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68
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 60 (20 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.
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 34 (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.
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 31 (5 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.
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...
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 25 (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.
Logical aspects of set constraints
 in Proc. 1993 Conf. Computer Science Logic
, 1993
"... Abstract. Set constraints are inclusion relations between sets of ground terms over a ranked alphabet. They have been used extensively in program analysis and type inference. Here we present an equational axiomatization of the algebra of set constraints. Models of these axioms are called termset alg ..."
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Cited by 25 (4 self)
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Abstract. Set constraints are inclusion relations between sets of ground terms over a ranked alphabet. They have been used extensively in program analysis and type inference. Here we present an equational axiomatization of the algebra of set constraints. Models of these axioms are called termset algebras. They are related to the Boolean algebras with operators of Jonsson and Tarski. We also de ne a family of combinatorial models called topological term automata, which are essentially the term automata studied by Kozen,Palsberg, and Schwartzbach endowed with a topology such that all relevant operations are continuous. These models are similar to Kripke frames for modal or dynamic logic. We establish a Stone duality between termset algebras and topological term automata, and use this to derive a completeness theorem for a related multidimensional modal logic. Finally, weproveasmall model property by ltration, and argue that this result contains the essence of several algorithms appearing in the literature on set constraints. 1
Complexity of Modal Logics of Relations
, 1997
"... We consider two families of modal logics of relations: arrow logic and cylindric modal logic and several natural expansions of these, interpreted on a range of (relativised) modelclasses. We give a systematic study of the complexity of the validity problem of these logics, obtaining price tags for ..."
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Cited by 24 (9 self)
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We consider two families of modal logics of relations: arrow logic and cylindric modal logic and several natural expansions of these, interpreted on a range of (relativised) modelclasses. We give a systematic study of the complexity of the validity problem of these logics, obtaining price tags for various features as assumptions on the universe of the models, similarity types, and number of variables involved. The general picture is that the process of relativisation turns an undecidable logic into one whose validity problem is exptimecomplete. There are interesting deviations to this though, which we also discuss. The numerous results in this paper are all directed to obtain a better understanding why relativisation can turn an undecidable modal logic of relations into a decidable one. We connect the semantic way of "taming logic" by relativisation with the syntactic approach of isolating decidable socalled guarded fragments by showing that validity of loosely guarded formulas is p...
On the Products of Linear Modal Logics
 JOURNAL OF LOGIC AND COMPUTATION
, 2001
"... We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz: ..."
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Cited by 24 (9 self)
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We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using nonstructural Gabbaytype inference rules.
Modal Logic, Transition Systems and Processes
, 1994
"... Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be mo ..."
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Cited by 22 (3 self)
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Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be motivated by operations on transition systems, but they can also be suggested by investigations of modal formalisms. To show that the equational view of processes from process algebra is closely related to modal logic, we consider various ways of looking at the relation between the calculus of basic process algebra and propositional dynamic logic. More concretely, the paper contains preservation results for various bisimulation notions, a result on the expressive power of propositional dynamic logic, and a definition of bisimulation which is the proper notion of invariance for concurrent propositional dynamic logic. Keywords: modal logic, transition systems, bisimulation, process algebra 1 In...