basis (4) of a correct CTS-system F, and let B be a F-atom. We define aterm {B, F} by induction on r .

(1) If B ^r Al for some i, then {B, F} a bl .

(2) If B begins with an i-place function symbol (l > 0) or an objectvariable f, then

{B,F}=f({B1,F},...,{B,,F},...,{B,,F}),where Bl is the i-argument of B.

Note that the terms constructed above do not depend, up to renaming ofobject variables, on the choice of basis in the CTS-system. Using Lemma6.5, we can prove (by induction on -<r) that, for any F-atoms A and B,if A r B, then {A, F} occurs in {B, F}. Similarly, using Lemma 6.1, wecan prove that, for any F-atom A and any solution a of a CTS-system F,

AC a {A , F}b ...b Al ... Ak.1 k

(s)

Let v1, ... , vq be the list of a-unknowns of a CTS-system F. We let{F} denote the list {t1, F}, ... , {t, F}, {v1 , F}, ... , {v, F}. Equality(5) implies the following assertion.

LEMMA 7.6. For any correct CTS-system r with a solution a,{T}bbA...A=cr.

1 k

Let A and B be r-atoms. The expression A mar, B will mean thatA <r B and

0a{A,r},si[{B, r}] >0.

The expression A B will mean: there is a r-atom A 1 such that A1 rA and Al <r, B or A r B. A correct CTS-system r will be calledS2-acyclic if there are no r-atoms B1 and B2 such that B1 r B2 andB1 <r, B2. The following assertion can be proved using Lemmas 6.5 and6.7.

LEMMA 7.7. If a correct CTS-system r has no a-unknowns as-hangingrelative to S2 in r, then

(1) for any r-atoms Al , A2, B1, B2 and any natural i, i > 1, if A2 isthe i-argument of Al , B2 is the i-argument of B1, Al begins witha function symbol from S2 , and A 1 r B1, then A2 r B2 ;

(2) for any r-atoms A, B, and C, if A B and B C, thenA -< C;

(3) r is c2-acycl ic.

Let A be a r-atom. Let 9 [A ; r] denote the set of r-atoms B such thatB A and B is distinguished in F, 9 [A ; r] the set of r-terms belong-ing to 9 [A ; F], y[ A; ; r] the number of r-atoms belonging to 9 [A ; r]that begin with a function symbol from S2 and are not terms. Below we willuse the following notation:

r] max {h[T]},r]

a0 [A ; r] = max {a° [T]},TE9 [A ; r]

nh*[A

r] U s [A ; r] ,r

1=1

where t1, ... , to is the list of all unknowns of F.

LEMMA 7.8. If a correct CTS-system r is c2-acycl ic, then

[r].

PROOF. Let r be an S2-acyclic correct CTS-system. Then it can be proved,by induction on .<r that, for any basic r-atom A,

r}] < r] + q [9 [A ; r]]Suppose that an unknown t is not a basic r-atom. Then, by Lemma 6.8,

{ t, r} is a variable, and consequently

h[{t, r}] = 0.

Now, in order to complete the proof, we need only use Lemma 7.5. 0

LEMMA 7.9. If a correct CTS-system I' has no a-unknowns as-hangingrelative to S2 in I', then

h[{F}] y[F] + h[F].

PROOF. Let I' be a correct CTS-system that has no a-unknowns of-hangingrelative to S2 in F. Using Lemma 7.7, we can prove, by induction onthat for any I'-atoms A and B, if A r B, then

h[{A, F}] = h[{B, F}].

Using this fact, equality (2) and the fact that r is S2-acyclic, we can prove,by induction on <r, that for any basic r-atom C,

hn[{c, r}] < r] r] r]. (6)

Lemma 7.9 now follows from this and Lemmas 6.8 and 7.5.In what follows, we will use the following notation:

-, r, it1+1r

LEMMA 7.I0. Let I' be any correct CTS-system and set m =or m = y[F] + Suppose that for any i (0 < i < m), there is ana-unknown as-hanging in ic'I' relative to S2, but there is no a-unknown /3-hanging relative to S2. Then we can construct a-unknowns v and w of theCTS-system n"'I' such that w is free in c"`I' and

v ,s W, v W.

PROOF. Assume that r and m satisfy the conditions of the lemma. Us-ing Lemma 6.7 and condition (5) of Lemma 7.2, construct a free chainv°, 'o , v1, j

1, ... , Jm _ 1 , vm of a-unknowns of m r such that, for all i

(0 < i < m), v1 is a-hanging in ir`r relative to S2 , but it is not /3-hanging in i`r . Using the correctness of i`r and Lemma 6.7, constructa sequence of r-atoms A0, A1, ... , Am such that for all i (0 < i < m) :

(1) v1 r'r Aln

(2) Al is distinguished in F, it is not unknown and begins with a func-tion symbol from 1;

(3) if i < m and Al has the form f(P1, ... , PI) , where f belongs to S2and P1, ... , PI are r-terms, then vl+ 1 n'+'r P, where 1 <j < l ,

and, if P is not an unknown, A,+1 coincides with P.

Since m = y[ F] + co [r] or m = y[ F] + h[ F],, we can find naturalnumbers i and j such that i + j + 1 < m and A, coincides withThe lemma clearly follows.

LEMMA 7.11. If a correct CTS-system I' has no as-hanging or /3-hanginga-unknowns, then a solution a of I' can be constructed such that

ar[Ql = 0, Ialo'l = a"II'l, h[a] = h[{r}],

PROOF. Assume that I' satisfies the conditions of the lemma, and let (4)be a basis of F. Construct a collection of natural numbers i 1, ... , i, (1 > 0)such that

(1)1i1k,...,1<i,<k;(2) there are no natural numbers j and s, 1 < j < 1, 1 < s < 1, j 5,

such that Al r A. ;S

(3) there are no natural number j, 1 < j < 1, and an object variable ora 0-place function symbol x such that x r Al ;

(4) for any j (1 <j <k) either there is an object variable or a 0-placefunction symbol x such that A r x, or there is a natural numbers, 1< s< l , such that Aj r Al .

S

Let B be a F-atom. Define a term {B, f}* as follows. If B r x forsome object variable or 0-place function symbol x, then {B, f}* a x. IfB r Al for some j (1 <j < 1), then {B, f} * a bl . If neither of these

> >

cases occurs, then

{B, f}* a {B, r}b ...b {A1 , F}* {Ak , f}*.1 k

We will denote the collection {F}b . b { A 1 ,f}*

{Ak,f}* by {f}*,

Ob-

viously,h[{r}]. (7)

Using induction on <r, we can prove that, for any a-unknown v,

{VaR, f}* a {v, r}Q{R, f}* ,

where R is a substituting term of F. This graphic equality implies that {f}*

is a solution of F. If the a-unknown v is minimal in F, then la[{v , F}*] =0. This, together with Lemma 6.11, implies that

r[{1}*]0, l[{F}*]=c*[F]. (8)

Note that for any I'-atoms A and B, if A r B, then {A, I'}` a{B, I'}" . Hence it follows that

co[{F}*] y[F] + o[F].

Proceeding as in Lemma 7.9, but using (3), we obtain

aQ,n[{r}*] < y[F] + h[F] + aQ,[r].

§8. REPRESENTATION OF ENUMERABLE SETS BY TS-SYSTEMS 51

The last two inequalities, equalities (7) and (8), and Lemma 7.9 implyLemma 7.11.

Any CT-system I' can obviously be considered as aCTS-system with noa-unknowns; I' is then admissible if and only if it is correct. A collectionof terms a will be called a universal solution of a CT-system I', if a is asolution of I' and any solution of I' can be obtained from a by simultaneoussubstitution of terms for those object variables that are not object variables ofF. Clearly, universal solutions of a CT-system differ only by the renamingof variables. It follows from the Unification Theorem (see [32]) that if aCT-system has a solution, then it has a universal solution. In what followswe will need certain bounds on the height of a universal solution.

LEMMA 7.12. Ifa CT-system I' in n unknowns has a solution, it is possibleto construct a universal solution a of I' such that

h[a]h[a] o[a] o[F], (9)

h[a] n.h[F], (10)

PROOF. It follows from Lemmas 6.2, 7.6, and 7.9 that {I'} is the requireduniversal solution. Note also that {I'} is just the solution constructed in theproof of Lemma 7.11. The bound (10) is obtained by using inequality (6).0

If I' is a T-system and S2 includes all the function symbols, inequality(10) was proved in [19]. Inequalities (9) and (10) were also proved in [45]for the same case.

§8. Representation of enumerable sets by TS-systems

We will assume that the list of function symbols used in this section in-cludes 0-place symbols 0 and b, a 1-place symbol ' , and a 2-place symbolf. Let be an enumerable 1-place predicate on the set of natural num-bers. We will say that a CTS-system F in n unknowns and q a-unknowns(q > 0) represents if:

(1) n>l;(2) for any natural numbers m1, ... , ml , if (m1, ... , ml) , then there

is a solution 81, ... , en , 0n+1 , ... , en+ of F such that 81 a 0(m,q

8 a 0(m!);

(3) if 81, ... , 0n , °n+1' ... , en+q is a solution of F, then there existnatural numbers m1, ... , ml such that 81 a Ohm' , ... , 81 a 0 /

and '(m1, ... , ml) .

Henceforth the expression T, where T is a term and k > 0, denotes


T with k primes. The purpose of this section is to prove that enumer-able predicates are representable by normal TS-systems. We first prove threelemmas.

Consider the following TS-system in one unknown t and one a-unknownv:

t=va0,Any collection of terms

t . (1)' = va 0'

O(n) a(n) (Z)

is obviously a solution of (1).

LEMMA 8.1. Any solution of a TS system (1) has the form (2).

This lemma is proved by induction on the height of a solution of (1).

LEMMA 8.2. The predicate of addition of natural numbers is representableby a normal TS-system.

PROOF. Consider the following TS-system in unknowns t1, t2, t3 and a-unknowns v, u :

t1 = vaO , t1 = va0' , t2 = ua0 , t2 = ua0' , t3 = uatl . (3)

Any collection of terms of the formO(n) 0(m), O(n+m), a, a(m)

(4)

is obviously a solution of (3). On the other hand, by Lemma 8.1, any solutionof (3) has the form (4). 0

LEMMA 8.3. The predicate of multiplication of natural numbers is repre-sentable by a normal TS-system.

PROOF. Consider the following TS-system in unknowns t1 , t2, ... , t 13

and a-unknowns r, v0, w0, uo , U0, t1°:

t2=ra0, 4 =raO', t5=rob, t6=rat3,

f(t8, f(t, t6)) = v°O = wa°0' , w°0 = ua°t2 ,

t9 = u°O, f(t9, f(t, t6)) = v° f (0 , f(O, 0)),1 a

f(t10, f(t4, t6)) = U O = wa°b' , w°b = au °t2 , (5)a

t11 = u°0, f(t, .f(t4, t6)) = v°f(0, f(b, 0)),a 11 a

f(t12, f(t4, t7)) = 3 0=ta°b', ivab=uat5,

t13 = u°b, .f(t13, .f(t4, t7)) = v°.f(0, f(b, b))a a

System (5) is not, strictly speaking, a CTS-system with a-unknowns, butit can be transformed into a normal TS-system by adding new unknowns.

Consider the following collection of terms:0(n) 0(m), 0(m), b', b(m), 0(nm+m)

b(nm+m),Tn , m (O , 0, 0), Tn' m (0 , 0), T n' m (0 , b, 0)

r'm(b, 0), Tn' m (0 , b, b), T n' m (b , b), a(m), (6)N0, 0), ?"m(a, 0), Tn' m (0 a)

Tn+l,m(a, b, 0), r,m(b a), Tn+l,m(a,b, b), Tn,m(a b),

where the terms T? ' m (a ,) , fn ' m (q' , yr) , T n ' m (q' , yr) are defined bythe following equalities:

To'm(a, SP yr) a aO'm(q,

vi) a f(0, f(q, y,(m)))

y/) a .f(0, f(Y' , y/)),Tn+l, m(d ,

SP , yr) a f(Tn,m(a, SP , yr) ,

f(So(n+1)

,yr(nm+m)))

,

T, n+1 , jn(, yr) a

f(T-n ,m(so , yr) ,

yr(nm+2m))),

Tn+1'm(SP, yr) a f(Tn,m(So,yr),

f(So(n+1), yr(nm+m)))

The following equalities are easily proved by induction on n :

n'm(so,,

yr), Tn'm(co,w)=T(m))

,

n+l ,m(f(0,f(co,yi)),So,yi).

These equalities imply that (6) is a solution of system (5).Denote system (5) by TO and introduce the following notation:

r1+1 irr` .fIf k> 1 , fk includes all equalities of the form

z` =

f(Z2

Z = ),

where 0 < t < k, 0 <j <k - 1, z a v, 3, w, u, U, Ti, ti'. In addition,I,k satisfies the following conditions:

UaO t9 UQ f(0, f(0, 0)), v 10 t w0', w0 uQ t2 ,a a 8

u10 v2f(0, f(0, 0)), v20 w20', w20 uQt2, ... ,a a a a a

k-iUa 0 vakf(0, f(0, 0)), vk0 wk0', wk0 uk t2;a a a a a

-0 1

Ua0 t11 vaf(O, f(b, 0)), U10 t10 wab', wab ' uat2,a a1 2 2 2 i 2 2

ua0 vaf(0, f(b, 0)), iT 0 wab , wab uat2, ... ,

ua -10 vakf(0, f(b, 0)), v 0 wakb/,

wakb uat2;a

auab t13 ^-'vaf(O f(b, b)), v10 t12 wab', wab uat5,a


v20 ^ iv2b/

a a', iv2b uat5, ... ,u1 b ^ v 1(0, f(b, b)), aa aa

' , wkb uat5;vk0 ^ ivkb aua -1 b ^ vak f(O, f(b, b)), aa

f(t, t) v°'2 f(0, f(0, 0)) v°,20 r wa,20,'1 6 a a

w°,20 u°, 2t u°,20 v1'2f(0, f(0, 0)), Z1a1'20wal'20',a a

2,

a a

1'20 u1'2 t2, ... , uk-2,20

vk1'2Wa a a a f(0, f(0, 0)),

vk-1,20wk-1,20' wk-1,20uk-1,2t;a a a a 2

f(t4,t6)Va '2f(0f(10))ta °'20w°'2b'aw°'2b u°a

2

'2t , ua°,20 v1 a'2f(0, f(b, 0)), a211'20 wl'2b',a a

w1'2b u1'2t , ... , -k-2'20 ^' -ak-1'2 f(0, f(b, 0)),a a 2 a

vk-1,20 wk-1'2b' k-1,2b uk-1,2ta a , wa a 2;

f(t4, t7) v°'2f(0, f(b, b)) 0,2 0 0,2 b' ,

a a a

w°,2b 0,2a

t5, u a

°,2b N vl a'2211'20 wl'2b',a f(0, f(b, b)), a a

w1'2b u1'2t , ... , -k-2'2b ^' vak-1'2f(0, f(b, b)),a a 5 a

21k-1,20 wk-1'2b' "uk-1'2b uk-1'2ta a , 2a a 5;

t' °'2'1f(0, f(0, 0)) v°'2'10 w°'2'10', w°'2'10 u°'2'1t2,...,1

va a a a a

uk-3,2, 10 N vk-2,2, 1f(0,f(0, 0)),a a

v t2;k-2,2, 10 N wk-2,2, l0i' wk-2,2, 10 N uk-2,2, la a a a

t °'2'1f(0f(b0))Va,w°21b u°21t2. . .

4va

a a a

(7)

uk-3,2,1

0k-2,2 ' 1 f(0,f(b,0)),a a

_k-2,2,1 N k-2,2, 1 i k-2,2, 1 N-k-2,2, 1 (8)v a 0Wa b, wa b- u a t2

t v0'2'2f(0,

f(b ,0)) v °'2'20 w°,2,2bi, wa'2'2b u a'2'2t2 , ... ,

6 a a a

uk-3,2,20 ti v k-2' 2'

2f (0 , f(b, 0)),a a

_k-2,2,2 Nva k-2,2,2 i k-2,2,2 N -k-2,2,2 (9)0 2a b , wa b ua t2,

°,2,1 °,2,1 °,2,1 °'2'1

'°'2'10 w 2u b ubf(0, f(b, ,t , ... ,b)) vv4 a a a a a 5

-3,2,1b N vk 2,2,1f(0, f(b, b)),a a

k-2,2,10 wk-2,2, lb' k-2,2,1

b uk-2,2, It5;, a aa a

t ^' v°'2' 2 f(0,

f(b, b)) v°'2'20 w°'2'2b' , w°'2'2b u°'2'2t ... ,7 a a a a a 5,

uk-3,2,2b N vk-2,2,2 f(0, f(b, b)),a a

k-2,2,2 Nva _ k-2,2,2 i k-2,2,2 N -k-2,2,2 (10)

0 wa b , wa b _ u a t5 .

For simplicity, we have omitted the index IT in .

In the TS-system IT", the a-unknowns v k , w k, uk , v k , u k , v k k ,

vk-1,2, wk-1,2, uk-1,2, 3.k-1,2 uk-1,2' vk-1,2' wk-1,2 are of-hangingrelative to f. Let z be an a-unknown which is a-hanging relative to f inIlk and is distinct from v k

, uk , v k , v k . Then the TS-system rk has nosolution. Indeed, otherwise some a-unknown would be /3-hanging in eachTS-system of the tree of a-extensions of f, contradicting Lemma 7.4.

Z

Let a be a solution of F0 . It follows from Lemma 8.1 that for somenatural number m

t2 a O(m),

ra =a(m),

ta a b(m).s

By Lemma 7.4, a singles out a finite branch U in the tree whichr°begins at the root, passes only through vertices that agree with a, and endsat a vertex O such that the TS-system T(O) contains no a-unknowns thatare a-hanging relative to f. For no natural number i does U pass through

,edges with assigned a-unknowns w', u ` , iv ',2

, iv 1, v i,2 , w i , 2 , u"2v`'2 `'2 2u v It is easy to show that, for some natural number n,(O) is Tl t'v +i vn+l vn+1 or ?Lr'un+ rvn+l vn+l n+1 In either case the followingconditions hold in (O) :

un-1'2' 1 vn'2' f(0, 0)), vn'2' l0 wn'2' 10/a a a awn,2, 10 un,2, lt2' un-1,2, l0 va n'2' lf(0, f(b, 0)),a a avn,2, 10 wn,2, 1b/ n,2' 1b un,2' 1ta a ,

waa 2,

un,2, l0 0, un,2, l pb,

a a a

{ua

n-1,2,2 r vn'2'2f(O,f(b,O)), vn'2'20 wn'2'2b/wn,2,2bun,2,2t -1,2,2bvn'2'2f(0

f(b, b)),a a

2'

a a

tin' 2' 20 2vn' 2'2bi

ivn' 2' 2b un' 2' 2ta a , a a 5,un,2,20

0, u,2,2b

b.a a

a a

(13)

Let T be a solution of E(O) whose restriction to the unknowns and a-unknowns of T° is a. Using Lemma 6.1, the graphic equalities (1), andconditions (7), (8), (12), we can use induction on n - i to prove that for alli (0<in)

i,2, 1 zUaa

aua

ui, 2, 1z b(n-i)avi,2, lz 0(n-i+1)a

tZ a O(n)1

Similarly, using conditions (9), (10), and (13), we can prove that for all j(1<j<n)

uj, 2, 2z a(mn-mj)av j , 2 , 2z 0(mn+m - jm)a

t6

vj , 2, 2z a b(mn+m- jm)

uJj , 2 , 2z (mn+m- jm)=0w j, 2, 2z a b(mn+m- jm)

a v 0, 2, 2z a v0, 2, 2z a 0(mn+m) a [ta](m)3 '

Thus, the values of the unknowns t 1 , t2, t3 in any solution of (5) havethe form 0, 0(m), 0(m), respectively. This proves Lemma 8.3. o

THEOREM 5.1. Any enumerable predicate on the set of natural numbers isrepresentable by a normal TS-system.

COROLLARY. There is no decision algorithm for the solvability of CTS-systems.

PROOF OF THEOREM 8.1. Let be an enumerable 1-place predicate onthe set of natural numbers. According to [14], is a Diophantine predicate.Hence, by adding new unknowns, can be represented by a system ofDiophantine equations of the form x y = z, x +y = z, or x = n, where nis a natural number and x, y, z are unknowns. Theorem 8.1 now followsfrom Lemmas 8.2 and 8.3. o

§9. Upper bounds on the height of natural solutionsof systems of linear Diophantine equations

Consider a homogeneous system of linear equations

AX+BY=0,

where A is an n x m matrix over integers, B an n x k matrix whose entriesare 0, -1 , and 1 with only one nonzero element in each column; X and Yare columns of unknowns of lengths m and k, respectively; n > 1, m > 0,k > 0, m + k > 1. Let H0 > 1 be a number that maj orizes the maximumEuclidean norm of the rows of A.

LEMMA 9.1. If the rank of system (1) is less than m + k, then there is anontrivial integer solution a, ... , 4, b, ... , bk of (1) such that for all i(1 <i< m)

IaI < Ho -1

and for all j (1 < j <k)IbI< H.

(2)

(3)

PROOF. Without loss of generality, we will assume that the rank of (1) isn and that the determinant of the matrix composed of the first r columns

§9. UPPER BOUNDS ON THE HEIGHT OF NATURAL SOLUTIONS 57

of A and first s columns of B is nonzero, where r + s = n, 0 < r < m,and 0 < s < k. Denote this matrix by C. Let q be the natural number

q=r+l, ifr<m,s+l, ifr=m.

Let Q denote the qth column of A if r < m and the qth column of B ifr = m. Let RP (1 <p < r) denote the matrix obtained from C by replacingthe pth column by Q and St (1 < t < s) denote the matrix obtained fromC by replacing the (r + t)th column by Q. For all i (1 <i < m) define

det R ,

a° _ - det C ,

to,and for all j (1 <j <k) define

det

o - det C ,

0,

ifi<r,if i=r+1 <m,ifr+1<i<m,

if j < s,ifr=m, j=q<k,ifr<m,j=s+1<k,ifs+1 <j<k.

Using Cramer's rule, one easily proves that a, ... , 4, b, ... , bk is anontrivial integer-valued solution of (1). The following inequalities are easilyproved using the Hadamard inequality and our assumption that each columnof B has only one nonzero element:

(1) IdetCI< H.(2) Forall p (1pr)

IdetRI <

(3) Forall t (1<t<s)

Ho, ifr<m,Ho-1, ifr = m.

IdetSI <Ho+ 1, ifr < m,Ho, ifr=m.

Inequalities (2) and (3) follow from these estimates. The lemma is proved.Consider a system of linear equations

AX+BY=C, (4)

where the matrices A, B and columns of unknowns X and Y are the sameas in (1), C is a nonzero column of natural numbers of length n. Let ddenote the greatest element of C. Let us add the column a C to A on theright; denote the resulting matrix by (Ad C). Let H be a real number suchthat H > 2, H > Ho , and H majorizes the Euclidean norms of all the rowsof (AaC) . If U is a column or a row of rational numbers, we denote byp(U) the number of indexes of nonzero elements of U.

We will say that the jth column U of B (1 < j < k) is regular in (4) ifthere is an index i, 1 < i < n, such that the ith element of U is 1 , noneof the elements in the ith row of A are positive, and none of the elementsin the ith row of B, except the jth one, are positive. Let U be a row ofrational numbers of length k. Let p (U ; A, B, s) denote the number ofindexes of nonzero elements of U other than s and the indexes of regularcolumns of B in (4).

LEMMA 9.2. The following conditions hold for any nonnegative rationalsolution a1, ... , am , bl , ... , bk of system (4).

(1) Suppose that for some i° (1 < i° <m), we have a, > d Hr + qHo-1,

0

where r=p(al,...,am) and q=p(b1,...,bk;A,B,k+1). Then thereis an integer-valued solution a, ... , a, b, ... , bk of system (1) such thata° >O,forall i (1<i<m)

0

< Hr-1- 0 '

and for all j (1 <j < k)

> 0,

0<b0 <Hr, (7)

b -b°>0. (8)

(2) Suppose that for some j° (1 <J° <k), we have b . > dHr+ 1 + q* Ho,o

where q* = p(b1, ... , bk ; A, B, j°) . Then there is an integer-valued solution

a, ... , a, b, ... , bk of (1) such that b°. > 0 and inequalities (5)-(6) ando(7)-(8) hold.

PROOF. By induction on p(a1, ... , am, bl , ... , bk). The induction baseis trivial. We now prove the inductive step. Let al , ... , am , b1, ... , bk bea nonnegative rational solution of (4); denote it by a. Delete from A and Ball columns with the same indexes as that of the zero elements of a. Denotethe resulting matrices by A° and B°, respectively, and consider the systems

A0X0+B0Y0=C, (9)

A0X0+B0Y0=O. (10)

Delete from a all zero elements and denote the result by a' . It is clearthat a' is a solution of (9). Without loss of generality, we will assume thatthe rank of system (10) is n.

Suppose that the rank of (10) is p(a). Then, applying Cramer's ruleto (9) and using the Hadamard inequality, one easily proves that for all i

(1<i<m)0 < a. dHr

and for all j (1< j< k)

0<b. <dHr+1

Suppose that the lemma is true for all nonnegative rational solutions cr'of systems of the required type such that p (a') < p(a). We wish to provethat a satisfies condition (1). Let io be a natural number such that 1 <io < m and al > dHr + qHo-1. Then the rank of (10) is less than p(a).

0

Applying Lemma 9.1 to (10), we construct a nontrivial integer-valued solutiona, ... , 4, b, ... , bk , ao say, such that a, > 0, IaI < Ho-1 for all i

0

(1 < i < m), IbI°. < Ho for all j (1 < j < k), and the result of deletingcertain zero elements from ao is a solution of (10). Denote this solution of(10) by cjo. It is obvious that one of the following cases holds:

(A) a° > 0 and ao satisfies inequalities (5) and (7);0

(B) one of the elements of ao is negative;(C) a° = 0 and ao satisfies inequalities (5) and (7).

0

Consider case (A). If a and ao satisfy inequalities (6) and (8), then aois the required solution of system (1). Suppose that a and ao do not satisfyone of inequalities (6) or (8). Then there is an s, 1 < s < m + k, such thatthe sth element of ao is greater than the sth element of a. It follows thatthere is a rational number , 0 <. < 1, such that a - ao is a nonnegativesolution of system (4) and p (a 1 - a) 2 and

a1o . - > dHr + qHr-1 - H0r-1 > d Hr-1 + qHo-1 > d Hr-1 + qHr-2

0 0 0

Now, applying the induction hypothesis to system (9) and the solution1 - , we obtain the required solution of system (1).Suppose that there is an sl , 1 < sl < k , such that bS 0, b- b° b° =0,

and the slth column of B is not regular in (4). Then q > 1 and

a, - a° > dHr + (q - 1)H'.0 10

Thus, in this case too the required solution of (1) is obtained by applyingthe induction hypothesis to a 1 - and (9).

