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MATHEMATICAL LOGIC
Editors
Sergey Goncharov 0 Rod Downey 0 Hiroakira Ono
M...

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roceeamgs OT me am Asian Logic uomerence

MATHEMATICAL LOGIC

Editors

Sergey Goncharov 0 Rod Downey 0 Hiroakira Ono

MATHEMATICAL LOGIC

IN

ASIA

Proceedings of the 9th Asian Logic Conference

MATHEMATICAL LOGIC

IN

ASI

Novosibirsk, Russia

1 6 - 1 9 August 2005

Editors S S Goncharov R Downey

Russian Academy of Sciences, Russia

Victoria University of Wellington, New Zealand

H Ono Japan Advanced Institute of Science and Technology, Japan

\jJ5 World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MATHEMATICAL LOGIC IN ASIA Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-270-045-5

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The Asian Logic Conference has occurred every three years since its inception in Singapore in 1981. It rotates among countries in the Asia Pacific region with interests in the broad area of logic including theoretical computer science. It is now considered a major conference in this field and is regularly sponsored by the Association for Symbolic Logic. This volume contains papers, many of them surveys by leading experts, of the 9th meeting in Novosibirsk, Russia. We were very pleased to find that World Scientific were enthusiastic to support this venture. Authors were invited to submit articles to the present volume, based around talks given at either meeting. The editors were very concerned to make sure that the planned volume was of high quality. We think the resulting volume is fairly representative of the thriving logic groups in the Asia-Pacific region, and also fairly representative of the meetings themselves. The Ninth Asian Logic Conference was organised by Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences and Novosibirsk State University under the sponsorship of Russian Foundation for Basic Research, Association for Symbolic Logic, Department of Mechanics and Mathematics of Novosibirsk State University, Siberian Foundation for Algebra and Logic, Novosibirsk Center of Information Technologies UniPro Co., Ltd., LLC Alekta, and Transtext Co. Ltd. The conference took place in Novosibirsk, Akademgorodok, Russia, from August 16 to August 19, 2005. The programme consisted of plenary lectures delivered by invited speakers and contributions in four sections. Plenary speakers were Pavel Alaev (Russia), Lev Beklemishev (Russia, Netherlands), Su Gao (USA), Yurii Ershov (Russia), Sanjay Jain (Singapore), Vladimir Kanovei (Russia), Bakhadyr Khoussainov (New Zealand), Andrei Mantsivoda (Russia), Joe Miller (USA), Hiroakira Ono (Japan), Vladimir Rybakov (Russia, Great Britain), Masahiko Sato (Japan), Moshe Vardi (USA), Andrei Voronkov (Great Britain), Xishun Zhao (China). The total number of plenary lectures was 15. Contributed lectures on recursion

vi

theory, set theory, proof theory, model theory and universal algebra, nonclassical logic, and logic in computer science were presented in the following sections: Computability theory, Model theory and Set theory, Non-classical logics, Proof theory, and universal algebra, and Applications of logic in computer science. The total number of contributed talks was 58. The geography of the event included Russia, China, Japan, Singapore, USA, New Zealand, Great Britain, Korea, Canada, Germany, Greece, Kazakhstan. The number of participants was about 100 scientists. We are grateful to Ekaterina Fokina for the great work with authors and referees while preparing the Proceedings and to Vladimir Vlasov for making the camera-ready manuscript. Preparing of the Proceedings was supported by the grant of President of the Russian Federation for Leading Scientific Schools 4413.2006.1. Sincerely yours, the editors: Rod Downey, Sergey Goncharov, and Hiroakira Ono.

