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66
Modal Languages And Bounded Fragments Of Predicate Logic
, 1996
"... Model Theory. These are nonempty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual BackandForth properties for extension with objects on either side  restricted to apply only to partial isomorphisms of size ..."
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Cited by 213 (12 self)
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Model Theory. These are nonempty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual BackandForth properties for extension with objects on either side  restricted to apply only to partial isomorphisms of size at most k . 'Invariance for kpartial isomorphism' means having the same truth value at tuples of objects in any two models that are connected by a partial isomorphism in such a set. The precise sense of this is spelt out in the following proof. 21 Proof (Outline.) kvariable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism IÎI which is defined on the Avalues for all variables x 1 , ..., x k , that M, A = f iff N , IoA = f . The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be res...
Fibring of logics as a categorial construction
 Journal of Logic and Computation
, 1999
"... Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the p ..."
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Cited by 51 (31 self)
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Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the prooftheoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both prooftheoretic and modeltheoretic fibring for logics without terms. To this end, we introduce the categories of Hilbert calculi, interpretation systems and logic system presentations. By choosing appropriate notions of morphism it is possible to obtain pure fibring as a coproduct. Fibring with shared symbols is then easily obtained by cocartesian lifting from the category of signatures. Soundness is shown to be preserved by these constructions. We illustrate the constructions within propositional modal logic.
Fibring: Completeness Preservation
 Journal of Symbolic Logic
, 2000
"... A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by bring logics with congruence provided that congruence is retained in the resulting logic. The class of logics ..."
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Cited by 45 (23 self)
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A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by bring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under bring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by bring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by bring logics endowed with standard (non general) semantics. A categorial characterization of bring is provided using coproducts and cocartesian liftings. 1 Introduction Much attention has been recently given to the problems of combining logics and obtaining transference results. Besides leading to very interesting applications whenever it is necessary to work with dierent logics at the same time, ...
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
Satisfiability problem in description logics with modal operators
 IN PROCEEDINGS OF THE SIXTH CONFERENCE ON PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING
, 1998
"... The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logi ..."
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Cited by 40 (21 self)
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The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.
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 36 (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.
Combining Temporal Logic Systems
 Notre Dame Journal of Formal Logic
, 1994
"... This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component onedimensional temporal logics to the resulting twodimensional tempora ..."
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Cited by 29 (2 self)
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This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component onedimensional temporal logics to the resulting twodimensional temporal logic. Three basic logical properties are analysed, namely soundness, completeness and decidability. Each combination method is composed of three submethods that combine the languages, the inference systems and the semantics of two onedimensional temporal logic systems, generating families of twodimensional temporal languages with varying expressivity and varying degree of transference of logical properties. The temporalisation method and the independent combination method are shown to transfer all three basic logical properties. The method of full interlacing of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several...