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14
Interpolation in Modal Logic
, 1999
"... The interpolation property and Robinson's consistency property are important tools for applying logic to software engineering. We provide a uniform technique for proving the Interpolation Property, using the notion of bisimulation. For modal logics, this leads to simple, easytocheck condit ..."
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Cited by 102 (8 self)
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The interpolation property and Robinson's consistency property are important tools for applying logic to software engineering. We provide a uniform technique for proving the Interpolation Property, using the notion of bisimulation. For modal logics, this leads to simple, easytocheck conditions on the logic which imply interpolation. We apply this result to fibering of modal logics and to modal logics of knowledge and belief.
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 42 (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.
Conservativity in Structured Ontologies
"... Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kind ..."
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Cited by 11 (5 self)
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Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kinds, based on the theory of institutions. This covers theory interpretations, (definitional) language extensions, symbol identifications, and conservative extensions. In particular, we study the problem of inheriting conservativity between subtheories in a diagram to its colimit ontology, and apply this to the problem of localisation of reasoning in ‘modular ontology languages’ such as DDLs or Econnections.
Modules in transition. Conservativity, Composition, and Colimits
 In Proceedings, Second International Workshop on Modular Ontologies
, 2007
"... Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing c ..."
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Cited by 5 (0 self)
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Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and the theory of institutions. In particular, we consider theory interpretations, language extensions, symbol identification, and conservative extensions. We study the problem of inheriting conservativity between subtheories in a diagram to its colimit ontology. Finally, we apply this to the problem of conservativity when composing DDLs or Econnections. 1
Products, or How to Create Modal Logics of High Complexity
"... The aim of this paper is to exemplify the complexity of the satisability problem of products of modal logics. Our main goal is to arouse interest for the main open problem in this area: a tight complexity bound for the satisability problem of the product KK.At present, only nonelementary decision p ..."
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Cited by 4 (0 self)
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The aim of this paper is to exemplify the complexity of the satisability problem of products of modal logics. Our main goal is to arouse interest for the main open problem in this area: a tight complexity bound for the satisability problem of the product KK.At present, only nonelementary decision procedures for this problem are known. Our modest contribution is twofold. We show that the problem of deciding KKsatisability of formulas of modal depth two is already hard for nondeterministic exponential time, and provide a matching upper bound. For the full language, a new proof for decidability is given which combines ltration and selective generation techniques from modal logic. We put products of modal logics into an historic perspective and review the most important results. Keywords: modal logic, computational complexity 1 Introduction Taking products of modal logics is one of the most straightforward ways to combine two or more modal logics. The construction is dened as ...
Interpolation in Guarded Fragments
, 2000
"... The guarded fragment (GF) was introduced by Andréka, van Benthem and Németi as a finestructure of first order logic which combines a great expressive power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. Slightly ..."
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Cited by 4 (2 self)
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The guarded fragment (GF) was introduced by Andréka, van Benthem and Németi as a finestructure of first order logic which combines a great expressive power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. Slightly generalizing the admissible relativizations yields the packed fragment (PF). In this paper we chart the behavior of these fragments with regard to interpolation. While GF and PF have been established as particularly wellbehaved fragments of first order logic in many respects, it will be shown that the interpolation property of first order logic fails in restriction to GF and PF. However, each of these fragments turns out to have an alternative interpolation property that closely resembles the interpolation property usually studied in modal logic. These results are strong enough to entail the Beth definability theorem for GF and PF. Even better, every nvariable guarded or packed fragm...
Interpolation and Bisimulation in Temporal Logic
 In Proceedings of WoLLIC'98. Workshop of Logic, Language, Information and Computation
, 1998
"... Building on recent model theoretic results for SinceUntil logics we define an adequate notion of bisimulation and establish general theorems concerning the interpolation property. Using these general results we prove that the basic SUlogic and any SUlogic whose class of frames can be defined ..."
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Building on recent model theoretic results for SinceUntil logics we define an adequate notion of bisimulation and establish general theorems concerning the interpolation property. Using these general results we prove that the basic SUlogic and any SUlogic whose class of frames can be defined by universal Horn formulas have interpolation. In particular, the SUlogic of branching time has interpolation, while linear time fails to have this property. 1 Introduction For many years, modal logic (ML) was viewed as an extension of propositional logic (PL) by the addition of new modalities 3 and 2. Nowadays the picture has changed in many directions. First, modal logic is no longer seen as just an extension of PL but also as a restriction of firstorder logic (FO)  when formulas are interpreted over models, or secondorder logic (SO)  when formulas are considered on frames. Furthermore, 3 and 2 have lost their privileged position as a wide variety of new modalities have been i...
EPIMORPHISMS IN CYLINDRIC ALGEBRAS AND DEFINABILITY IN FINITE VARIABLE LOGIC
, 2008
"... Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of ..."
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Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of representable algebras in CAn for finite n> 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Bethdefinability property fails for the finitevariable fragments of first order logic as long as the number n of variables available is> 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CAn are best possible. We raise several open problems. §0. INTRODUCTION AND THE MAIN RESULTS In algebra, the properties of epimorphisms (in the categorial sense) being surjective and the amalgamation property in a class of algebras are well investigated, see e.g. [1] and [37]. In algebraic logic these properties turn out
Oliver Kutz Notes on Logics of Metric Spaces
"... Abstract. In [14], we studied the computational behaviour of various firstorder and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representin ..."
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Abstract. In [14], we studied the computational behaviour of various firstorder and modal languages interpreted in metric or weaker distance spaces. [13] gave an axiomatisation of an expressive and decidable metric logic. The main result of this paper is in showing that the technique of representing metric spaces by means of Kripke frames can be extended to cover the modal (hybrid) language that is expressively complete over metric spaces for the (undecidable) twovariable fragment of firstorder logic with binary predicates interpreting the metric. The frame conditions needed correspond rather directly with a Boolean modal logic that is, again, of the same expressivity as the twovariable fragment. We use this representation to derive an axiomatisation of the modal hybrid variant of the twovariable fragment, discuss the compactness property in distance logics, and derive some results on (the failure of) interpolation in distance logics of various expressive power.