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102
Temporalizing description logics
, 1998
"... Traditional rst order predicate logic is known to be designed for representing and manipulating static knowledge (e.g. mathematical theories). So are manyof its applications. Knowledge representation systems based on concept description logics are not exceptions. ..."
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Cited by 78 (19 self)
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Traditional rst order predicate logic is known to be designed for representing and manipulating static knowledge (e.g. mathematical theories). So are manyof its applications. Knowledge representation systems based on concept description logics are not exceptions.
Hybrid Logics
"... This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur ..."
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Cited by 62 (18 self)
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This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur
Derivation rules as antiaxioms in modal logic
 Journal of Symbolic Logic
, 1993
"... Abstract. We discuss a ‘negative ’ way of defining frame classes in (multi)modal logic, and address the question whether these classes can be axiomatized by derivation rules, the ‘nonξ rules’, styled after Gabbay’s Irreflexivity Rule. The main result of this paper is a metatheorem on completeness ..."
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Cited by 45 (3 self)
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Abstract. We discuss a ‘negative ’ way of defining frame classes in (multi)modal logic, and address the question whether these classes can be axiomatized by derivation rules, the ‘nonξ rules’, styled after Gabbay’s Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas, and Λ+ is the extension of Λ with a set of nonξ rules, then Λ+ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.
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 41 (11 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.
A modal walk through space
 JOURNAL OF APPLIED NONCLASSICAL LOGICS
, 2002
"... We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and ..."
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Cited by 40 (6 self)
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We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.
A MultiDimensional Terminological Knowledge Representation Language
, 1995
"... An extension of the concept description language ALC used in klonelike terminological reasoning is presented. The extension includes multimodal operators that can either stand for the usual role quantifications or for modalities such as belief, time etc. The modal operators can be used at all lev ..."
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Cited by 36 (2 self)
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An extension of the concept description language ALC used in klonelike terminological reasoning is presented. The extension includes multimodal operators that can either stand for the usual role quantifications or for modalities such as belief, time etc. The modal operators can be used at all levels of the concept terms, and they can be used to modify both concepts and roles. This is an instance of a new kind of combination of modal logics where the modal operators of one logic may operate directly on the operators of the other logic. Different versions of this logic are investigated and various results about decidability and undecidability are presented. The main problem, however, decidability of the basic version of the logic, remains open.
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 32 (17 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...
Peirce Algebras
, 1992
"... We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming o ..."
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Cited by 28 (10 self)
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We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a setforming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the socalled terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.
Y.: Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Studia Logica 54
, 1995
"... with an Application ..."
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