Suppose that there is an index j, 1 < j < k, such that b 0, b -_0 and the j th column of B is regular in (4). Then one of the equations of(4) can be written in the form

y = c1 + a1, p xp + ... + a1 xpu + + ... + b1, li,y1t

where 1 <l <n, 0<u<m, 0<v <k-1, 1 <pl <... <pu <m,1 <ti a1 >0, ..., a1 >0, b11 >0, ..., b1, >0,

and columns of B with indexes t,, ... , to are not regular in (4). Note that

0 = c = a - a° _ = a - = b1 - b° _ = b -P1 P1 P« Pu 1 1 v v

a1 a a +bt j bj +...+bt b >0.P1 P1 ' Pu P« 1 1 ' v v

Hence one of the numbers aPl , ... , aP« , b1v is nonzero. It followsthat one of the two previously considered cases holds. Case (A) has alreadybeen considered.

Consider case (B). In this case, for sufficiently large positive , one of theelements of will be negative. Consequently, there is a positive rational

o such that a + is a nonnegative solution of (4) and p (a 1 + moo) <p(a). Clearly,

a1p . + 0a° > a1p . > dHr + qHo-1.1p

Applying the induction hypothesis to system (9) and the solution a 1we obtain a solution of system (1) for which case (A) holds.

Case (C) is reduced to (B) by replacing o with -moo . Thus, condition (1)of the lemma is proved. Condition (2) can be proved in a similar way. El

A natural solution a of a system of linear equations will be called minimalif there is no natural solution of the system, all of whose elements are notgreater than the corresponding elements of a, and one of them is strictly lessthan the corresponding element of a. By the height of a natural solution of asystem of linear equations we mean its maximal element. Lemma 9.2 impliesthe following stronger version of Lemma 1.1 in [13].

LEMMA 9.3. If all the columns of the matrix B are regular in system (4)and the system has a natural solution, then it has a minimal solution of heightat most dHm+l.

Consider the system of linear equations

AX=C, (11)

where the matrix A, column of unknowns X, and column C are the sameas in (1) and (4), and consider the system of linear equations

y=y

Y=Y +4+cN1

,N N

(12)

where 1 <ii <k, 1 < jl <k,..., 1 <N <k, 1 < jN<k; L?,...,LOONare homogeneous linear forms with integer coefficients in unknowns from thelist xl , ... , xm (some of these forms may be zero); c, ... , cN are naturalnumbers. System (12) can be represented as a directed multigraph as follows:the vertices are the unknowns y,, ... , y, its edges the equations in (12).The edge

(13)ip Jp P P

where 1 <p < N, connects vertex to y,. Denote this multigraph byP

F.

Let S be a path in F with beginning y1 and end y. We define itsweight p(S) as follows. If S consists of one edge (13), l = j, s = ip , then

p (S) = L° + c°; if l = iP, s = jr,, then p (S) = -L° - c°. If the length ofP P P P

S is zero, then p (S) = 0. In the general case, the weight of S is the sumof the weights of the edges composing S. We partition F into connectedcomponents T1, ... , rr and single out in each Tj (1 < t < r) a vertex y,and a tree with root yp whose set of vertices is the set of vertices of I'1.Let yl be a vertex of I,1. Let 5(y1) denote the unique simple chain inwith beginning yp and end y,. Let T1 denote the list of all equalities ofthe form p (S) = 0, where S is a simple cycle in IT.

Consider the system of linear equations

Y91 =L1 +cl,(14)

YqM

where 1 < q1 < k , ... , 1 < qM < k ; L, ... , LM are homogeneous lin-ear forms with integer coefficients in unknowns from the list x1, ... , xm ;

1 1c1 , ... , cM are integers. Assume that the unknowns y, ... , y (s > 0)P1 ps

occur in the list Y91 , ... , y9M , while the components TS+1, ... , rr do notcontain vertices from this list. Let T2 denote the list of all equalities

1 1 1 1L + c - Lu - cu = p (S)

where 1 < l <M, 1 <u < M, and S is a simple chain with beginning y9«and end y9 . Let T3 denote the list of all equalities

Lu + cu + p (S (Y)) = Yv ,

where 1 < v < k, yv is in a component I,1, 1 < t < s, such that yprcoincides with y9 . Let T4 denote the list of all equalities

u

y, + p (S (Yv)) = y,

where 1 < v <k, s + 1 <t < r, yv is in T; and T* the union of T, IT;,T, and T.

LEMMA 9.4. Any integer-valued solution of system (12), (14) is a solutionof T* ; any integer-valued solution of T* is a solution of (12), (14).

PROOF. Let(15)

be a collection of integers. For any path S in F, we let p(S) denote theresult of replacing the unknowns xl , ... , xm in p(S) by al , ... , a,, re-spectively. Let (15) be a solution of (12), (14). It is easy to prove, byinduction on the number of edges in S, that b1 - bu = p (S) , where yu isthe beginning and y1 the end of S. It follows that (15) is a solution of F.

Now let (15) be a solution of F. If the beginning and the end of Scoincide, then p(S) = 0. If yqu is the beginning and y9! the end of S, thenb9! bqu = p(S). Both these assertions can be proved by partitioning S intosimple cycles and simple chains. Now, using equalities from F3 and F,one easily shows that (15) is a solution of (12), (14). 0

Suppose that the Euclidean norms of the coefficient vectors of each ofthe forms L, ... , LN , L, ... LM do not exceed Ho . We introduce thefollowing notation:

1<1<N 1<1<M

LEMMA 9.5. For any natural solution ai , ... , am , bl , ... , bk of system(11), (12), (14), the following conditions hold.

(1) Suppose that for some to (1 <lo < m),

ago > max(d, kc° + 21c'I)[(k + 1)Ho + lj"' + k[(k + 1)Ho]"'-I;

then there is an integer-valued solution a, ... , a, b, ... , bk of the homo-geneous system corresponding to (11), (12), (14) such that a° > 0, for all l

0

(1<l<m),0 < a° < [(k + 1)Ho]"'-' , a1- a° > 0 ' (16)

and for all u (1<u< k),

0 < b° <[(k + 1)Ho]"' , b - b° > 0. (17)

(2) Suppose that for some uo (1 < uo <k),

buo > max(d, kc° + 21c'I)[(k + 1)H0]m+' + (k - 1)[(k + 1)Ho]"' ; (18)

then there is an integer-valued solution a, ... , a,°, b, ... , bk of the ho-mogeneous system corresponding to (11), (12), (14) such that b° > 0 and

0

inequalities (16) and (17) are true.

PROOF. Suppose that the collection of natural numbers

al,...,am,bl,...,bk (19)

is a solution of system (11), (12), (14). Without loss of generality, we willassume that, for each component F1 (s + 1 < t < r), the least vertex ofFj in (19) is b. Subtract by from all vertices of By Lemma 9.4,

r s+i

§10. UPPER BOUND ON THE PERIODICITY INDEX 63

we obtain a natural solution of (11), (12), (14). Doing the same for all thecomponents rs+1, ... , Fr , we obtain a natural solution

a1,...,am,b1,...,bk (20)

of the union of F* and (11), such that by = 0 for all t (s + 1 < t < r)and bu < bu for all u (1 < u < k). Substitute zeroes for the unknownsyp , ... , yp in the union of F* and (11) and call the resulting system F.

It is obvious that the solution (20) generates a natural solution of V andthat F+ is a system of the form (4). Hence the conditions of Lemma 9.2 aresatisfied.

It remains to consider the case when bu but not bu satisfies (18) for0 0

some u° (1 < u° < k). In that case there is a component Fj such thats+1 < t < r , bpi > 0 , and yuo is in Fj . In order to obtain a natural solutionof the homogeneous system as required in condition (2), it is sufficient to setthe unknowns in JT equal to 1 and the rest of unknowns equal to 0. ci

§10. Upper bound on the periodicity index of solutions of CTS-systems

Let b, b1, b2 , ... be a potentially infinite list of pairwise distinct objectvariables. We will assume that the variables in this list do not occur in CTS-systems under consideration or in their solutions. Terms containing only oneoccurrence of a variable b will be called b-singular. We will say that a termT involves a periodicity of length n if there are terms V, U, and W suchthat V and U are b-singular, b does not occur in W, V is distinct fromb, and

T=UbVb...VbW.n times

(1)

The representation of T in the form (1) will be called a V-periodicity inT. In what follows, we will stipulate that for n = 0 the expression Vb Vb

(n times) is equal to b.By the periodicity index of a solution of a CTS-system we mean the max-

imal length of the periodicities occurring in the terms of the solution. Ouraim in this section is to establish an upper bound on the periodicity index forsolutions of CTS-systems with restricted substitution width. Essentially, wewill extend Bulitko's lemma [4] to CTS-systems. We follow the exposition ofthe proof of Bulitko's lemma in [13].

LEMMA 10.1. For any terms U, T1, ... , T, W and any b-singular termV, if the variables b1, ... , bn do not occur in V and

Ubi ...bn Tl ... Tn a yb yV

then one of the following conditions holds.(a) There are a term W° and a b-singular term V ° such that

V = V°,.. T T , U =V °W° , W = VV°... T ...l

T .b bn

1 n b bl b'1 1 n

(b) There are a natural number i, 1 O, T a Vb1 W , U a V°bbiI , V a [V°bV Tn.

This lemma is proved by induction on ab [ V ] . Sometimes it will be moreconvenient to use the following special case of Lemma 10.1.

LEMMA 10.2. For any terms T, W and b-singular terms U and V, ifUb T = Vb W and T contains b or has the form Xb Vb T° , where X is a b-singular term, then there is a b-singular term Q such that one of the followingconditions holds:

(a) W a Qb T and U a Vb Q;(b) ab[Q] >0, TaQbW, and V a UbQ.

A term V will be called b-simple, if a[V] > 0, V is b-singular, novariables from the list b1, b2, ... occur in V, and there are no b-singularterm U and natural number n such that

V a Ub Ub ... U .

n+2 times

LEMMA 10.3. If a term V is b-simple, then there are no b-singular termsW 1 and W 2 such that

8b[W]>O, 8b[W2]>0, v w w2 a w, w1. (2)

PROOF. Let V be a b-simple term and W 1 and W 2 be b-singular termssatisfying (2). Note that ab [ V ] = ab [ W 1 ] + ab [ W2] . Using Lemma 10.2,proceeding by induction on ab [ W 1 ] + ab [ W 2 ] , and following the proof ofLemma 2.2 in [2], we construct a b-singular term U such that

W 1 a Ub Ub... U, W 2 a Ub 1b U.

n+1 times m+1 times

But this contradicts the b-simplicity of V. 0We will say that a V-periodicity (1) in a term T is stable if its length is

greater than 1 and there are neither a b-singular term U° such that U = Ub Vnor a term W ° such that W a Vb W°.

A term Q will be called 1--regular (1 > 0) if for any i (1 < i < 1) thevariable b, occurs in Q only once, for all j, s (1 < j <s < 1) b occursin Q to the left of the occurrence of b, and the variables b, b11 , b1+2 ,

do not occur in Q.

LEMMA 10.4. For any stable V-periodicity (1) in T, if

T a Ub yb ... yb yy* , (3)

m times


V is b-simple, and ab [ U* ] > ab [ U] , then there is a term U° such that oneof the following conditions holds:

(a) U a U, n= m, and W a W*;(b) U° is 2-regular and

1 b

m times

UaUbbV...VWb;*

1 2 b b

n times

(c) U° is 2-regular and

U= Ubb Vb...VbW*b,1 2

m times

U*=UbbbVb...VbW;12

n times

(d) U° is b-singular; ab [ U°] > 0, and

U* a UbV ...V U°,b b

n times

W a UbVb VbW*;

m times

(e) U° is b-singular, 8b[U°] > 0, and there are b-singular terms W°

and R such that 8b[W°] > 0, 8b[R] > 0,

VaUb°RaRb

W a W° V ... V W*.b b b

m-1 times

U* a UbVb...VbU°,

n-1 times

PROOF. Let V be a b-simple term and (1) and (3) stable periodicities thatsatisfy the conditions of the lemma. Let b1 be a variable that does not occurin U, U, V, W, or W * . Then

Ub bl Vb ... Vb yy a Ub Vb ... Vb W * .bl

n times m times

Applying Lemma 10.1 to this equality, we see that one of the following

two cases holds:

(1) there are a term R and a b-singular term U° such that

UbblaUb°R, U*aUb°Vb...VbW,l

n times

Ub...VbW* =Rb Vb...V,Wi

m times n times

(2) There are b-singular terms Q and Z° such that b does not occurin Z°, 8b[Z]>O,

Ubbl a Qbbl, U* a [Q Z°]1b Ub ... Ub W

b

Zb Ub ... V Wb

n times

a Ub ... Ub W.

m times n times

(a)

In case (1), one of conditions (a), (b), or (c) holds. Consider case (2).Since U and Q are b-singular, we have U a Q and U` a UbZ°. Startingfrom (4) and applying Lemma 10.2 consecutively an appropriate number oftimes, we either conclude that condition (d) holds, or construct, at the i thstep (0 0,

U* = Ub Vb... VbZ`

i times

Ub ... Ub W*a X b

Ub ... Vb W.

Note thatm times n - i -1 times

(s)

ob[V] _ ob[Z'] + ob[X'] > ob[X']. (6)

Rewrite (5) as follows:

Vb Zb Xb Vb ... VbW * a Xb Ub ... Vb W.

m-2 times n - i -1 times

Applying Lemma 10.2 to this equality and using inequality (6), we canconstruct a b-singular term Yl such that V a Xb Yl and

yb Ub ... Ub W * a Ub ... Ub W. (7)

m-1 times n- i-1 times

It is clear that condition (e) is satisfied if i = n - 1. Consider the last casewhen i < n - 1. Rewrite (7) as

YbZ`Xi V ...Vb W* azixi Vb ...V W.b b b b b b

m-2 times n-i-2 times

Applying Lemma 10.2 to this graphic equality and using (6) and the factthat

8b[V] = 8b[X] + 8b[Y`]

we obtain V a ZbX ` a Xb Z ` . But by Lemma 10.3, this contradicts theb-simplicity of V. 0

Let V be a b-singular term. By a parametric tree of V-periodicities inunknowns x1, ... , xl (l > 0) we mean a rooted tree, with terms assigned toeach vertex and a variable from the list b1, b2, ... and a linear polynomialin x1, ... , xl with positive integer coefficients assigned to each edge. If Ois a vertex of a parametric tree of V -periodicities, -r[O] will denote the termassigned to O. The linear polynomials assigned to the edges will be calledthe degrees of the edges. Let deg E denote the degree of an edge E. Aparametric tree of V-periodicities will be called a tree of V-periodicities ifthe degrees of all its edges are natural numbers. A parametric tree of V-periodicities will be called stable if, for any its vertex O :

(1) i[O] does not contain stable V-periodicities;(2) if E1, ... , ES (s > 0) is a complete list without repetitions of all

edges that begin at 0, then i[O] is s-regular, one variable fromthe list b1, ... , bs is assigned to each edge E1, ... , ES , and eachvariable from the list is assigned to only one edge that begins at O ;

(3) if bl is assigned to an edge that begins at 0, then there is a b-singularterm W' such that

i[O] a Wb Vbb1 ;

(4) if there is an edge ending at 0, then there is a term U such that[O]= U;

(5) if O has both ingoing and outgoing edges, then i[O] is distinct fromVbb1 .

Let be a stable tree of V -periodicities. We define a term i [] byinduction on the number of vertices of 7.

(1) If consists of one vertex 0, then i [](2) If there are s edges beginning at the root O of , then

a i[O]bl ...bs T1 ... Ts ,

where, for all i (1 < i < s)

n. times

is a tree of V-periodicities such that an edge from O with thevariable bl ends at the root, and n. is the degree of this edge.

Let T be a term. We will call a stable tree of V-periodicities a V-treeof T if Tai[ 7] . The following assertion can be proved using Lemma10.4.


LEMMA 10.5. For any terms T and V, if T does not contain the variablesb, bi , b2, ... and V is b-simple, then

(1) there is a unique V-tree of T ;(2) each edge of degree n of the V-tree of T generates a stable V-

periodicity of length n + 2 in T ;(3) each stable V-periodicity in T is generated exactly one edge of the

V-tree of T.

Let be a stable parametric tree of V-periodicities in unknowns x1, ... ,xl , and n1, ... , nl natural numbers. Let r[; ; n1, ... , nl ] denote the termr[*], where is the stable tree of V -periodicities obtained from bysimultaneously replacing x1 with n1 , ... , xl with nl in the degrees of theedges. Let r°[] denote the term r[°], where is obtained fromby replacing the degrees of all edges with 0.

Let H be a (possibly empty) list of different object variables distinct fromb, b1, b2, ... , V a b-simple term which does not contain variables fromH, a stable parametric tree of V-periodicities in unknowns x1, ... , xl .A vertex O of will be called fl-critical if r[O] contains a variable fromH. We will say that an edge E of is fl-outgoing at 0, if E begins atO and there are b-singular terms U, W, a natural number i, and a list ofterms a such that the length of a is equal to the length of fl, bl is assignedto E , Wna a V , and

r[0] a Ub Wb Vb bl .

We will say that E is H-ingoing at O if E enters into O and there areterms W, Q and a list of terms a such that the length of a is equal to thelength of H, W11 a a V, and

r[O] a VbWbQ.

An edge will be called H-critical if it is either II-ingoing or H-outgoing atsome vertex. An edge E is said to be initial if E begins at the root O° of

and there is a b-singular term W such that

Wr[O°] a VbVbb1.

It is clear that the root may have no more than one initial outgoing edge.Let 111[T] denote the number of occurrences of variables from fl in a

term T, In [T] the number of occurrences S of variables from II in Tsuch that there are b-singular terms W 1 , and W2, a term Q, and a listof terms a such that (i) the length of a is equal to the length of II ; (ii)T a Wb Wb Q ; (111) Wna a V ; (iv) S originates from an occurrence of avariable from H in W2.

LEMMA 10.6. The number of edges that are II-outgoing at a vertex O isat most 111[r[0]].

§ 10. UPPER BOUND ON THE PERIODICITY INDEX 69

PROOF. Suppose, on the contrary, that the number of edges that are H-outgoing at some vertex O is greater than In [r[O]] . Construct b-singularterms U', U2, W', and W2, natural numbers i, j, i j, and lists ofterms a 1 and a2 such that

V a WHa1 a WHa2 , (8)

r[O] a U' W' Vbbi a Ub WI )2 , (9)

and some occurrence of a variable from H in r[O] originates from oc-currences of this variable in both W 1 and W2

. Equality (8) and the b-simplicity of V imply that variables b 1 , b2, ... do not occur in W', W2,and V. Hence it follows from (9) that the terms Ub bi and Ub b are 1-regular, where l is the number of edges beginning at 0. Consequently nooccurrence of a variable from H in r[O] can originate from occurrences inboth W 1 and W 2, a contradiction. o

Suppose that the list H has the form a1, ... , ak. The following assertioncan be proved using Lemma 10.1.

LEMMA 10.7. For any terms Q, T1 , ... , T, and V -periodicity

QH T 1 ... Tk a Ub Vb ... Vb W

n times

if the term U does not contain variables from H and Q, T1 , ... , Tk do notcontain stable V -periodicities, then there are b-singular terms U°, X, Y, w..., W n , a term Q", and a natural number s, 1 <s < k, such that

8b[Y]>0, Ua UnT1...Tk,

yaWnT1...Tka...aWnT1...Tk,

and one of the following conditions holds:

;(a) n < 1, Q a Xbas , 1 a Yb Vb ... Vb W and U a [Xb Y]H T1 ..T k;

n times(b) there is a natural number i, 0 < n - i - 1 < 1, such that

i < 2l v[Q] + 1, iOb[V] + 0b[X] < h[QlH

Vb...VbWa[Xbas]HT1...Tk, T'saYb Vb...Vb W

n-i times n-i-1 times

(c) n8b[V] < h[Q], n < 2![Q] + 1, where

Q=UbWb1...WbQn, WaQnTI...Tk

Let where k is the length of H, be stable parametric treesof V-periodicities in unknowns x1 ,... , xl , and 0 , ... , 0 be the rootsof these trees. Consider a H-critical vertex O in 7. Rename the variablesb1, b2, ... in each term r[0], r[0], ... , r[0] and in the edges begin-ning at the corresponding vertices, to get new variables c1 , c2, ... (different

variables in each term). Call the resulting terms r* [O] , r* [0 ] , ... ,Construct the V-tree 011001 Ok of the term r* [O]nr* [01 ] r* [Ok ] ; byo o 0

Lemma 10.5 there is a unique such tree.Replace each H-critical vertex O in by 0H0 0. The places at

which the edges beginning at O are reconnected are determined by the oc-currences of the variables c1 , c2, ... assigned to the edges. The places atwhich the edges that begin at 0, ... , 0 are reconnected are determinedin a similar way. Rename the variables c1, c2, ... in the resulting paramet-ric tree, again calling them b1, b2, ... in such a way as to meet condition(2) of the definition of a stable parametric tree. Denote the resulting tree by

Using Lemma 10.4, one can prove that it satisfies conditions(1), (3), and (4) of the definition of stable parametric trees. Now delete from

all vertices for which condition (5) does not hold using thefollowing transformation.

Suppose that a vertex O of k} has an ingoing edge E ofdegree f with assigned variable b. and an outgoing edge E* of degree gwith assigned variable b1. Let r[0] = Vbb1 . Now delete O and merge Eand E* to form a single edge, of degree f+ g + 1 , assigned the variable b,.Denote the stable parametric tree of V-periodicities in unknowns x1 , ... , xlobtained by deleting all such vertices by

LEMMA 10.8. For any l -tuple of natural numbers a,

a] r[111 k' a].

This lemma is proved by induction on the number of vertices in 7.A sequence 0, E1, 01, ... , E, On (n > 0) of vertices and edges of a

stable parametric tree will be called a H-chain if, when n = 0, 00 is aH-critical vertex, and when n> 1 , the following conditions hold.

(1) E1 is a H-critical edge going from 00 to 01.(2) For all i (2 < i < n), El is a H-outgoing edge at 0_

1and an

ingoing edge at 0,.

Let 00, E1, 0,, ... , E, On be a H-chain. We introduce the followingnotation:

h[i[00]] - 1, if n = 0,n. h [i[O ]] - 1, if n> 1 and E1 is=o

h11[00, 01, ... , On ] H-outgoing at O,n

h[r[O3]], if n> 1 and E1 is notj=1

H-outgoing at 00.

The following assertion can be proved using Lemma 10.7.

LEMMA 10.9. Any edge E of the stable parametric tree sat-isfies one of the following conditions:

(1) the degree of E equals the degree of some edge of , , ... or(2) there are a H-chain 00, E1, 01, ... , E, On in and a natural

number c such thatn

deg E = deg E j +c,j=1

3n-3<c<min h11[00, 01,..., On],n-1

N

2 In [r[0;]] + n + 2ln[i[On]] -1 ;

=o

(3) there are a H-chain 0, E1, 0, ... , E, On in , an initial edge E°of one of the trees , and a natural number c such that In [i[0 ]] <ln[i[On]] , On is a leaf of , En is H-ingoing at On ,

n

deg E = deg E . + deg E° + c,j=1

n

3n-1< c < min hn[0o, 01 , ... , On] + 1 , 2i [r[0j]] + n + 2 .

1=0

Let T be a normal CTS-system in unknowns t1, ... , to and a-unknownsv', ... , v"; V a b-simple term that does not contain unknowns, a-unknowns, or a ; ...n , , ... , stable parametric trees of V -periodicities in unknowns z1, ... , zl . Assume that the list of term equalitiesof T has the form:

T1 =®1, ... , TN =®N

and that the list of equalities with substitutions has the form

S1 =v1aR1, ... ,SM=v1MQ RM.

We wish to find necessary and sufficient conditions for the trees ... ,n , 1 , ... , "q to determine a solution of f.

Let T, ®d , ... , TN , ®N , Sd , Rd , ... , SM , RM be the V-trees of the1 1

terms T1, ®1 , ... , TN, ®N , s, R 1 , ... , SM, RM, respectively. Such V -trees can be constructed using Lemma 10.5. Assume that

lt, t 1 n lt. t__ 1 n

Ntl tn 1 n Ntl t, 1 n '

ltl tn 1 n ila ltl t,,1 n

Mtl tn 1 n] r jMa Mtl t,,1 n

It follows from these equalities and from Lemma 10.5 that for all j (1 << N), the tree Td " 92J coincides with ®d " " up to the. tl...tn 1 n tl...tn 1 n

degrees of the edges, and for all s (1 <s < M), the tree "l*a R t 1 ns 1 n

coincides with Sd ...t "1 2lJn up to the degrees of the edges.1 n

Let be a list of all equalities of the degrees of edges similarly placed inthese trees. Let us call an unknown zl (1 < i < l) inessential if every linearpolynomial in in which the coefficient of z, is not zero equals z,. Close

with respect to transitivity and symmetry and write the resulting list ofequalities of linear polynomials as follows:

{f1f1l = 2

ftm1

m2

1 1 1 1=fkl=ul=u2=...=up1,

m m m m=fk =u1 =u2 =...=um p»:

where k1>0,p1>_0, kl+p1>0,...,km>0, pm>_0, km+pm>0;u i , ... , u 1 , ... , um , ... , um are pairwise distinct inessential unknowns;

p l pmfl. . . , f,', ... , fm, ... , f m are pairwise distinct linear polynomials1 , 1 k

that are not inessential unknowns. Let L'{T ; , ... , 92Jn, , ... , q }

denote the system of linear equations defined by all equalities of the form

.1 9

will denote the system defined by all equalities of the form fj' = u, where

the system defined by all equalities of the form u 1 = u, where 1 < j <m, k = 0, 1 < s < p3. Denote the union of these three systems byL{T ; ... 92J, ... , q }. We can now formulate the required nec-essary and sufficient condition.

LEMMA 10.10. For any l-tuple a of natural numbers, the list

1 ; a], ... , r[; a], r[; Q] , ... , T [ ti ; a]

is a solution of the normal CTS system T if and only if equalities (1) are satis-fied, all the constraints of r hold for the list r°[1], ... , r°[] , r°[] , ... ,

q n g

This follows from Lemmas 10.5 and 10.8.Let a be a solution of a normal CTS-system T. By the notation length of

a we mean the length a written out as a list of terms, that is, the total numberof occurrences of function symbols, object variables, and parentheses. Asolution a will be called r-minimal if 1[a] < r and there is no solutionaof T of lower notation length whose substitution width is at most r. Anunknown t3 (1 < j < n) will be called essential in T if it is either asubsituting variable or 8t [f] > 0.

J

LEMMA 10.11. The periodicity index of any r-minimal solution of a normalCTS system T is not greater than

(3r + h[r]) (v'2r + 3)2rq+no+1 +2,

where n° is the number of essential unknowns in T.

PROOF. Let T1 , ... , T, W1 , ... , Wq be an r-minimal solution of f,V a b-simple term that does not contain unknowns or a-unknowns. Lettl , t2, ... , t, be the essential unknowns in T and tno+1 , ... , to all theothers. It is obvious that, to prove the lemma, we need only find an upperbound for the lengths of stable V-periodicities.

