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PSPACE bounds for rank 1 modal logics
 IN LICSâ€™06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
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Cited by 34 (19 self)
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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
Prime implicates and prime implicants: From propositional to modal logic
 Journal of Artificial Intelligence Research
"... Prime implicates and prime implicants have proven relevant to a number of areas of artificial intelligence, most notably abductive reasoning and knowledge compilation. The purpose of this paper is to examine how these notions might be appropriately extended from propositional logic to the modal logi ..."
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Cited by 7 (0 self)
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Prime implicates and prime implicants have proven relevant to a number of areas of artificial intelligence, most notably abductive reasoning and knowledge compilation. The purpose of this paper is to examine how these notions might be appropriately extended from propositional logic to the modal logic K. We begin the paper by considering a number of potential definitions of clauses and terms for K. The different definitions are evaluated with respect to a set of syntactic, semantic, and complexitytheoretic properties characteristic of the propositional definition. We then compare the definitions with respect to the properties of the notions of prime implicates and prime implicants that they induce. While there is no definition that perfectly generalizes the propositional notions, we show that there does exist one definition which satisfies many of the desirable properties of the propositional case. In the second half of the paper, we consider the computational properties of the selected definition. To this end, we provide sound and complete algorithms for generating and recognizing prime implicates, and we show the prime implicate recognition task to be Pspacecomplete. We also prove upper and lower bounds on the size and number of prime implicates. While the paper focuses on the logic K, all of our results hold equally well for multimodal K and for concept expressions in the description logic ALC. 1.
Deciding the guarded fragments by resolution
 Journal of Symbolic Computation
, 2003
"... The guarded fragment is a fragment of firstorder logic that has been introduced for two main reasons: First, to explain the good computational and logical behavior of propositional modal logics. Second, to serve as a breeding ground for wellbehaved process logics. In this paper we give resolution ..."
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Cited by 4 (2 self)
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The guarded fragment is a fragment of firstorder logic that has been introduced for two main reasons: First, to explain the good computational and logical behavior of propositional modal logics. Second, to serve as a breeding ground for wellbehaved process logics. In this paper we give resolutionbased decision procedures for the guarded fragment and for the loosely guarded fragment (sometimes also called pairwise guarded fragment). By constructing an implementable decision procedure for the guarded fragment and for the loosely guarded fragment, we obtain an effective procedure for deciding modal logics that can be embedded into these fragments. The procedures have been implemented in the theorem prover Bliksem. 1.
A Does Treewidth Help in Modal Satisfiability?
"... Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence g ..."
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Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulas in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we show either that modal satisfiability is Fixed Parameter Tractable (Fpt), or that it is W[1]hard. In particular, modal satisfiability in general models is Fpt, while it is W[1]hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is Fpt.