We present a new method for deciding Gödel-Dummett logic. Starting from a formula, it proceeds in three steps. First build a conditional graph based on the decomposition tree of the formula. Then try to remove some cycles in this graph by instantiating these boolean conditions. In case this is possible, extract a countermodel from such an instance graph. Otherwise the initial formula is provabl...

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a con...

In this paper we show that the quantified Boolean formula problem can be solved in polynomial time by an alternating Turing machine with 2 alternations, if in Boolean formula there exist no more than order of logarithm clauses in which all variables with the existential quantifier.

This report describes the new features of our four SAT solvers that entered the SAT’06 Race affiliated to the SAT’06 conference in Seattle. Two core solvers were submitted, CompSAT and PicoSAT. We also integrated these two solvers as back end into our QBF solver Quantor [1]. Quantor can be used for purely propositional problems. Then it is used as preprocessor very similar to SATeLite [6]. The ...

Quantifiers like “all” and “every” are frequent in daily language. However, when a sentence contains two quantifiers or is “doubly-quantified” (e.g., “Every boy ate a cookie”) the meaning can often be ambiguous since there are two potential interpretations: collective (e.g., several boys sharing a single cookie) and distributive (e.g., several boys each eating their own cookie). Psycholinguisti...

In this article, we deal with the following problem which we call Boolean unification with predicates: For a given formula F[X ] in first-order logic with equality containing an n-ary predicate variable X , is there a quantifier-free formula G[x1,...,xn] such that the formula F[G] is valid in first-order logic with equality? We obtain the following results. Boolean unification with predicates f...

The Boolean satisfiability (SAT) problem asks whether the variables in a given Boolean formula can be assigned in such a way that the formula evaluates to true. Given a Boolean formula, a SAT solver tries to find a set of satisfying assignments or declares that the formula is unsatisfiable. In this project, we implemented a parallel SAT solver using the Cilk multithreaded programming language. ...

Various extensions of public announcement logic have been proposed with quantification over announcements. The best-known extension is called arbitrary public announcement logic, APAL. It contains a primitive language construct ✷φ intuitively expressing that “after every public announcement of a formula, formula φ is true.” The logic APAL is undecidable and it has an infinitary axiomatization. ...

We examine the complexity of the formula value problem for Boolean formulas, which is the following decision problem: Given a Boolean formula without variables, does it evaluate to true? We show that the complexity of this problem is determined by certain closure properties of the connectives allowed to build the formula, and achieve a complete classification: The formula value problem is eithe...