CONTENTS Another Characterization of the Deduction-Detachment Theorem S. V. Babyonyshev

1

Computable Numberings in the Hierarchy of Ershov S. A. Badaev and Zh. T. Talasbaeva

17

On Behaviour of 2-Formulas in Weakly o-Minimal Theories B. S. Baizhanov and B. Sh. Kulpeshov

31

Proofs about Folklore: Why Model Checking = Reachability? K. Choe, H. Eo, S. O, N. V. Shilov and K. Yi

41

A Note on Ai Induction C. Dimitracopoulos and A. Sirokofskich

51

Arithmetic Turing Degrees and Categorical Theories of Computable Models E. Fokina

58

Equivalence Relations and Classical Banach Spaces S. Gao

70

Negative Data in Learning Languages S. Jain and E. Kinber

90

Effective Cardinals in the Nonstandard Universe V. Kanovei and M. Reeken

113

Model-Theoretic Methods of Analysis of Computer Arithmetic S. P. Kovalyov

145

The Functional Completeness of Lesniewski's Systems F. Lepage

156

Analysis of a New Reduction Calculus for the Satisfiability Problem S. Noureddine

166

Elementary Type Semigroup for Boolean Algebras with Distinguished Ideals D. Pal'chunov

175

Interval Fuzzy Algebraic Systems D. E. Pal'chunov and G. E. Yakhyaeva On Orientability and Degeneration of Boolean Binary Relation on a Finite Set V. Poplavski Hierarchies of Randomness Tests J. Reimann and F. Stephan Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions V. V. Rybakov

191

203 215

233

Isomorphisms and Definable Relations on Rings and Lattices J. A. Tussupov

254

The Logic of Prediction E. Vityaev

263

The Choice of Standards for a Reporting Language M. Walicki, U. Wolter and J. Stecher

277

Conceptual Semantic Systems Theory and Applications K. E. Wolff

288

Complexity Results on Minimal Unsatisfiable Formulas X. Zhao

302

1

A N O T H E R CHARACTERIZATION OF T H E DEDUCTION-DETACHMENT THEOREM SERGEI V. BABYONYSHEV Krasnoyarsk State University Krasnoyarsk, Russia E-mail: [email protected] In Abstract Algebraic Logic, a Hilbert-style deductive system is identified with the set of its theories. This set of theories must be algebraic and must be closed under arbitrary intersections and inverse substitutions. Similarly, a Gentzenstyle deductive system can be defined by providing a set of theories with similar properties, but now each theory must be a set of sequents, not just formulas. There are various kinds of Gentzen-style structures that naturally arise in connection with Hilbert systems, but in generally they fall short of being Gentzen systems. One of such structures is a family of axiomatic closure relations. Each of axiomatic closure relations is defined as a set of consequences that can be derived in the Hilbert system by modulo of some its theory, taken as the set of additional axioms. T h e main result of this work is the proof t h a t a Hilbert system S admits the Deduction-Detachment Theorem if and only if the set of all axiomatic closure relations for S forms a Gentzen system.

1. Introduction In Algebraic Logic, an abstract Hilbert-style deductive system <S is identified with a family of sets, called <S-theories, of formulas of a given propositional language, such that this family, usually denoted by Th<S, is 1) closed under arbitrary intersections, i.e., Th<S is a closure system, 2) closed under unions of upward-directed families, i.e., ThiS is algebraic, 3) closed under inverse substitutions, i.e., a preimage of any <S-theory under an arbitrary substitution is an <S-theory again. We call <S "abstract" because its definition does not refer to any particular axiomatization. It is easy to see that the closure operator {.f :X^d{TeThS\X

CT},

associated with such closure system Th<S, defines a finitary, substitutional consequence relation as follows X \-s a -£=> a G Xs. It was suggested by researchers of Barcelona group to treat the Gentzen

