Towards a proof theory for Henkin quantifiers

This paper presents a methodology to construct globally sound but possibly locally unsound analytic calculi for partial theories of Henkin quantifiers. It is demonstrated that usual locally sound analytic calculi do not exist for any reasonable fragment of the full theory of Henkin quantifiers. This is due to the combination of strong and weak quantifier inferences in one quantifier rule.

iDQ: Instantiation-Based DQBF Solving

Dependency Quantified Boolean Formulas (DQBF) are obtained by adding Henkin quantifiers to Boolean formulas and have seen growing interest in the last years. Since deciding DQBF is NEXPTIME-complete, efficient ways of solving it would have many practical applications. Still, there is only few work on solving this kind of formulas in practice. In this paper, we present an instantiation-based technique to solve DQBF efficiently. Apart from providing a theoretical foundation, we also propose a concrete implementation of our algorithm. Finally, we give a detailed experimental analysis evaluating our prototype iDQ on several DQBF as well as QBF benchmarks.

Relativized logspace and generalized quantifiers over finite ordered structures

We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures LC, i.e., logarithmic space relativized to an oracle in C. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures LC. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures LNP. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform, vol. 18, 1993].