If a occurs in V, the length of the V-periodicity is at most r. Assume... ,that a does not occur in V. Construct the V-trees T, ... , T,', d,

W d of the terms T1 , ... , T, W1 , ... , W by Lemma 10.5, and let l denoteq q

the total number of edges in these V -trees. Let z 1 , ... , zl be pairwisedistinct variables for rational numbers and replace the degrees of the edgesin all the V -trees Td , ... , T,, Wd , ... , w" by unknowns from z1, ... , zl

9(in such a way that each edge is assigned its "own" unknown). As a result, weobtain stable parametric trees ... of V-periodicitiesin the unknowns z 1 , ... , zl .

Let a be an l -tuple of natural numbers such that T1 = r[1; a], ..., Tn =r[; a], W1 r[; , Wgar[q;a]. By Lemma 10.10, a isaminimal solution of L{T ; , ... , n , , ... , q }. Now delete fromthis system all equations belonging to L{T; , ... , n , , Jq }(this is possible, since all the values of the unknowns in the equations of thelatter system are zero in a). Write the resulting system as

AX+BY=C,where the inessential variables are included in the column Y, the degreesof the a-critical edges of the trees ... , and of the initial edges of1, ... , 2"n in the column X, and C is a column of natural numbers. It

0

follows from Lemma 10.6 that the length of X is not greater than 2rq + no .Lemma 10.9 implies, in turn, that a maximal element d of C is not greaterthan 3r + h [F] and that the Euclidean norms of the rows of A are not greaterthan 2r +2. Consequently, the Euclidean norms of the rows of the matrix(A a C) do not exceed 2r + 3. To complete the proof, we need only applyLemmas 9.3 and 10.5. 0

Let Q be a term. We will say that s is an a-final degree of Q, if thereis an occurrence of the term ads+2) in Q which does not arise from anoccurrence of the term ads+3) in Q. We will say that an a-unknown v'(1 <j < q) is b'-essential in T if the list of equalities with substitutionsof T includes an equality with the right-hand side v' R , where R is a termthat contains an unknown for terms or h[R] > 0. Let o' be a solution of anormal CTS-system T. The number of distinct a-final degrees of the valuesin o' of a-unknowns that are b'-essential in T will be denoted by va[o ; IT].

Let be the b'-tree of Q. Clearly, no edge of can be a-outgoing atsome vertex. If an edge is a-ingoing at a vertex O of , then O is a leafof and i[O] a a'. It follows that a natural number s is an a-final degreeof Q if and only if s is the degree of an edge in which is a-ingoing atsome leaf of . Note that has an initial edge of degree m if and onlyif h[Q]=m+2.

A list , ... , PlJn, ... ,. of stable parametric trees of b'-periodi-

cities in an unknown u will be called a u' parametrized solution of a normalCTS-system IT if, for any natural number m, the list T[1; m], ... , m],T[; m], ... , m] is a solution of IT that contains no occurrences ofu.

LEMMA 10.12. For any solution aof a normal CTS system r and anynatural number 1, 1 <1 < n, one can construct a u'-parametrized solution1 , ... , sn

,i , ... , q of IT such that(1) the list r['1 ; 1], ... , r[' ; 1], r['1 ; 1], ... , 1] coincides

with a;(2)if

h[T!] > (h +2)(v +2)(vv+ 3)n0 + 2, (11)

where h = h[IT], v = vQ[v ; IT], T, is the ith element of o, andno is the number of unknowns for terms that are essential in I', thenthere is a natural number d, such that, for all natural numbers m,

h[z[; m]] = h[T,] +d,(m - 1),0< d< < ((ii +

1)V)"0_ 1

, h[I] > d,+2;(3)if

h,°[T,] > (h+2)(n0+2)(n0v+3)'' +2, (12)

§ 10. UPPER BOUND ON THE PERIODICITY INDEX 75

then there is a natural number b1 such that, for all natural numbersm,

m]] = h[I] + b.(fn - 1)0 < b1 < ((n° + 1)V)v , h[T ] > b1+2.

PROOF. Let obe a solution of a normal CTS-system T. Construct thelist d of the b'-trees of the terms in oin accordance with Lemma 10.5 andlet l denote the number of distinct degrees of edges in the trees of 0.d . Letz 1 , ... , zl be pairwise distinct unknowns for rational numbers. Replace thedegrees of the edges in all the trees of d by z1, ... , zl , in such a way thatidentical degrees are replaced by identical unknowns and different degrees bydifferent unknowns. As a result, we obtain a list 0'd* of stable parametrictrees of b'-periodicities in the unknowns z1, ... , zl .

Consider the systems of linear equations L{I' ; o,d* } and L' {f; ad * }. Letbe a natural-valued solution of L' {f; o,d* }. Clearly, can be extended to

a natural-valued solution + of L{I' ; 0.d*}. By Lemma 10.10, in turn,

determines a solution z[ +] of IT. Let L0 {I ; o,d*} denote the homoge-

neous system corresponding to L' {IT; 0 d* } and 1 a natural-valued solutionof L'{I; o,d*} such that z[1] is o. Let° be a natural-valued solution ofL' { IT;

0.d*} such that 1 - ° is a collection of natural numbers. It followsfrom Lemma 10.10 that, for all natural numbers m, T[(1 + (m - 1) ° )+ ]

is a solution of IT. Let " [ ( 1 + (u - 1)°) + ] denote the list of stable para-metric trees of b'-periodicities in u obtained by replacing the unknownsz1, ... , zl in the trees of o* by their values in the list1 + (u - 1)0 ,whereo is the natural extension of° to a solution of the homogeneoussystem corresponding to L{IT; 0d*

} .

It follows from Lemmas 10.7 and 10.9 that the equations in L' {I' ; a*}}

may be written°y. +k °x. = 1p y + k1x + kp

o J0 1 J l

where y., y. are the degrees of the a-critical edges of the trees in 0d*o i

corresponding to b'-essential a-unknowns in IT, x, x the degrees of theo

initial edges of the trees corresponding to essential unknowns for terms in° k° , p 1 , k 1 , k are natural numbers, 0 < p° < 1, 0 < k°IT, p , < 1,

0<p1 < 1, 0<k1 < 1, 0<k<h[T]+2.Let v = va[o ; IT], and let i be a natural number, 1 < i < n, T, the

i th element of o. We will show that [(1 + (u - 1)°)+] is the requiredu'-parametrized solution for an appropriate choice of° . Indeed, it fol-lows from the foregoing that condition (1) of Lemma 10.12 is satisfied. Ifh,,°[I ] satisfies neither (11) nor (12), we define ° to be the trivial solution

of L' {F; od * } If (11) holds but (12) does not, the required solution ofLo{I'; od"} can be obtained by applying part 1 of Lemma 9.5. If (12) holdsbut (11) does not, we apply part 2 of the same lemma. Assume now thatboth (11) and (12) hold. Then, if

[(V + I)V L]n°- [(np + l)v]V

we apply part 1 of Lemma 9.5, otherwise, part 2 of the same lemma.

§11. An algorithm deciding the existence of solutionsof restricted substitution width

Let I' be a correct CTS-system. Recall that 7c0I' t, 7c[n`I'].

LEMMA 11.1. For any correct CTS-system t and any natural number k,if k > y[F] + h[F] and for any i (0 < i < k), n`I' has an a-hanginga-unknown but no /3-hanging a-unknowns, then any solution o of nki' con-tains aperiodicity of length at least

k + /3[f] - a[o]+(fl[f]+h[f]+y[f]-1)a[] ' (1)

where c is the restriction of ato the unknowns and a-unknowns of T.

PROOF. Let T and k satisfy the conditions of the lemma. By Lemma7.2, 7c1 f , ... , ikr are correct CTS-systems. Let r = y[f] + h+[f] . UsingLemma 7.10, construct a list u00, uo , 1' , uo of pairwise distinct freea-unknowns of crf such that l > 0, the list connects uo , o with uo in

and uo 1 nrj u0 , 0 We will assume that there are no natural numbersi and j , 0 < i < 1, 0 < j < 1, 0 < Ii - ii <1, such that u0 1 nrr u0.

Now, using Lemma 6.5, construct an so-place function symbol g0, a nat-ural number po , and a-unknowns wo , o ,1 ' w0050 of 7ZT such that

1 po so u0,1 nrr W000

U00 _nrr g0 w0,0, 1' ... , W000,,0 , ... , w00so).

Now, using Lemma 6.7 and condition (2) of Lemma 7.2, construct, fork> r a-unknowns wo,l,i, ... , wo,j,so of r+1

such that, for all d (1 < d < so ), the list wo , o , d' wo, 1 ,d' , wo ,1, d con-

nects woo d to wo 1 d in cr+1f and, for all j (1 <j < 1),

u0, -nr+ir go w01' ... ,We introduce the following notation:

u1,0 ± w0,0,pa,

u1,1 ± w0,1,po L... , u1,1 w0, j, po.

Note that u1 1 < r+I r u 1 , o . Hence the above considerations are also appli-cable to the list u1 , 0' ... , u1 1Performing k - r - 1 more steps, we obtain

§11. EXISTENCE OF SOLUTIONS OF RESTRICTED WIDTH 77

natural numbers s1, ... , Sk-r-1 pl , ' pk-r-1 a-unknownsuk_r,o, ... , uk_r,1, w1,0,1 , ... , wl,o,sl , ... , wk_r_1,1,1

that sat-wk-r-1 1 s of 7ZrI' and function symbols gl , . . gk-r-1k-r- Iisfy the following conditions for all i (0 < i < k - r -1) and j (Oj < l):

(1) gi is an si-place function symbol, 1 < pi < si and

u1,l ~nkr g1(w1 , .. , wi,j,P .0

(2) ui+1,1 nkr ui+l,o' ui+1,0 a w1,o, ui+1, 1 a Wi, 1,P;' ... , u1+1,1 a

(3) for any d (1 < d < Si ), the list W1, o , d' wi , 1 , d' ... , W, d connects

wi,o,d to Wild in 2 r.Let v be one of the a-variables of irT just constructed. Then v is free

in 7Ck r by condition (6) of Lemma 7.2, and hence ankr [v ] = 0.Let o be a solution of T. Then expression (1) is meaningful in view of

Lemma 6.4 and the correctness of T. Without loss of generality, we willassume that (1) is greater than 1. It follows that

k> /J[f]+2r-1. (2)

Using Lemma 6.5, construct a chain SZ in nrr with beginning u0,0 andend uo and let m denote the length of SZ . Construct also a chain Qin kr with beginning uk-r 0 and end uk-r 1

It follows from Lemma6.10 that the union of the chains u0 , 0' p0, U1 , o p1, , uk - r , 0 and Qcoincides with the union of the chains u01, p0, u1 , 1 , p1, ... , uk-r

,and,l

SZ . Condition (3) of Lemma 7.2 implies that

m fl[f]+r. (3)

Hence, it follows from (2) that uk-r, o coincides with uk-m-r+1 ,1 Weintroduce the following notation for all i (k - r < i < k - r + m - 2):

ui+l , 0 ui-m+2,1 gi gi-m+1 si Si-m+1 '

Pi pi-m+1 , wi, o, d wi_m+l, l , d (1 d < Si).

If 0 < i < k - r - 1, we let Di denote the list of all a-unknowns wi d ,where 0< j <l, 1 <d <si, d dpi; if k-r< i <k-r+m-2, itwilldenote the list of all a-unknowns W1,o , d' where 1 <d < si and d pi .

Lemmas 6.4 and 6.10 imply that the following conditions hold for all i

(O<i<k-r) and d (1<d<S1):(1) ui,1 a ui+m-1 , 0 g1 a gi+m-1 Si = si+m-1 ' p1 - pi+m-1 ' wi , l , d

ai+m-1 ,0,d'

(2) the list wi , o , d , wi , 1, d , wi , l - l , d , wi+m- l , 0 , d connects wi , o , d to

? r',wi+m ,

-1,0,d ink

(3) if d p1, >vEL, la[v ]>vEi1+,,_1

la[v ] 0, then

a aWild Wi+m_l,O,d

Using Lemmas 6.10 and 6.11, one easily proves thatI k-r+m-2

la[uk-r,j] + la[ves] < ankr[ 7]. (4)j=1 i=0 vEO;

Note that the list uk-r, 0' uk-r, 1 ' uk-r, I connects uk-r, 0 to uk-r, I

in Hence there is a substituting term R of r such that uk-r,1 1'

uk_r I connects uk-r oaR to uk_r IaR in k Hence, using Lemmas 6.1and 6.4, we obtain

1 _Qla[uk_r j] >0.

j=1

This inequality, taken together with inequality (4) and condition (4) ofLemma 7.2, implies

k-r+m-2i >la[Va]<ar[Jr].i=0 VEO;

Using this inequality, we construct natural numbers c and d, 0 < c <d < k - r + m - 1 , such that

k - r + m - ar[crr] d-'- -

[o' ]ar r i=c v E01

Hence, by the proven properties of the chain

u0,0 p0' u1,0' P1 "'it follows that the term uo , 0 begins with a periodicity of the length at least

k-r+m-ar[°r](m - 1)ar[o'r]

To obtain the required lower bound on the periodicity index of a, weneed only use inequality (3). 0

LEMMA 11.2. Any solution o' of a normal CTS-system T can be trans-formed into a solution a° of T such that J"[o°] < l r[a] anda a

h[a°] < h+[f] (p + (lc[f] + ar[o']! , (5)

where p is the periodicity index of

PROOF. Let T be a normal CTS-system and oa solution of F'. We firstconstruct a tree "r of a-extensions of F. By condition (2) of Lemma 7.4,

§11. EXISTENCE OF SOLUTIONS OF RESTRICTED WIDTH 79

o singles out a finite branch ' in r that begins at the root and ends ata vertex O such that E(O) has no a-hanging or /3-hanging a-unknowns.Using Lemma 7.11, we can construct a solution o of E(O) such that

iE(O)[o.l]_a*[:(o)] h[o'] <y[(0)]+h[f]. (6)

Let ° be the restriction of Q1 to the unknowns and a-unknowns of F'.

Since v can be extended to a solution of E(O) , it follows from Lemma 6.12

that a`[E(O)]<lQ[Q]. Hence-r o -E(O) 1 -r

We now prove inequality (5). It is obvious that h[Q°] = h[Q']. Letp be the periodicity index of o. Denote the root of r by Oo and letOl , ... '°k be a list of all vertices through which ' passes and which haveingoing edges with assigned a-unknowns. It follows from Lemmas 7.2 and7.3 that k = a[E(O)] ,

aryl >_ k, Y[(Ok)]= Y[(°)] (7)

Let of (0 1,

o

p> 1 ,and K[f] > 1. Using Lemmas 7.2, 7.3, and 11.1, we easily prove that

fl[(O)] <_ (fl[(O)] + h[f] -1)aE(oo)[oo](P+

h[f](p + 2)ar[o],i> < h+[r](p+2)(K[r]+l)ar[o].

Similarly, we can prove that, for all i (1 < i < k),(fl[(0)] 1)

h+[I'](P +2)`(K[I'] + l)i+'(p + l)ar[o]... (ar[Q] _ ] )

h[F] + y[(O1)] <h[f] + y[(O1)] + K[r]/3[:(011)]

< 2)t (K[I'] + 1)i+iar[7] ... (ar[Q] - i).

Consequently,

h+[r] + h+[r](P + 2)k(x[I'] + 1)kar[Q] ... (ar[o] - k + 1).

Inequality (3) follows from this and from (6) and (7). o

THEOREM 11.1. For any natural number r, there is an algorithm that de-cides whether a given CTS-system I' has a solution of substitution width atmost r.

This follows from Lemmas 6.9, 6.12, 10.11, and 11.2.

CHAPTER III

Logical Deduction Schematain Axiomatized Theories

A proof schema in an axiomatized theory is a sequence of analyses ofaxioms and inference rules. We will write proof schemata in KGL(2() andIGL(2() as planar rooted trees, with analyses of axioms assigned to the leavesand analyses of k-premise rules assigned to vertices with exactly k ingoingedges (k > 0). Schemata in KH(2l) and IH(2l) will be written as linear se-quences of analyses of axioms and applications of inference rules. In additionto KH(2I) and IH(2I), we will also consider other axiomatized Hilbert-typetheories, where axioms and inference rules will be defined using enumerablepredicates on the set of formulas of the predicate calculus. Proofs in thesetheories will be written in linear form with analyses of applications of ax-ioms and inference rules. The analyses will indicate the code of the axiomor inference rule and the indexes of the premises used. Throughout, axiomsare treated as 0-premise rules.

Let U be a logical deduction schema. A deduction in accordance with U isany proof whose list of analyses of axioms and inference rules applications isprecisely U. A schema U is admissible if there exists a proof in accordancewith U. A formula A is said to be deducible in accordance with a schemaU if there is a deduction in accordance with U which constitutes a proof ofA. In this chapter we construct an algorithm that decides whether a logicaldeduction scheme is admissible in the calculi KGL, I GL , KH, and I H .We will also prove that the problem of recognizing deducibility by a givenschema is algorithmically undecidable for a broad class of extensions of KHand IH.

In this chapter we will also prove that deduction schemata are sufficient toaccomplish all the transformations described in Chapter I.

The main results of this chapter were announced in [23] and published in[25, 27, 29].

§12. Systems of equations in formulas

We will need a potentially infinite list X, Y, Z, ... of rrtetavariables(unknowns) for formulas. The notion of a formula schema is defined by

81

82 III. LOGICAL DEDUCTION SCHEMATA IN AXIOMATIZED THEORIES

induction: (a) formulas of the predicate calculus are formula schemata; (b)metavariables for formulas are formula schemata; (c) if d and are for-mula schemata, then the expressions (d ), (d & ), (d V a) , and-id are formula schemata; (d) if d is a formula schema and x an objectvariable, then the expressions dxd and xd are formula schemata.

Let A be a formula of the predicate calculus, x an object variable, andt a term. The expression xEA will mean that x does not occur free in A,the expression t * x * A will mean that term t is free for x in A.

Let X 1, X2, ... , Xn be a list of pairwise distinct metavariables for for-mulas; t1 , t2, .. , tm , a 1 , a2 , ... , ap pairwise distinct object variables. Asystem of equations and constraints in formulas (a CF system) in unknownsfor formulas X1, X2, ... , X,, unknowns for terms t 1 , t2, ... , tm , and un-knowns for object variables a1, a2, ... , ap consists of a list of equalities

a` 1, ... '"N

1 a []T , ... , M a[M]T , (2)I M

a list of expressionsx12, ... , (3)

and a list of expressions

R1*y1*3'1,...,RL*yL*9'L. (4)

Here N+M>0; S+L>0; T!,..., TM,R1,...,RL are terms; al, ...,aM, xl , ... , xs , y 1, ... , yL are object variables distinct from t 1 , ... , tm ;

are formula schemata that do not contain metavariables forformulas other than X1, ... , X,, and in which the variables t 1, ... , tm arenot bound. We will assume that the list (4) includes the expressions

TM*aM* M.

Let Cl , ... , Cn be a list of formulas, 8l , ... , em a list of terms, z 1 , .

zp a list of object variables. The tuple

(Ci,...,Cn;Oi,...,Om;zi,...,zp) (5)

will be called a solution of the CF system (1)-(4), if, upon simultaneous re-placement of all occurrences of the unknowns X1, ... , Xn in (1)-(4) byC1, ... , C, respectively, of the unknowns t 1 , ... , tm by 01 , ... , °m' andof the unknowns a1, ... , a, by z1, ... , zp , followed by performance ofthe required substitutions for free occurrences of variables in (2), equalities(1)-(2) become true graphic equalities and (3)-(4) valid expressions. By theheight of a solution (5) we mean the maximal height of the subterms occur-ring in formulas and terms in (5).

§12. SYSTEMS OF EQUATIONS IN FORMULAS 83

Let x be an object variable, A a formula. The maximal value of lX [ T]

for all terms T whose occurrence in A generates a free occurrence of x inA will be called the rank of A relative to x. Natural numbers s for whichthere is a free occurrence of the term in A that does not arise froman occurrence of x(s+3) in A will be called free x -final degrees of A. Leto be a solution of a CF-system (1)-(4). For all i (1 < i < M), we let 9(respectively, a') denote the result of simultaneous replacement in s (a1)of the unknowns by their values in o. The maximum rank of S relativeto a' for all i (1 < i < M) will be called the rank of o. The union of thesets of free ar-final degrees of for all i (1 0) we mean a rootedtree with a formula assigned to the root, terms assigned to the other vertices,and a variable from the list b1, b2, ... , together with a linear polynomialin u1, ... , ul with natural coefficients, assigned to each edge. The linearpolynomial assigned to an edge will be called the degree of the edge. Aparametric c0-tree will be called a c0-tree if the degrees of all its edges arenatural numbers.

Let be a parametric cD-tree in unknowns u1, ... , ul and O the rootof . Suppose that a formula A is assigned to O ; E1, ... , ES (s > 0) arethe edges beginning at O ; for all j (1 <j <s), a variable b. is assignedto E ; and E3 ends at a vertex O . Assume that 1 < i

1< i2 <_ <_ is.

A parametric cP-tree will be called stable if, for any j (1 < j < s), thefollowing conditions hold.

(1)(2) A has no bound occurrences of b.(3) b occurs free in A exactly once.(4) The free occurrence of b in A arises from an occurrence of a term

b in A.(5) For any natural number k, if k + 1 > s, then bk+

1does not occur

in A ; but if k + 1 <j, then bk+1

occurs to the left of b in A.(6) For any term Q, the term Q" does not occur in A.(7) The subtree of with root O is a stable tree of b'-periodicities in

the unknowns u1, ... , ul .(8) The term T[O] begins with '.

Let be a stable c0-tree. Let y'[] ] denote the formula obtained whenthe variables b 1, ... , bs are replaced in A by

t[1 ... , ,s

respectively, where, for all j (1 <j <s), is the subtree of with rootO and f is the degree of E.

Let cb be a stable parametric c0-tree in unknowns u1, ... , ul and f1, ... ,

f natural numbers. Let lr [ ; f, ... , f] denote the formula lr [c* ] , whereis the stable cP-tree obtained from by simultaneous replacement of u 1

by f, ... , ul by f in the degrees of the edges.Let ... , Jn be a list of stable parametric cP-trees in an unknown u,

a list of stable parametric trees of b'-periodicities in u, andz1, ... , zp a list of object variables. The tuple

(,...,c1,,; '. zm, 1,...,zp

will be called a u parametrized solution of the CF-system (1)-(4) if, for anynatural number f, the tuple

(v[I1 ; f], .. , v[I ;f];does not contain occurrences of u and is a solution of system (1)-(4).

Let T be the CF-system consisting of the lists of equalities (1)-(2) andlists of constraints (3)-(4); o the list of all object variables other thana1, ... , ap and t1, ... , tm occurring in the terms T1, ... , T, R1, ... ,RL, a1,... ,

a, x1,... , xs , y 1, , yL and in the formula schemata `/1,,...,t' ,...,S

the list of all function symbols occurring in the above terms andformula schemata; 2 the list of all predicates occurring in them; II a listof terms which is the union of the lists a 1 , ... , ap , t 1 , ... , tm , o , andthe list of all terms occurring in T1, ... , TM, , ... , dN, 1 , ... , MN ,

The expression Ax Q , where A is a formula, x an object variable, andQ a term, will denote the result of replacing all occurrences of x by Qin A (including occurrences in quantifier complexes). Let a be an objectvariable distinct from a1, ... , ap , t 1 , ... , tm , and the variables in L. Weintroduce new variables for formulas, V', ... , VM ,and let I' be the systemconsisting of the list of equalities (1), the list of equalities with substitutions

aJ/ T1, T1 aV'a1,..., 'MaVQ TM, 7 VaMaM,

and a list of constraints containing exactly the expressions of the form®a1, faa,, PEa1, ®t3, PEt3, aryl, anda, where 1 <i <p, 1 <j <m, 1 <k S, 1 <1<_L; ®= ,

&, V, , V, ; f is any function symbol in1; P a predicate in 2

We will consider F as a CTS-system in unknowns t 1 , ... ,tm , X1, ... , X,,, a-unknowns V 1 , ... , V

M, function symbols , &, v, ,

V, , , and object variables a, o . Here , &, v, V, and will betreated as 2-place function symbols, as a 1-place function symbol, and1-place predicates as 1-place function symbols.

Let o be a solution of a CF-system F. If a occurs in a, we replace it bya new object variable. Then we add to a the formulas [ 9 ]Q' , ... , [9]',

'

where , ... , M are the results of replacing the unknowns in1 , ... ,M , respectively, by their values in a. In this way we obviously obtain asolution of the CTS-system T, which we denote by &.

Let A and B be formulas. We will say that A is a q*-type of B, if Ais a q+-type of B and A is transformable into B. Clearly, a q*-type canbe constructed for any formula, and q*-types of the same formula coincideup to renaming of free variables. We will say that A is q*-equivalent to Bif some formula is a q * -type of both A and B.

Let A denote the list , &, v, - , V, . Let k be a natural number suchthat, for all i (1 < i < k), there is an a-unknown that is a-hanging relativeto A in 7TA, but there is no a-unknown that is /3-hanging relative to A.The following assertion can be proved using Lemma 6.1, by induction on k.

LEMMA 12.1. For any solution a of a CF-system T, there exists a solutioncJk of the CTS-system lr TA that satisfies the following conditions.

(1) The restriction of cJk to the unknowns and a-unknowns of r coincideswith &.

(2) For any lc"I'A-atom A, if A°k is a formula, it does not contain abound occurrence of a.

(3) For any lc"TA-atoms A and B, if Ask is a formula and A n I'n

then B° is q*-equivalent to ACk and consequently st[B°k ] = st[A°k ] .

We introduce the following notation:

sv* [r] S°A[t'] ' h* [r] h [IT], h* [r] h [F].A CF-system F will be called admissible if there is a natural number k

such that

(1) k<_ sv* [r](2) for all i (1 < i < k) there is an a-unknown that is a-hanging

relative to A in fl FA , but there is no a-unknown that is /3-hangingrelative to A ;

(3) FA has neither a-unknowns /3-hanging relative to A nor a-unknowns a-hanging relative to A,

1;

(4) r is a correct (hence also admissible) CTS-system;(5) there exist no natural number i, 1 < i < n, and term Q such that

Q is included in fl and Xl nkr Q ;n

(6) for any a-unknown w of 7r rA and any term Q in fl, if w nkrn

Q, then Q is included in the list a1, ... , ap , t 1 , ... , tm , and

there is an a-unknown w° that begins with d or , such that w isthe 1-argument of w°;

(7) there are no lr FA-atoms A and B such that A nkr B andn

A :nk r

B.n

The following assertion can be proved using Lemmas 7.2, 7.7, 7.10, and12.1.

LEMMA 12.2. If a CF-system has a solution, it is admissible.

Let k be a natural number satisfying conditions (1)-(7) of the definitionof admissible CF-systems. Let SZ1 denote the list of all minimal a-unknownsin lr rA that are not a-hanging, and SZ2 the list of all a-unknowns that area-hanging in lrkrA . If SZ2 is empty, we will denote Tt"TA by F. OtherwiseF will denote ltltkTA . Let SZ2 denote the list of the minimal a-unknownsin F that are not a-unknowns in lc"FA.