2

case similarly, and identify an abstract Gentzen-style deductive system with the set of its theories with two distinctive features 1) a theory is a set of sequents, i.e., sequences of formulas, 2) a substitution acts on sequents componentwise. Out of numerous and intricate connections between Hilbert- and Gentzen-style deductive systems we will consider in this paper just one: the deduction-detachment theorem, discovered independently by Tarski and Herbrand. We define it in a slightly more general form. A Hilbert-style deductive system admits the multiterm deductiondetachment theorem if there is a finite set of formulas A = {Si(x, y)}i T^A) such that for any X, Y C A, X C C(X) = C ( C p Q ) C C(X U 7 ) . A set X £ V{A) such that C(X) = X is called a closed set of C. A closure operator C is finitary if for any X C A, C(X) = L K C ( 7 ) I Y ^u X}. The following relations between closure systems and closure operators are well known: 1) if C is a closure operator on A, then the family of its closed sets is a closure system on A; 2) if C ia a closure system on A, then the mapping C c : V(A) - • P(A) defined for each XCAasCcX:= f){Y £C\XCY} is a closure operator on A; 3) C is algebraic iff C c is finitary. We use interchangeably the exponential and prefix notations for closure operators, thusXc = CcX. Every closure system C, as a family of subsets ordered under setinclusion, is a complete lattice. The infimum of a family {Xi}, 6 / C C is its intersection f]ieI Xi, and its supremum is \JiGlXi := Cc(\JieI Xi); its largest element is A, and its smallest element is Cc(0) = H^A Hilbert system is a pair S — ( F m £ , T h S ) such that T h S C P(Fm£) is an algebraic invariant closure system on Fm,c. A Gentzen system is a pair Q — ( F m £ , T h £ ) such that T h S Q T^FmJ) is an algebraic invariant closure system on F m J . For a Hilbert system S and all T £ Th <S, [T)xh s := {U £ T h S | T C U} denote a principal filter of the lattice T h S generated by T. If V, is a Hilbert or Gentzen system, we denote Thm7?. := p|Th7?.. We take a Cantor-style approach towards Gentzen rules: we view a rule not as a "rule"— description of an action, but as a list of all its applications. A Gentzen sequent is a sequence s > s of sequents. A Gentzen rule s\- s is a set of all substitution instances of the Gentzen sequent s > s, i.e., s I- s := {a(s> s) \ a : Fni£ —> F i n e } . A Gentzen rule si,...,sn

\- s can also be written as ———'—^-. s Let x, y, z be variables. Standard rules (sometimes called structural) are rules of the form (Ax) r - T , x , S > x (Ex) r , a ; , y , S > z \- T,y, £,£t> z (W) r , £ t > y \-T,x,T,>y

Axioms Exchange Weakening

6

(Con) r , x, x, £ > y h T, x, S > y (Cut) r,x,Z>y;G>x\-r,Q,Z>y

Contraction Cut

where T, S, 0 range over the set of finite, possibly empty, sequences of variables of Fm^. Suppose Q = ( F m £ , T h £ ) is a Gentzen system. We say that a Gentzen rule s\- s holds in Q (we write it as s \-g s) if for every substitution a and every ^-theory T a{s}

CT => u s e r .

3. Closure Relations Definition 3.1. Let C be a closure system on Fm£. Define RcC = {a>ae

Fm+ \ (VX £ C) {a} C X =^> a € X}.

Definition 3.2. Let S be a Hilbert system. If C C Th<S is an algebraic closure system on Fni£, then Tic C is called a general finite closure relation for S or simply a general closure relation for S. The set of all general closure relations for S will be denoted by Gcr S. • For every Hilbert system <S of type £ there is a distinguished general closure relation R/;Th<S, which in its turn defines a Gentzen axiomatization for a Gentzen system: hR£Th5:=U{^a>a I

a>a a | a \-s a}, RcThS is invariant, Gcr<S can be axiomatized by standard rules and h i?£Th<S Gcr S forms a Gentzen system on Fmc, JR£Th«S = Thm(GcrcS).

