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36
Products of Modal Logics, Part 1
 LOGIC JOURNAL OF THE IGPL
, 1998
"... The paper studies manydimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: pmorphisms, the finite depth method, normal forms, ..."
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Cited by 36 (1 self)
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The paper studies manydimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: pmorphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.
MultiDimensional Modal Logic as a Framework for SpatioTemporal Reasoning
 APPLIED INTELLIGENCE
, 2000
"... In this paper we advocate the use of multidimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a twodimensional logic capable of describing topological relationships that change over time. This logic, ca ..."
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Cited by 35 (6 self)
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In this paper we advocate the use of multidimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a twodimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional SpatioTemporal Logic) is the Cartesian product of the wellknown temporal logic PTL and the modal logic S4u , which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both pointbased and interval based) of the spatial logic RCC8 can be embedded. We consider known decidability and complexity results that are relevant to computation with mulidimensional formalisms and discuss possible directions for further research.
Modal description logics: Modalizing roles
 Fundam. Inform
, 1999
"... In this paper, we construct a new concept description language intended for representing dynamic and intensional knowledge. The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic term ..."
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Cited by 31 (14 self)
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In this paper, we construct a new concept description language intended for representing dynamic and intensional knowledge. The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic terms: concepts, roles and formulas. Moreover, the language may contain both local (i.e., statedependent) and global (i.e., stateindependent) concepts, roles and objects. All this provides us with the most complete and natural means for re ecting the dynamic and intensional behaviour of application domains. We construct a satis ability checking (mosaictype) algorithm for this language (based on ALC) in(i) arbitrary multimodal frames, (ii) frames with universal accessibility relations (for knowledge) and (iii) frames with transitive, symmetrical and euclidean relations (for beliefs). On the other hand, it is shown that the satisfaction problem becomes undecidable if the underlying frames are arbitrary strict linear orders, hN; <i, or the language contains the common knowledge operator for n 2 agents. 1
Decidable Fragments of FirstOrder Modal Logics
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... The paper considers the set ML1 of firstorder polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML1, which reduces the modal satisfiability to the classica ..."
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Cited by 28 (8 self)
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The paper considers the set ML1 of firstorder polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML1, which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.
On the Products of Linear Modal Logics
 JOURNAL OF LOGIC AND COMPUTATION
, 2001
"... We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz: ..."
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Cited by 24 (9 self)
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We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using nonstructural Gabbaytype inference rules.
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]
A Survey of Decidable FirstOrder Fragments and Description Logics
 Journal of Relational Methods in Computer Science
, 2004
"... The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard tran ..."
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Cited by 9 (2 self)
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The guarded fragment and its extensions and subfragments are often considered as a framework for investigating the properties of description logics. There are also other, some less wellknown, decidable fragments of firstorder logic which all have in common that they generalise the standard translation of to firstorder logic. We provide a short survey of some of these fragments and motivate why they are interesting with respect to description logics, mentioning also connections to other nonclassical logics.
Reactive Kripke semantics and arc accessibility
 In Pillars of Computer Science: Essays dedicated to Boris (Boaz) Trakhtenbrot on the occasion of his 85th birthday, A.Avron, N.Dershowitz, A.Rabinovich eds., LNCS
, 2008
"... Ordinary Kripke models are not reactive. When we evaluate (test/measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the ..."
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Cited by 8 (5 self)
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Ordinary Kripke models are not reactive. When we evaluate (test/measure) a formula A at a model m, the model does not react, respond or change while we evaluate. The model is static and unchanged. This paper studies Kripke models which react to the evaluation process and change themselves during the process. This is reminiscent of game theoretic semantics where the two sides react to each other. However, reactive Kripke models do not go as far as that. The only additional device we add to Kripke semantics to make it reactive is to allow the accessibility relation to access itself. Thus the accessibility relation R of a reactive Kripke model contains not only pairs (a, b) ∈Rof possible worlds (b is accessible to a, i.e. there is an accessibility arc from a to b) but also pairs of the form (t, (a, b)) ∈R, the arc (a, b) is accessible to t. This new kind of Kripke semantics allows us to characterise more axiomatic modal logics (with one modality □) by a class of reactive frames. There are logics which cannot be characterised by ordinary frames but which can be characterised by reactive frames. We use such models to fibre logics which disagree on their common language. 1