{hustadt, schmidt} topi-sb.mpg.de This paper investigates the evaluation method of decision procedures for multi-modal logic proposed by Giunchiglia and Sebastiani as an adaptation from the evaluation method of Mitchell et al of decision procedures for propositional logic. We compare three different theorem proving approaches, namely the Davis-Putnam-based procedure KSAT, the tableaux-based system KTUS and a translation approach combined with first-order resolution. Our results do not support the claims of Giunchiglia and Sebastiani concerning the computational superiority of KSAT over KRIS, and an easy-hard-easy pattern for randomly generated modal formulae. 1

We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulae X the "superformulae" involved are always bounded by a finite set of formulae X L depending only on X and the logic L. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hi...

This paper reports on an empirical performance analysis of four modal theorem provers on benchmark suites of randomly generated formulae. The theorem provers tested are the Davis-Putnam-based procedure Ksat, the tableaux-based system KRIS, the sequent-based Logics Workbench, and a translation approach combined with the first-order theorem prover SPASS.

The description logic SHI extends the basic description logic ALC with transitive roles, role hierarchies and inverse roles. The known tableau-based decision procedure [9] for SHI exhibit (at least) NEXP-TIME behaviour even though SHI is known to be EXPTIME-complete. The automata-based algorithms for SHI often yield optimal worst-case complexity results, but do not behave well in practice since good optimisations for them have yet to be found. We extend our method for global caching in ALC to SHI by adding analytic cut rules, thereby giving the first EXPTIME tableau-based decision procedure for SHI, and showing one way to incorporate global caching and inverse roles.

We give algorithms to construct the least L-model for a given positive modal logic program P , where L can be one of the modal logics KD, T , KDB, B, KD4, S4, KD5, KD45, and S5. If L 2 fKD5;KD45;S5g, or L 2 fKD;T ; KDB;Bg and the modal depth of P is finitely bounded, then the least L-model of P can be constructed in PTIME and coded in polynomial space. We also show that if P has no flat models then it has the least models in KB, K5, K45, and KB5. As a consequence, the problem of checking the satisfiability of a set of modal Horn formulae with finitely bounded modal depth in KD, T , KB, KDB, or B is decidable in PTIME. The known result that the problem of checking the satisfiability of a set of Horn formulae in K5, KD5, K45, KD45, KB5, or S5 is decidable in PTIME is also studied in this work via a different method. 1.

Abstract. We show that global caching can be used with propagation of both satisfiability and unsatisfiability in a sound manner to give an EXPTIME algorithm for checking satisfiability w.r.t. a TBox in the basic description logic ALC. Our algorithm is based on a simple traditional tableau calculus which builds an and-or graph where no two nodes of the graph contain the same formula set. When a duplicate node is about to be created, we use the pre-existing node as a proxy, even if the proxy is from a different branch of the tableau, thereby building global caching into the algorithm from the start. Doing so is important since it allows us to reason explicitly about the correctness of global caching. We then show that propagating both satisfiability and unsatisfiability via the and-or structure of the graph remains sound. In the longer paper, by combining global caching, propagation and cutoffs, our framework reduces the search space more significantly than the framework of [1]. Also, the freedom to use arbitrary search heuristics significantly increases its application potential. A longer version with all optimisations is currently under review for a journal. An extension for SHI will appear in TABLEAUX 2007.

The paper shows satisfiability in many propositional modal systems can be decided by ordinary resolution procedures. This follows from a general result that resolution and condensing is a decision procedure for the satisfiability problem of formulae in so-called path logics. Path logics arise from propositional and normal uni- and multi-modal logics by the optimised functional translation method. The decision result provides an alternative decision proof for the relevant modal logics (including K, KD, KT and KB, their combinations ...

Abstract. We give complete sequent-like tableau systems for the modal logics KB, KDB, K 5, and KD5. Analytic cut rules are used to obtain the completeness. Our systems have the analytic superformula property and can thus give a decision procedure. Using the systems, we prove the Craig interpolation lemma for the mentioned logics. 1

We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional logics CK and CK Id.

Craig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.