Let B1, ... , Bq be a list without repetitions of all the basic r-atoms of the

form wQ T, where w * belongs to S and T is a substituting term of F. Weintroduce new unknowns for terms, t i , ..., t. Let I'* be the TS-systemwith unknowns t1, ... , tm , t, ... , tq , al , ... , ap and a-unknowns SZ2consisting of the list of term equalities containing just those equalities thathave the form Q 1 = Q2 or tl = t , where Q1, Q2 are terms in fl, Q 1 :nk r

nQ2, 1 < i < q, 1 < j < q, and B. ^-r B3, the list of equalities withsubstitutions containing just those equalities that have the form Q = Bl ort = B3, where 1 < i < q, 1 < j < q, Q is a term in fl, and Q ^-rB.. Lemmas 6.1 and 12.1 imply that any solution of F of rank r can betransformed into a solution of T* of substitution width r.

Let {A1, ... , A1; b1, ... , bl } be a basis of F. Fix some 0-place predicatePo and define Ql for all i (1 < i < l) as follows. If Al is a lr rA-atomand there is a natural number j, 1 <j < n, such that

X ^ nk? Al ,n

put Ql a Po . If Al is a lr TA-atom and there is an object variable x ino such that x k? A1, put Q. a x. If Al is a 7 FA-atom, the previous

ncondition does not hold and there is a natural number j, 1 <j <p, suchthat

a j~nk?n

Al , (6)

put Ql a ado , where jo is the smallest j satisfying condition (6). If thereis a natural number s, 1 < s < q, such that

BS ^r Al (7)

put Ql a t, where so is the least s satisfying condition (7). If there is an0

a-unknown w * in S such that Al a w * or A, a wQ a , put QI a w * . Ifnone of the above five cases holds, put Q, a A.. It follows from Lemma6.8 that either Ql is included in the list Po , o , a 1 , ... , ap , t 1 , ... , tm ,

ti , ... , t, SZ2 , or A is not a basic f-atom and Ql has the form wQ T,where w * belongs to SZ2 and T is a substituting term of F.

Let C be a F-atom and a a solution of F*. In what follows we will needthe following notation:

{C}a {C , F'}b ...b Q1 ... QI

where Po a Po by definition.

LEMMA 12.3. For any solution a of the TS-system r* , if a does not occuran the terms t1 , ... , t, al , ... , a;, and a, ... , ap are object variables,then

(a,...,a,t,...,t,{Xi}0,...,{Xn}0,{Vl}0,...,{VM}c7) (8)P

is a solution of f.

PROOF. Let a be a solution of r* that satisfies the conditions of thelemma. It can be proved by induction on - that for any a-unknown w ofF

{WaT}° {w}{T}°,

where T is a substituting term of F. It follows from this graphic equalityand from condition (4) of Lemma 7.2 that (8) is a solution of F.

We will say that a CS-system I' is equivalent to a collection of CTS-systems

1 , ... , "S (s >_ 1) with a-unknowns if the following conditions hold.

(1) The unknowns t1 , ... , tm are included in the list of unknowns ofEl for all i (1 < i <s).

(2) For any solution a of F there is a solution a* of one of the sys-tems E1, ... , S such that the restriction of a to the unknownst 1 , ... , tm coincides with the restriction of a* to the same un-knowns, the rank of a is equal to the substitution width of a*,the height of a is equal to the height of a*, and the sets of all a-final degrees of the values of the a-unknowns in a* are subsets ofthe set of free final degrees of a.

(3) Any solution T of one of the systems 1 , ... can be transformedinto a solution T' of F such that the restrictions of T and T' tot 1 , ... , tm are the same, T and T' have the same height, the substi-tution width of T is equal to the rank of T' , and for any formula Ain T ,

st[A] < cP* [r] + 1.

'(4) Any - u -parametrized solution T* of one of the systemscan be transformed into a u'-parametrized solution yr* of F suchthat the restrictions of t* and yr* to t1 , ... , t,n are the same.

In what follows we let arg[f] denote the maximal dimension of the pred-icates occurring in the equalities and constraints of the CF-system F, m [F']the number of unknowns for terms in IT, M[f] the number of equalitieswith substitutions in F, and M°[r] the number of equalities with substitu-tions in r in which the substituting term contains an unknown for terms orbegins with symbol '.

LEMMA 12.4. For any admissible CF-system F there is an equivalent col-lection of CTS-systems E, ... , S such that, for any i (1 < i < s), thefollowing conditions hold.

(1) l is a normal CTS-system.(2) l has no more than arg[F] M[r] a-unknowns.

(3) l has no more than arg[I,] M°[r] a-unknowns which areb'-essential in l .

(4) The list of function symbols of El is included in the list of functionsymbols occurring in the equalities and constraints of F.

(5) The number of essential unknowns in l is not greater than m[F].(6) ,c[,] = 2, h[E1] = h*[F], h* [r].

PROOF. Let us add to I'* all expressions of the form:

}b...b,Q1 ... QI ,

R j * y j * }b, ... b, Q 1 ... QIu

auE{V , Q1

where 1 < i < S, 1 < j < L, 1 < u < M, denoting the resulting systemby . We now transform into the required collection of CTS-systemso 0

in two steps.At the first step we use the following transformations:(a) If the system contains a constraint of the form

xEQuX, (9)

where Q a V, , it is transformed into a pair of systems, in one of whichthe constraint (9) is replaced by the equality x = u, and in the other by theconstraint x EX .

(b) If the system contains a constraint of the form

(10)

where Q a V, , it is transformed into a pair of systems, in one of whichthe constraint (10) is replaced by the constraint yEQu9', and in the otherby the two constraints and R * y * .

(c) If the system contains a constraint of the form

xEP(81, ... , O) , (11)

where P is an l-place predicate in 2' it is transformed into a system inwhich (11) is replaced by the list of constraints x81 , ... , x81.

(d) If the system contains a constraint of the form

R*y*C, (12)

where C is a quantifier-free formula, it is replaced by the system obtainedby omitting the constraint (12).

The transformations corresponding to the connectives j, &, v, andare trivial. We apply the above transformations successively to Eo and tothe systems thus obtained, until we obtain a collection of systems E, ... ,which do not include symbols in A, 2 .

At the second step we delete from , ... , E° the unknowns for ob-ject variables a1, ... , ap . To this end we introduce new object variables

°, ... ,at,... , a;. The final collection of systems is obtained by applying to

E° all possible substitutions of the variables in the list a i , ... , a, o forP

a1, ... , ap . The resulting collection of CTS-systems clearly satisfies condi-tions (1)-(5), and it is equivalent to F by Lemmas 12.1 and 12.3

We now define the notion of a q-type schema inductively: (a) predicatesymbols and metavariables for formulas are q-type schemata; (b) if Q1 andQ2 are q-type schemata, then

(Q1Q2), (Q1&Q2), (Q1vQ2), -Q1, VQ1 , Q1are q-type schemata. A q-type schema will be called a q-type if it does notcontain metavariables for formulas.

Let d be a formula schema. We let q [d ] denote the q-type schemaobtained from d by deleting the variables in all occurrences of quantifiersin d and deleting the terms and parentheses in all occurrences of elementaryformulas in d. Let Q be a q-type. We let coq [Q] denote the number ofdifferent q-types occurring in Q that are not predicate symbols. Clearly, forany formula A,

st[A] < coq[q[A]] + 1.

We will say that a list of q-types Q1, ... , Qn is transformable into alist of q-types R1, ... , Rm if n = m and R1, ... , Rm can be obtainedfrom Q1, ... , Qn by simultaneous substitution of q-types for predicate sym-bols. We will say that Q1, ... , Qn is transformable into a list of formulasA1 , ... , Am if Q1, ... , Qn is transformable into the list q [A1 ] , ... , q [Arn ]

Let F be a CF-system with unknowns for formulas X1, ... , X,,, con-sisting of lists of equalities (1)-(2) and lists of constraints (3)-(4). Letq[F] denote the following system of equations in q-types with unknowns

1 , "' , n '

J q[ ] a a q[`N]a a q[f].

By a solution of q[T] we mean an n-tuple of q-types (Q1, ... , Q,) suchthat, upon simultaneous replacement of X1, ... , Xn by Q1 , ... , Q, re-spectively, the equalities of q[r] become true graphic equalities.

Let a be a solution of r of the form (5). Let q[a] denote the tuple(q[C1}, ... , q[Cn]) . Clearly, q[a] is a solution of q[I,]. A solution ofq[T] will be called r-universal if, for any solution p of F, the tuple q[p]can be obtained from by simultaneous replacement of q-types for thepredicate symbols not occurring in the equalities of q [r] .

Let z be a solution of q[f]. Let cpq[z] denote the number of differentq-types occurring in T that are not predicate symbols.

LEMMA 12.5. If q[r] has a solution, we can construct a r-universal solutioni of q [f] such that

coq [z0] < cv* [r].

PROOF. Assume that q[f] has a solution. Let us treat system q[f] as aT-system, with , &, and V considered as 2-place function symbols, , d ,and as 1-place function symbols, predicate symbols considered as objectvariables, and the unknowns X1, ... , Xn as unknowns for terms. UsingLemma 7.12, we construct a universal solution i of the T-system q [f]such that

coq[z0] = SPA[zo] <_ SPn[q[r]] <_ cP*[r]

t is obviously a r-universal solution.Let A be a formula, P an 1-place predicate, B a q * -type of A. Let

bl , ... , bs be the list of all free variables in B, arranged so that if i <j,1 < i, j < s, then bl occurs free in B to the left of a free occurrence ofb3. Let Tl , ... , TS be terms such that A is obtainable from B via simul-taneous replacement of the free occurrences of b1, ... , bs by T1, ... , T,respectively. (It is assumed that Tl is free for b1 in B, ... , TS is free forbs in B.) We introduce the following notation:

P(Tl , ... , TI) , if l < s,P(T1,...,TS,...,TS), ifl>s.

Let Q be a q-type that is transformable into A. We define a formula[A/Q] by induction on the construction of Q, as follows. If Q is a predicatesymbol, [A/Q] has already been defined. Assume that Q is (Q1 O Q2),where O j, &, V. Then A has the form (A1 O A2) and the list of q-typesQ1 , Q2 is transformable into Al , A2. We then define

[(A1 0 A2)/(Q1 O Q2)] a ([A1/Q1]®[A2/Q2]).

The case in which Q has the form SZQ1 , where S = , V, , is similar.It is obvious that q[[A/Q]] coincides with Q and [A/q[A]] with A.

LEMMA 12.6. For any formula A, q-type Q, and term T, if Q is trans-formable into A, then the following conditions hold.

(1) xEA if and only if xE[A/Q].

(2)IfT*x*A,thenT*x*[A/Q].(3) If T * x * A, then [[A]/Q] a [[A/Q]]T

This lemma is proved by induction on the construction of Q.Let a be a collection of formulas, terms, and object variables of the form

(5), and T a collection of q-types (Q1, ... , Q). Assume that T is trans-formable into a. We let [a/T] denote the collection

([C1/Q1], ... , [Cn/Qn]; 81 , ... , Bm; z1 , ... , zp).

Let d be a formula schema occurring in some equality or constraint ofIT and Si be a q-type schema occurring in some equality of q [r] . Let ddenote the result of simultaneous replacement of the unknowns X 1, ... , Xn

by C1, ... , C,,, of the unknowns t1, ..., tm by O, ... , 8m , of the un-knowns a1, ... , ap by z1, ... , zp , and by T the result of simultaneousreplacement of the unknowns X1, ... , Xn by Q1, ... , Q. Clearly, q [d Q

]

coincides with q [d ]q['] .

LEMMA 12.7. For any solution a of a CF-system F and any solution T ofq[f], if r is transformable into a, then [a/T] is a solution of F.

PROOF. Let a be a solution of F and r a solution of q [r] . Assume thatr is transformable into a, and let P1, ... , PI be the list of all predicates inr replaced by q-types distinct from themselves under the transformation ofT into a. Let P, ... , PI be predicates that do not occur in T and q[r]such that, for all i (1 < i < 1), the dimensions of Pl and Pl coincide. LetR be a q-type. Denote the result of replacing P1, ... , PI by P, ... , PI inR by R*, the result of replacing P, ... , P7 by P1, ... , PI in R by R1 ,

the result of replacing P, ... , PI in R by the values of P1, ... , P, underthe mapping of r into a by R. Let r' (respectively, T 1 , T ° ) denotethe result of termwise application of the operation * (1, ° ) to T . Clearly,T*1 coincides with T and T*° with q[a]. The following proposition canbe proved by induction on the construction of q-types:

For any q-types R 1 and R2, if R 1 a R2 and Rf a R° , then R1

a R2.This fact and the equalities

[`J T*]J a `J T,

[`J T*]° 9 Q]

where S is a q-type schema occurring in some equality of q [r] , imply thatT* is a solution of q [r] .

Let d be a formula schema occurring in some equality of F. The fol-lowing graphic equality can be proved by induction on the construction of

[dc/q[d]T*]It follows from this equality and from Lemma 12.6 that [a/T*] is a so-

lution of F. In order to complete the proof, we need only notice that


replacement of P, ... , P by P1, ... , P, transforms [Qli*] into [Q/r]and solutions of IT into solutions of F. o

We will say that a CF-system t is q -free if every object variable thatundergoes substitution in some equality with substitution in I' is an unknownfor object variables, and every object variable that occurs explicitly in thesubstituting terms, equalities, equalities with substitutions or constraints ofI' is either an unknown for terms or an unknown for object variables. It iseasy to prove the following assertion:

LEMMA 12.8. For any q -free CF-system I' with a solution a, the followingtransformations map a into a solution of I':

(1) replacement of all occurrences of the value of an unknown for objectvariables in a by a new variable;

(2) replacement of all bound occurrences of an object variable, which is notthe value in a of an unknown for object variables, by a new variable;

(3) substitution of a term, in which there are no variables that are boundin a or are values in a of unknowns for object variables, for all freeoccurrences of an object variable in a that is not bound in a and isnot a value in a of an unknown for object variables.

Let T be a CF-system with n unknowns for formulas, m unknownsfor terms, and p unknowns for object variables. Fix two different objectvariables, x and y. Let r be a solution of q[F]. We let r[T] denote thetuple

+,(r , x,...,x, y,...,y),m times p times

where T+ is obtained from r by appending y to all occurrences of d and2 and adding x to all occurrences of predicates as many times as necessary.

We will say that a CF-system IT is degenerate if F is q-free and no ele-mentary formula that occurs explicitly in some equality, equality with sub-stitution, or constraint of IT includes function symbols or object variablesother than unknowns for terms. An occurrence W of a quantifier complexQu in a formula A, where Q a V, , is called degenerate if u does notoccur free in the scope of W. A formula A is called degenerate if all itsquantifier complexes are degenerate. A solution a of a CF-system F will becalled degenerate if a includes only degenerate formulas. It is easy to provethe following assertion.

LEMMA 12.9. For any CF-system F and any solution T of q [r] , if F isdegenerate, then r[T] is a degenerate solution of F.

Let S be a formula or a sequent, xl , ... , xl a list without repetitionsof object variables, Tl , ... , TI a list of terms. Let [S]T ' T denote theresult of simultaneous replacement in S of all free occurrences of xl byTl , ... , all free occurrences of xl by 1. Let P be an s-place predicate

and V1 , ... , Vr a list without repetitions of all occurrences of elementaryformulas in a formula A that begin with P. Suppose that for all i (1 <i < r), V is an occurrence of a formula P (T1 , ... , T'). We will say thata formula B is obtainable from A by substitution of a formula C withdistinguished parameters x1, ... , xl for P, if l < s, B can be obtainedfrom A through simultaneous replacement of V by a formula [C]x'i'

TJ ,Tl !

for all i (1 < i < r), in such a way that the following constraints are satisfied:

`. is free for x3(1) for all i and j (1 < i < r, 1 <J < 1), the term TJin C ;

(2) for any i (1 < i < r), V is not in the scope of occurrences ofquantifier complexes in A that involve object variables which occurfree in C and are distinct from x1, ... , x1.

LEMMA 12.10. For any admissible CF-system T, there exists a r-universalsolution zo of q[f] such that, for any solution a of F, there is a solution T

1

of q[T] with the following properties:

(1) T1

is obtained from zo by renaming certain predicates not occurringin equalities of q[r] as predicates of higher dimensions;

(2) a is obtained from [aIz1 ] through simultaneous substitution of q*

types of certain formulas, all of whose free variables are distinguishedparameters, for predicates.

PROOF. Let F be an admissible CF-system with unknowns X1, ... , Xnfor formulas, consisting of lists (1)-(4). Let k be a natural number satisfyingconditions (1)-(7) of the definition of admissible CF-systems. Let denotethe system lr rA , and let {C1, ... , Cu ; z 1, ... , zu } be a basis of . Fix alist without repetitions of all basic i-terms that are basic i-atoms and beginwith predicate symbols. Now append to this list from the right the lists of0-place function symbols of , object variables of , unknowns for objectvariables of F, unknowns for terms of F', and unknowns for formulas of F.Denote the resulting list by Or . For any i (1 < i < u), we let Cl denotethe leftmost i-term among all i-terms S of O t for which S C1. Forany i-atom A, we introduce the following notation:

.. ,For any natural number s, we introduce n new s-place predicates Pr,.P. If {A,}+ is a formula schema, we will use the notation

{A ,}q's

q[{A ,}+]x

...X Pl ... pn,

For any i-atoms A 1 and A2, if A 1 A2 and {A1, }+ is a formulaschema, {A2 ,}+ is also a formula schema and

{A1, }q,s ={A2, }(1S.

This assertion is proved by induction on -<i, using Lemmas 7.2 and 7.7.Let rS denote the tuple of q-types

({X1, ... , {X, }S).

It follows from the previous assertion that rS is a solution of q[F]. Let abe a solution of F. Using Lemma 12.1, we can transform a into a solution

of . Let A be a E-atom and suppose that Aa is a formula. Then{ A, }+ is a formula schema and

q[Aa ] a q[{A, E}Z ...Z Ci .C] ] a {A, }.q[a].Hence rS is transformable into a.

We now fix some s such that for all j (1 < j < n), s is greater thanthe number of free occurrences of object variables in a q * -type of X7 . We

claim that i° and rS are the required solutions of q[F]. Indeed, it followsfrom the above that T° is F-universal. It is also obvious that condition (1)is satisfied, while condition (2) follows from the following proposition:

For any i-atom A, if Aa* is a formula, then ADO TS]* is also a formula,and is obtained from by simultaneous substitution of q*-typesof the formulas X' , ... , Xn for the predicates P, ... , Pn, respectively.

This proposition is proved by induction on -<i , using Lemma 12.1. Thesolution [Q/is]* is constructed from [Q/is] by Lemma 12.1. By Lemma12.7, [Q/zs] is a solution of F. 0

§13. Deduction schemata in axiomatized Hilbert-type theories

Let 3 be a k-place predicate defined on the set of all formulas of thepredicate calculus. We will say that a CF-system F represents 3 if, for anyformulas A1, ... , A, 3 is satisfied on A1, ... , Ak if and only if this k-tuple can be extended to a solution of F. Assume that F represents 3.The first k unknowns for formulas in F will be called the distinguishedunknowns; the other unknowns for formulas, the unknowns for terms, andthe unknowns for object variables will be called auxiliary unknowns.

The set of formulas defined by the axiom schema (la) of KH is repre-sentable by the CF-system

X1 a (X2 (X3 X2))

with a distinguished unknown X1 and auxiliary unknowns for formulasX2, X. For this system So* = 2, M° = h* = 0.

The rule of generalization Gen in KH is representable by the CF-system

X2 a daX 1

with distinguished unknowns X1, X2 and one auxiliary unknown a for ob-ject variables. For this system So* = 1, h* = 0.

§13. DEDUCTION SCHEMATA IN HILBERT-TYPE THEORIES

The set of formulas defined by the induction schema Ind

((A(O) &Vx(A(x) A(xe))) A(t)),

is representable by the CF-system

X1 a ((X2&Va(X3 X4)) X5),X2 a [X3], X4 a [X3]1, X5 a [X3],0*a*X3, a'*a*X3, t*a*X3

95

with a distinguished unknown XI and auxiliary unknowns X2, X3, X4, XS(for formulas), t (for terms), and « (for object variables). For this system

The set of formulas defined by the axiom schema for equality Eq

(t, = t2 (A(t1) A(t2))),


XI a (Z1 = t2 (XZ

XZ a [X4], X3 a

t1*a*X4, t2*a*X4with a distinguished unknown XI and auxiliary unknowns XZ , X3, X4 (forformulas), t1, t2 (for terms), and a (for object variables). Incidentally, thelast two CF-systems are degenerate.

The set of formulas defined by the axiom schema for equality Eq*

b'xy(x = y (A(x) A(y)))


X1 = b'a1a2(a1 = a2 (X2 D X3)),

X3 a

a2 * a1 * X2.

This system is q-free but not degenerate.Let 91 be an axiomatic Hilbert-type theory. We will say that a set of CF-

systems Erepresents 91 if each CF-system in E represents an axiom or in-ference rule of 91, and all axioms and inference rules of 91 are representableby CF-systems in E. The calculi KH and IH are obviously representableby sets of CF-systems. The calculi KH(2() and IH(2() are representable bysets of CF-systems if 2i is a union of finitely many sets representable by CF-systems. This condition is satisfied if 2( consists of a finite list of particularformulas and a finite list of formula schemata.

A deduction schema U in 91 is said to be correct if the indexes of thepremises in each occurrence of an analysis of a rule application are smallerthan the index of the occurrence itself. Admissible deduction schemata areclearly correct.

Fix a set of CF-systems T, ... , Tn representing a theory 9t. Let usassume that, for each j (1 < j < n), Tj has mj unknowns for terms andincludes Mj equalities with substitutions. Let U be a correct deductionschema in 9t, l the number of occurrences of analyses of applications ofaxioms and inference rules in U, X1, ... , Xl pairwise distinct unknownsfor formulas. Suppose that the ith analysis in U (1 < i < l) has theform [K1 ; ki

1 , ... , ki , s. ] and that the axiom or inference rule with codeK is representable by rr By renaming the unknowns, transform the setcr , ... , Tr into a set of CF-systems r', ... , r such that, for all i, the

1

distinguished unknowns of r1r are those unknowns in the list X1, ... , Xlindexed kl

1 , ... , ki i and r', ... , F'r will have no auxiliary unknowns

in common. Let y{ U} denote the result of combining r', ... , r`'r into asingle CF-system. We introduce the following notation:

m[B] max m . , M[B] max M.,1<j<n J 1<j<n J

arg[9't] max arg[F . ] , M°[9't] max M° [F . ] ,1<j<n 1<j<n

c* [fl] max c* [F ] , h* [9't] max h* [T ].1<j<n 1<j<n

The following assertion is obvious.

LEMMA 13.1. The CF-system y { U} has at most l m[R] unknowns forterms, includes no more than l M[9t] equalities with substitutions, and

arg[y{U}] < arg[ot], rp*[y{U}] < 1. rp*[yt],

h*[y{U}] < h*[9t], M°[y{U}] < 1. M°[9I],

h* [y{U}] < (1 + 1. m[gt]) h,[91].

Let a be a solution of y{U}. For all i (1 < i < 1), insert the valueof Xt in o before the ith analysis in U and denote the resulting figure by[a: U]. The following assertion is easily proved by induction on the numberof analyses in U.

LEMMA 13.2. For any correct deduction schema U in 91 the followingconditions hold.

(1) For any solution o of y{U}, [a: U] is a deduction in yt in accor-dance with U.

(2) For any deduction in 9I in accordance with U, there is a solution0 of y{U} such that is [Q: U].

Lemmas 12.5, 12.9, 13.1, and 13.2 imply the following proposition.

THEOREM 13.1. If an axiomatic Hilbert-type theory 91 is representable bya set of degenerate CF-systems, then the admissibility of deduction schematain 9't is algorithmically decidable.

§ 13. DEDUCTION SCHEMATA IN HILBERT-TYPE THEORIES 97

KH, IH, and Hilbert-type predicate calculi with equality in which theaxiom schemata and rules for equality contain no explicit occurrences ofquantifiers are representable by sets of degenerate CF-systems.

We will say that a q-type Q is deducible in a theory 91 if there is a formulaA such that A is deducible in B and Q a q[A]. If B is representableby a set of degenerate CF-systems and the set of degenerate and deducibleformulas in B is recursive, then by Lemma 12.9, the set of deducible Q-types in B is recursive. Consequently, the sets of q-types deducible in KHand IH are also recursive. Predicates and function symbols will be calledsupplementary in 9't if they do not participate in a set of CF-systems whichrepresent 9t.

We will say that a formula A is irreducible in 9t, if

(1) A is deducible in 9t;(2) for any q-type Q, if Q is deducible in 91 and is transformable into

A, then q[A] coincides with Q up to renaming of supplementarypredicates in 9t.

The following formulas are obviously irreducible in both KH and I H :

(P° P°), (P° xl ...

((A1) DP)

where

k

(P:D (\/P,9))i=o=0

(P° Kk), (K; P°),

Ko P°, K,1 (K, &_ K,),* P o

Ko , Ki+1

(Ki V Ki) ,

P° , Po , ... , Pk are pairwise distinct 0-place predicates, x1, ... , xk areobject variables.

LEtKMa. 13.3. If a theory 33 is represerctable by a set of CF-systems, A isan irreducible formula in 9t, and the formulas A and B are deducible in 9tin accordance with the same schema U, then there is a q-type Q such thatthe following conditions hold.

(1) q[A] coincides with Q up to renaming of supplementary predicatesin fit.

(2) Q is transformable into B.(3) Both [B/q[A]] and [BlQ] are deducible in 91 in accordance with

U.(4) B is obtained from [B/Q] by simultaneous substitution for predicates

of q*-types of certain formulas, all of whose distinguished parametersare free variables.

PROOF. Let A be an irreducible formula in R and let A and B bededucible in 9t in accordance with U. It follows from Lemmas 12.2 and13.2 that the CF-system y{U} is admissible. Using Lemma 13.2, constructa solution o of y{U} such that [Q U] is a deduction of B in 91 inaccordance with U. Then, applying Lemma 12.10 to y{ U} and a, constructsolutions io and zl of q[y{ U}]. Let Q be the value in i1 of the unknownfor formulas corresponding to the last rule application in U. Then Q is therequired q-type.

Indeed, the fact that r0 is y{ U}-universal and condition (1) of Lemma12.10 imply that Q is transformable into both A and B. The deducibilityof Q follows from Lemmas 12.7 and 13.2. Hence, since A is irreducible,condition (1) is satisfied. Let T be the solution of q[y{U}] obtained fromT1 via the same renaming of predicates as in the transformation of Q intoq[A]. It is obvious that T is transformable into a. Hence it follows byLemmas 12.7 and 13.2 that [B/q[A]] is deducible in accordance with U.The proof that [B/Q] is deducible in accordance with U is similar. Condi-tion (4) follows from condition (2) of Lemma 12.10.

LEMMA 13.4. Let U be a deduction schema in 33 and A and B anyformulas. Assume that 91 is representable by a set of q -free CF-systems;A is deducible in 91 in accordance with U ; B is obtained from A by si-multaneous substitution of formulas with distinguished parameters for certainsupplementary predicates in 91; no variables with bound occurrences in substi-tuting formulas or free occurrences in substituting formulas that are not theirdistinguished parameters occur as bound or free variables in A. Then B isdeducible in 91 in accordance with U.

