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.

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.

Abstract In the last two decades, modal and description logics have been applied to numerous areas of computer science, including knowledge representation, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In particular, many approaches have been proposed for efficiently handling the satisfiability of the core normal modal logic K m , and of its notational variant, the description logic ALC. Although simple in structure, K m /ALC is computationally very hard to reason on, its satisfiability being PSpace-complete. In this paper we start exploring the idea of performing automated reasoning tasks in modal and description logics by encoding them into SAT, so that to be handled by stateof-the-art SAT tools; as with most previous approaches, we begin our investigation from the satisfiability in K m . We propose an efficient encoding, and we test it on an extensive set of benchmarks, comparing the approach with the main state-of-the-art tools available. Although the encoding is necessarily worst-case exponential, from our experiments we notice that, in practice, this approach can handle most or all the problems which are at the reach of the other approaches, with performances which are comparable with, or even better than, those of the current state-of-the-art tools. Motivations and Goals In the last two decades, modal and description logics have provided an essential framework for many applications in numerous areas of computer science, including artificial intelligence, formal verification, database theory, distributed computing and, more recently, semantic web and ontologies. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated (e.g., Fitting, 1983; Many approaches have been proposed for efficiently reasoning in modal and description logics, starting from the problem of checking the satisfiability in the core normal modal logic K m and in its notational variant, the description logic ALC (hereafter simply "K m "). We classify them as follows. 343 Sebastiani & Vescovi • The "classic" tableau-based approach • The DPLL-based approach These two approaches merged into the "modern" tableaux-based approach, which has been extended to work with more expressive description logics and to provide more sophisticate reasoning functions. Among the tools employing this approach, we recall • The CSP-based approach • In the Inverse-method approach • In the Automata-theoretic approach, (a symbolic representation based on BDDsBinary Decision Diagrams -of) a tree automaton accepting all the tree models of the input formula is implicitly built and checked for emptiness 1. Notice that there is not an universal agreement on the terminology "tableaux-based" and "DPLL-based". E.g., tools like FaCT, DLP, and Racer are most often called "tableau-based", although they use a DPLL-like algorithm instead of propositional tableaux for handling the propositional component of reasoning 344 Automated Reasoning in Modal and Description Logics via SAT Encoding • Pan and Vardi To the best of our knowledge, the last four approaches so far are restricted to the satisfiability in K m only, whilst the translational approach has been applied to numerous modal and description logics (e.g. traditional modal logics like T m and S4 m , and dynamic modal logics) and to the relational calculus. A significant amount of benchmarks formulas have been produced for testing the effectiveness of the different techniques In the last two decades we have also witnessed an impressive advance in the efficiency of propositional satisfiability techniques (SAT), which has brought large and previouslyintractable problems at the reach of state-of-the-art SAT solvers. Most of the success of SAT technologies is motivated by the impressive efficiency reached by current implementations of the DPLL procedure, As a consequence, many hard real-world problems have been successfully solved by encoding into SAT (including, e.g., circuit verification and synthesis, scheduling, planning, model checking, automatic test pattern generation , cryptanalysis, gene mapping). Effective encodings into SAT have been proposed also for the satisfiability problems in quantifier-free FOL theories which are of interest for formal verification In this paper we start exploring the idea of performing automated reasoning tasks in modal and description logics by encoding them into SAT, so that to be handled by state-ofthe-art SAT tools; as with most previous approaches, we begin our investigation from the satisfiability in K m . In theory, the task may look hopeless because of worst-case complexity issues: in fact, with few exceptions, the satisfiability problem in most modal and description logics is not in NP, typically being PSpace-complete or even harder -PSpace-complete for K m 345 Sebastiani & Vescovi memory in current computers is in the order of the GBytes and current SAT solvers can successfully handle huge formulas, the encoding of many modal formulas (at least of those which are not too hard to solve also for the competitors) may be at the reach of a SAT solver. Finally, even for PSpace-complete logics like K m , also other state-of-the-art approaches are not guaranteed to use polynomial memory. In this paper we show that, at least for the satisfiability K m , by exploiting some smart optimizations in the encoding the SAT-encoding approach becomes competitive in practice with previous approaches. To this extent, the contributions of this paper are manyfold. • We propose a basic encoding of K m formulas into purely-propositional ones, and prove that the encoding is satisfiability-preserving. • We describe some optimizations of the encoding, both in form of preprocessing and of on-the-fly simplification. These techniques allow for significant (and in some cases dramatic) reductions in the size of the resulting Boolean formulas, and in performances of the SAT solver thereafter. • We perform a very extensive empirical comparison against the main state-of-the-art tools available. We show that, despite the NP-vs.-PSpace issue, this approach can handle most or all the problems which are at the reach of the other approaches, with performances which are comparable with, and sometimes even better than, those of the current state-of-the-art tools. In our perspective, this is the most surprising contribution of the paper. • As a byproduct of our work, we obtain an empirical evaluation of current tools for K msatisfiability available, which is very extensive in terms of both amount and variety of benchmarks and of number and representativeness of the tools evaluated. We are not aware of any other such evaluation in the recent literature. We also stress the fact that with our approach the encoder can be interfaced with every SAT solver in a plug-and-play manner, so that to benefit for free of every improvement in the technology of SAT solvers which has been or will be made available. Content. The paper is structured as follows. In Section 2 we provide the necessary background notions on modal logics and SAT. In Section 3 we describe the basic encoding from K m to SAT. In Section 4 we describe and discuss the main optimizations, and provide many examples. In Section 5 we present the empirical evaluation, and discuss the results. In Section 6 we present some related work and current research trends. In Section 7 we conclude, and describe some possible future evolutions.

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 CNF2m 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.

. The recent emergence of heavily-optimised modal decision procedures has lead to a number of generation methods for modal formulae. However, the generation methods developed so far are not satisfactory. To cope with this fact, we propose a much improved version of one of the best-known methods, the random CNF2m test. The new method drastically reduces the influence of a major flaw of the previous method, and allows us to generate a wider variety of problems covering much more of the input space. This produces more interesting test sets for the current generation of modal decision procedures. 1 Motivation and goals Heavily-optimised systems for determining satisfiability of formulae in propositional modal logics are becoming available. These systems, including DLP [11], FaCT [6], *SAT [3], and MSPASS [8], have more optimisations and are much faster than the previous generation of modal decision procedures. As with most theorem proving problems, neither computational complexit...