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Econnections of abstract description systems
"... Combining knowledge representation and reasoning formalisms is an important and challenging task. It is important because nontrivial AI applications often comprise different aspects of the world, thus requiring suitable combinations of available formalisms modeling each of these aspects. It is chal ..."
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Cited by 95 (25 self)
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Combining knowledge representation and reasoning formalisms is an important and challenging task. It is important because nontrivial AI applications often comprise different aspects of the world, thus requiring suitable combinations of available formalisms modeling each of these aspects. It is challenging because the computational behavior of the resulting hybrids is often much worse than the behavior of their components. In this paper, we propose a new combination method which is computationally robust in the sense that the combination of decidable formalisms is again decidable, and which, nonetheless, allows nontrivial interactions between the combined components. The new method, called Econnection, is defined in terms of abstract description systems (ADSs), a common generalization of description logics, many logics of time and space, as well as modal and epistemic logics. The basic idea of Econnections is that the interpretation domains of n combined systems are disjoint, and that these domains are connected by means of nary ‘link relations. ’ We define several natural variants of Econnections and study indepth the transfer of decidability from the component systems to their Econnections. Key words: description logics, temporal logics, spatial logics, combining logics, decidability.
Reasoning over Extended ER Models
 PROCEEDINGS OF ER 2007
, 2007
"... Abstract. We investigate the computational complexity of reasoning over various fragments of the Extended EntityRelationship (EER) language, which includes a number of constructs: ISA between entities and relationships, disjointness and covering of entities and relationships, cardinality constraint ..."
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Cited by 26 (11 self)
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Abstract. We investigate the computational complexity of reasoning over various fragments of the Extended EntityRelationship (EER) language, which includes a number of constructs: ISA between entities and relationships, disjointness and covering of entities and relationships, cardinality constraints for entities in relationships and their refinements as well as multiplicity constraints for attributes. We extend the known EXPTIMEcompleteness result for UML class diagrams [5] and show that reasoning over EER diagrams with ISA between relationships is EXPTIMEcomplete even without relationship covering. Surprisingly, reasoning becomes NPcomplete when we drop ISA between relationships (while still allowing all types of constraints on entities). If we further omit disjointness and covering over entities, reasoning becomes polynomial. Our lower complexity bound results are proved by direct reductions, while the upper bounds follow from the correspondences with expressive variants of the description logic DLLite, which we establish in this paper. These correspondences also show the usefulness of DLLite as a language for reasoning over conceptual models and ontologies.
On the Computational Complexity of SpatioTemporal Logics
 Proceedings of the 16th AAAI International FLAIRS Conference
, 2003
"... Recently, a hierarchy of spatiotemporal languages based on the propositional temporal logic PTL and the spatial languages RCC8, BRCC8 and S4u has been introduced. Although a number of results on their computational properties were obtained, the most important questions were left open. ..."
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Cited by 21 (0 self)
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Recently, a hierarchy of spatiotemporal languages based on the propositional temporal logic PTL and the spatial languages RCC8, BRCC8 and S4u has been introduced. Although a number of results on their computational properties were obtained, the most important questions were left open.
A logic for metric and topology
 Journal of Symbolic Logic
, 2005
"... Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the inten ..."
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Cited by 13 (11 self)
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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘εdefinitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIMEcompleteness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘wellbehaved ’ common denominator of logical systems constructed in temporal, spatial, and similaritybased quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of realtime systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial
A New Combination Procedure for the Word Problem that Generalizes Fusion Decidability Results in Modal Logics
 In David A. Basin and Michaël Rusinowitch, editors, IJCAR ’04
, 2004
"... Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of mod ..."
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Cited by 13 (7 self)
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Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other equational theories.
Modal languages for topology: Expressivity and definability, in "Annals of Pure and
 n o 12, 2009, p. 146170, http://hal.inria.fr/inria00424693/en/ GENL. International PeerReviewed Conference/Proceedings
"... In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the GoldblattThomason definability theorem in terms of the well established firstorder top ..."
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Cited by 12 (3 self)
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In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the GoldblattThomason definability theorem in terms of the well established firstorder topological language Lt.
A Note on Relativised Products of Modal Logics
 Advances in Modal Logic
, 2003
"... this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cu ..."
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Cited by 10 (6 self)
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this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cubic universal product S5 frames. Note that the `ireduct' F U 1 U n ; R i of F 1 F n is a union of n disjoint copies of F i . Thus, F and F i validate the same formulas, and so L n L 1 L n : There is a strong interaction between the modal operators of product logics. Every nary product frame satis es the following two properties, for each pair i 6= j, i; j = 1; : : : ; n: Commutativity : 8x8y8z xR i y ^ yR j z ! 9u (xR j u ^ uR i z) ^ xR j y ^ yR i z ! 9u (xR i u ^ uR j z) Church{Rosser property : 8x8y8z xR i y ^ xR j z ! 9u (yR j u ^ zR i u) This means that the corresponding modal interaction formulas 2 i 2 j p $ 2 j 2 i p and 3 i 2 j p ! 2 j 3 i p belong to every ndimensional product logic. The geometrically intuitive manydimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatiotemporal representation and reasoning (see e.g. [33, 34]) or reasoning about the behaviour of multiagent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexityeven the product of two NPcomplete logics can be nonrecursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non nitely axiomatisable [16]
Local Variations on a Loose Theme: Modal Logic and Decidability
"... This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of ..."
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Cited by 8 (1 self)
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This chapter is about decidability and complexity issues in modal logic; more specifically, we confine ourselves to satisfiability (and the complementary validity) problems. The satisfiability problem is the following: for a fixed class of
Spatial Logics with Connectedness Predicates
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2010
"... We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of thes ..."
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Cited by 7 (2 self)
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We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of these logics and show that the connectedness constraints can increase complexity from NP to PSpace, ExpTime and, if component counting is allowed, to NExpTime.
SCAN is complete for all Sahlqvist formulae
 In Relational and KleeneAlgebraic Methods in Computer Science (RelMiCS 7
, 2004
"... Abstract. SCAN is an algorithm for reducing monadic existential secondorder logic formulae to equivalent simpler formulae, often firstorder logic formulae. It is provably impossible for such a reduction to firstorder logic to be always successful, even if there is an equivalent firstorder formul ..."
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Cited by 6 (3 self)
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Abstract. SCAN is an algorithm for reducing monadic existential secondorder logic formulae to equivalent simpler formulae, often firstorder logic formulae. It is provably impossible for such a reduction to firstorder logic to be always successful, even if there is an equivalent firstorder formula for a secondorder logic formula. In this paper we show that SCAN successfully computes the firstorder equivalents of all Sahlqvist formulae in the classical (multi)modal language. 1