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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 44 (7 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 37 (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 absence of th ..."
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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
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 propositional ..."
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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 theories. Key words: Intuitionistic Logic, Heyting Arithmetic, Kripke models, admissible rules MSC codes: Primary: 03F25, 03F30, Secondary: 0302, 03B20, 03F50, 03F40 Contents 1 Introduction 3 2 Theories and Logics 3 2.1 Propositional Logics of Theories . . . . . . . . . . . . . . . . . . 4 2.2 Predicate Logics of Theories . . . . . . . . . . . . . . . . . . . . . 5 2.3 A Brief History of de Jongh's Theorem . . . . . . . . . . . . . . . 7 2.4 Markov's Principle and Church's Thesis . . . . . . . . . . . . . . 9 2.5 Exactness and Extension . . . . ...
Strong cutelimination systems for Hudelmaier’s depthbounded sequent calculus for implicational logic
 PROCEEDINGS OF THE 3RD INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR’06), VOLUME 4130 OF LNAI
, 2006
"... Inspired by the CurryHoward correspondence, we study normalisation procedures in the depthbounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cutadmissibility. We decorate proofs with proofterms and introduce various ..."
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Cited by 2 (1 self)
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Inspired by the CurryHoward correspondence, we study normalisation procedures in the depthbounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cutadmissibility. We decorate proofs with proofterms and introduce various termreduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cutadmissibility suggesting weakly normalising procedures for Cutelimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence.
Modularity and Interpolation in a Development Workspace.
, 1997
"... The potential benefits of a uniform version of interpolation are hindered by its absence from many expressive logics. This motivates a thorough investigation of appropriate expansions of known entailments so that an easytoderive, uniform presentation of the interpolants is supported. This paper p ..."
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Cited by 1 (1 self)
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The potential benefits of a uniform version of interpolation are hindered by its absence from many expressive logics. This motivates a thorough investigation of appropriate expansions of known entailments so that an easytoderive, uniform presentation of the interpolants is supported. This paper presents the skeleton of a general construction and indicates that a potentially large class of entailments can be extended so that a uniform presentation of the interpolants is available. 1 Introduction There is a well established relation between interpolation [8] and modularity properties of refinements [23, 24, 29, 5, 32, 40, 39, 12, 11] and databases [25]. On the other hand, many logics that have been used in refinement or databases lack the desirable interpolation properties. To compensate for this inadequacy, several groups of researchers have proposed techniques to restrict these logics to fragments that have the desirable modularity properties. Some of these enterprises have focuse...
Uniformity, Interpolation and Module specification in a Development Workspace
 the proceedings of the TFM'98 workshop
, 1997
"... . Interpolation and Schematic Reasoning are shown to underlie critical and somewhat complementary aspects of designing and (syntactically) manipulating specification modules. In addition, the presence of a Uniform presentation of interpolants facilitates the specification of modules. Also, the a ..."
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Cited by 1 (1 self)
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. Interpolation and Schematic Reasoning are shown to underlie critical and somewhat complementary aspects of designing and (syntactically) manipulating specification modules. In addition, the presence of a Uniform presentation of interpolants facilitates the specification of modules. Also, the ability to encapsulate and manipulate Uniform Schemata may assist us in reasoning with (abstractions of) hidden data. Unfortunately, most formalisms that have been used in fundamental approaches to software engineering lack uniform interpolation and do not directly support schematic reasoning. This paper reveals the critical role of uniform interpolants and uniform schemata from the perspective of modularity, and quotes a general construction indicating that a potentially large class of calculi can be extended conservatively so that a uniform presentation of the critical interpolants becomes available and the manipulation of uniform schemata is supported. 1 Introduction There is a ...
Computing Interpolants in Implicational Logics
"... I present a new syntactical method for proving the Interpolation Theorem for the implicational fragment of intuitionistic logic and its substructural subsystems. This method, like Prawitz’s, works on natural deductions rather than sequent derivations, and, unlike existing methods, always finds a ‘st ..."
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Cited by 1 (0 self)
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I present a new syntactical method for proving the Interpolation Theorem for the implicational fragment of intuitionistic logic and its substructural subsystems. This method, like Prawitz’s, works on natural deductions rather than sequent derivations, and, unlike existing methods, always finds a ‘strongest ’ interpolant under a certain restricted but reasonable notion of what counts as an ‘interpolant’.