Über die Erfüllbarkeit derjenigen Zählausdrücke, welche in der Normalform zwei benachbarte Allzeichen enthalten

Kalmár, László

doi:10.1007/bf01452848

Cited by 33 publications

(14 citation statements)

References 3 publications

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“…Let L 2 denote the class of rst order sentences with two variables over a relational vocabulary, and let C 2 p denote L 2 extended with additional quanti ers \there exists exactly (at most, at least) i", for i p. Finally, let C 2 be the union of C 2 p taken over all integers p. We prove that the problem of satis ability of sentences of C 2 1 is NEXPTIME-complete. Problems concerning decidability of restricted classes of quanti cational formulas have been studied since the second decade of this century by many logicians including W. Ackermann, P. Bernays, K. G odel, L. Kalm ar, M. Sch on nkel, T. Skolem, H. Wang 1,2,5,11,12,24,33,34,35,37] and many others. In the late twenties and in the thirties (see 7] and 19] for more informations) the study of classi cation of solvable classes of prenex formulas was one of the most active areas of logic.…”

Section: Introductionmentioning

confidence: 99%

Complexity Results for First-Order Two-Variable Logic with Counting

Pacholski¹,

Szwast²,

Tendera³

2000

SIAM J. Comput.

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Let C 2 p denote the class of rst order sentences with two variables and with additional quanti ers \there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satis ability problem for C 2 1 sentences is NEXPTIME-complete. This strengthens the results by E. Gr adel, Ph. Kolaitis and M. Vardi 15] who showed that the satis ability problem for the rst order two-variable logic L 2 is NEXPTIME-complete and by E. Gr adel, M. Otto and E. Rosen 16] who proved the decidability of C 2 . Our result easily implies that the satis ability problem for C 2 is in non-deterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size.It is worth noticing, that by a recent result of E. Gr adel, M. Otto and E. Rosen 17], extensions of two-variables logic L 2 by a week access to cardinalities through the H artig (or equicardinality) quanti er is undecidable. The same is true for extensions of L 2 by very week forms of recursion.The satis ability problem for logics with a bounded number of variables has applications in arti cial intelligence, notably in modal logics (see e.g. 22]), where counting comes in the context of graded modalities and in description logics, where counting can be used to express so-called number restrictions (see e.g. 8]).

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Section: Introductionmentioning

confidence: 99%

Complexity Results for First-Order Two-Variable Logic with Counting

Pacholski¹,

Szwast²,

Tendera³

2000

SIAM J. Comput.

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“…In [26] formulae with prefixes in ∃ * ∀ * and without functions are considered. Formulae with prefixes in ∃ * ∀ 2 ∃ * and without functions and equality predicates are tackled in [11], [24] and [27]. Formulae without equality predicate, and with unary predicate and functions are studied in [19] and [12].…”

Section: The Function Elimination Methodsmentioning

confidence: 99%

A Function Elimination Method for Checking Satisfiability of Arithmetical Logics

Castiglioni

Lanotte

Tini

2016

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“…(The decidability proofs of Kalmir [10], Schfitte [12], and G6del [8] for prenex formulas with prefixes W3 ---3 are of this form, as are many proofs in [6].) must verify PalfA(ala2) if d falsifies Paja2, and must falsify Pajf1(aja2) if it verifies Paja2.…”

Section: Predicate Letters Of F and Arguments From U Under Which The mentioning

confidence: 99%

The finite controllability of the Maslov case

Aanderaa

Goldfarb

1974

J. symb. log.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic.In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form Al v A2, where Al and A2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is MVaslov if its prefix is 3*-3V. --... * 3 and its matrix is Krom. A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes v... * *-* *3, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes 3V3V, or prefixes V33V, or prefixes V3VV are all reduction classes, while formulas with Krom matrices and prefixes V3V comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].The basis of our proof is the construction of finite universes with certain combinatorial properties; this is done in ?1 by generalizing two lemmas devised by G6del in his finite controllability proof for prenex formulas with prefixes V3 ... 3 [9]. In ?2 we show that we need only consider those Maslov formulas in Skolem form whose matrices satisfy two special truth-functional conditions. Finally, in ?3 we define, for any formula of this form whose matrix is truth-functionally consistent, a model over a finite universe with the combinatorial properties given in ?1. Since consistency of the matrix is necessary for the satisfiability of a formula, the finite controllability of the full Maslov case is thereby established.?1. A model for an arbitrary satisfiable Skolem formula F = Vy1 --Vyn3x, *.. 3xmM can be obtained by starting with a suitable universe U and functionsfj: Un U for j = 1, --*, m, and then defining a truth-assignment s' to atomic formulas with Received July 11, 1973.

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Über die Erfüllbarkeit derjenigen Zählausdrücke, welche in der Normalform zwei benachbarte Allzeichen enthalten

Abstract: U b e r d i e E r f i i U b a r k e i t d e~j e n i g e n Z~h l a u s d r f i c k e , w e l c h e i n d e r N o r m a l f o r m z w e i b e n a c h b a r t e A l l z e i e h e n e n t h a l t e n . Von L~szl6 K a l m s in Szeged (Ungarn).Einleitung.

Cited by 33 publications

References 3 publications

Complexity Results for First-Order Two-Variable Logic with Counting

Complexity Results for First-Order Two-Variable Logic with Counting

A Function Elimination Method for Checking Satisfiability of Arithmetical Logics

The finite controllability of the Maslov case

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