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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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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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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
Foundations of Modal Logic Programming: The Direct Approach (release 2.0)”, manuscript (provided as a technical report), available at http://www.mimuw.edu. pl/~nguyen/papers.html
"... 1.1 Classical Logic Programming............................ 5 1.2 Previous Works on Modal Logic Programming.................. 7 ..."
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1.1 Classical Logic Programming............................ 5 1.2 Previous Works on Modal Logic Programming.................. 7
Weakening Horn knowledge bases in regular description logics to have PTIME data complexity
 Proceedings of ADDCT’2007
, 2007
"... This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of ..."
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This work is a continuation of our previous works [4,5]. We assume that the reader is familiar with description logics (DLs). A knowledge base in a description logic is a tuple (R,T,A) consisting of an RBox R of assertions about roles, a TBox T of global assumptions about concepts, and an ABox A of facts about individuals (objects) and roles. The instance checking problem in a DL is to check whether a given individual a is an instance of a concept C w.r.t. a knowledge base (R,T,A), written as (R,T,A)  = C(a). This problem in DLs including the basic description logic ALC (with R = /0) is EXPTIMEhard. From the point of view of deductive databases, A is assumed to be much larger than R and T, and it makes sense to consider the data complexity, which is measured when the query consisting of R, T, C, a is fixed while A varies as input data. It is desirable to find and study fragments of DLs with PTIME data complexity. Several authors have recently introduced a number of Horn fragments of DLs with PTIME data complexity [2,1,3]. The most expressive fragment from those is HornSHIQ introduced by Hustadt et al. [3]. It assumes, however, that the constructor ∀R.C does not occur in bodies of program clauses and goals. The data complexity of the “general Horn fragment of ALC ” is coNPhard [6]. So, to obtain PTIME data complexity one has to adopt some restrictions for the “general Horn fragments of DLs”. The goal is to find as less restrictive conditions as possible. A RBox is a finite set of assertions of the form Rs1 ◦... ◦ Rs ⊑ Rt, whereRs1,..., k Rs, Rt are role names. A regular RBox is an RBox whose set of corresponding grammar k rules t → s1...sk forms a grammar such that the set of words derivable from any symbol s using the grammar is a regular language specified by a finite automaton. We assume that the corresponding finite automata specifying R are given when R is considered. By R eg we denote ALC extended with regular RBoxes. We extend the language of ALC and R eg with the concept constructor ∀∃, which creates a concept ∀∃Rt.C from a role name Rt and a concept C.LetSem1(∀∃Rt.C)={∀Rt.C,∃Rt.⊤} and Sem2,R (∀∃Rt.C)= {∀Rt.C} ∪{∀Rs1...∀Rsi−1∃Rsi.⊤Rs1 ◦ ···◦Rs ⊑ Rt is a consequence of R and 1 ≤
Modal logic programming revisited
"... ABSTRACT. We present optimizations for the modal logic programming system MProlog, including the standard form for resolution cycles, optimized sets of rules used as metaclauses, optimizations for the version of MProlog without existential modal operators, as well as iterative deepening search and ..."
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ABSTRACT. We present optimizations for the modal logic programming system MProlog, including the standard form for resolution cycles, optimized sets of rules used as metaclauses, optimizations for the version of MProlog without existential modal operators, as well as iterative deepening search and tabulation. Our SLDresolution calculi for MProlog in a number of modal logics are still strongly complete when resolution cycles are in the standard form and optimized sets of rules are used. We also show that the labelling technique used in our direct approach is relatively better than the Skolemization technique used in the translation approaches for modal logic programming. KEYWORDS: modal logic, logic programming, MProlog. DOI:10.3166/JANCL.19.167–181 c ○ 2009 Lavoisier, Paris
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.
Declarative Programming for Agent Applications
"... This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capa ..."
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This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capabilities. The need for these features is motivated and examples are given to illustrate the central ideas.
A Logical Foundations for More Expressive Declarative Temporal Logic Programming Languages
"... In this paper, we present a declarative propositional temporal logic programming language called TeDiLog that is a combination of the temporal and disjunctive paradigms in Logic Programming. TeDiLog is, syntactically, a sublanguage of the wellknown Propositional Lineartime Temporal Logic (PLTL). T ..."
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In this paper, we present a declarative propositional temporal logic programming language called TeDiLog that is a combination of the temporal and disjunctive paradigms in Logic Programming. TeDiLog is, syntactically, a sublanguage of the wellknown Propositional Lineartime Temporal Logic (PLTL). TeDiLog allows both eventualities and alwaysformulas to occur in clause heads and also in clause bodies. To the best of our knowledge, TeDiLog is the first declarative temporal logic programming language that achieves this high degree of expressiveness. We establish the logical foundations of our proposal by formally defining operational and logical semantics for TeDiLog and by proving their equivalence. The operational semantics of TeDiLog relies on a restriction of the invariantfree temporal resolution procedure for PLTL that was introduced by Gaintzarain et al. in 2013. We define a fixpoint semantics that captures the reverse (bottomup) operational mechanism and prove its equivalence with the logical semantics. We also provide illustrative examples and comparison with other proposals.
Abstract Draft – Do not distribute Modal Functional Logic Programming
"... This paper introduces aspects of a novel modal functional logic programming language called Bach that is an extension of the existing functional logic language Escher. Language facilities available in Bach but not in Escher include (1) support for modalities and (2) an improved theoremproving capab ..."
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This paper introduces aspects of a novel modal functional logic programming language called Bach that is an extension of the existing functional logic language Escher. Language facilities available in Bach but not in Escher include (1) support for modalities and (2) an improved theoremproving capability. We show how the increased expressiveness of Bach can be exploited to produce easytounderstand programs for solving a variety of computational problems that arise in applications, especially agents applications.