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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...
Relation Algebras of Intervals
 ARTIFICIAL INTELLIGENCE
, 1994
"... Given a representation of a relation algebra we construct relation algebras of pairs and of intervals. If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original rel ..."
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Cited by 15 (3 self)
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Given a representation of a relation algebra we construct relation algebras of pairs and of intervals. If the representation happens to be complete, homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra. If, further, the original relation algebra is !categorical we show that the interval algebra is too. The complexity of relation algebras is studied and it is shown that every pair algebra with infinite representations is intractable. Applications include constructing an interval algebra that combines metric and interval expressivity.
Completely Representable Relation Algebras
 Bull. IGPL
, 1995
"... A boolean algebra is shown to be completely representable if and only if it is atomic whereas it is shown that the class of completely representable relation algebras is not elementary. 1 Introduction There are two types of representation in algebraic logic: ordinary and complete representations. O ..."
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Cited by 2 (2 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 the class of completely representable relation algebras is not elementary. 1 Introduction There are two types of representation in algebraic logic: ordinary and complete representations. Ordinary representations, or just representations, have been studied extensively [J'onsson and Tarski1948, Lyndon1950, Lyndon1956, McKenzie1970, Maddux1978, Maddux1982, Henkin et al.1985, Andr'eka et al.1991, Venema1992, Monk1993] and are isomorphisms from a boolean algebra with operators to a more concrete structure where the boolean operators and \Gamma are replaced by [ and n, and the other operators have certain settheoretically definable interpretations. For example, in relation algebra the binary operation ; gets interpreted as composition of binary relations. Complete representations have the additional property that they preserve arbitrary unions (hence intersections too), wherever...
A Decision Procedure for MultiModal Logics Specified with Relational Algebra Axioms
, 2001
"... Normal Modal Logics can be characterized either by Hilbert axioms or by the properties of the accessibility relations. The properties of the accessibility relations can for example be expressed in predicate logic. They can, however, also be expressed with Relational Algebra formulae. In this pap ..."
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Normal Modal Logics can be characterized either by Hilbert axioms or by the properties of the accessibility relations. The properties of the accessibility relations can for example be expressed in predicate logic. They can, however, also be expressed with Relational Algebra formulae. In this paper it is shown that satisability of a propositional multimodal logic with Relational Algebra terms as modal parameters is decidable if the properties of the accessibility relations can be expressed with Relational Algebra formulae. A tableaux decision procedure is presented which takes as input the formula to be checked together with the Relational Algebra axioms which specify the frame class. It is shown that the satisability problem in this general form is NEXPTIMEcomplete. 1