Closure relations were introduced in [9] as a framework for studying metatheoretical properties of Hilbert systems. The fact that Gcr 5 form a Gentzen system was first observed also in [9]. The Gentzen system Gcr S formalizes a metalogic over the Hilbert system S. This metalogic is quite weak and equivalent in expressive power to the strict universal Horn logic without equality [4]. Although Gcr S is almost trivial, since can be axiomatized by only taking all proper sequents of Th<S and standard Gentzen

7

rules, by Proposition 3.3(3), it is proved to be useful as a framework for working with other kinds of closure relations like full or axiomatic [1]. We make a distinction between an element a and a vector (a) of length one with this element as its only component. This approach requires the following set of technical operators, that would allow us smooth transitions from formulas to sequents and back, and also between the theories of Hilbert and Gentzen systems. Define for every X C Fiti£ and every A C F m J >X := {\>a | a G X}, Thm.4 := {a G F m £ | >a G -4}, &A:={>ae

Fni£ | >a G A}.

Thus we obtain operators (>) : P ( F m £ ) -> ^(FmJ;), Thm : P ( F m J ) -» P ( F m £ ) , 0 : P(Fm+) -+ P(Fm^). Mnemonically, the Greek letter 0 above stands for "Theorems". Reminder. In the following proofs we rely heavily on, so called, "exponential" notation for closures of sets. Namely, if C is a closure system on some set X, then for all Y C X: Yc = ( F ) c : = r V c F € c ^ Definition 3.4. For a Hilbert system S, define the set of axiomatic closure relations of S as follows: AcrS := { ( > T ) G c r S | T G Th<S}. An element of Acr<S is called an axiomatic closure relation for S.

•

Note, that, in the definition above, the set > T contains sequents of the form t>a, a G T, where T C Fni£, and we take the closure of > T in the family of Gentzen theories, each of them is a set of sequents itself. Proposition 3.5. For every Hilbert system S of type C (1) (2) (3) (4)

A c r S C GcrS, .4 G A c r S => A = (&A)GcrS, GcrS A c r S = {(t>X) | X C Fm£}, A c r S = {R£[T) T h S | T G T h S } .

(5) For every X C Fm£, a>ae{>X)GcrS

< = • ae{a}sVXs

«=»

X,ahsa.

Proof. (1) By definition. (2) If A € Acr S, then .4 = ( > T ) G c r 5 for some T <E Th<S. Then Gcr5 Gcr5 Gcr5

> r c 9 i a =>• ^ = (>T)

c(0^)

c^

=^

=> ^ = (0^)GcrS. (3) If ^ e A c r 5 , then .4 = ( 0 . 4 ) G c r 5 = ( > T h m > l ) G c r 5 . For the other direction, suppose A = (>X)GcrS, for some X C Frri£. Then A = (©_4)Gcr5 = ( o T h m ^ ) 0 " 5 , because Q)&ACA

= > ( 0 ^ ) G c r S C ^ G c r S = A,

(C) A = {t>X)GcrS

=>

> l C i = > > l C 0 y (

==» .4 = ( > X ) G c r S C ( 0 . 4 ) G c r 5 . (4) Suppose A e Acr S. Then, by (3), A = (t>T)GcrS, where T = Thm.4 € Th<S. Let C = [T^xhs- Being a general closure relation for <S, A = R,cX>, for some algebraic closure system V C T h 6 \ Then A = R ^ C , because

(D)T(^riV^VC[T)ThS

=C =• R£CCR£P = i,

( C ) 0 R £ P = >(p|X>) = i>T = c>(nC) = 0 R £ C = = >

^ i = (>r)

GcrS

>TCR£C

cR : c.

