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Fusions of modal logics revisited
 In Advances in modal logic
, 1998
"... The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of n ..."
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Cited by 47 (8 self)
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The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht &Wolter [10]]. The paper de nes the fusion `l `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics. Given two logical system L1 and L2 it is natural to ask whether the fusion (or join) L1 L2 of them inherits the common properties of both L1 and L2. Let us consider some examples: (i) It is known that the rst order theory of one equivalence relation has the nite model property and is decidable. However, the rst order theory of two equivalence relations does not have the nite model property and is in fact undecidable (see Janiczak [7]). This result shows that even if we know the rst order properties of the individual relations of a theory, there may be no algorithm to determine the purely logical consequences of these properties. (ii) Various positive and negative results are known for joins of term rewriting systems (TRSs) whose vocabularies are disjoint. For example, the join of two TRSs is con uent i the two TRSs are con uent but there are complete TRSs whose join is not complete (see e.g. Klop [8]). In fact, the literature on TRSs shows how useful the study of joins of systems can be. (iii) In contrast to rst order theories the join of two decidable equational theories in disjoint languages is decidable as well. This was proved by Pigozzi in [12]. So we observe interesting di erences between logical systems by investigating the behavior of joins. To form the join of two modal logics (in languages with disjoint sets of modal operators) is { in a sense { a generalization of forming the join of two equational theories in disjoint languages. Namely, it is wellknown that each modal logic corresponds to an equational theory of boolean algebras with operators. So the join of two modal logics corresponds to
Modularity and Web Ontologies
 In Proc. KR2006
, 2006
"... Modularity in ontologies is key both for large scale ontology development and for distributed ontology reuse on the Web. However, the problems of formally characterizing a modular representation, on the one hand, and of automatically identifying modules within an OWL ontology, on the other, has not ..."
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Cited by 41 (9 self)
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Modularity in ontologies is key both for large scale ontology development and for distributed ontology reuse on the Web. However, the problems of formally characterizing a modular representation, on the one hand, and of automatically identifying modules within an OWL ontology, on the other, has not been satisfactorily addressed, although their relevance has been widely accepted by the Ontology Engineering and Semantic Web communities. In this paper, we provide a notion of modularity grounded on the semantics of OWLDL. We present an algorithm for automatically identifying and extracting modules from OWLDL ontologies, an implementation and some promising empirical results on realworld ontologies.
Notes on Refinement, Interpolation and Uniformity.
"... The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The abs ..."
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Cited by 6 (5 self)
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The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The absence of the desirable interpolation properties from many logics that have been used in refinement, motivates a thorough investigation of methods to expand a specification formalism orthogonally, so that the critical uniform interpolants become available. A potential breakthrough is outlined in this paper. 1. A refinement paradigm Let us consider program development by means of stepwise refinements. One postulates some abstract data typelike specification 1 (ADT), suitable for the problem at hand, which has to be implemented on the available system. The end product consists of (the text of) an abstract program manipulating the postulated ADT, together with a suite of (texts of) modules implementin...
Propositional quantification in the topological semantics for S4
 Notre Dame Journal of Formal Logic
, 1997
"... quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S ..."
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Cited by 4 (1 self)
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quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub S4πt, isstrictly weaker than its Kripkean counterpart. I prove here that secondorder arithmetic can be recursively embedded in S4πt. Inthe course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to secondorder logic. 1Introduction One way to extend a propositional logic to a language with propositional quantifiers is to begin with a semantics for the logic; extract from the semantics a notion of a proposition; and interpret the quantifiers as ranging over the propositions. Thus, Fine [4] extends the Kripke semantics for modal logics to propositionally quantified systems S5π+, S4π+, S4.2π+, and such: given a Kripke frame, the quantifiers range over all sets of possible worlds. S5π+ is decidable ([4] and Kaplan [14]). In later unpublished work, Fine and Kripke independently showed that S4π+, S4.2π+, K4π+, Tπ+, Kπ+, and Bπ+ and others are recursively isomorphic to full secondorder classical logic. (Fine informs me that he later proved this stronger result. Kripke informs me that he too proved this stronger result in the early 1970s. A proof of this result occurs in Kaminski and Tiomkin [13], who use techniques similar to those used in Kremer [16] and to those used below. These techniques do not apply to S4.3π+. But according to Kaminski and Tiomkin, work of Gurevich and Shelah ([9], [10], and [39]) implies that secondorder arithmetic is interpretable in S4.3π+ and furthermore that, under
Substitutions of Σ 0 1Sentences  explorations between intuitionistic propositional logic and . . .
, 2002
"... ..."
On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond
, 2008
"... Département de mathématiques Faculté des sciences ..."
Dynamic change evaluation for ontology evolution in the semantic web
 In Proc. of (WIIAT
, 2008
"... Changes in an ontology may have a disruptive impact on any system using it. This impact may depend on structural changes such as introduction or removal of concept definitions, or it may be related to a change in the expected performance of the reasoning tasks. As the number of systems using ontol ..."
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Cited by 2 (0 self)
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Changes in an ontology may have a disruptive impact on any system using it. This impact may depend on structural changes such as introduction or removal of concept definitions, or it may be related to a change in the expected performance of the reasoning tasks. As the number of systems using ontologies is expected to increase, and given the open nature of the Semantic Web, introduction of new ontologies and modifications to existing ones are to be expected. Dynamically handling such changes, without requiring human intervention, becomes crucial. This paper presents a framework that isolates groups of related axioms in an OWL ontology, so that a change in one or more axioms can be automatically localised to a part of the ontology. 1
Investigations on the dual calculus
, 2004
"... The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: A. Efforts to extend the CurryHoward isomorphism, established between the simplytyped lambda calculus and intuitionistic logic, to classical logic. B. Efforts to establ ..."
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Cited by 2 (0 self)
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The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: A. Efforts to extend the CurryHoward isomorphism, established between the simplytyped lambda calculus and intuitionistic logic, to classical logic. B. Efforts to establish the tacit conjecture that callbyvalue reduction in lambda calculus is dual to callbyname reduction. This paper initially investigates relations of the Dual Calculus to other calculi, namely the simplytyped lambda calculus and the Symmetric lambda calculus. Moreover, ChurchRosser and Strong Normalization properties are proven for the calculus ’ callbyvalue reduction relation. Finally, extensions of the calculus to secondorder types are briefly introduced. 1
Rules and Arithmetics
, 1998
"... This paper is concerned with the `logical structure ' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible proposi ..."
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Cited by 2 (1 self)
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This paper is concerned with the `logical structure ' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical