Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP (...) 2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE +, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. (shrink)

This paper introduces a new Ehrenfeucht-Fraïssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating moves played in different models. The game allows Duplicator to delay her choices of the models till the very end of the game, making it easier for her to win. This adds on the toolkit of winning strategies for Duplicator in Ehrenfeucht-Fraïssé type (...) games and opens up new methods for tackling some open problems in descriptive complexity theory. As an application of the class game, it is shown that, if m is a power of a prime, then first order logic augmented with function quantifiers F1n of arity 1 and height n < m can not express that the size of the model is divisible by m. This, together with some new expressibility results for Henkin quantifiers H1n gives some new separation results on the class of finite models among various F1n on one hand and between F1n and H1n on the other. Since function quantifiers involve a bounded type of second order existential quantifiers, the class game solves an open problem raised by Fagin, which asks for some inexpressibility result using a winning strategy by Duplicator in a game that involves more than one coloring round. (shrink)

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures.In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy.As applications of this technique it is shown that • — Graph Connectivity is not expressible in existential monadic second-order (...) logic , even in the presence of a built-in linear order,• — Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n0, and• — the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation. (shrink)

We report two experiments which tested whether cognitive capacities are limited to those functions that are computationally tractable (PTIME-Cognition Hypothesis). In particular, we investigated the semantic processing of reciprocal sentences with generalized quantifiers, i.e., sentences of the form Q dots are directly connected to each other, where Q stands for a generalized quantifier, e.g. all or most. Sentences of this type are notoriously ambiguous and it has been claimed in the semantic literature that the logically strongest reading is preferred (Strongest (...) Meaning Hypothesis). Depending on the quantifier, the verification of their strongest interpretations is computationally intractable whereas the verification of the weaker readings is tractable. We conducted a picture completion experiment and a picture verification experiment to investigate whether comprehenders shift from an intractable reading to a tractable reading which should be dispreferred according to the Strongest Meaning Hypothesis. The results from the picture completion experiment suggest that intractable readings occur in language comprehension. Their verification, however, rapidly exceeds cognitive capacities in case the verification problem cannot be solved using simple heuristics. In particular, we argue that during verification, guessing strategies are used to reduce computational complexity. (shrink)

We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA ) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of second-order quantification is bounded by n . We show that this imposes a proper hierarchy on second-order logic, i.e. for every k , n there are problems not definable in AA but definable in AA for some c (...) 1 , d 1 . The method to show this is to introduce the set AUTOSAT of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second-order loeic. For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT is PSpace complete. For first-order logic FOL n with alternation of quantifiers bounded by n , AUTOSAT is definable in AA . AUTOSAT) is definable in AA for some c 1 , d 1. (shrink)

This paper introduces a new Ehrenfeucht‐Fraïssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai‐Fagin game to the case when there are several alternating (coloring) moves played in different models. The game allows Duplicator to delay her choices of the models till (practically) the very end of the game, making it easier for her to win. This adds on the toolkit of winning strategies for Duplicator in (...) Ehrenfeucht‐Fraïssé type games and opens up new methods for tackling some open problems in descriptive complexity theory. As an application of the class game, it is shown that, if m is a power of a prime, then first order logic augmented with function quantifiers F1n of arity 1 and height n < m can not express that the size of the model is divisible by m. This, together with some new expressibility results for Henkin quantifiers H1n gives some new separation results on the class of finite models among various F1n on one hand and between F1n and H1n on the other. Since function quantifiers involve a bounded type of second order existential quantifiers, the class game (in a sense) solves an open problem raised by Fagin, which asks for some inexpressibility result using a winning strategy by Duplicator in a game that involves more than one coloring round. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L∞ωω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L∞ωω to obtain the logics L∞ωω, where Q = {Qi: iε I} is a family (...) of generalized quantifiers on finite structures. Using the logics L∞ωω, we can express polynomial-time properties that are not definable in L∞ωω, such as “there is an even number of x” and “there exists at least n/2 x” , without going to second-order logic. We show that equivalence of finite structures relative to L∞ωω can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L∞ωω and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L∞ωω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L∞ωω. We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L∞ωω of L∞ωω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers. (shrink)

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to (...) the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. (shrink)