(5) a > a G ( > T ) G « s ( i ) R [ T ) T h 5 4=> a e { a } [ T ' T h s = ( T U { a } ) 5 = T V { a } 5 «=»

T,Qhsa.n

Lemma 3.6. Acr S is a closure system iff for all families {Ai}i^i Acr S

C

a e 7 A = (n ie/ €>A) Gcr5 . Proof. It follows directly from the implications

(=0 ©(nie/A) = a e /©A

=>• a e /A 3^2) (e(a e /A) G c r 5 = (n ie7 eA) Gcr5 . (

MATHEMATICAL LOGIC

Editors

Sergey Goncharov 0 Rod Downey 0 Hiroakira Ono

MATHEMATICAL LOGIC

IN

ASIA

Proceedings of the 9th Asian Logic Conference

MATHEMATICAL LOGIC

IN

ASI

Novosibirsk, Russia

1 6 - 1 9 August 2005

Editors S S Goncharov R Downey

Russian Academy of Sciences, Russia

Victoria University of Wellington, New Zealand

H Ono Japan Advanced Institute of Science and Technology, Japan

\jJ5 World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MATHEMATICAL LOGIC IN ASIA Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-270-045-5

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The Asian Logic Conference has occurred every three years since its inception in Singapore in 1981. It rotates among countries in the Asia Pacific region with interests in the broad area of logic including theoretical computer science. It is now considered a major conference in this field and is regularly sponsored by the Association for Symbolic Logic. This volume contains papers, many of them surveys by leading experts, of the 9th meeting in Novosibirsk, Russia. We were very pleased to find that World Scientific were enthusiastic to support this venture. Authors were invited to submit articles to the present volume, based around talks given at either meeting. The editors were very concerned to make sure that the planned volume was of high quality. We think the resulting volume is fairly representative of the thriving logic groups in the Asia-Pacific region, and also fairly representative of the meetings themselves. The Ninth Asian Logic Conference was organised by Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences and Novosibirsk State University under the sponsorship of Russian Foundation for Basic Research, Association for Symbolic Logic, Department of Mechanics and Mathematics of Novosibirsk State University, Siberian Foundation for Algebra and Logic, Novosibirsk Center of Information Technologies UniPro Co., Ltd., LLC Alekta, and Transtext Co. Ltd. The conference took place in Novosibirsk, Akademgorodok, Russia, from August 16 to August 19, 2005. The programme consisted of plenary lectures delivered by invited speakers and contributions in four sections. Plenary speakers were Pavel Alaev (Russia), Lev Beklemishev (Russia, Netherlands), Su Gao (USA), Yurii Ershov (Russia), Sanjay Jain (Singapore), Vladimir Kanovei (Russia), Bakhadyr Khoussainov (New Zealand), Andrei Mantsivoda (Russia), Joe Miller (USA), Hiroakira Ono (Japan), Vladimir Rybakov (Russia, Great Britain), Masahiko Sato (Japan), Moshe Vardi (USA), Andrei Voronkov (Great Britain), Xishun Zhao (China). The total number of plenary lectures was 15. Contributed lectures on recursion

vi

theory, set theory, proof theory, model theory and universal algebra, nonclassical logic, and logic in computer science were presented in the following sections: Computability theory, Model theory and Set theory, Non-classical logics, Proof theory, and universal algebra, and Applications of logic in computer science. The total number of contributed talks was 58. The geography of the event included Russia, China, Japan, Singapore, USA, New Zealand, Great Britain, Korea, Canada, Germany, Greece, Kazakhstan. The number of participants was about 100 scientists. We are grateful to Ekaterina Fokina for the great work with authors and referees while preparing the Proceedings and to Vladimir Vlasov for making the camera-ready manuscript. Preparing of the Proceedings was supported by the grant of President of the Russian Federation for Leading Scientific Schools 4413.2006.1. Sincerely yours, the editors: Rod Downey, Sergey Goncharov, and Hiroakira Ono.