PROOF. Let R, U, A, and B satisfy the conditions of the lemma. Us-ing Lemma 13.2, construct a solution ci of the CF-system y{U} such that[Q : U] is a deduction of A. Let SZ be the list of all variables which areeither bound in some substituting formula or occur free in some substitutingformula but are not its distinguished parameters. Using Lemma 12.8, trans-form a into a solution o * of y{ U} such that the values in ? of unknownsfor object variables do not occur in SZ , the variables of SZ do not occur asbound variables in formulas in a*, and the value in a* of the unknown forformulas corresponding to the last rule application in U is just A. Now, inorder to obtain a deduction of B in 9't in accordance with U, we need onlysubstitute formulas for predicates in a* and then use Lemma 13.2. 0

Let T and Q be terms, xl , ... , x1 (1 > 0) a list without repetitionsof object variables, f an s-place function symbol (s > 0). Assuming thatl < s, we define a term T f1 xl x1Q by induction on the construction ofT.

(1) If T is an object variable, then

TI f1.x1... x1Q = T.

§14. DEDUCTION IN A GIVEN SCHEMA 99

(2) If T has the form g(T1, ... , Tk), where k > 0, T1, ... , Tk areterms, and g a k-place function symbol distinct from f, then

(3) If T has the form f(T1, ..., Ts) ,where T1, ... , TS are terms, then

TI X1... x!Q a Qx1 IjiXi ...

Let P be a k-place predicate. We introduce the following notation:

P(Tl,... , Tk)I f2xl ... xjQ a p(TI If2xl... xjQ, ... , TkIf%xl... xjQ),

Let A and B be formulas. We will say that B is obtained from A bysubstituting a term Q with distinguished parameters x1, ... , x for a functionsymbol f, if 1 < s, B is obtained from A via simultaneous replacement ofeach occurrence of an elementary formula E in A by EIfI,x1 and fdoes not occur in A in the scope of quantifiers that involve object variablesother than xl , ... , xl and occurring in Q.

LEMMA 13.5. Let U be any deduction schema in 91 and A and B anyformulas. Assume that 9t is representable by a set of q -free CF-systems; A isdeducible in 9I in accordance with U ; B is obtained from A by simultaneoussubstitution of terms with distinguished parameters for certain supplementaryfunction symbols in 9; no variables that occur in a substituting term but arenot its distinguished parameters occur as bound or free variables in A. ThenB is deducible in 9t in accordance with U.

The proof is similar to that of Lemma 13.4.

§14. Deducibility of a formula in accordance with a given schema

An axiomatic Hilbert-type theory 33 will be called I-regular if the follow-ing conditions hold.

(1) 9t is representable by a set of q-free CF-systems.(2) If a formula without and v is deducible in IH, it is deducible

in R.(3) If a formula without and v is deducible in 33, it is deducible in

KH.(4) 9t contains the modus ponens rule.

LEMMA 14.1. The following formulas are irreducible in any I-regulartheory

Cpo (1)

C(° b'zQ°) (po1 I t& Q° I & `dx3yQ0/

/ / '(2)

where k > 0, l > 0, s > 1, P°, Q°, P, ... , Pk are pairwise distinct 0-place predicates, x, y, z, x1, ... , xl are object variables.

PROOF. Let A be one of formulas (1) or (2). A is clearly deducible inI H and hence in any I -regular theory. Let Q be a q-type, transformableinto A, that is not obtained from q [A] by renaming predicates. Let B be aformula such that Q a q[B]. Then we can find a predicate occurring in Qsuch that when this predicate is replaced in B by the formula (P° & -iP°)the result is equivalent to (P° & in KH. It follows that Q is notdeducible in KH or, therefore, in any I-regular theory. o

LEMMA 14.2. For any formula A, if A and (1) are deducible in someI-regular theory in accordance with the same schema, then A has the form

k([B0]::: T [B1]T.. T ... [Bk]T ...T xl ...xl (L.B))...))

where, for all i and j (0 < i < k, 0 < j < 1), the term T3 is free for x3 inB1.

PROOF. Assume that both A and (1) are deducible in an I-regular theoryB in accordance with a schema U. Then, in view of Lemma 14.1, (1) isirreducible in R. Let Q be a q-type constructed on the basis of U, (1) andA as in Lemma 13.3. Then the formula [A/Q] is deducible in 91 and hasthe form

...,R°) (P'(R, ...,R1 ... pgk(Rk,...,Rk)q0 1 1 q k 1 qk

kD..xl (PI(Si ... , S`)

i =0 q

where P'°, Pl' , ... , Pkk are distinct predicates. This formula is deduciblein KH. Hence we can construct terms Tl , ... , Tl such that, for all i

(0<_i<k)[Pq` (Sl , ... , s q)]T ...T a Pq` (R 1 , ... , Rq ).

r l ! r

In order to complete the proof, we need only use condition (4) of Lemma13.3. 0

The following assertion is proved similarly.

LEMMA 14.3. For any formulas A and B, if formula (2) is deducible insome I-regular theory in accordance with the same schema as the formula

sA (B. l&((R(7' , ei) & x)

then there is a term S such that

A a (B b'zR(S, a))

and for all i (1 < i < s), the term T, coincides with SZ &, .

Let 9't be an axiomatic Hilbert-type theory, U a deduction schema in9t, A a formula, and x an object variable. We will say that U and Awith a distinguished parameter x represent a set £ of natural numbers ifany natural number n belongs to £ if and only if the formula isdeducible in 9't in accordance with U.

LEMMA 14.4. For any theory 9't and any set 4' of natural numbers, if 9is I-regular and 4' is enumerable, then there exist a deduction schema Uin 9t, a formula A, and an object variable x such that U and A with adistinguished parameter x represent 4'.

PROOF. Let 9't and 4' satisfy the conditions of the lemma. Suppose that4' is representable by a normal TS-system F with unknowns t1, ... , to ,a-unknowns v 1, ... , v1', and equalities

T1 =®1,..., TN=®N,1 1 1 1

S1 = vQ R 1 , ... , Ski = vQ

q q .q qS1 =vaR1,...,Sk =v9 4

By Theorem 8.1, 4' is representable by a normal TS-system. We intro-duce the following notation:

A°[a, la] (P(a) P(/3))1

Ak [a 1 ,

2Ak [a 1 ,

...,ak,fk] ((LR(aj , /31) & x)

... , ak , 3k] (dxyR(x Y) Ak[al , 1 , ... , ak , 3k]) ,

Bl = A°[T, , ®1] (1<iN),BN Ak[S1,R1,...,Sk,Rk] (1<jq),+ J J J

N+qA 3 t2 ... to &, B

s=1 s

M±max{n,k1,...,kq}.Let us assume that the object variables x, y, z and 0-place function

symbols e1 , e2, ... , eM do not occur in (3) and that the 0-place predicateP°, 1-place predicate P', 2-place predicate R, (n - 1)-place predicates

Q1 , Q2 , ... , QN+q , 0-place function symbols e1, e2, ... , e, and 1-placefunction symbol f are supplementary in 9t.

Let U° be the schema of some deduction in 91 of the formula A°[e1, e1],Uk the schema of some deduction in 9't of the formula

((P° dzR(f(z), z)) (P° Ak[f(e1), el, ... , f(ek), ek]))

1 1

Rk ,(3)

q qaRk.

U2 the schema of some deduction in 9I of the formula

(VxyR(x, Y) b'zR(f(z), z)),

U3 the schema of some deduction in 9t of the formula

(Qi(e2 ... , en) (Q2(e2 ...

(N+qD 312...ln 8G Qi(C2, ... , ln) . .

i=1

We construct a schema U4 as follows:

U0;

U0; U3m. p.

m. p.

U2; U,

U2; Ukgmp.

m. p.;

U2; Uk U0;' m .p.; m.

m. p.

m. p.

m. p.

Then U4 and A3 with distinguished parameter tl represent £. Indeed,let m E 4' and let o be a solution of F such that ti a 0(m. Let us assumethat o does not contain the variables x, y, z and the 0-place functionsymbols el , ... , e,,1. By Lemma 13.4, the formula

([Bl]'(m)..e2...en

([B2]°(m) e2...en D ... ([BN+q]'(m) e2...en [A ]°(m)) ... / /

is deducible according to U3. Consequently, it follows from Lemma 13.5that the formula

t t t 12tn t

t° D [A3],n)) ... ) )([Bl](m)"r to to D2 ... n 2 n 2 n

is also deducible according to U3. It now follows from Lemmas 13.4 and13.5 that, for all i (1 < i < N), the formula A°[T' , 8'] , which coincideswith [Bl.]t'm) is deducible in accordance with U°; for all j (1 <j <q),°c t2 tnthe formula

(VxyR(x, y) `dzR(v'a, z))

is deducible according to U2 ; and the formulajQ 2 jQ jQ jQ jQ

((VxyR(x, y) `dzR(v, , z)) R1 , ... , Ski , Rk])

is deducible according to U1. But Ak [S1 Q , R iQ , ... , Sk' , R'] coincides

J J J J

with [BN+ .]t Consequently, [A3]t (m) is deducible in accordance with0 2 n o

U4 and hence our assertion has been proved in one direction.Let m be a natural number such that [A3]t (m) is deducible in B in accor-

11

4dance with U . Then there are formulas C1,...,CN+q, C1,...,Cq suchthat the following conditions are satisfied.

(1) The formula

(C1 (C2 (Cq [A3]m)) "))

is deducible in accordance with U3.(2) For all i (1 < i < N), the formula C, is deducible in accordance

(3) is deducible in accordance

is deducible in accordance

with Uk.J

By Lemma 13.4, a formula of the form (1), where k = N + q - 1 andl = n -1 , is deducible in accordance with U3. Hence it follows from Lemma14.2 that there are terms 02, ... , Bn such that for all s (1 < s < N + q),

CS a [BS](m) a , a .0 2 n

By Lemma 13.4, formula (2) is deducible in acordance with U. UsingLemma 14.3, we construct terms W1, ... , Wq such that for all j and l(1<j<_q, 1 <l <k;),

..t0(m)o ... 0 a W3R3 O(m)8 ... en,S3

1 n 2 n 2lt tnSimilarly, using Lemmas 13.4 and 14.2, we see that for all i (1 < i < N),

Tit t 0(m) 82 ... 0 a ® 0(m) a ... en,n its tn 2

Thus, the list 0(m), 02 , ... , Bn , W 1 a , ... , W q a is a solution of r andconsequently m E £. 0

Let 9t1 and2 be axiomatic Hilbert-type theories. We will say that 9t1is an extension of 2 if

1can be obtained from 2 by adding axioms

and inference rules.

THEOREM 14.1. For any axiomatic Hilbert-type theory 9t, if 9 is an ex-tension of some I-regular theory, then there is a deduction schema U in 91such that the deducibility of predicate formulas in 9't in accordance with Uis not algorithmically decidable.

PROOF. Let 9't be an extension of an I-regular theory 3. Using Lemma14.4, we construct a deduction schema U in 3 such that the deducibilityof predicate formulas in 3 in accordance with U is not algorithmicallydecidable. We need only note now that U is also a deduction schema in 91

with U° .

For all j (1 < j < q), the formula C)with U2 and the formula (C) CN+f)

i

and that a predicate formula is deducible in 9t in accordance with U if andonly if it is deducible in 3 in accordance with U.

Let A be a formula. The maximal height of the terms occurring in Awill be called the height of A, denoted by h* [A] . We let arg[A] denote themaximal dimension of the predicates in A.

Let 9 be an axiomatic Hilbert-type theory, U a deduction schema in R,k a natural number. Assume that 9t is representable by a set of CF-systems.A proof in 9t in accordance with U will be called k-restricted if thereis a solution of the CF-system y{ U} such that the rank of is at mostk and coincides with [o: U].

THEOREM 14.2. For any axiomatic Hilbert-type theory R and any formulaA, if 9 is representable by a set of CF-systems, then any k-restricted proof

of A in 9 can be transformed into a k-restricted proof of A in R,in accordance with the same schema as , such that the maximal height ofthe terms occurring in is at most

(m01 + 1 hc12O1+1 (/4k + 6 c212(mo+5)x°01

and for any formula B in £ 5

where

c = k max{arg[A], arg[9t]} M[B]ho = max{h* [A], h* [9']}, oo = SO* [ ] mo = m [Yt] ,

l is the length of I, and X01 the number of different subformulas of A thatbegin with , &, V, - , V. or .

COROLLARY 1. For any formula A, if A contains only 0 -place and 1-placefunction symbols, then any proof of A in KH or IH can be transformedinto a proof of A, in accordance with the same schema as l" , such thatthe maximal height of the terms occurring in £"* is at most

l+ 1 h A +1 1 o3(arg[A]1)2461+"

and for any formula B in £ 5

st[B] < 61+ co1 + 1.

COROLLARY 2. If an axiomatic Hilbert-type theory R is representably by aset of CF-systems, then there is an algorithm that will decide whether formulasare k-deducible in 9 in accordance with a deduction schema.

PROOF OF THEOREM 14.2. Let 9 be a theory representable by a set ofCF-systems, a k-restricted proof of a formula A in 9t in accordancewith a schema U, and suppose that the last analysis in y{ U} correspondsto an unknown for formulas X. Let F be the CF-system obtained from

y{ U} by adding the equality X a A. It follows from Lemmas 12.2 and 13.2that F is admissible. Using Lemmas 6.12, 10.11, 11.2, and 12.4, constructa solution of r with height at most

h* [I']{(3k + h*[F])(2k + 3)kq+(m+1)/2 + 4}kg 3kq (kq)!

< h[F] kg+1m + 1 4k + 6 2k2g2+(m+3)kq

where q = arg[T] M. 2q' fir'] , m is the number of unknowns for terms inT, and M the number of equalities with substitutions in F. Using this andLemmas 13.1 and 13.2, we conclude that [o: U] is the required proof ofA.

LEMMA 14.5. For any axiomatic Hilbert-type theory R, deduction schemaU in R, formula A, and natural numbers d and n, if 9 is representableby a set of CF-systems, the CF-systems representing 9 and A involve only0 -place and 1-place function symbols, the formula [A]o) is deducible in 9in accordance with U, arg[A] > 0, M°[R] > 0, and

l< loge n

V(omo+1)(d+1)' (4)

4(co0m0+ 1)(d + 1)B2(log2 B)2 < log2 n , (5)

then there is a natural number c such that 0 < c < n11 (d+ 1) and for allnatural numbers i, the formula [A]n+cci- )) is deducible in 9t in accordancewith U, where l is the number of analyses in U, c 1 the number of differentsubformulas of A that begin with , &, v, - , V. or ,

= * [] ' m0 = m [R] , M0 = M°[9'],0

h = max{h* [A] , h* [B]} , s = max{arg[A], arg[9t]},

B = (mo + 2)(co1 + log2 6sMo) + log2(h + 3).

PROOF. Let R be a theory, U a schema, A a formula, and d, n naturalnumbers satisfying the conditions of the lemma. Let us assume that theunknowns for terms of the CF-system y{ U} are distinct from x and donot occur in A, and that the last analysis in U corresponds to an unknownfor formulas X. Let T be the CF-system obtained from y { U } by' addingX a A to the list of equalities and x to the list of unknowns for terms. Itfollows from Lemmas 12.2 and 13.2 that F is admissible. Using Lemmas12.4, 13.1, and 13.2, construct a normal CTS-system E with a solutionsuch that , has at most b'-essential a-unknowns and at mostm0l + 1 essential unknowns, the value of x in is 0(n), and

h[E] = h, va[o; ] < (6)

It follows from (5) that4<B<L, (7)

whereloge nloge n

1)(d+ 1)'Inequalities (4), (6), and (7) imply that

log2[(h+2)(va[Q; ]+2)(va[Q; E]/+3)mo1+1+2] < rp0m0L2+BL1og2L. (8)

Using (5) and (7), one easily proves that

B loge L < L.

From this and from (8) we have

n > nl/(d+i)> (h + 2)(va[Q ; :.] + 2)(va[Q ; ]v' + 3)mol+l + 2.

Similarly, we obtain

((va[o; =] + 1)/)m°1 < n1/(d+1).

To complete the proof, we need only use Lemmas 10.12 and 13.2.

§15. Deduction schemata in Gentzen calculi

Let U be a deduction schema in K GL (2t) or I GL (2t) , V an occurrencein U of an analysis of some axiom or inference rule. The number of occur-rences of analyses below V in U will be called the level of V in U. Wewill define natural numbers Ka(V) and xC ( V) by induction on the level ofVinU.

If the level of V in U is 0, then xa(V) = ,cc(V) = 0.Assume that the level of V in U is not zero and let Vl be an occurrence

of an analysis in U with V as a premise. If Vl is an occurrence of theanalysis of an application of the rule 1 , V -p, & - 1 , &V -p, 3 -p, or 2t-cut, or if V is the right premise of Vl and Vl is anoccurrence of an analysis of an application of the rule -- or cut, then

Ka(V) = xa(V) + 1, KC(V) = KC(v ).

If Vl is an occurrence of an analysis of an application of the rulevl , -- v2 , --* &, -p, --> V, or - in U or if V is the left premise

of Vl and Vl is an occurrence of an analysis of an application of the ruleor cut, then

C(V)C(V)+1Define

,c(V) K"(V) - C(V)

Let k and s be natural numbers. A schema U will be called a (k, s)-correct schema in KGL(2() (in IGL(2t)) if it is a deduction schema inKGL(2() (IGL(2t)) and, for any occurrence V of an analysis in U,

(1) if V is an occurrence of an analysis of an axiom with principaloccurrences n i and n2, then 1 < n2 < s + x` (V) (respectively,n2=1)and -k-xa(V)<nl <-1;

§15. DEDUCTION SCHEMATA IN GENTZEN CALCULI 107

(2) if V is an occurrence of an analysis of an application of a rule ofintroduction of a logical connective in the antecedent with principaloccurrence m, then -k - xa (V) <m < -1;

(3) if V is an occurrence of the analysis of an application of a rule ofintroduction of a logical connective in the succedent with principaloccurrence m, then 1 < m < s + x`( V) (respectively, m = 1).

The maximal level of the occurrences of analyses of axioms in a schema Uwill be called the height of U (notation: h(U) ). Let 2( be the union of sets2t1, 2' 'r representable by CF-systems I'1, F2, ... , I',., respectively.It is obvious that, for any rule in KGL or IGL, the relation between sideformulas and the main formula is representable by a degenerate CF-system.

Let U be a (k, s)-correct schema in KGL(2i) (IGL(2t)). Let us assign toeach occurrence V of an analysis in U a list of pairwise distinct unknownsfor formulas X k_Ka(y), X", ... , ,where p(V) = s+x`(V)(respectively, p(V) =min{ 1, s+x`(V)} ). We will assume that such lists fordifferent occurrences of analyses in U have no variables in common. LetS2{ U} denote the union of these lists for all occurrences of analyses in U.

Let V be an occurrence of the analysis of an 1-premise rule in U. If nis the index of the principal occurrence of this analysis, we will say that theunknown Xn corresponds to the principal occurrence of V.

Let l = 1 and let W be an occurrence of the premise of V. We will saythat Xv)-x(w) corresponds to the side occurrence of V. Let y"{ V} denotethe list of all equalities

w VXl-Ka(W)+Ka(V) a X1

where -k - ,Ca (V) < i < -1. Let C{ V} denote the list of all equalitiesw V

X j+Kc(W)-Kc(V) a X j ,

where 1 < j < p (V) (respectively, 1 < j < p (V) and j + K` (W) - K` (V) <1)

Assume that l = 2, W1 is the occurrence of the left premise of V, andW2 the occurrence of the right premise of V. We will say that the unknown

X WV -K W corresponds to the left side occurrence of V and the unknown

X W V -K W corresponds to the right side occurrence of V. Let ya { V} denotethe union of the list of all equalities

x w1a

a XV

i-K (Wi)+Ka(V) !

and the list of all equalities

XW2a aXVr K

(W2)+Ka(V)

where -k - K"(V) < i <-1 . Let yC{ V} denote the union of the list of allequalities

XWI a Xvj+Kc(Wi)-Kc(V) '

where 1 <j < p( V) (respectively, 1 < j < p(V) and j + ,cc( W) - x`( V) <1), and the list of all equalities

2

X +KC(W2)-KC(V)axJ

where 1 < j < p(V) (respectively, 1 <j < p(V) and j + ,cc(W2) - x`( V) <1)

Let V be an occurrence in U of an analysis of an axiom with principaloccurrences n 1 and n2. In that case we stipulate that the lists a{ V} and

C{ V} are graphically equal to the empty list and let y{ V} denote theequality Xn a Xn

z

Let V be an occurrence of an analysis of an application of the cut rule inU. Then we let y{ V} denote the equality Y1 a Y2, where the unknownY1 corresponds to the left side occurrence of V and Y2 to its right sideoccurrence.

Let V be an occurrence in U of an analysis of an application of a ruleL of introduction of some logical connective in the succedent or antecedent.Then y{ V} will denote a CF-system representing the relation between theside formulas of L and its principal formula, in which the distinguishedunknowns are the unknowns in 1{ U} corresponding to the side occurrences,and the principal occurrence of V and its auxiliary unknowns do not belongto 1{U}.

Let V be an occurrence in U of an analysis of an application of the2t-cut rule, whose side formula belongs to 2t. (1 < i < r). Then we lety{ V} denote a CF-system, obtained from F. by renaming unknowns, whosedistinguished unknown is the unknown in 1{ U} corresponding to the sideoccurrence of V and its auxiliary unknowns do not belong to 1{ U } .

Let us assume that, for any different occurrences W1 and W2 of analysesin U, the CF-systems y{ W1 } and y{ W2 } have no unknowns in commonthat belong to SZ{ U}. Let y{ U ; k, s} denote the union of the CF-systemsya { V } ,

C{V } , y{ V} for all occurrences V of analyses in U, m [2t] the

maximal number of unknowns for terms in F1, ... , F',., and M[2t] the maxi-mal number of equalities with substitution in T, ... , F'. We also introducethe following notation:

arg[2t] max arg[Ti] , c* [2t] max1<i<r 1<i<r

h* [2t] max h* [F1].1<t<r

The following assertion is easily proved by induction on h(U).

LEMMA 15.1. y{ U ; k, s} has at most max { 1, m [2t] } 2h U -1 unknownsfor terms, includes at most max{1, equalities with substitution,

arg[y{ U ; k, s}] < arg[2t], h* [y{ U ; k, s}] < h* [2t] ,

0*[y{U; k, s}] < max{1, *[Qj]}2h(u)-1

Let Q be a solution of y{U; k, s}. For any unknown X in SZ{U} , welet X° denote the value of X in o. To the left of each occurrence V ofan analysis in U, we add a sequent

V°Vo

v° XV0 X XX

Denote the resulting figure by [Q : U ; k, s]. The following assertion iseasily proved by induction on h(U).

LEMMA 15.2. For any (k, s)-correct deduction schema U in KGL(2t)(IGL(2t)) the following statemnts hold.

(1) For any solution Q of y{ U ; k, s}, [Q : U ; k, s] is a deduction inaccordance with U in KGL(2t) (respectively in IGL(2(), ifs < 1).

(2) For any deduction in KGL(2t) (IGL(2t)) with a schema U inwhich the antecedent of the last sequent contains k formulas and thesuccedent s formulas, we can construct a solution Q of y{T.I ; k, s}such that is precisely [Q : U ; k, s].

Let U be a deduction schema in KGL(2t) or IGL(2t),Van occurrenceof some analysis of an axiom or an inference rule in U. We will definenatural numbers x* (V) and x* (V) as follows:

If V is an occurrence of an analysis of an axiom with principal occurrencesnl and n2, nl <0 < n2, then x*(V) _ -nt - xa(V) and x(V) _ -n2 -ICC(V)

If V is an occurrence of an analysis of an application of a rule of introduc-tion of some logical connective in the antecedent with principal occurrencen, then x*(V) _ -n - xa(V) and x*(V) = 0.

If V is an occurrence of an analysis of an application of a rule of intro-duction of some logical connective in the succedent with principal occurrencem, then x* (V) = 0 and x;(V) = m - x`(V).

If V is an occurrence of an analysis of an application of the cut rule or2(-cut rule, then x* (V) = x* (V) = 0.

Let x*{U} denote the maximum of x*( V) for all occurrences V ofanalyses in U, and x* { U} the maximum of x* (V) for all occurrences Vof analyses in U.

LEMMA 15.3. For any deduction schema U in KGL(2t) (IGL(2t)) andany sequent S. if S is deducible in KGL(2t) (IGL(2t)) in accordance withU, then

(1) x; { U} <k and x* {U} < s, where k is the number of occurrencesof formulas in the antecedent of S and s the number of occurrencesof formulas in the succedent of S ;

(2) U is a (k, s)-correct schema in KGL(2t) (IGL(2t));(3) the sequent obtained from S by deleting all occurrences of formulas

from the antecedent with indexes less than -x; {U} and all


occurrences of formulas from the succedent with indexes greater thanx* { U} is deducible in KGL(2t) (IGL(2t)) in accordance with U.

This lemma is proved using Lemma 1.1 by induction on h(U).

THEOREM 15.1. If f 2t is the union of finitely many sets representable by de-generate CF-systems, then the problem of admissibility of deduction schematais decidable for both KGL(2t) and IGL(2().

This theorem follows from Lemmas 12.5, 12.9, 15.1, 15.2, and 15.3.Let x be an object variable and A a formula. The natural numbers s for

which there is an occurrence of a term in A that does not arise froman occurrence of a term in A and which generates a bound occurrenceof x in A will be called the bound x-final degrees of A. We will say that anatural number k is a bound final degree of A if there is an object variabley such that y has a bound occurrence in A and k is a bound y-final degreeof A.

Let 2( be the union of finitely many sets represented by CF-systems, andlet U be a (k, s)-correct deduction schema in KGL(2t) or in IGL(2i), Qa solution of y{ U ; k, s}. Then, by Lemma 15.2, [0: U ; k, s] is a proof.A natural number n will be called a bound final degree of Q if n is a boundfinal degree of some formula in the last sequent of [o: U ; k, s] or of a sideformula of some application of the cut rule or 2t-cut rule in [0: U ; k, s].

Let L be an application of the 2(-cut rule in [a: U ; k, s] correspondingto a schema of applied axioms that is represented by a CF-system I', T asolution of t generated by a solution a of y { U ; k, s } . We will say thata natural number n is a free final degree of L relative to Q if n is a freefinal degree of i . A natural number m will be called a free final degree ofan applied axiom in a if m is a free final degree relative to Q of someapplication of the 2(-cut rule in [0: U ; k, s].

LEMMA 15.4. For any (k, s)-correct deduction schema U in KGL(2t) orIGL(2O and any solution a of y{ U ; k, s}, the set of free final degrees of ais a subset of the union of the set of all bound final degrees of Q and the setof all free final degrees of applied axioms in 0.

This lemma follows from Lemmas 1.2 and 1.3.A figure obtained from an arbitrary deduction schema in KGL(2t) (resp.,

IGL(2i)) by adding a formula to any analysis of an application of the cutor 2(-cut rule will be called a deduction schema with fixed cut types. Let Ube a deduction schema with fixed cut types in KGL(2t) or IGL(2[), L theanalysis of an application of the cut or 2(-cut rule in U. The formula in Lwill be called the type of L. A schema U will be called a correct deductionschema with fixed cut types if the following conditions hold.

