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 Km, and of its notational variant, the description logic ALC. Although simple in structure, Km/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 Km. 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. 1. 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; Ladner, 1977; Baader & Hollunder, 1991; Halpern & Moses, 1992; Baader, Franconi, Hollunder, Nebel, & Profitlich, 1994; Massacci, 2000). In particular, the research in modal and description logics has followed two parallel routes until the seminal work of Schild (1991), which proved that the core modal logic Km and the core description logic ALC are one a notational variant of the other. Since then, analogous results have been produced for a bunch of other logics, so that, nowadays the two research lines have mostly merged into one research flow. 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 Km and in its notational variant, the description logic ALC (hereafter simply “Km”). We classify them as follows. c ©2009 AI Access Foundation. All rights reserved. Sebastiani & Vescovi • The “classic” tableau-based approach (Fitting, 1983; Baader & Hollunder, 1991; Massacci, 2000) is based on the construction of propositional tableau branches, which are recursively expanded on demand by generating successor nodes in a candidate Kripke model. Kris (Baader & Hollunder, 1991; Baader et al., 1994), Crack (Franconi, 1998), LWB (Balsiger, Heuerding, & Schwendimann, 1998) were among the main representative tools of this approach. • The DPLL-based approach (Giunchiglia & Sebastiani, 1996, 2000) differs from the previous one mostly in the fact that a Davis-Putnam-Logemann-Loveland (DPLL) procedure, which treats the modal subformulas as propositions, is used instead of the classic propositional tableaux procedure at each nesting level of the modal operators. KSAT (Giunchiglia & Sebastiani, 1996), ESAT (Giunchiglia, Giunchiglia, & Tacchella, 2002) and *SAT (Tacchella, 1999), are the representative tools of this 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 FaCT/FaCT++ and DLP (Horrocks & Patel-Schneider, 1999), and Racer (Haarslev & Moeller, 2001). 1 • In the translational approach (Hustadt & Schmidt, 1999; Areces, Gennari, Heguiabehere, & de Rijke, 2000) the modal formula is encoded into first-order logic (FOL), and the encoded formula can be decided efficiently by a FOL theorem prover (Areces et al., 2000). Mspass (Hustadt, Schmidt, & Weidenbach, 1999) is the most representative tool of this approach. • The CSP-based approach (Brand, Gennari, & de Rijke, 2003) differs from the tableauxbased and DPLL-based ones mostly in the fact that a CSP (Constraint Satisfaction Problem) engine is used instead of tableaux/DPLL. KCSP is the only representative tool of this approach. • In the Inverse-method approach (Voronkov, 1999, 2001), a search procedure is based on the “inverted” version of a sequent calculus (which can be seen as a modalized version of propositional resolution). K K(Voronkov, 1999) is the only representative tool of this approach. • In the Automata-theoretic approach, (a symbolic representation based on BDDs – Binary Decision Diagrams – of) a tree automaton accepting all the tree models of the input formula is implicitly built and checked for emptiness (Pan, Sattler, & Vardi, 2002; Pan & Vardi, 2003). KBDD (Pan & Vardi, 2003) is the only representative tool of this approach. 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 (Horrocks, 1998; Patel-Schneider, 1998; Horrocks & Patel-Schneider, 1999; Haarslev & Moeller, 2001).
Output: Automated Reasoning in Modal and Description Logics via SAT Encoding: the Case Study of Km/ALC-Satisfiability