CONTENTS Another Characterization of the Deduction-Detachment Theorem S. V. Babyonyshev

1

Computable Numberings in the Hierarchy of Ershov S. A. Badaev and Zh. T. Talasbaeva

17

On Behaviour of 2-Formulas in Weakly o-Minimal Theories B. S. Baizhanov and B. Sh. Kulpeshov

31

Proofs about Folklore: Why Model Checking = Reachability? K. Choe, H. Eo, S. O, N. V. Shilov and K. Yi

41

A Note on Ai Induction C. Dimitracopoulos and A. Sirokofskich

51

Arithmetic Turing Degrees and Categorical Theories of Computable Models E. Fokina

58

Equivalence Relations and Classical Banach Spaces S. Gao

70

Negative Data in Learning Languages S. Jain and E. Kinber

90

Effective Cardinals in the Nonstandard Universe V. Kanovei and M. Reeken

113

Model-Theoretic Methods of Analysis of Computer Arithmetic S. P. Kovalyov

145

The Functional Completeness of Lesniewski's Systems F. Lepage

156

Analysis of a New Reduction Calculus for the Satisfiability Problem S. Noureddine

166

Elementary Type Semigroup for Boolean Algebras with Distinguished Ideals D. Pal'chunov

175

Interval Fuzzy Algebraic Systems D. E. Pal'chunov and G. E. Yakhyaeva On Orientability and Degeneration of Boolean Binary Relation on a Finite Set V. Poplavski Hierarchies of Randomness Tests J. Reimann and F. Stephan Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions V. V. Rybakov

191

203 215

233

Isomorphisms and Definable Relations on Rings and Lattices J. A. Tussupov

254

The Logic of Prediction E. Vityaev

263

The Choice of Standards for a Reporting Language M. Walicki, U. Wolter and J. Stecher

277

Conceptual Semantic Systems Theory and Applications K. E. Wolff

288

Complexity Results on Minimal Unsatisfiable Formulas X. Zhao

302

1

A N O T H E R CHARACTERIZATION OF T H E DEDUCTION-DETACHMENT THEOREM SERGEI V. BABYONYSHEV Krasnoyarsk State University Krasnoyarsk, Russia E-mail: [email protected] In Abstract Algebraic Logic, a Hilbert-style deductive system is identified with the set of its theories. This set of theories must be algebraic and must be closed under arbitrary intersections and inverse substitutions. Similarly, a Gentzenstyle deductive system can be defined by providing a set of theories with similar properties, but now each theory must be a set of sequents, not just formulas. There are various kinds of Gentzen-style structures that naturally arise in connection with Hilbert systems, but in generally they fall short of being Gentzen systems. One of such structures is a family of axiomatic closure relations. Each of axiomatic closure relations is defined as a set of consequences that can be derived in the Hilbert system by modulo of some its theory, taken as the set of additional axioms. T h e main result of this work is the proof t h a t a Hilbert system S admits the Deduction-Detachment Theorem if and only if the set of all axiomatic closure relations for S forms a Gentzen system.

1. Introduction In Algebraic Logic, an abstract Hilbert-style deductive system <S is identified with a family of sets, called <S-theories, of formulas of a given propositional language, such that this family, usually denoted by Th<S, is 1) closed under arbitrary intersections, i.e., Th<S is a closure system, 2) closed under unions of upward-directed families, i.e., ThiS is algebraic, 3) closed under inverse substitutions, i.e., a preimage of any <S-theory under an arbitrary substitution is an <S-theory again. We call <S "abstract" because its definition does not refer to any particular axiomatization. It is easy to see that the closure operator {.f :X^d{TeThS\X

CT},

associated with such closure system Th<S, defines a finitary, substitutional consequence relation as follows X \-s a -£=> a G Xs. It was suggested by researchers of Barcelona group to treat the Gentzen