(1) The types of analyses of applications of the cut or 2[-cut rule in Uhave no free variables in common.

(2) The type of any analysis of an application of the 2t-cut rule in Ubelongs to the subset of 2t with the index specified in the analysis.

Let S be a sequent, t1, ... , to (n > 0) a list of pairwise distinct objectvariables, U a correct deduction schema in KGL(2t) or I GL(2t) with fixedcut types, a proof in KGL(2t) (IGL(2t)). We will say that is a(ti, ... , tn)-deduction of S in accordance with U, if the following conditionshold.

(1) is a deduction in accordance with the schema obtained from Uby deleting all formulas.

(2) The type of the analysis of any application of the cut or 2t-cut rulein is transformable into its side formula.

(3) t 1, ..., to do not occur free in types of analyses of applications ofthe cut or 2t-cut rule in U.

(4) There are terms T1, ... , Tn such that, for all i (1 < i < n), Tl isfree for tl in S and the last sequent of is

A proof will be called a universal (ti, ... , to )-deduction of a sequentS in accordance with a schema U if is a (ti, ... , tn) -deduction of Sin accordance with U and any (t1, ... , to)-deduction of S in accordancewith U can be obtained from by substituting terms for free occurrencesof variables.

Let be a (t1, ... , to )-deduction of a sequent S in accordance witha schema U, L an application of the cut or 2t-cut rule in " . The termsinvolved in the substitution transforming the type of the analysis of L intothe side formula of L will be called the proper terms of L. The termsinvolved in the substitution transforming S into the last sequent of willbe called the proper terms of £Y.

Let SZ be a list of function symbols, Q a term, S a sequent or a formula.Let h[ S] denote the maximal value of h[T] for all terms T occurringin 5, of [S] the maximal value of o° [ T ] for all terms T such thatsome occurrence of T in S generates a free occurrence of Q in S, o [S](o [S]) the maximal value of o° [T] for all object variables x and terms Tsuch that some occurrence of T in S generates a free (respectively, bound)occurrence of x in 5, and icy [S] the number of object variables x suchthat x occurs free in S and o [S] > 0.

Let U be a correct deduction schema in KGL(2t) or I GL(2t) with fixedcut types. We let q[ U] U] denote the number of formulas A that are men-tioned in some analysis in U such that 0, q[U] the numberof analyses of applications of the rules - and d - in U, oj[ U] (re-spectively, o [ U] , of [ U] , h[ U] , is [ U]) the maximal value of o[ A]Q,

(o[A], of [A] , h[ A], , is [A]) for all formulas A occurring in U andQ,in analyses of applications of the cut or 2t-cut rule in U. If SZ is the set of all

function symbols, we will omit the symbol SZ in h[ S], , a f [S], [S],Q,

a [S], x [S] , q [ U] , aj [ U] , a [ U] , of M[ U] , h[ U] , and icy [ U] .

LEMMA 15.5. For any correct deduction schema U in KGL(2t) or I GL(2t)with fixed cut types, any sequent S, and any term Q that does not containfree variables of types of analyses from U and object variables t1if the subsets into which 2t is divided are closed with respect to substitutionof terms for free variables, and there exists a (t1, ... , tn)-deduction of S inaccordance with U, then there exists a universal (t1, ... , tn)-deductionof S in accordance with U such that, for any term T that is a proper termof , - , V -*, cut or 2t-cut in I, the following inqualities hold :

hQ[T] < [U ; S] + max{h[U] , h[S]},[T] < 0 [U ; S] + max{of [U], of [S]} ,

wheren

0 [ U ; S] q [ U] max{o [ U] , ac [S]} + x [ U] q* [ U] f [ U] + of [S].1

i=1

PROOF. Let S be a sequent and U a correct deduction schema withfixed cut types in KGL(2t) or in I GL(2t) , satisfying the conditions of thelemma. It follows from Lemmas 1.2 and 1.3 that, in order to obtain a(t1, ... , to)-deduction of S in accordance with U, it is sufficient to constructits proper terms and the proper terms of applications of the rules -p , V -p,cut, or 2t-cut. In order to find these terms, we will construct a CT-systemI'{ U ; S ; t 1 , ... , t,} by induction on h (U) . Let II { S ; t 1, ... , to } denotethe list of all expressions 1 < j < n, where x is an object variableoccurring in a quantifier complex governing some free occurrence of t3 inS.

If h(U) = 0, the only analysis in U is an analysis of an axiom. Supposethat the indexes of the principal occurrences in this analysis are n 1 and n2.We let A 1 denote the formula whose occurrence in S has index n 1 , and A2the formula whose occurrence in S has index n2. In view of the conditionsof the lemma, A 1 is q * -equivalent to A2. Let B be a q * -type of both A 1

and A2. Let us write the list of all free variables of B as b1, ... , b. LetT1 , ... , TS and R1, ... , RS be terms such that

A a [B]bl ....bs, A a [B]b ..: bs1 T TS 2 Ri..Rs

Define T{ U ; S ; t1, ... , to } to be the CT-system

J T1=R1,...,I=RS,II{S;t1,...,tn}

with unknowns for terms t 1 , ... , t,.

Let h (U) > 0. Then U ends with an analysis of an application ofan l-premise rule L (1 < l < 2). Let Ul (1 < i < l) denote thededuction schema occurring in U above the ith premise of the lastanalysis. Suppose that L is an introduction rule of some propositionalconnective. Let S1, ... , SI be the sequents from which S is obtainedby L. Define a CT-system I'{ U ; S ; t 1 , ..., to } to be the union of theCT-systems T { U1; S1 ; t 1, ... , to } , ... , T { UI ; SI ; t 1, ... , to } and listII{S ; t1, ... , to } , first renaming the unknowns for terms in these CT-systemsso that they will have no common unknowns other than t

1, ... , to .

Suppose that L is the 2t-cut rule. Then l = 1. Let A be the type ofthe last analysis in U and a list of all free variables of A. Let S1 be asequent from which S follows by the 2t-cut rule with A as side formula, anddefine F{ U1 ; S1; t1, ... , to , } to be the CT-system T{ U ; S ; t1, ... , to}.

Suppose that L is the cut rule. Then l = 2. Let A be the type ofthe last analysis in U and a list of all free variables of A. Let S1and S2 be sequents from which S follows by the cut rule with A as sideformula, and define a CT-system T{ U ; S ; t 1 , ... , to } to be the union ofF{ U1 ; S1 ; t1, ... , to , E} and T{ U2 ; S2 ; t1, ... , with variables forterms renamed so that they have no common unknowns in the list t

1, ... ,

In, `r.Suppose that L is the rule -p d or -p. Then l = 1. Let b be a

1variable not occurring in S and Q and distinct from t1, ... , to . Let 5be a sequent from which S follows by L with b as proper variable. AsI'{ U ; S ; t1 , ... , to } we take the CT-system

r'{U; S; t1 , ... , to},{S; t1 , ... , tn} ,

l r

where t1 , ... , tar is the list of all of the variables t1 , ... , to that occur freein S.

Suppose that L is the rule d --* or --> I. Then l = 1. Let t be a variablenot occurring in S and Q and distinct from t

1, ... , to . Let S1 be a sequent

from which S follows by L with t as proper term. As I'{ U ; S ; t 1 , ... , to }

we take the union of I'{ U1 ; S1 ; t1 , ... , to , t} and the list II{S ; t1 , ... , to } .Using Lemma 3.2, one easily proves by induction on h(U) that a proofin KGL(2t) or in IGL(2t) is a (ti, ... , tn)-deduction of a sequent S

in accordance with a schema U if and only if the list of proper terms ofand of proper terms of applications of the rules cut, 2t-cut, -p , and

d - is a solution of T{ U ; S ; t1, ... , to } , and that a universal solution of

I'{ U ; S ; t1, ... , to} generates a universal (t1, ... , to)-deduction of S inaccordance with U.

Note that

S; t1, ... , to}] < S],h[ T{ U ; S ; t 1, ... , to } ] = max { h[ U] , h[S] } ,

QSZ[r{ U ; S ; t1, ... , t }] = max{oQfSZ[ U] , oQfSZ[S]}.

Now, in order to complete the proof, we need only use Lemma 7.12. 0Let be a finite list of closed formulas, a proof in KGL(2.) or

I GL(9i) . Let q[] denote the number of applications of the rules - * andV -> in , and OC[PuJ] the maximal value of oc[A] for all side formulas Aof applications of the cut rule in .. The following proposition strengthensLemma 4 of [15].

LEMMA 15.6. For any finite list of closed formulas and any proofof a sequent S in KGL(Y) (IGL(Y.)), if oc[] = 0, then, by replacingcertain free terms by variables, we can transform into a proof of S inKGL(Y) (I GL(Y) ), such that the height of the proper terms of applicationsof the rules and d in and the height of the terms occurring freein side formulas of applications of the cut rule in is at most

q- [Pl7] + h* [YS] < h* [YS] 2h.

PROOF. Let be a proof in KGL(Y) or I GL(Y) and U the deductionschema of . Add a q * -type of the side formula to each analysis of anapplication of the cut rule in U and the side formula itself to each analysis ofan application of the i-cut rule in , renaming, if necessary, free variablesin the types of the analyses. Denote the resulting correct deduction schemawith fixed cut types by U*. It is clear that is a ( )-deduction of S inaccordance with U*.

Note that

max{oC[U*], o`[5]} = max{oC[. T] , OC[YS]} ,

max{h*[U*], h*[S]} = max{oC[. T] ,

q [ U* ] = q [j7] <

of [U*] < oC[r].

Consequently, we can define to be the ()-deduction of S in accor-dance with U * constructed in Lemma 15.5. 0

§16. Rebuilding of proofs on the level of schemata

Let p 1 and p2 be axiomatic Hilbert-type theories or sequent calculi, aan algorithm. We will say that a transformation a of deductions in q1 intodeductions in2 is applicable to a proof in 31, if a transformsinto a proof in P2 . A pair of algorithms (fi, y) will be called a rebuildingof deductions in1 into deductions in2 on the schema level if, for anydeduction schema U in 1 and any formula or sequent S deducible in 1

§16. REBUILDING OF PROOFS ON THE LEVEL OF SCHEMATA 115

in accordance with U, which fi transforms into a formula or sequent, ytransforms U into a deduction schema in X32 such that fl(S) is deduciblein X32 in accordance with y(U). We will say that a transformation a ofdeductions in X31 into deductions in X32 agrees with a rebuilding (/3, y) ofdeductions in X31 into deductions in X32 on the schema level if, for anydeduction schema U in X31 and any formula or sequent S the followingconditions hold.

(1) If S is deducible in X31 in accordance with Uand fi transformsS into a formula or sequent, then we can construct a deduction ofS in X31 in accordance with U, to which a is applicable.

(2) For any deduction of a formula or sequent S in P1 in accor-dance with U, if a is applicable to ,then fi transforms S intoa formula or sequent and a() is a deduction of /3(S) in X32 inaccordance with y(U) .

LEMMA 16.1. A transformation a of deductions in X31 into deductions inX32 agrees with some rebuilding of deductions in X31 into deductions in X32on the schema level if and only if, for any deduction schema U in X31 andany formula or sequent S, the following conditions hold.

(1) If S is a proof in p1 to which « is applicable, such that S is de-ducible in X31 in accordance with a schema U, then we can constructa deduction of S in X32 in accordance with U to which a is appli-cable.

(2) For any proofs £1 and 2 of a formula or sequent S in q31, if« is applicable to both and 2' then the last sequent of a(1)coincides with the last sequent of

(3) For any deductions £T and 2 in accordance with a schema U, ifa is applicable to both and 2' then a deduction inX32 in accordance with the same deduction schema as a(2).

PROOF. If a agrees with some rebuilding of deductions on the schemalevel, then conditions (1)-(3) are obviously satisfied. Assume now that con-ditions (1)-(3) are satisfied. The axioms and inference rules in 131 and X32are represented by recursive predicates on the set of formulas or sequents.Hence we can construct algorithms and that satisfy the following con-ditions for any deduction schema U in 31 and any formula or sequentS:

(a) is applicable to S if and only if we can construct a proof of S inX31 to which a is applicable;

(b) if is applicable to 5, then transforms S into a proof of S into which a is applicable;

(c) is applicable to U if and only if there is a deduction in inaccordance with U to which a is applicable;


(d) if ' is applicable to U, then ' transforms U into a deduction inP 1 in accordance with U to which a is applicable.

Let be an algorithm that selects the last formula or sequent in proofs ofP2 , and C an algorithm that transforms any proof in q32 into its deductionschema in P2 . Then the required rebuilding of deductions on the schemalevel is a pair of algorithms (/3, y) such that, for any formula or sequent Sand any deduction schema U in 31,

Q(S) ((a((S))), y(U) C(a((U))).LEMMA 16.2. For any rebuilding of deductions in q31 into deductions in

X32 on the schema level we can construct a transformation of deductions thatagrees with it.

The proof is similar to that of Lemma 16.1.

LEMMA 16.3. The following transformations of deductions in KGL(2() orin IGL(2() into deductions in the same calculus agree with rebuilding of de-ductions on the schema level:

(1) transformation of proofs of 2(-pure sequents into pure variable proofs;(2) transformation of a proof of the conclusion of an application of some

structural rule or the rule of substitution (see §3) into a proof of thepremise.

The proof, which is similar to that of Lemma 16.1, proceeds by inductionon the height of the given deduction. In the same way we can prove thefollowing assertion:

LEMMA 16.4. The transformation of deductions in KGL(2[) (IGL(2t))into deductions in KH(2[) (IH(2t)) described in the proof of Lemma 5.6agrees with rebuilding of deductions on the schema level.

LEMMA 16.5. The transformation of deductions in KH(2() (IH(2t)) intodeductions in KGL(2[) (IGL(2t)), described in the proof of Lemma 5.3,agrees with a rebuilding of deductions. on the schema level.

This assertion is proved, using Lemma 16.1, by induction on the height ofthe given deduction.

Let p be an axiomatic Hilbert-type theory or sequent calculus, a and/3 algorithms. Let denote an algorithm, applicable only to proofs in3, which transforms any proof in 3 into . Construct an algorithm

a o Q such that, for any proof in 3,

a o /3

The next assertion follows from Lemma 16.1.

LEMMA 16.6. For any transformation « of deductions in 1tI into deduc-tions in q3 and transformation /3 of deductions in 3 into deductions in X32 ,

if both wand /3 agree with rebuildings of deductions on the schema level,

§16. REBUILDING OF PROOFS ON THE LEVEL OF SCHEMATA 117

then a o fi is a transformation of deductions in X31 into deductions in X32that agrees with some rebuilding of deductions in p1 into deductions in X32on the schema level.

Let 2i be the union of sets ail , 2' ... , i . Let i° denote the union ofr copies of the set of all formulas of the predicate calculus. It is obvious thatany deduction schema in KGL(2t) (IGL(2t)) is also a deduction schema inKGL(2[°) (IGL(2t°)), and that, for any deduction schema U in KGL(2t)(IGL(2t)), a deduction in KGL(2t) (IGL(2t)) in accordance with U is alsoa deduction in KGL(2[°) (IGL(9J°)) in accordance with U.

Let 3 be one of the calculi KGL(2t) or IGL(2i), q3° the correspondingcalculus KGL(2i°) or IGL(2[°). Let Q denote the transformation, definedin the proof of Lemma 4.1, of pure variable deductions in 3 which end withan application of the cut rule into deductions in 3.

LEMMA 16.7. Transformation Q agrees with a rebuilding of deductionson the schema level, and for any proof in 3 with pure variables which isended with an application of the cut rule, Q () coincides with Quo ().

This is proved, using Lemma 16.1, by induction on the sum of the heightsof the proofs of the premises of the last rule application in the initial proof.

LEMMA 16.8. For any deduction schema in 3 which is admissible in 3° ,we can construct a deduction schema V in 3 such that

(1) V does not include analyses of applications of the cut rule;(2) for any 2[ pure sequent S. if S is deducible in j3 in accordance with

U, then S is also deducible in j3 in accordance with V.

PROOF. Let U be a deduction schema in 3 which is admissible in 3° .Using Lemmas 12.9 and 15.2, construct a deduction o in 3° in accordancewith U which involves only degenerate formulas. o is obviously a purevariable proof. Applying Theorem 4.1 to we obtain a proof o in 3°without applications of the cut rule. Let V be the deduction schema of o .Then V clearly satisfies condition (1). In order to prove condition (2), weconsider the transformation of o into o in more detail.

Let be a proof in p. Recall that proofs in KGL(2i) and IGL(2() arewritten as planar rooted trees. We assign a word in the alphabet {O, 1 } toeach occurrence of a sequent in , as follows.

The empty word is assigned to the last sequent. If a word P is assignedto an occurrence of a sequent in and the sequent is obtained by somerule, then in the case of aone-premise rule, we assign the word PO to thepremise; in the case of atwo-premise rule, we assign PO to the left premiseand P1 to the right premise.

Note that the occurrence of a sequent in is uniquely determined by theword assigned to it.


Let P be a word in the alphabet {O, 1}. Construct an algorithm P suchthat, for any proof in 3, the following conditions are satisfied.

(1) P is applicable to if and only if is a pure variable proof andP is assigned to an occurrence of a sequent in which is obtainedby the cut rule in £?T.

(2) If P is applicable to , then P () is obtained from by re-placing the subdeduction with the end sequent to which P is assignedby the result of applying a to that subdeduction.

The process of cut elimination in the proofs of Lemmas 4.2 and 4.4amounted in fact to successive applications of P with different P. Conse-quently, we can write the transformation of into as

Plo o Po o ... o P1o

where l and the words P1, P2, ... , PI are determined by and conse-quently by the schema U alone.

Let be a deduction of an 2.-pure sequent S in 3 in accordance withU. Using Lemmas 3.3 and 3.4, we rebuild into a pure variable proof £ Tof S in 3. Clearly, the schema of is U. By Lemmas 16.6 and 16.7,the transformation

Pi o P2 o ... o Pq q3 q

rebuilds "1 into a deduction of S in 3 in accordance with V. 0

CHAPTER IV

Bounds for the Complexity of TermsOccurring in Proofs

In the preceding chapter we obtained bounds on the height of terms oc-curring in proofs in the predicate calculus (see Lemmas 14.5 and 15.6). Inthis chapter we will consider two applications of these bounds.

(A) Comparison of the lengths of direct and indirect proofs of existencetheorems in the predicate calculus. It will be shown that the length of adirect proof of an existence theorem cannot be bounded from above by aKalmar elementary function of the length of an indirect proof of the sametheorem. In other words, direct existence proofs in the predicate calculusmay be essentially longer than indirect ones.

(B) Complexity version of the existence property of the constructive predi-cate calculus. While rebuilding derivations of existence theorems into directproofs, we will establish upper bounds both on the length of the new proofsand on the maximal height of the terms occurring in them.

The main results of this chapter were published in [15, 18, 20, 19, 26].

§17. Comparison of the lengths of direct and indirect proofsof existence theorems in the predicate calculus

In this section we will consider deductions of the following formulas inthe predicate calculus:

wk(`dx(P(x, o, x, v)&P(v, x, u)) P(y, x', u))))

0, wo)&P(0, wo, w1))& P(0, wi, w2))...

&P(0, Wk_l, Wk))),

(1)

where k > 0, x, y, u, v, wo , w1, ... , wk _ 1 , wk are pairwise distinct ob-ject variables, P a three-place predicate.

Let A and B be formulas, R an s-place predicate, x1, ... , x1 a listwithout repetitions of object variables, l < s. Let A f R ¶x 1 x1 B denote theformula obtained from A by substituting B with distinguished parameters

119

120 IV. BOUNDS FOR THE COMPLEXITY OF TERMS

xl , ... , x1 for R. In what follows we will use the following notation:

Co b'x(P(x, 0, x, v)&P(v, x, u)) P(y, x', u))),

P+(a, /3, Y) =(Co P(a, /3, y)), Bo (a) = P(0, 0, a),B( a) , w1(R(w1)&P(0, w1, a)), Bo (a) P+(0, 0, a),

Bi+1 (a) B («)Itt¶aBi (a) ,Eo(a) b'vo3uoP+(vp , a, up)

A; («) (Co B; (a)), 0,B+1(a) B,°(«) IQ¶«B; (a) , Ei+t (a) Ei+t (a) I T¶aE, (a)

B.(a) W1(Q(w,)&P (0, W,, a)),

E, (a) b'vi(T(vi) 3ui(P+(vi, a, ui)&T(u;))),

where i > 0, R, Q, T are one-place predicates, vo , uo , v 1, u 1, ... differ-ent object variables other than x, y, u, v, wo , W1 ..... In this notation (1)can be written as wkAk (Wk).

LEMMA 17.1. The following formulas are deducible in KH.

(1) VxP(x, 0, x').(2) b'xyu(2v(P+(v, x, v)&P+(v, x, u)) P+(v, x', u))(3) (b'xP+(x> b, f(x)) b'xP+(x, b', f(f(x)))).(4) ((Co R(b)) (b'xP+(x, b, f(x)) (Co Bo(f(0)))))(5) A(0).(6) Eo(0).

(7) E1(0).

(8) Ei (0)IT1f«Eo(«)(9)

(10) (E(0) (T(0) wo(Bo ('wo) &T(Wo))))

(11)(wo) Ao (wo)) ,

(13) (dwo(Q(wo) (Co R(wo))) Db'w1(Bo(w1) (Co Bo(ws))))(14) (dwo(Q(wo) (Co DR(wo))) R(wo))))

Here b is a 0-place function symbol other than 0, f a one-place functionsymbol other than '.

It will be convenient to divide the proof of the above formulas in KHinto two stages. We first proceed from bottom to top, finding cut-free deduc-tions of the corresponding sequents in IGL, and then rebuild the resultingdeductions into proofs in KH.

LEMMA 17.2. There is a natural number c such that, for all k > 0,

IH Fc2 Ak (2k).k-1

§17. COMPARISON OF DIRECT AND INDIRECT PROOFS 121

PROOF. Let {O}, , F, Fo be deductions of (1), (3), (4), (5) ofLemma 17.1, respectively, in I H . Let us assume that the lengths of {O},., F* , Fo are less than a natural number co . Let denote the result ofreplacing all occurrences of b by Oand all occurrences of f by the term

n

with distinguished parameter x. Consider the figure

+ m. p.VxP (x,O ,x ) ,

m p, .

dxP+(x , 0",

dxP+(x , x(2 )) ; n-1

dxP (x , 0 , x2)This figure is a tree representation of a deduction of dxP+(x , o,

in I H , which we denote by . { n } . Clearly,

lH[{n}] < (c + 1)n + CO. (2)

Let P7 denote the result of renaming of wo as w1 in F* and replacingin the resulting deduction all occurrences of b by 21 , all occurrences of fby the term

l+ 1with distinguished parameter x, and all occurrences of R

by the formula B1 (a) with distinguished parameter a. It is obvious thatFl* is a proof of the formula

(AT() (VxP(x, 21 , xi+1) Al+1(2i+1)))

in I H . We can now write the required proof of Ak (2k) as

Fo ; Fo

{ }; (VxP(x 2 x A- 2 m.p.

A (21)

`4k-1(2k-i); Fk-t

m. p.

m.p.{2k-1}; (VxP(x, 2k-1' xk) Ak (2k))

Using inequality (2) and Lemma 4.3, one easily shows that the length ofthis proof is less than 4 (c0 + 1) 2k _ 1 + co .

LEMMA 17.3. For any terms 0, ... , Bn , if the formula V1A(0) is

deducible in KH, then one of 81 , ... , Bn is 2k .

PROOF. Let 81, ... , Bn satisfy the condition of the lemma. Let us in-terpret the predicate P(x, y, z) in primitive recursive arithmetic as the

formula x + 2y = z and all function symbols other than 0 and / as theterm 2k+

1 . In our case we can treat the deducibility in primitive recursivearithmetic as truth in a given interpretation. It is clear that C° will be trueunder our interpretation and that Ak (0) will be true if and only if 0 is kConsequently, V 1 1 A(01) will be true if and only if one of 0, ... , Bn

1S 2k .

LEMMA 17.4. There is a natural number c such that, for all k,

I H F-c(k+1)

Wk Ak (wk) .

PROOF. Lemma 17.1 implies that

IH F- w0Ao (w0). (3)

Assume that k> 1 and let 01 ,

* , F 0 , and F be deductions offormulas (6), (7), (8), (10), and (11) of Lemma 17.1, respectively, in IH.Let Phil (n > 2) denote the result of renaming the variables v1 , u1 ,

and u0 in as vn , un , v,_1, and u_1 1 , respectively, and then replacingall occurrences of T by the formula E_2 (a) with distinguished parametera. Obviously, for all natural numbers i, is a proof of the formula E,(0)in I H . Let F k denote the result of renaming the variables v0 and u0 inF ° as Vk and uk, respectively, and then replacing all occurrences of T bythe formula E_1 (a) with distinguished parameter a. Obviously, F k is aproof of the formula

(Ek(O) (Ek_l (0) w0(B0 (w0) &Ek_1(w0))))

in I H . Let F ` (0 < i < k) denote the result of renaming the variablesk _w0, w1, v0, and u0 in F as w,_1, w,, vk _, , and uk_ 1 , respectively, andthen replacing all occurrences of Q and T by the formulas Bl 1(a) andEk- ._1 (a), respectively, with distinguished parameter a. Obviously, F` is

i ka proof of the formula

(w,_1 (B&Ek_I(wj_1)) (E_1_ (0) wj(B! (wj)&Ek_1_1(w1))))

in I H . Consider the figure

Ek-i(0) 3wo(Bo (wo) &Ek-l(wo))) rwo(Bo(wo) &Ek-l (wo));

m.p.Fk

(Ek_2(o) wl(Bi (w,)&Ek-2(wl)))m.p.

moo; (E0(O) wk-i(Bk i(wk-i)&Eo(wk-i))) --- -wk_1 (B±1 (wk_i) & E0(2uk_l)) m p

k


This figure is a tree representation of a proof in I H , which we denote by*Fk-1 'Let S and S* be deductions of formulas (12) and (13), respectively, in

I H . Let Sm (m > 1) denote the result of renaming the variables w° andw 1 in S* as wm _ 1 and wm , respectively, and then replacing all occurrencesof Q and R by the formulas B_1 (a) and B_1 (a)respectively, withdistinguished parameter a. Obviously, Sm is a proof of the formula

(VWm (Wmi) Am_1(wm_1)) dwm(Bm(wm) Am(wm)))

in IH. Consider the figure

'so;

'st

b'w1(Bl A( w1))m.p.

b'wk-i(Bk i(wk-i) DAk-l(wk-i)); Sk

b'wk(Bk (wk) Ak (wk))m.p.

This figure is a tree representation of a proof in I H , which we denote bySk.

