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22
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 conditions ..."
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Cited by 82 (6 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.
Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 28 (15 self)
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Complete Representations in Algebraic Logic
 JOURNAL OF SYMBOLIC LOGIC
"... A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary. ..."
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Cited by 19 (8 self)
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A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
COMPLEXITY OF EQUATIONS VALID IN ALGEBRAS OF RELATIONS  Part II: Finite axiomatizations.
"... We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known exam ..."
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Cited by 17 (2 self)
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We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of nary relations, RPEAn of polyadic equality algebras of nary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 ! n ! !. Completely analogous statement holds for the case n !. This improves Monk's famous nonfinitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...
A Note on Graded Modal Logic
 STUDIA LOGICA
, 2000
"... We introduce a notion of bisimulation for graded modal logic. Using these bisimulations the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property, and proving invariance and definability results. ..."
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Cited by 12 (0 self)
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We introduce a notion of bisimulation for graded modal logic. Using these bisimulations the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property, and proving invariance and definability results.
Monotonic Modal Logics
, 2003
"... Monotonic modal logics form a generalization of normal modal logics... ..."
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Cited by 11 (0 self)
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Monotonic modal logics form a generalization of normal modal logics...
Beth Definability in the Guarded Fragment
, 1999
"... The guarded fragment (GF) was introduced in [1] as a fragment 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. While GF has been established as a particul ..."
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Cited by 10 (0 self)
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The guarded fragment (GF) was introduced in [1] as a fragment 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. While GF has been established as a particularly wellbehaved fragment of first order logic in many respects, interpolation fails in restriction to GF, [9]. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics. ¿From this we obtain that, more specifically, even every nvariable guarded fragment with up to nary
Relativized Relation Algebras
 Journal of Symbolic Logic
, 1999
"... Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by rel ..."
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Cited by 6 (2 self)
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Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which is relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. In algebraic logic, relativized cylindric and relation algebras have been studied relatively deeply (cf. e.g., Henkin et al. (Henkin et al., 1981), Maddux (Maddux, 1982) and Resek Thompson (Resek and Thompson, 1991)). The emphasis, however was different from the perspective we will take here. As the name "relativization" indicates, the non...
Modal Logic and nonwellfounded Set Theory: translation, bisimulation, interpolation.
, 1998
"... Contents Acknowledgments vii 1 Introduction. 1 2 General preliminaries. 7 2.1 Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Languages and structures. . . . . . . . . . . . . . . . . . . 7 2.1.2 Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 ..."
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Cited by 6 (1 self)
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Contents Acknowledgments vii 1 Introduction. 1 2 General preliminaries. 7 2.1 Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Languages and structures. . . . . . . . . . . . . . . . . . . 7 2.1.2 Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Semantics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.4 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.5 Derivability in Basic Modal Logic. . . . . . . . . . . . . . . 10 2.2 Bisimulation and the like. . . . . . . . . . . . . . . . . . . . . . . 12 2.3 A brief introduction to the Calculus. . . . . . . . . . . . . . . . 17 2.4 The family of graded modal logics and their semantics. . . . . . . 22 2.5 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Uniform interpolation. . . . . . . . . . . . . . . . . . . . . 29 2.5.2 Elementary interpolation. . . . . . . . . . . . . . . . . . . 30 2.6 N