2

case similarly, and identify an abstract Gentzen-style deductive system with the set of its theories with two distinctive features 1) a theory is a set of sequents, i.e., sequences of formulas, 2) a substitution acts on sequents componentwise. Out of numerous and intricate connections between Hilbert- and Gentzen-style deductive systems we will consider in this paper just one: the deduction-detachment theorem, discovered independently by Tarski and Herbrand. We define it in a slightly more general form. A Hilbert-style deductive system admits the multiterm deductiondetachment theorem if there is a finite set of formulas A = {Si(x, y)}i T^A) such that for any X, Y C A, X C C(X) = C ( C p Q ) C C(X U 7 ) . A set X £ V{A) such that C(X) = X is called a closed set of C. A closure operator C is finitary if for any X C A, C(X) = L K C ( 7 ) I Y ^u X}. The following relations between closure systems and closure operators are well known: 1) if C is a closure operator on A, then the family of its closed sets is a closure system on A; 2) if C ia a closure system on A, then the mapping C c : V(A) - • P(A) defined for each XCAasCcX:= f){Y £C\XCY} is a closure operator on A; 3) C is algebraic iff C c is finitary. We use interchangeably the exponential and prefix notations for closure operators, thusXc = CcX. Every closure system C, as a family of subsets ordered under setinclusion, is a complete lattice. The infimum of a family {Xi}, 6 / C C is its intersection f]ieI Xi, and its supremum is \JiGlXi := Cc(\JieI Xi); its largest element is A, and its smallest element is Cc(0) = H^A Hilbert system is a pair S — ( F m £ , T h S ) such that T h S C P(Fm£) is an algebraic invariant closure system on Fm,c. A Gentzen system is a pair Q — ( F m £ , T h £ ) such that T h S Q T^FmJ) is an algebraic invariant closure system on F m J . For a Hilbert system S and all T £ Th <S, [T)xh s := {U £ T h S | T C U} denote a principal filter of the lattice T h S generated by T. If V, is a Hilbert or Gentzen system, we denote Thm7?. := p|Th7?.. We take a Cantor-style approach towards Gentzen rules: we view a rule not as a "rule"— description of an action, but as a list of all its applications. A Gentzen sequent is a sequence s > s of sequents. A Gentzen rule s\- s is a set of all substitution instances of the Gentzen sequent s > s, i.e., s I- s := {a(s> s) \ a : Fni£ —> F i n e } . A Gentzen rule si,...,sn

\- s can also be written as ———'—^-. s Let x, y, z be variables. Standard rules (sometimes called structural) are rules of the form (Ax) r - T , x , S > x (Ex) r , a ; , y , S > z \- T,y, £,£t> z (W) r , £ t > y \-T,x,T,>y

Axioms Exchange Weakening

6

(Con) r , x, x, £ > y h T, x, S > y (Cut) r,x,Z>y;G>x\-r,Q,Z>y

Contraction Cut

where T, S, 0 range over the set of finite, possibly empty, sequences of variables of Fm^. Suppose Q = ( F m £ , T h £ ) is a Gentzen system. We say that a Gentzen rule s\- s holds in Q (we write it as s \-g s) if for every substitution a and every ^-theory T a{s}

CT => u s e r .

3. Closure Relations Definition 3.1. Let C be a closure system on Fm£. Define RcC = {a>ae

Fm+ \ (VX £ C) {a} C X =^> a € X}.

Definition 3.2. Let S be a Hilbert system. If C C Th<S is an algebraic closure system on Fni£, then Tic C is called a general finite closure relation for S or simply a general closure relation for S. The set of all general closure relations for S will be denoted by Gcr S. • For every Hilbert system <S of type £ there is a distinguished general closure relation R/;Th<S, which in its turn defines a Gentzen axiomatization for a Gentzen system: hR£Th5:=U{^a>a I

a>a a | a \-s a}, RcThS is invariant, Gcr<S can be axiomatized by standard rules and h i?£Th<S Gcr S forms a Gentzen system on Fmc, JR£Th«S = Thm(GcrcS).