Let F* and S* be deductions of formulas (9) and (14), respectively, inI H . Let Fm (m > 1) denote the result of renaming the variables w° andw 1 in F* as wm_

1and wm , respectively, and then replacing all occurrences

of Q by the formula B_1 (a) with distinguished parameter a. Clearly, F,is a proof of the formula

(Wmi (B_1 (Wm 1) & E0( Wm 1)) WmBm(wm))

in IH. Let Sl denote the result of renaming the variable w° in S* as w1and then replacing all occurrences of Q and R by the formulas B(a) andBl (a) , respectively, with distinguished parameter a. Clearly, S1 is a proofof the formula .

A,(w,)) ( w1AT(w1)))in IH. Consider the figure

N * *

!-i' Fk m r S Sk m r

3wkAk(wk)m.p.

This figure is a tree representation of a proof in I H , which we denote byN

In order to complete the proof of Lemma 17.4, we will find an upper boundN

on the length of k' Suppose that the lengths of °, " 1 , "* , F F ,

S ° , S* , F* , andare less than a natural number c1. Then

k1N[Fk-l] < 2(cl + 1)k + cl , 1H[S ] <(cl + 1)k + cl ,

1H[9k] < 3(cl + 1)k + 4c1 +3.

Lemma 17.4 now follows from these inequalities and from condition3).

LEMMA 17.5. For any natural number k, k > 6, and any term 0, thelength of any proof of A(0) in KH or IH is greater than 3 2k-1

PROOF. Let :t be either I H or KH. Let 8 be a term and k, 1 naturalnumbers such that k > 6 and

3 I-1 A(0)..

Then it follows from Lemma 17.3 that 0 is 2k . Note that 3 is repre-sentable by a set of CF-systems such that

c° * [3] = 6, arg[3] = h* [3] = 0, m[3] = M°[q3] = 1.

On the other hand, A(0) has exactly 2k +8 subformulas beginning withsymbols , &, V, , V, or 2, and

h* [Ak (a)] = 1, arg[Ak (a)] = 3.

Denote

Bn 28B2(log2 Bn)2.

It is easy to show that B6 <2 5 and that, for any n,

Bn 1 < 2lB' < 2Bn .

Consequently, Bn <2n for all n > 6.Suppose that 712 <2) _ 1 . Then, by Lemma 1 4.5, there is a natural number

c, 0 <c < 2k , such that

I- Ak ( )By Lemma 17.3, 2k - c = 2, and we have obtained a contradiction.

Hence

l> 21 7> 1 21- k-1 3 k-1

THEOREM 17.1. There exist a sequence of formulas A1, A2, ... and a con-stant c such that, for any natural number k,

(1) I H F C(k+ 1) 2xAk+ 1x(2) I H Hc2k [Ak+ 1 ]2k+ '

(3) for any term 8, if [Ak+l] e is deducible in KH, then h[B] > 2k+1and the length of any proof of [Ak+11 e in IH or KH is greater than

1 Zk - C.3

This theorem follows from Lemmas 17.2, 17.3, 17.4, and 17.5.

LEMMA 17.6. For any proof of a sequent -> 3wkAk (wk) in KGL orin IGL, if the side formulas of the applications of the cut rule in do notinclude quantifiers, then 2k is the proper term of some application of the rule-+2 in .

PROOF. Let 3 be either IGL or KGL, a deduction of - wk Ak (wk)in p. Assume that the side formulas of cuts in do not include quantifiers.Then it follows from Lemmas 1.2 and 1.3 that includes no applicationsof the rules -* d and 2 -> and that the principal formulas of the axioms in

have no quantifiers. Let 01, 02 , ... , em be the list of proper terms of allapplications of the rule -> 2 in whose principal formula is wkAk (wk) .

Clearly, m > 0. We replace 2wk Ak (wk) by Vm1A(01) in all sequents of

and, if necessary, insert applications of the rules -* V and thinning in thesuccedent. This yields a proof of the sequent - Vm

1Ak (0k) in + . But by

Lemma 17.3, one of the terms 01, 02, ... , em is 2k .

Lemmas 15.6 and 17.6 imply the following assertion.

LEMMA 17.7. For any proof of a sequent -* wkAk (wk) in KGL or inIGL, if the side formulas of applications of the cut rule in do not includequantifiers, then

q h[am]>2k_1.

THEOREM 17.2. There exists no upper bound for the length of a proof inIGL or in KGL obtained from a given proof by eliminating all cuts, whichis a Kalmdr elementary function of the height of the given proof.

PROOF. Assume the contrary. Then, using Lemma 2 and Theorem 2 of[31], we can construct natural numbers K and m such that the height ofthe proof after cut elimination is at most 1 ) , where h is the heightof the proof before cut elimination. It follows from Lemmas 5.3, 17.4, and17.7 that, for some constant c and all natural numbers j,

21 < 2Kc(j+ 1)-1 - m+1

Consequently, for all j > m + 2,122 < Kc(j + 1)

and we get a contradiction.

§18. Complexity version of the existence propertyof the constructive predicate calculus

Let I' be a list of formulas, A a formula, x an object variable, aproof of the sequent t -> 2xA in IGL. An application L of an inferencerule in will be called E-critical if L is an application of the rule -an occurrence of some formula in the succedent of the end sequent of isa successor of the principal occurrence of L, and the principal formula itselfis 3xA.

LEMMA 18.1. If neither the formulas of I' nor the side formulas of theapplications of the cut rule in include occurrences of 2 or disjunction,then has exactly one E-critical rule application.

This lemma is proved by induction on h[am].

LEMMA 18.2. For any list of object variables o, if neither the formulasof t nor the side formulas of the applications of the cut rule in includeoccurrences of 2, then can be rebuilt, by renaming the proper variables ofthe applications of the rules -> d and 3 -> in QJ, into a proof in IGL withthe same end sequent and the same proper terms of E-critical rule applications,such that the proper variables of the applications of the rules -* d and 3 -do not occur in a and are not bound.

This lemma is also proved by induction on h[QJ].Let and 2 be proofs of t - 2xA in 1 GL . We will say that is

E-embeddable in '2 if:(1) is g+-embeddable in(2) for any E-critical application Ll of -* in there is an E-

critical application of -> 2 in 2 whose proper term is the same asthe proper term of L1.

LEMMA 15.3. Let D be a q+-closed set of formulas, a proof of a sequentI' - 2xA in IGL, and a proof in IGL, such that is q+-embeddablein is a pure variable proof which ends with a O-application of the cutrule, and neither the formulas of I' nor the side formulas of the applicationsof the cut rule in include occurrences of 2. Then we can construct a proofJ' of I' -> 3xA in IGL such that the following conditions are satisfied:

(1) 2J' is E-embeddable in ;

(2) 2J' is a pure variable proof;(3) the side formulas of the applications of the cut rule in ' contain no

occurrences of ;(4) h''['] h''[?I] + hV[912];

(5) h[PIJ'] <h[9J1] + h[QJ2];(6) StV[Y!, *j < StV[B, #] - 1).

Here and 2 are the subproofs of the left and right premise, respectively,of the last rule application in QJ, B the side formula of this application.

§18. COMPLEXITY VERSION OF THE EXISTENCE PROPERTY 127

This assertion is proved, using Lemma 18.2, by induction on h[PIJ1] +h[2], as in Lemma 4.1.

LEMMA 18.4. Let D be a q+-closed set of formulas, a proof of a sequentI' -> 3xA in IGL, and a proof in IGL, such that is q+-embeddablein , is a pure variable proof, st°[.", '] > 0, and neither the formu-las of I' nor the side formulas of the applications of the cut rule in includeoccurrences of . Then we can construct a proof P27' of I' -> 3xA in IGLsuch that the following conditions are satisfied:

(1) P27' is E-embeddable in ;

(2) ' is a pure variable proof;(3) the side formulas of the applications of the cut rule in ' contain no

occurrences of 2;(4) hv[!] <2hV[]_1;

v(5) h['] h[](6) stV[1, *] <stV[, *]

This assertion is proved in the same way as Lemma 4.2, by induction onh[], using Lemma 18.3.

LEMMA 15.5. Let O be a q+-closed set of formulas, a proof of a sequentI' - 3xA in IGL, and a proof in IGL, such that is q+-embeddablein , is a pure variable proof including a O-application of the cut rule,and neither the formulas of I' nor the side formulas of the applications of thecut rule in include occurrences of . Then we can construct a proof 'of t -p 3xA in IGL such that the following conditions are satisfied:

(1) 2J' is E-embeddable in ;

(2) 2J' has no D-applications of the cut rule;(3) the side formulas of the applications of the cut rule in 2J' have no

occurrences of 2;

(4) h[.QJ'] < h[am] 2 to , .This assertion is proved in the same way as Lemma 4.4, by induction on

st° [ , zi" * ], using Lemma 18.4.Let E.1 denote the set of formulas A of the predicate calculus such that

either A contains 2 or 8`[A] > 0, and E.2 denote the union of E.1 andthe set of formulas containing disjunctions.

THEOREM 18.1. For any proof in IH of a formula 2xA, if no boundvariable of A occurs free in 2xA, then there is a term T free for x in Asuch that

h[T] < h[A] M"' ' , (1)

IH HL,H[Y1 [A], (2)

where

M = 221X' ]+ 12 L =221X.2 [ ]+c

31Q I [ ]+2

and the constant c does not depend on the choice of and A.

PROOF. Let be a deduction of 2xA in I H . Using Lemma 5.3, rebuildinto a deduction of the sequent -* 2xA in IGL such that

h[P1J*] < 7 , (3)hE.l[QJ*]<21X.1[9f]+9, (4)

hE.2\E.1 21E.2\E.1 [95] + 9,H

1E.2\E.1[*] < 31E.2\E.1(5)1 a

Suppose that no bound variable of A occurs free in 2xA. Then, elimi-nating all E.1-applications of the cut rule from 95 (Lemma 4.4), we obtaina pure variable deduction of -> 2xA in IGL such that the side formu-las of the applications of the cut rule in contain no occurrences of 2,OC[91] =0,

EI

h[951] < h[*]. 2hE.l[PJ*]

*

E.2\E.1 E.2\E.1 *

(6)

(7)

Using the last inequality, apply Lemma 15.6 to rebuild (replacingcertain free terms by variables) into a deduction 2 of -* 2xA in IGLsuch that, for any proper term T of an application of the rule - in 2'

h[T] < h[A] 1 h[PJI].

Inequality (1) now follows from this inequality, taken together with in-equalities (3), (4), and (6) and Lemmas 2.7 and 4.3.

Eliminating all (E.2\E. 1)-applications of the cut rule in 2 (Lemma18.5), we obtain a deduction 93 of - xA in IGL such that £3 is E-embeddable in 2' the side formulas of the applications of the cut rule in953 include no occurrences of disjunction or , and

E.2 \ E. I

h [ ] h [" ] 2E2\ E [(8)1 st

I'I ]

Let T be the proper term of the unique E-critical application of the rule2 in 953 (see Lemma 18.1). Replace every occurrence of xA as a mem-

ber of a sequent succeeded by an occurrence of a formula in the succedentof the end sequent of 93 by [A], and delete from the conclusion ofthe E-critical application of the rule -* 2. This yields a deduction ofthe sequent - [A]T in IGL. Clearly,

h[9J4] <_ h[913] (9)

§18. COMPLEXITY VERSION OF THE EXISTENCE PROPERTY 129

Now, to prove condition (2), we can use inequalities (3)-(9), Lemmas 2.1,2.3, 2.5, 2.7, 4.3, 5.5, 5.6, and the equalities

1E.1 1E.2\E.1H H H

= f1[] + F2\E.l[ii].a a a

Let A and B be formulas of the predicate calculus. We will say that Bis a ic'-type of A if B is transformable into A, ic, [-* B] = 0, and

of ,[B] < o; [A].o,

Clearly, a q*-type of a formula A will also be a lc'-type of A. An enu-merable set of formulas 2 will be called ic'-closed if any formula in 2has a ic'-type belonging to 2'. We will now show that the set of formulasdefined by the schema Ind* :

((A(0)&Vx(A(x) A(x'))) VxA(x)) (10)

is ic'-closed. Indeed, let VxA* (x) be a q*-type of VxA(x). Then the fol-lowing formula will be a lc'-type of (10):

((A*(0) &Vx(A*(x) A*(x'))) `dxA*(x)).

Similarly, it can be proved that the set of formulas defined by the schemaEq* is ic'-closed.

Let be a proof in KH(2t) or I H(Qt) . Let o denote the maximumof o, [A] for all applied axioms A in that are not particular appliedaxioms. Let K.m denote the set of formulas B such that o, [B] > m.

THEOREM 18.2. For any natural number n and formula A, if 2t is theunion of the list of particular applied axioms and finitely many ic'-closedenumerable sets, and there is a proof of the formula [A]ocn, in KH(2)(IH(2J)) such that

n > max{m, 8C[ E - 2xA]} (Mm)e"[] +max{m, 80 [E -> A]}, (11)

then for some k <n the formula `dx[A]X(k) is deducible in KH(2i) (IH(2J)),where

[ ] , m 31 K.ma

PROOF. Let be a deduction of [A]ocn, in KH(2[) or IH(2) that sat-isfies the conditions of the theorem. Using Lemma 5.3, rebuild into adeduction of the Sequent -* [A]ocn, such that

hK.m [91i ] < 21K.m [g,] + 9, IK.m [gJ' ] < 31K.m [g, ] . (13)H 1 a

Now, using Lemmas 3.1 and 3.2, rename the free variables in 7' so as toobtain a pure variable proof ' * of the sequent [A * ]ocn, . Then, deleting all

K.m-applications of the cut rule in (Lemma 4.4), we obtain a deductionof -* [A*](n) such that

]

'1] < 2StK.4 i] ' h[91'].

It now follows from inequalities (12) and (13) and Lemmas 2.7 and 4.3that

(Mm)lH[l (14)

Let U1 be the deduction schema of . Add q * -types of the side formulasto all analyses of applications of the cut rule and the side formulas themselvesto all analyses of applications of the 2[-cut rule, if it is a particular appliedaxiom; otherwise add a ic'-type of the side formula which is an applied axiomwith the same index. Now rename the free variables in such a way that thetypes of analyses do not include free occurrences of x and have no freevariables in common. Denote the resulting correct deduction schema withfixed cut types by U. Clearly, is an (x)-deduction of the sequent A*

with proper term Oin accordance with U. Note that

max{o; [Ui ] , o[-4 A*]} < max{m, o; [ 2xA]},

max{bf ' [U*] , o1[- > A*]} < max{m, o1[ -* A]},o, 1 0, o,

ic, [ U1* ] = 0 , 2xA], q [ U* ] <]-1.

x, 1

Using Lemma 15.5, construct a universal (x)-deduction i of A* inaccordance with Ul such that

bo, [T] < max{m, o; [ -> 2xA]} max{m , b1[ - A]}, (15)

where T is the proper term of 3. Obviously, T has the form z, whereeither z is 0 and k = n, or z is an object variable and k < n. It followsfrom inequalities (11), (15), and (14) that z is not 0. Hence z is an objectvariable. It is also clear that z does not occur free in Vx[A* ]X(k) .

Applying the rule d s + 1 times, extend to a deduction of thesequent

yyl ...

y (s > 0) is a list of all the free variables of [A]x n) . Tos 0complete the proof, we need only use Lemma 5.6 and the fact that

IH F- (Vy1 ... ysx[A]x(k) D Vx[A]x(k)).

CHAPTER V

Proof Strengthening 't'heorems

Theorem 18.2 is an example of a proof strengthening theorem. There, theset of natural numbers for which the deducibility of a formula is assumedcontains only one element. In the present chapter the cardinality of such setsof natural numbers in the proof strengthening theorems will not be restricted(except for Theorem 19.1).

In §19 we will prove two proof strengthening theorems for finitely ax-iomatized Hilbert-type theories based on KH or I H . These results werepublished in [16, 20, 27].

In §20 we will establish proof strengthening theorems for a few versionsof formal arithmetic, both when the language of arithmetic contains 0-placeand 1-place function symbols only and when it contains 2-place functionsymbols as well. In the latter case we will use other complexity measures ofproofs in addition to length. The main results of §20 were published in [26].

In §21 it will be proved that in a certain finitely axiomatized fragmentof Parikh arithmetic PA*, and in all its consistent extensions, the rule ofinfinite induction is not admissible if, for any i, the length of a deduction ofthe i th premise is at most c loge (i + 1). The last section of the chapter willestablish upper and lower bounds on the length of deductions of the followingpropositions:

(1) The natural number n is even.(2) The natural number n is composite.

The main results of §21 were published in [16, 26, 30].

§19. Proof strengthening theorems in finitely axiomatized theories

LEMMA 19.1. Let E be a finite list of closed formulas, A a formula, andU a deduction schema in KH(E) (JH()). If there are natural numbers nand m, n m, such that both [A]ocn, and [A]n) are deducible in(IH()) in accordance with U, then there is a natural number k such thatk < min(n , m) and VX[A]x(k) is deducible in KH(E) (IH()).

PROOF. Assume that n m and both [A]ocn, and [A]n) are deduciblein accordance with U. Using Lemma 16.5, construct a deduction schema Vin the sequent calculus such that both -* [A]ocn, and -* [A]ocn>) are deducible

131

132 V. PROOF STRENGTHENING THEOREMS

in accordance with V. It follows from Lemmas 3.1 and 3.2 that the sequents- [A* and -> [A* ]ocm) are also deducible in accordance with V, whereA* is a formula obtained from A by renaming all free variables other thanx that are bound in A or E . Using Lemma 16.8, rebuild V into a schemaW without analyses of cut applications and then add the corresponding sideformulas to each analysis of an application of E-cut in W. Denote theresulting deduction schema with fixed cut types by W. Using Lemma 15.5,construct a universal (x)-deduction of the sequent A* in accordancewith W * . Clearly, the last sequent in has the form -* [A* ]yck , where

y(k) is a term free for x in A*, y does not occur free in Vx[A*]x(k) , andk < min (n , m). Applying the rule -* V, extend to a deduction of

x-+ dzl ... zsx[A]xck

where z1, ... , zs (s > 0) is the list of all free variables in In orderto complete the proof of the lemma, we use Lemma 5.6 and the fact that

I H I- (Vz1 ... zsx [A]x(k) D Vx [A]x(k) ).

THEOREM 19.1. For any finite list E of closed formulas and any formulaA, if A and E contain only 0-place and 1-place function symbols and for asufficiently large natural number n there is a proof of the formulain KH(E) (IH()) such that

5 3 /log2n, (1)

then for some k < n the formula Vx[A]x(k) is deducible in KH(E) (IH()).

PROOF. Let E be a list of closed formulas, A a formula, and 3 eitherKH(E) or IH(E). Assume that A and E contain only 0-place and 1-placefunction symbols. Let c1 denote the number of different subformulas in thesequent E - A that begin with , &, V, -, V, or and s the maximaldimension of predicates in E -> A. Let n be a natural number and aproof of [A]o(n) in 3 such that (1) holds and

28B2(log2 B)2 < 1og2 n,

where

B = 3 (cp 1 + 1og2 6s) + 12 + 1og2 (h + 3),

h =h*[E-*A].

Without loss of generality, we will assume that each applied axiom inhas at most one occurrence in . Using Lemma 14.5, construct a naturalnumber k, k < n, such that both [A]o(n) and [A]o(k) are deducible in 3 inaccordance with the same schema. Now, in order to complete the proof, weneed only use Lemma 19.1.

§19. CASE OF FINITELY AXIOMATIZED THEORIES 133

LEMMA 19.2. Let E be any finite list of closed formulas, n a sufficientlylarge natural number, r a natural number, and A a formula. If there arenatural numbers k1, k2, ... , k, k1 < k2 < <ks <n, s n2' 2r+3) , andproofs

, 2 , ... , s in KH(E) (IH()) such that for all i, 1 < i < s,the last formula of . is [A]k1) and

loge n

(2r + 3)1og21og2 n'

then for some m < n the formula b'x[A]x(m) is deducible in KH(E) (IH()).

PROOF. Let E be a list of closed formulas, A a formula, and n a naturalnumber such that

NZ + N + IEI+ 15 < (log2 n)2 , (2)

whereloge n

N (2r+ 3)1og21og2 n'

Assume that E , n, and A satisfy the conditions of the lemma. Let LK(L1 ) denote the number of correct deduction schemata in KH(E) (IH())that contain at most 1 occurrences of analyses of axioms and applicationsof inference rules. It is easy to prove that

max(LK , L11) < (l2+l+I,+ 15)x.

Hence it follows from (2) that

max(LN, LN) <ns.

Consequently, there are natural numbers i and j, i j, 1 < i < 5,1 <j <s, such that both and [A]k) are deducible in accordancewith the same schema. Then, applying Lemma 19.1, we obtain the requirednatural number.

THEOREM 19.2. Let E be any finite list of closed formulas, m, d, r naturalnumbers, d > 1, n a sufficiently large natural number, A a formula. Supposethat for any i (0 < i < n) there is a proof in KH(E) (IH()) whoselast formula is [A]I1), where

ll = (m + di)r+1

andlog2(m + dn)r+1

N (2r+3)log2log2(m+dn)r+1

Then, for some k < (m+dn)r+1 the formula `dx[A]x,k, is deducible in KH(E)(IH()).

This theorem follows from Lemma 19.2 and the fact that for all naturalnumbers m, d, r, and sufficiently large n,

(m + (d +1)n)(2r+2)/(2.+3) <n + 1.

COROLLARY (Kreisel's conjecture for finitely axiomatized theories). Letbe any finite list of closed formulas, m a natural number, n a sufficiently largenatural number, and A a formula. Suppose that, for any i (0 < i < n),

KH(E) I log2(n+m) [A]o(m+;).3 log2 log2 (n+m)

Then, for some k < m + n,

KH(E) F `dx [A]X(k) .

§20. Proof strengthening theorems in formal arithmetic

The following formulas will be taken as particular applied axioms of for-mal arithmetic:

(Z.1)

(Z.2)

Vx(x + 0 = x) , (Z.3)

Vxy(x + y' = (x + y)'), (Z.4)

Vx(x 0 = 0) , (Z.5)

dxy(x y' = (x y)+x). (Z.6)

Denote this list of formulas by Z. The schemata I nd * and Eq * willbe considered as axiom schemata of formal arithmetic. In what follows,we denote the calculus KH(Ind* , Eq* , Z) by PA and IH(Ind* , Eq* , Z)by HA. Obviously, PA is equivalent to Kleene's system of classical formalarithmetic [8]; HA is equivalent to Kleene's system of intuitionistic formalarithmetic [8].

LEMMA 20.1. For any sufficiently large natural number d, any formula A,and natural numbers n, s, do , d1, ... , d, n> 1, 0 < do <d, 0 < d1 <d,... , 0 < ds <d and s > 22d , we have

HA F ((4vx[A])" `dx[A]X(n) (1)

where, for all i (O<i<s),

L a ((x 0(d1)) +t

PROOF. By the asymptotic law of distribution of prime numbers (see, e.g.,[36]),

w(y) ---+1'y-'ooy

(2)

where yr is the Chebyshev function and e" the least common multiple ofall natural numbers j such that 0 <j < y.

§20. PROOF STRENGTHENING THEOREMS IN FORMAL ARITHMETIC 135

Let d be a natural number and d0, d1, ... , ds natural numbers such that,for all i (0< i< s),

0<d1< d. (3)

Let m denote the least common multiple of do , d1, ... , d. It followsfrom (2) and (3) that, for sufficiently large d,

m e" <22d (4)

Note that

HA F- z 0("')) + u & u < 0("')), (5)s- i

HA F- du u < v u = 0(1) (6)=o

Assume that s > 22d . Then (1) follows from (5), (6), and (4).Let n and m be natural numbers, a and u object variables. We will use

the following notation:

a(m), if n =0,{nu+m}(a)=< (n)(((u.0)+a)m, ifn0.

Let be a stable parametric tree of b'-periodicities in the unknown u.We will define a term t [. ] by induction on the number of vertices in £.

(1) If consists of one vertex 0, then r[] ] a r[0].(2) If l edges go from the root O of to vertices Ol , ... , O! , then

a T[O]b ...b {Pl }(zu[.1]) ... {P,}(r[.]),

where, for all i (1 < i < 1), is a stable parametric tree of b'-periodicities in u whose root is the ingoing vertex of the edge withvariable b. that begins at 0, and P, is the degree of that edge.

LEMMA 20.2. Let t1 = vat2 be any normal TS-equation in unknowns t1, t2and an a-unknown v, and W) any u' parametrized solution of thisequation.

(1) If the degrees of all the a-critical edges of the parametric tree W arenatural numbers and the term r[2] does not begin with ' , then

1] a zu[W]azu[pJ2].

(2) HA F Vu(r [. 1 ] = zu[W

This lemma is proved, using Lemmas 10.5 and 10.8, by induction on thenumber of vertices in W.

Let c be a stable parametric SP-tree in u, O the root of c I. Assume thata formula A is assigned to O', E1, ... , ES (s > 0) are the edges beginningat 0 for all j (1 < j < s), a variable b3 is assigned to E, and E ends at

a vertex O,. Let y/[1)] denote the formula obtained from A by replacingthe variables b1, ... , bs by

{P1}(r[g,T1]), ... ,respectively, where, for all j (1 < j < s), is the subtree of c with rootO and P the degree of E.

The following assertion can be obtained from Lemma 20.2 via the equiv-alent replacement theorem (see, e.g., Theorem 23 in [10]).

LEMMA 20.3. Let Xl a [X2]i be any formula equation with formula vari-ables X1, X2 and a term unknown t and (c1, (1)2; ) any u'-parametrizedsolution of this equation.

(1) If is a SP-tree or r[] does not begin with the symbol ', then

a [ u

(2) HA H Vu(Wu [Wu

Let F be a CF-system with n formula unknowns, m term unknowns,and p unknowns for object variables. We will say that F is HA-closed if,for any u'-parametrized solution

of T, there is a solution

(Ci,...,Cn;Oi,...,Om;ui,...,up)of T such that, for all i (1 < i < n),

HA H Vu(C1

for all j (1< j< m),HA H `du(O3 =

and, for all s (1 < s < p), us coincides with z..

LEMMA 20.4. The following CF-systems are HA-closed:

(1) CF-systems representing the propositional logical axioms schemata ofKH and IH, the logical axiom schemata (5a) and (6a) of KH, therules of inference of KH ;

(2) a CF-system representing the schema Eq* ;(3) CF-systems representing the axioms and the rules --

- V 1 , . V2, V 4) - &, & - 1 , &. -2' - -, , -, -- , - y , -4,and the cut rule of KGL and I GL ;

(4) a CF-system representing the schema I nd * ;(5) CF-systems representing the logical axioms schemata (Sb) and (6b) of

KH ;(6) CF-systems representing the rules V -- and - of KGL and I GL .

PROOF. The substituting terms of all the CF-systems listed in (1)-(3) con-tain neither term unknowns nor the symbol '. Consequently, they are HA-closed, by condition (1) of Lemma 20.3.

