Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are search-based. In this work we explore an alternative approach to QBF solving, based on symbolic quantifier elimination. We extend some recent symbolic approaches for SAT solving to symbolic QBF solving, using various decision-diagram formalisms such as OBDDs and ZDDs. In both approaches, QBF formulas are solved by eliminating all their quantifiers. Our first solver, QMRES, maintains a set of clauses represented by a ZDD and eliminates quantifiers via multi-resolution. Our second solver, QBDD, maintains a set of OBDDs, and eliminate quantifier by applying them to the underlying OBDDs. We compare our symbolic solvers to several competitive search-based solvers. We show that QBDD is not competitive, but QMRES compares favorably with search-based solvers on various benchmarks consisting of non-random formulas.

We use a strong form of the tree model property to boost the performance of resolution-based first-order theorem provers on the so-called relational translations of modal formulas. We provide both the mathematical underpinnings and experimental results concerning our improved translation method.

The tradeoff between expressiveness and tractability in Description Logics (DLs) has long been recognised [9]. For DL system designers and implementors there are several possible approaches to this problem [3]. These

We present a novel reasoning calculus for the description logic SHOIQ + —a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions—a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies.

We describe BDD-based decision procedures for the modal logic K. Our approach is inspired by the automata-theoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types---which can be viewed as an on-the-fly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a "level-based" representation scheme. The latter turns out to speed up construction and reduce memory consumption considerably. We also study the effect of formula simplification on our decision procedures. To proof the viability of our approach, we compare our approach with a representative selection of other approaches, including a translation of to QBF. Our results indicate that the BDD-based approach dominates for modally heavy formulae, while search-based approaches dominate for propositionally heavy formulae.

Abstract. Effective optimisation techniques can make a dramatic difference in the performance of knowledge representation systems based on expressive description logics. Because of the correspondence between description logics and propositional modal logic many of these techniques carry over into propositional modal logic satisfiability checking. Currently-implemented representation systems that employ these techniques, such as FaCT and DLP, make effective satisfiable checkers for various propositional modal logics. 1

In mathematics there exist powerful symbolic computation packages which have drawn considerable attention, also because of their easy-to-use interfaces and graphical capabilities. In computer logic, however, there were mainly complex tools for experts or purely didactic proof assistants. The Logics Workbench LWB is an attempt to fill this gap in the area of propositional logics. On the one hand, the LWB is intended for being used as an educational tool, especially for non-classical logics, and on the other hand as a programmable logic platform for more experienced users. The present thesis comprises three major parts: a system design, an empirical study, and a theoretical contribution. (a) Beside a general introduction to the LWB, we will present design aspects of its key components in the first part: the kernel, the parser, the programming language and the user interface. It is shown that the adopted solutions are adequate for doing logic on the computer. The design goal is to provid...

The recent emergence of heavily-optimized modal decision procedures has highlighted the key role of empirical testing in this domain. Unfortunately, the introduction of extensive empirical tests for modal logics is recent, and so far none of the proposed test generators is very satisfactory. To cope with this fact, we present a new random generation method that provides benefits over previous methods for generating empirical tests. It fixes and much generalizes one of the best-known methods, the random CNF test, allowing for generating a much wider variety of problems, covering in principle the whole input space. Our new method produces much more suitable test sets for the current generation of modal decision procedures. We analyze the features of the new method by means of an extensive collection of empirical tests.

In an earlier work we showed how a competitive satisfiability solver for the modal logic K can be built on top of a BDD package. In this work we study optimization issues for such solvers. We focus on two types of optimizations.

We provide an empirical evaluation of one of the main test sets that is currently in use for testing modal satisfiability solvers, viz. the random modal QBF test set. We first discuss some of the background underlying the test set, and then evaluate the test set using criteria set forth by Horrocks, Patel-Schneider, and Sebastiani. We also present some guidelines for the use of the test set.