Closure relations were introduced in [9] as a framework for studying metatheoretical properties of Hilbert systems. The fact that Gcr 5 form a Gentzen system was first observed also in [9]. The Gentzen system Gcr S formalizes a metalogic over the Hilbert system S. This metalogic is quite weak and equivalent in expressive power to the strict universal Horn logic without equality [4]. Although Gcr S is almost trivial, since can be axiomatized by only taking all proper sequents of Th<S and standard Gentzen

7

rules, by Proposition 3.3(3), it is proved to be useful as a framework for working with other kinds of closure relations like full or axiomatic [1]. We make a distinction between an element a and a vector (a) of length one with this element as its only component. This approach requires the following set of technical operators, that would allow us smooth transitions from formulas to sequents and back, and also between the theories of Hilbert and Gentzen systems. Define for every X C Fiti£ and every A C F m J >X := {\>a | a G X}, Thm.4 := {a G F m £ | >a G -4}, &A:={>ae

Fni£ | >a G A}.

Thus we obtain operators (>) : P ( F m £ ) -> ^(FmJ;), Thm : P ( F m J ) -» P ( F m £ ) , 0 : P(Fm+) -+ P(Fm^). Mnemonically, the Greek letter 0 above stands for "Theorems". Reminder. In the following proofs we rely heavily on, so called, "exponential" notation for closures of sets. Namely, if C is a closure system on some set X, then for all Y C X: Yc = ( F ) c : = r V c F € c ^ Definition 3.4. For a Hilbert system S, define the set of axiomatic closure relations of S as follows: AcrS := { ( > T ) G c r S | T G Th<S}. An element of Acr<S is called an axiomatic closure relation for S.

•

Note, that, in the definition above, the set > T contains sequents of the form t>a, a G T, where T C Fni£, and we take the closure of > T in the family of Gentzen theories, each of them is a set of sequents itself. Proposition 3.5. For every Hilbert system S of type C (1) (2) (3) (4)

A c r S C GcrS, .4 G A c r S => A = (&A)GcrS, GcrS A c r S = {(t>X) | X C Fm£}, A c r S = {R£[T) T h S | T G T h S } .

(5) For every X C Fm£, a>ae{>X)GcrS

< = • ae{a}sVXs

«=»

X,ahsa.

Proof. (1) By definition. (2) If A € Acr S, then .4 = ( > T ) G c r 5 for some T <E Th<S. Then Gcr5 Gcr5 Gcr5

> r c 9 i a =>• ^ = (>T)

c(0^)

c^

=^

=> ^ = (0^)GcrS. (3) If ^ e A c r 5 , then .4 = ( 0 . 4 ) G c r 5 = ( > T h m > l ) G c r 5 . For the other direction, suppose A = (>X)GcrS, for some X C Frri£. Then A = (©_4)Gcr5 = ( o T h m ^ ) 0 " 5 , because Q)&ACA

= > ( 0 ^ ) G c r S C ^ G c r S = A,

(C) A = {t>X)GcrS

=>

> l C i = > > l C 0 y (

==» .4 = ( > X ) G c r S C ( 0 . 4 ) G c r 5 . (4) Suppose A e Acr S. Then, by (3), A = (t>T)GcrS, where T = Thm.4 € Th<S. Let C = [T^xhs- Being a general closure relation for <S, A = R,cX>, for some algebraic closure system V C T h 6 \ Then A = R ^ C , because

(D)T(^riV^VC[T)ThS

=C =• R£CCR£P = i,

( C ) 0 R £ P = >(p|X>) = i>T = c>(nC) = 0 R £ C = = >

^ i = (>r)

GcrS

>TCR£C

cR : c.

(5) a > a G ( > T ) G « s ( i ) R [ T ) T h 5 4=> a e { a } [ T ' T h s = ( T U { a } ) 5 = T V { a } 5 «=»

T,Qhsa.n

Lemma 3.6. Acr S is a closure system iff for all families {Ai}i^i Acr S

C

a e 7 A = (n ie/ €>A) Gcr5 . Proof. It follows directly from the implications

(=0 ©(nie/A) = a e /©A

=>• a e /A 3^2) (e(a e /A) G c r 5 = (n ie7 eA) Gcr5 . (

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