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Multimodal logic programming
 Theoretical Computer Science
, 2006
"... We give a framework for developing the least model semantics, fixpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn formulas. Our approach is dir ..."
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Cited by 12 (7 self)
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We give a framework for developing the least model semantics, fixpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn formulas. Our approach is direct and no special restriction on occurrences of ✷i and ✸i is required. We apply our framework for a large class of basic serial multimodal logics, which are parameterized by an arbitrary combination of generalized versions of axioms T, B, 4, 5 (in the form, e.g., 4: ✷iϕ → ✷j✷kϕ) and I: ✷iϕ → ✷jϕ. Another part of the work is devoted to programming in multimodal logics intended for reasoning about multidegree belief, for use in distributed systems of belief, or for reasoning about epistemic states of agents in multiagent systems. For that we also use the framework, and although these latter logics belong to the mentioned class of basic serial multimodal logics, the special SLDresolution calculi proposed for them are more efficient.
The modal logic programming system MProlog
 Proceedings of JELIA 2004, LNCS 3229
, 2004
"... Abstract. We present a general framework for developing fixpoint semantics, the least model semantics, and SLDresolution calculi for logic programs in normal multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn fo ..."
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Cited by 10 (6 self)
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Abstract. We present a general framework for developing fixpoint semantics, the least model semantics, and SLDresolution calculi for logic programs in normal multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn formulas. Our approach is direct and no special restriction on occurrences of ✷i and ✸i is assumed. We prove that under certain expected properties of a concrete instantiation of the framework for a specific multimodal logic, the SLDresolution calculus is sound and complete. Based on the framework, we have developed and implemented the modal logic programming system MProlog. Our system is written in Prolog as a module for Prolog. Codes, libraries, and most features of Prolog can be used in MProlog programs in a pure way. The system contains a number of builtin SLDresolution calculi for modal logics, including calculi for useful multimodal logics of belief. It is a tool to experiment with applications of modal logic programming to AI. We also present the design and implementation of the MProlog system and give formulations of the wise men puzzle in MProlog. 1
Constructing finite least Kripke models for positive logic programs in serial regular grammar logics
 Logic Journal of the IGPL
"... A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G( ..."
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Cited by 6 (4 self)
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A serial contextfree grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form ✷tϕ → ✷s1... ✷skϕ. Such an inclusion axiom corresponds to the grammar rule t → s1... sk. Thus the inclusion axioms of L capture a contextfree grammar G(L). If for every modal index t, the set of words derivable from t using G(L) is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positive multimodal logic program P and a set of finite automata specifying a serial regular grammar logic L, constructs a finite least Lmodel of P. (A model M is less than or equal to model M ′ if for every positive formula ϕ, if M  = ϕ then M ′  = ϕ.) A least Lmodel M of P has the property that for every positive formula ϕ, P  = ϕ iff M  = ϕ. The algorithm runs in exponential time and returns a model with size 2 O(n3). We give examples of P and L, for both of the case when L is fixed or P is fixed, such that every finite least Lmodel of P must have size 2 Ω(n). We also prove that if G is a contextfree grammar and L is the serial grammar logic corresponding to G then there exists a finite least Lmodel of ✷sp iff the set of words derivable from s using G is a regular language. 1
Analytic cutfree tableaux for regular modal logics of agent beliefs
 Proceedings of CLIMA VIII, vol. 5056 of LNAI
, 2008
"... Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound ..."
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Cited by 5 (4 self)
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Abstract. We present a sound and complete tableau calculus for a class BReg of extended regular modal logics which contains useful epistemic logics for reasoning about agent beliefs. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. Applying sound global caching to the calculus, we obtain the first optimal (EXPTime) tableau decision procedure for BReg. We demonstrate the usefulness of BReg logics and our tableau calculus using the wise men puzzle and its modified version, which requires axiom (5) for single agents. 1
Clausal Tableaux for Multimodal Logics of Belief
"... Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutf ..."
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Abstract. We develop clausal tableau calculi for seven multimodal logics variously designed for reasoning about multidegree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multiagent systems. Our tableau calculi are sound, complete, cutfree and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of one was known and that of the seventh remains open.