Ind* is represented by the CF-systemXl a ((XZ &da(X3 X4)) daX3) ,

X2 a [X3]Q , X4 a (7)

0*a*X3, ai*a*X3

with distinguished unknown Xl , formula unknowns X2, X3, X4, and an un-known a for object variables. Let (cI 2, (1)3, (1)4, z) be a u'-parametrizedsolution of (7). Then, by condition (2) of Lemma 20.3, the solution of (7)required in the definition of an HA-closed system will be((([y/[cI3]]&Vz(y.i[ct'3]

dz jiru [c3]) , [u w[1)3], [w[1)3]]1 ; z).Similar arguments show that the CF-systems listed in (5) and (6) are HA-

closed. oLet Z* denote the union of the lists Z.1 , Z.2, and the following formu-

las:

VxP(x, 0, x). (Z*.3)

b'xYz(P(x,Y , z) Pox,Y , z')) (Z*.4)

VxQ(x, 0, 0). (Z*.5)

tlxyzu(Q(x, v, z) (P(z, x, u) Q(x, v', u))) (Z*.6)

dxyzu(P(x, v, z) (P(x, v, u) z = u)) (Z*.7)

`dxyzu(Q(x, v, z) (Q(x, v, u) z = u)) (Z*.8)

Y , z) (Z*.9)

b'xydzQ(x,Y , z), (Z*.10)

where P and Q are three-place predicates. Let PA" denote the calculusKH(Ind*, Eq", Z"`) and HA" the calculus IH(Ind*, Eq", Z*). Elimi-nating the function symbols + and from the proofs (see, e.g., §74 in [8]),we can prove the following assertion:

LEMMA 20.5. Let A be any formula that does not contain the symbols+ and . Then A is deducible in PA* (HA*) if and only if the formulaobtained by simultaneous substitution of the formulas x+ y = z and x y = zwith distinguished parameters x, y, z for P and Q, respectively, in A isdeducible in PA (HA).

THEOREM 20.1. For any formula A and sufficiently large n and k, if Acontains only 0-place and 1-place function symbols and for all i (0 < i < k)there is a proof of in PA" (HA") such that

3 /ig2 log2(n + i), (8)

then dx[A]x(n) is deducible in PA* (HA" ).

PROOF. Letq3* be either PA* or HA* and assume that A contains only

0-place and 1-place function symbols. Fix a natural number No such that,for all N> No ,

h+ 2 s 2 s 3)N+1 <27N2(9)

2((sN26N, +

1)) N < 27N2 (10)

where P 1 is the number of different subformulas occurring in the sequentZ * - A that begin with , &, V, , V, or ;

s = arg[Z * _ A] , h = h* [Z * -4 A].

Let n and k be natural numbers such that:

4 < n, No < 3 Jlog2 log2 n, (11)

91og21og2(n + k) < 1og21og2 k. (12)

Let 0 < i < k and let be a proof of in q3* for which (8) istrue. Assume that the term unknowns of the CF-system y{ U1}, where U1 isthe deduction schema of , are distinct from x and do not occur in A,and that the formula unknown X is assigned to the last analysis in U1. Addthe equality X a A to the list of equalities of y { U1 } and the unknown xto the list of term unknowns; denote the resulting CF-system by T1. Then,using Lemmas 12.4, 13.1, and 13.2, construct a normal CTS-system , with

a solution o, such that -1 has at most s11261;+(0, b'-essential a-unknowns(where 11 = 1H[] ), the number of essential unknowns in E1 is at most1, + 1 , the value of x in ci1 is and

.] < s11.261'+s°' . (13)h[1] = h, uQ[1 . ;1

It follows from (8), (9), (11), and (13) that

n + 1> (h + 2)(vQ [°i " I] + 2)(vQ [°j ; ]s/ + 3)"' + 2.1

Consequently, condition (2) of Lemma 10.12 is applicable.Using Lemma 10.12, construct a u'-parametrized solution and a nat-

ural number di such that the polynomial diu + n + i - dl - 2 is the degreeof the unique edge of the value of x in c and

0 < d. < ((vQ[°I .; .] + 1)V)".I 1

Using this inequality and (8), (10), (11), (12), and (13), we obtain

0 < diloge k

< 2

The system ri is equivalent to a set of CTS-systems containing ul . Hencecan be transformed into a u'-parametrized solution of IT. in which x has

the same value as in .

Let 3 be PA, if q3* is PA*, and HA otherwise. Using Lemmas 13.2,20.3, and 20.4, use induction on ll to construct a proof of the formula

du[A*]{d u+n+i-d -2}(0')'

in 3, where A* is obtained by substitution of x + y = z and x y = z forP and Q in A. It follows that

q3 F- vu[A*]L , (15)

whereL a ((u. 0(d1)) + 0')(n+i-1).

By Lemma 20.1, formulas (14) and (15) imply that 3 F `dx [A* ]X(n) for suf-ficiently large k. To complete the proof, we need only use Lemma 20.5. 0

Let us denote by GPA the calculus KGL(I nd * , Eq*, Z) and by GHAthe calculus I GL(I nd * , Eq* , Z). Let be a proof in GPA or GHA.Let v [Y] denote the number of different bound final degrees of formulas inthe end sequent of and in side formulas of applications of the rules cut,I n d * -cut and E q * -cut in .

LEMMA 20.6. Let n be any natural number, A a formula, and a proofof the sequent -4 [A]o(n) in GPA (GHA). If

(h[Z -+ A] + 3)2(h[271+3)(11[27]+2) < n, (16)

then there is a natural number d such that

0 < d < 2(h[J]+3)v[2Y] (17)

and the sequentVx[A]((X.o(d))+o')(n-d-1) (18)

is deducible in GPA (GHA).

PROOF. Let 3 be either GPA or GHA, a proof of - [A]n) in 3,and U the schema of . By Lemma 15.3, U is a (0, 1)-correct schemain 3. Assume that the term unknowns of the CF-system y { U ; 0, 1 } aredistinct from x and do not occur in A, and that the formula unknown Xcorresponds to an occurrence of a formula in the succedent of the sequentcorresponding to the last occurrence of an analysis in U. Add the equalityX a A to the list of equalities of y{ U ; 0, 1} and the unknown x to the listof term unknowns; denote the resulting CF-system by F.

Using Lemma 15.2, construct a solution oo of y{ U ; 0, 1 } such thatcoincides with [moo : U ; 0, 1]. Adding the term 0(n) to Qo as a value of x,we obtain a solution o of F. Note that all free final degrees of applied


axioms in °o are at the same time bound degrees of Qo . Hence, by Lemma15.4, the cardinality of the set of free final degrees of o does not exceedu[7]. By Lemma 15.1, T has at most 2h[]-1 + 1 term unknowns.

Assume that inequality (16) is satisfied. Using Lemma 12.4 and condition(3) of Lemma 10.12, construct a u'-parametrized solution a* of r and anatural number d such that d satisfies (17) and the polynomial d u+ n -d-2is the degree of the unique edge of the value of x in 7. Now, using Lemmas15.2, 20.3, and 20.4, we can rebuild o.* into a deduction of (18) in 3, byinduction on h [ U] . o

A theorem similar to Lemma 20.6 was proved in [57].Let A be a formula, x a bound variable of A, V an occurrence of an

elementary formula in A, s a natural number. We will say that s is a boundx-final degree of A relative to V if there is an occurrence W of the termx (5+2 in A such that W generates a bound occurrence of x in A, W isgenerated by V and does not arise from an occurrence of in A. Thenumber of different bound x-final degrees of A relative to an occurrence Vwill be called the (A, x)-rank of V. Let slfl[A ; m] denote the union of thesets of bound x-final degrees of A relative to V over all x and V suchthat the (A, x)-rank of V is at most m. The greatest (A, x)-rank of anoccurrence of an elementary formula in A will be called the x-rank of Aand the maximum x-rank of A over all bound variables x of A will becalled the s-rank of A.

LEMMA 20.7. For any formulas A and B and natural number n the fol-lowing statements hold:

(1) the cardinality of 7t[A ; m] is at most

m(st[A] - 1)2St[A]_1

(2) if the s-rank of A is at most m, then 7t[A ; rra] is the set of all boundfinal degrees of A ;

(3) if B is contained in A, then the s-rank of B is at most that of Aand

f7t[B ; m] c s7t[A ; m] ;

(4) if A is q+-equivalent to B, then the s-rank of A is equal to that ofB and the sets f7t[A ; m] and sJ't[B ; m] coincide;

(5) if q[B] can be obtained from q[A] by simultaneous substitution of q-types for 0-place predicate symbols, then the s-rank of [Blq[A]] doesnot exceed that of B.

This lemma is proved by induction on st[A].Let be a proof in PA or HA. Let u[] denote the maximum of

the s-ranks of applied axioms in and the s-rank of the last formula in" , and N. m the set of formulas A of s-rank greater than m.

LEMMA 20.8. FOYCIiZyfOYi7ZtlIQ A, proof 91 of A in PA (HA) and naturalnumber m, if u[ Y] < m, then there is a proof 91" of the sequent -- A inGPA (GHA) such that

YYl(lHh Pi '* < l P] , 22111 [1+12

[ ] - H [ 1

H

(19)

(20)

where c° 1 is the number of different subformulas occurring in the sequentZ - A that begin with , &, V, , V, or .

PROOF. Let m be a natural number and be a deduction of A in PAor HA in accordance with a schema U such that v[] <m. Assume thatthe formula unknown X corresponds to the last analysis in U. Add theequality X a A to the list of equalities of the CF-system y{ U} and denotethe resulting CF-system by T. By Lemma 13.1,

c * [r] < 6lH [91 ] + co1. (21)

Using Lemma 13.2, construct a solution o of IT such that 9" coincideswith [o : U]. Then, using Lemma 12.5, construct a f-universal solution zof q [T] such that

svq[Z] < SP*[r] (22)Without loss of generality, we may assume that all the predicates occurring

in z but not in Z -p A are 0-place predicates. Introduce the followingnotation:

[[cr/t]: U].It follows from Lemmas 12.7 and 13.2 that o is a proof of A in accor-

dance with U. Using inequalities (21) and (22), we see that for any formulaB in

st[B] < SP 1 + 1.

Condition (5) of Lemma 20.7 implies that

l .m[ J]

vs <m.

(23)

(2a)

(2s)

Using Lemma 5.3, rebuild £o into a proof £ of the sequent - A suchthat the side formulas of the cuts in are q+-equivalent to formulas in

, the side formulas of the I n d * -cuts and Eq * -cuts are q+-equivalent toapplied axioms in ,

h[91] < 2lH[9] + 7, hN.m[91] < 21Z.m[910] + 9,2

11 .m 31N m [J] < 31H.m [9o].

The last inequality and Lemma 2.7 imply that5tN.m[911 , C 1] < 31NI.m[910] + 1. (27)

Eliminating all N. m-applications of cuts in £ in accordance with Lemma4.4, we obtain a proof of -p A such that is q+-embeddable in £jand

Inequality (20) now follows from (24), (26), (27), and Lemma 4.3.Let SM* denote the union of all the sets SJ't[B1 ; m], where Bj is either

A or a side formula of an application of one of the rules cut, I n d * -cut, orEq * -cut in * , and 01° the union of all the sets 0[B2 ; m], where B2 is aformula occurring in . It follows from (25) and condition (4) of Lemma20.7 that the s-ranks of the side formulas of applications of the rules I n d * -cut and Eq*-cut in are at most m. Consequently, by condition (2) ofLemma 20.7, the cardinality of sJrt* is v[*]. On the other hand, using (23)and condition (1) from Lemma 20.7, we see that the cardinality of 0° is atmost

Finally, by Lemmas 1.2 and 1.3, for any formula B which is the sideformula of an application of the rule cut, I n d * -cut or Eq * -cut in , thereare formulas B' and B" such that B is q+-equivalent to B' , B' occurs inB", and B" is q+-equivalent to a formula in . Consequently, 0* C 0°by conditions (3) and (4) of Lemma 20.7, which implies (19). 0

THEOREM 20.2. Let m and p be natural numbers, A a formula, n andk sufficiently large natural numbers. Suppose that, for all i (0 < i < k),there is a proof of [A]o(n+i) in PA (HA) such that

N

(28)

l ! ", < 1 to to to n + i . (29)

Then is deducible in PA (HA).

PROOF. Let m and p be natural numbers, A a formula, 3 either PAor HA. Fix a natural number N° such that, for all N > N°,

h + 3)2kMN+3)(mN226N+29o,+s+2) <2

27N

(30)

22(MN+3)(mN226N+299 5) < 227N

(31)

where c°j is the number of different subformulas occurring in the sequentZ -p A that begin with , &, V, -i, V, or ,

h =h M-22p+12

* * - 3p+ 1 '

Let n and k be natural numbers, n> 16, such that

16 < n N < 1 to to lo g2°8

g2 g2 g2 (32)

71og21og21og2(n + k) < logZ1og21og2 k. (33)

Let 0 < i <k, a proof of in 3 such that (28) and (29) aresatisfied. Using Lemma 20.8, rebuild into a proof £ Y$ of the sequent-p such that

v[ *] <t t '

h[PiY1*] < M1H[PiY1].

It follows from (29), (30), (32), (34), and (35) that

(h + 3)2(h[1+3)(v[]+2) < n + i.*

Hence Lemma 20.6 is applicable to .

Using Lemma 20.6, construct a natural number d1 such that

(34)

(35)

0 <d1 <2(h[27,*]+3)v[2';*] (36)

t

and the sequent-- -1)

is deducible. Then, by Lemma 5.6,

q3 H (37)

It now follows from (29), (31), (32), (33), (34), (35), and (36) that

0< dlloge k

< 2

Consequently, it follows from (37) and Lemma 20.1 that, for sufficientlylarge k,

Let A be a formula. The maximum of i[T] over all bound variables xin A and all terms T whose occurrences in A generate bound occurrencesof x in A will be called the µ-rank of A. The expression

PA F- p A

will mean that there is a proof of A in PA satisfying the followingconditions.

(1) The length of £ is at most 1.(2) The maximum of the µ-ranks of applied axioms in and the µ-

rank of A is at most k.(3) The number of formulas in of µ-rank greater than k does not

exceed p.It is easy to prove the following assertion.

LEMMA 20.9. For any natural numbers k, 1, p, and any formula A,(1) if PA F'o A, then there exists a k-restricted proof £"i of A in PA ;(2) if PA A, then there exists a proof 2 of A in PA such that

u[T.2 ] < m, 1H. m

Pwhere m = k arg[Z -* A].

THEOREM 20.3. For any natural numbers m and p. formula A, and suf-ficiently large n and k, if

PA F'.1og21og21og2(n+i)>P [`4]0(n+i)

for all i (0 .

This theorem is a corollary of Theorem 20.2 and Lemma 20.9.

§21. Upper and lower bounds on lengths of deductionsin formal arithmetics

Let R" denote the following list of formulas:

Vxyz(x=y(x=zy=z)), (R*.1)

'c/xy(x=y x' =y'), (R*.2)

`dxyzx1Y1Z1 (x = xl (Y = Y1 CZ = Z (R* 3)

`dxyzx1Y1Z. (x = x. (Y = Y1 CZ = Zi(Q(x, y, z) Q(x1 ,y1, zl)))))

(R* 4)

b'xyzu(P(x, y, z) (P(x, y, u) z = u)), (R*.5)

dxyzu(Q(x, v, z) (Q(x, v, u) z = u)), (R*.6)

\/xy-iP(x, y', 0), (R*.7)

b'xyz(P(x, Pox, y, z)), (R*.8)

b'x(x = 0 V 3y(x =y')), (R*.9)

\/xP(x, 0, x), (R*.10)

b'xYz(P(x, y, z) Pox, y', z')), (R*. 11)

\/xQ(x, 0, 0), (R*.12)

b'xyzu(Q(x, y, z) (P(z, x, u) Q(X, y', u))). (R*.13)

In what follows, we will denote the calculus IH(R") by RHA`. Elimi-nating function symbols from proofs, one easily shows that, for any formulaA that does not contain the symbols + and ,the formula

(VxyzP(x, y, z) (VxyazQ(x, y, z) A))

is deducible in RHA` if and only if the result of simultaneous substitution ofthe formulas x + y = z and x y = z with distinguished parameters x, y, zfor P and Q in A is deducible in the intuitionistic variant of Robinson'sarithmetic (see, e.g., [8] or [10]).

LEMMA 21.1. The following formulas are deducible in RHA*

'c/x(x=x), 'c/xy(x=yDy=x),= 0), `dxy(x' = Y' x = Y).

§21. LENGTHS OF DEDUCTIONS IN FORMAL ARITHMETICS 145

PRooF. R* . 5 and R*.10 are used in the deduction of Vx (x = x) ;R* .1 and `/x (x = x) in the deduction of `/x y (x = y y = x) ; R* . 3 ,

* . 7 , R* .10, R* .11 and Vx (x = x) in the deduction of `dx -i(x'R = 0) ;R* . 3 , R* . 5 , R* .8 , R* .10 , . R* .11 and Vx (x = x) in the deduction of`/xy(x'=y' x=y). 0

We introduce the following notation:

Fn =? `/xyz(P(x, y, z) P(x, y(n) , z(n)))

Hn ? exy(x(n) = y(n) x = y),

Gm dyz(Q(0(n) , y, z) Q(0, (m),z(njn)))

LEMMA 21.2. There are natural numbers c1 and c2 such that the followingconditions hold:

(1) RHA ["c, [log2(n+ 1)]+c2 Fn*(2) RHA ["c, [loge (n+ 1)]+c2 Hn* n

(3) RHA c,[log2(n+1)+log2(m+1)]+c2 Gm

PROOF. Construct a natural number c such that for all i we have

IH I-c F0, IH I-c (F1 F21), RHA* I-c (F, F21+1) ;

IH I-c Ho , IH I-c (H1 H21), RHA* I-c (H, : H21)

I H I-c G, I H I-c (G G2i) ,

RHA* I-c (Fn (Gn G2i+1))

Conditions (1) and (2) of the lemma are proved by backward inductionon n. The following assertion can also be proved by backward induction onm:

RHA* He3[log2(m+ 1)]+c4 (Fn G).Now, use condition (1) to obtain condition (3). 0

LEMMA 21.3. There are natural numbers cl and c2 such that the followingconditions hold for all natural numbers n, m, and k :

(1) if n = m, then RHA* I- 0(n) = 0(m) ;c,

(2) if n m, then RHA* I- ,(0(n) = 0');(3) if n + in = k, then RHA* I-c to m 1 c P(0", 0(m), 0(k)) ;

(4) if n m = k, then RHA* ["c to nm 1 c Q(0, 0(m), 0(k)) ;

(5) if n + m k, then RHA* ["c to nm 1 +c -iP(0(n),

0(m)' 0);

(6) if n m k, then RHA* He to nm 1 c-iQ(0(n)

,0(m,

0(k))

This lemma follows from Lemmas 21.1 and 21.2.A system of Diophantine equations will be called an A-system if

includes only equations of the following three types:

(a) k=a,(b) a+f3 = y,(c)

where k is an arbitrary natural number and a, Ii, y are unknowns for nat-ural numbers. Let be an A-system of Diophantine equations. Assign anatomic formula to each equation of as follows: an equation of type (a) willbe assigned 0(k) = a, an equation of type (b) will be assigned P(a, fi , y),an equation of type (c) will be assigned Q(a, fi , y). Let denote theconjunction of the atomic formulas assigned to the equations of . Lemma21.3 implies the following assertion.

LEMMA 21.4. For any A-system of Diophantine equations in unknownsz 1, z2, ... , zl , there are natural numbers c1 and c2 such that the followingconditions hold for all natural numbers n1, n2... , nl :

(1) if the list n 1, n2, ... , nl is a solution of E , then

RHA* F--c [log2(n+1)]+c20 0

(2) if n1, n2, ... , nl is not a solution of , then2

RHA Hcl[log2(n+l)]+c2 {}]n)2):::n,)where n = max{ n1, n2, ... , nl } .

LEMMA 21.5. For any natural number n,

RHA* F dxy (P(x, y, O(n)) V (x = 0(i) & y = 0(j))i+j=n

PROOF. The following formulas are deducible in RHA* :

exy(y = 0 (P(x, y, o(n)) x = O), exyz- (y = z'&P(x, y, 0))

`dxyz y = z (( V (x = 0&z = p) \ / (x =0(J+1))

I I

i+j=n j i+i=n JJThe lemma is proved by induction on n, using these formulas.Recall that the Cantor number of a pair of natural numbers (x, y) is

computed by the formulan_(x+y)(x+y+1)+x.

2We know that x + y < n, the Cantor numbering of pairs is one-to-one, andany natural number is the Cantor number of some pair of natural numbers.

We will use the following notation:

12(a1, a2, y) au1 u2u3u4u5(P(a1 , a2, u1) & Q(u1 , u1, u2)

& P(a1 , a2, u3) & P(u2 , u3, u4) &P(u4 , u1, u5) & P(y , y, u5)),

Ik+l (a1 , ... , ak+1 , y) auk+4(Ik(a1 , ... , ak , uk+4) & I2(ak+1 , uk+4, y))Lemma 21.5 implies the following assertion:

LEMMA 21.6. If n is the Cantor number of (i, j) , then

RHA* H `dxy (I2(x , y, 0(n)) (x = 0(`) & y = o(j)) )

The following assertion can be proved by induction on k, using Lemma21.6.

LEMMA 21.7. For any natural numbers k, i2 , ... , ik , k > 2, we canconstruct n such that

kRHA* H Vx1 ... xk (Ik(x1 , ... , xk , O(n)) & x = o(1i)

LEMMA 21.8. For any natural number k, k > 2, there exist c1 and c2such that, for any natural number n, there are natural numbers i1, i2 , ... , iksatisfying the following conditions:

(2) RHA* Fc to n+1 +c Ik(O(`') , ... , 0(`k) , 0(n)) .

This lemma follows from Lemma 21.3.

THEOREM 21.1. There is a natural number c1 such that for any consistentaxiomatizable extension 2 of RHA*, there exist a one-parameter formulaA and a natural number c2 such that

(1) for any natural number n,*

RHA F'c1[log2(n+l)]+c2 [A]o(n)

(2) for no natural number k is the formula dx[A]x(k) deducible in 21.

PROOF. Following [38, §5], let us construct a polynomial U in unknownsz 1, z2, ... , zk (k > 15) with integer coefficients, such that U = 0 is auniversal Diophantine equation. By "universal" we mean that for any enu-merable set 931 of natural numbers there is a natural number m such thatn E 931 if and only if there are p3, ... , pk for which m, n, p3, ... , pk is asolution of U = 0 in natural numbers. Such a number m will be called anindex of 931 relative to the equation U = 0.

Introducing new unknowns zk+ l , ... , zl , transform the equation U = 0into an A-system of Diophantine equations U* in unknowns zl , z2, ... , zlwith the following property: a list of natural numbers p

1 , p2, ... , pk is asolution of U = 0 if and only if for some natural numbers pk+

1 , ...the list p1, ... , pk ... , pl is a solution of U* in natural numbers.We introduce the following notation:

A1(a) z3 .. z1(I1-2(z3, ... , z1, a) & [{ U*}]Z(;) Z(r) ).

0 0

(1) max{il, i2, ... , ik} < n ;

Let 21 be a consistent axiomatizable extension of RHA". Let m denotean index of the set of natural numbers n such that b'xAn(x) is deducible


in Qt relative to U = 0. Consider the formula `dxAm (x) . (Note that thisformula coincides with the corresponding formula from [55].)

Suppose that `dxAm (x) is deducible in Qt. Then there are natural numbers

p3, ... , pl such that m, m, p3, ... , pl is a solution of U*. It follows fromLemma 21.4 that

RHA* I- [{U* }]ZZ2 ". Z/

]o o 0lOn the other hand, using Lemma 21.7, we can construct a natural number

n such thatRHA* F (Am(O(m)) -1[{ Z(,n) ).

Thus, we have obtained a contradiction in 21. Hence VxA`n (x) is notdeducible in 21.

Let n be an arbitrary natural number. Using Lemma 21.8, constructnatural numbers i3, ... , it such that, for appropriate c1 and c2,

RHA* I- I (O(`3) ... , 0(`') , 0(n)) (1)c1[log2 (n+ 1)]+c2 1_2 ,

The list m, m, i3, ... , it is not a solution of U*, since bxA. m (x) is notdeducible in 21. Hence, by Lemma 21.4, for appropriate c3 and c4,

RHA p,).* rc3[log2(n+1)+log2(m+1)]+c4

Hence it follows from (1) that, for an appropriate constant c5,

RHA* I-Am (O(n) ).

(2)(c, +c2)[1og2(n+ 1)]+c3[tog2(m+ 1 )]+ca+cs

Thus, condition (1) of the theorem is satisfied.Note that the following formula is deducible in RHA* :

((VXAm(X) &n-18L Am (0(`)) VXAm (x) .i=0

Consequently, condition (2) follows from (2) and from the fact VxAm (x )is not deducible in 21. 0

Let A be a formula, 91 an axiomatic Hilbert-type theory. The expression91 A will mean that there is no proof of A in 91 of length at most 1.

THEOREM 21.2. For all sufficiently large n, the following conditions hold.

(1) If n is even, then

RHA* axP(x, x, 0(n)).3 loge n

(2) If n is composite, then

RHA* !xyQ(xii , y.. , 0(n) ).

3 loge n

This theorem follows from Theorem 19.1 and from the fact that for all nthe formulas

x, y(n)), yii , z(n))

are not deducible in RHA*.

THEOREM 21.3. For infinitely many natural numbers n,

xP(x (2n)RHA* log2 2n , x, 0 ).3log2 log2 2n

This theorem follows from Corollary 2 of Theorem 14.2 and from Theo-rem 19.2.

THEOREM 21.4. For infinitely many natural numbers n,2

RHA* # log n2xyQ(x , y , 0(n )).

S log2 log2 n2

This theorem also follows from Corollary 2 of Theorem 14.2 and fromTheorem 19.2.

THEOREM 21.5. For all sufficiently large prime numbers p.

PA* ax x" 0(P2))yQ( , y",

4 V 2

PROOF. Let p be a prime and suppose that2

PA* H2

axyQ(xii , y.. , 0(p )).a 1og2 P

If p is sufficiently large, we can use Lemma 14.5 to construct a naturalnumber d such that 0 <d <p and for all i

PA* H ax x" 0(p2+di) .

yQ( , y ) (3)

By Dirichlet's theorem for arithmetic progressions, there is a j such thatp 2 + d j is a prime. Thus, we have obtained a contradiction with (3). 0

THEOREM 21.6. There is a natural number c such that, for all naturalnumbers n, the following conditions hold.


RHA ).* H l 2 axP(x, x, 0(n)

(2) If n is composite, then

RHA).

* He[tog2 nJ

axyQ(xii , yii , O(n)

This theorem follows from Lemma 21.3.

THEOREM 21.7. There is a natural number c such that, for all naturalnumbers n, the following conditions hold.


PA* I- axP(x, x, O").c

(2) If n is composite, then//

(PA* Hc[log2 n]

axyQ(x

,y

, 0n) )


PROOF. Consider a particular case of the schema Ind * :

((p(0(m), 0, 0(m)) &VX(P(O(m) , x , x(m))P(0(m), xi , x(m+l)))) VXP(O(m), x, x(m)))

This formula and axioms Z* . 3 and Z*.4 imply the formula

VXP(O(m), x , x(m) ).

Hence condition (1) holds. A similar device is used in [50, 56] to provethe formulas

`dx(x + 0(m) = x(m)), O(n) + 0(m) _ O(n+m)

in a fixed number of steps.Condition (2) is a consequence of condition (2) of Theorem 21.6. 0

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