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54
On Binary Constraint Problems
 Journal of the ACM
, 1994
"... The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of pathconsistency plays a central role. Algorithms for pathconsistency can be implemented on matrices of relations and on matrices of elements from a relation algeb ..."
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Cited by 87 (2 self)
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The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of pathconsistency plays a central role. Algorithms for pathconsistency can be implemented on matrices of relations and on matrices of elements from a relation algebra. We give an example of a 4by4 matrix of infinite relations on which no iterative local pathconsistency algorithm terminates. We give a class of examples over a fixed finite algebra on which all iterative local algorithms, whether parallel or sequential, must take quadratic time. Specific relation algebras arising from interval constraint problems are also studied: the Interval Algebra, the Point Algebra, and the Containment Algebra. 1 Introduction The logical study of binary relations is classical [8], [9], [51], [52], [56], [53], [54]. Following this tradition, Tarski formulated the theory of binary relations as an algebraic theory called relation algebra [59] 1 . Constraint satis...
PairDense Relation Algebras
 Transactions of the American Mathematical Society
, 1991
"... The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big ..."
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Cited by 62 (8 self)
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The central result of this paper is that every pairdense relation algebra is completely representable. A relation algebra is said to be pairdense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form fha; ai ; hb; big (with a = b allowed). In a simple pairdense relation algebra, every pair is either a "point" (an algebraic analogue of fha; aig) or a "twin" (a pair which contains no point). In fact, every simple pairdense relation algebra A is completely representable over a set U iff jU j = + 2, where is the number of points of A and is the number of twins of A.
The Second Calculus of Binary Relations
 In Proceedings of MFCS'93
, 1993
"... We view the Chu space interpretation of linear logic as an alternative interpretation of the language of the Peirce calculus of binary relations. Chu spaces amount to Kvalued binary relations, which for K = 2 n we show generalize nary relational structures. We also exhibit a fourstage unique fa ..."
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Cited by 55 (18 self)
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We view the Chu space interpretation of linear logic as an alternative interpretation of the language of the Peirce calculus of binary relations. Chu spaces amount to Kvalued binary relations, which for K = 2 n we show generalize nary relational structures. We also exhibit a fourstage unique factorization system for Chu transforms that illuminates their operation. 1 Introduction In 1860 A. De Morgan [DM60] introduced a calculus of binary relations equivalent in expressive power to one whose formulas, written in today's notation, are inequalities a b between terms a; b; . . . built up from variables with the operations of composition a; b, converse a, and complement a \Gamma . In 1870 C.S. Peirce [Pei33] extended De Morgan's calculus with Boolean connectives a + b and ab, Boolean constants 0 and 1, and an identity 1 0 for composition. In 1895 E. Schroder devoted a book [Sch95] to the calculus, and further extended it with the operations of reflexive transitive closure, a ...
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
"... ..."
On Binary Constraint Networks
, 1988
"... It is wellknown that general constraint satisfaction problems (CSPs) may be reduced to the binary case (BCSPs) [Pei92]. CSPs may be represented by binary constraint networks (BCNs), which can be represented by a graph with nodes for variables for which values are to be found in the domain of intere ..."
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Cited by 39 (5 self)
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It is wellknown that general constraint satisfaction problems (CSPs) may be reduced to the binary case (BCSPs) [Pei92]. CSPs may be represented by binary constraint networks (BCNs), which can be represented by a graph with nodes for variables for which values are to be found in the domain of interest, and edges labelled with binary relations between the values, which constrain the choice of solutions to those which satisfy the relations (e.g. [Mac77]). We formulate networks and algorithms in a general algebraic setting, that of Tarski's relation algebra [JonTar52], and obtain a parallel O(n log n) upper bound for pathconsistency, and give a class of examples on which reductiontype algorithms (which include the standard serial algorithms [Mac77, MacFre85, MohHen86] and all possible parallelisations of them) are O(n ). We then consider BCNs over various classes of relations that arise from an underlying linearly ordered set, the most wellknown being the interval algebra [All83, LadMad88.1]. There are three main consequences of the algebraic approach. Firstly, it puts the theory of BCNs on a firm (and classical) theoretical footing, enabling, for example, the complexity results. Secondly, we can apply techniques from relation algebra to show that consistency checking for a large class of relations on intervals ([All83]) is serial cubic, or parallel log time, significantly extending previous results (the problem is NPhard in general [VilKau86]). Thirdly, results are obtained via a new construction of relation algebras from other algebras which is of independent mathematical interest.
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
Relation algebras in qualitative spatial reasoning
 Fundamenta Informaticae
, 1999
"... The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various ..."
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Cited by 34 (13 self)
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The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra. 1
The Logic of Time Representation
, 1987
"... This investigation concerns representations of time by means of intervals, stemming from work of Allen [All83] and van Benthem [vBen83]. Allen described an Interval Calculus of thirteen binary relations on convex intervals over a linear order (the real numbers). He gave a practical algorithm for che ..."
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Cited by 29 (1 self)
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This investigation concerns representations of time by means of intervals, stemming from work of Allen [All83] and van Benthem [vBen83]. Allen described an Interval Calculus of thirteen binary relations on convex intervals over a linear order (the real numbers). He gave a practical algorithm for checking the consistency of a subclass of Boolean constraints. First, we describe a completeness theorem for Allen's calculus, in its corresponding formulation as a firstorder theory LM . LM is countably categorical, and axiomatises the complete theory of intervals over a dense unbounded linear order. Its only countable model up to isomorphism is the nontrivial intervals over the rational numbers. Algorithms are given for quantiferelimination, consistency checking, and satisfaction of arbitrary firstorder formulas in the Interval Calculus. A natural countable model of the calculus is presented, the TUS , in which clock and calendartime may be represented in a straightforward way. Allen an...
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...
Introductory Course on Relation Algebras, FiniteDimensional Cylindric Algebras, and Their Interconnections
 Algebraic Logic
, 1990
"... These are notes for a short course on relation algebras, finitedimensional cylindric algebras, and their interconnections, delivered at the Conference on Algebraic Logic, Budapest, Hungary, August 814, 1988, sponsored by the the Janos Bolyai Mathematical Society. ..."
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Cited by 24 (3 self)
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These are notes for a short course on relation algebras, finitedimensional cylindric algebras, and their interconnections, delivered at the Conference on Algebraic Logic, Budapest, Hungary, August 814, 1988, sponsored by the the Janos Bolyai Mathematical Society.