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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 61 (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.
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 38 (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.
Some Algebras And Algorithms For Reasoning About Time And Space
, 1990
"... Constraint networks over relation algebras are defined. Compass algebras are introduced for reasoning about space. They are related to the interval algebras, which are used for reasoning about time. The problem of determining whether a network has a closed zeroless reduction is shown to NPcomplete ..."
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Cited by 7 (0 self)
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Constraint networks over relation algebras are defined. Compass algebras are introduced for reasoning about space. They are related to the interval algebras, which are used for reasoning about time. The problem of determining whether a network has a closed zeroless reduction is shown to NPcomplete for almost all compass and interval algebras. This implies constraint satisfaction for these algebras is NPcomplete. x1.
Representations for Small Relation Algebras
 Notre Dame Journal of Formal Logic
, 1994
"... There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations. ..."
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Cited by 6 (2 self)
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There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations.
Relation algebras for reasoning about time and space
 Algebraic Methodology and Software Technology, Enschede 1993, Workshops in Computing Series
, 1994
"... This paper presents a brief introduction to relation algebras, including some examples motivated by work in computer science, namely, the ‘interval algebras’, relation algebras that arose from James Allen’s work on temporal reasoning, and by ‘compass algebras’, which are designed for similar reasoni ..."
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Cited by 5 (0 self)
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This paper presents a brief introduction to relation algebras, including some examples motivated by work in computer science, namely, the ‘interval algebras’, relation algebras that arose from James Allen’s work on temporal reasoning, and by ‘compass algebras’, which are designed for similar reasoning about space. One kind of reasoning problem, called a ‘constraint satisfaction problem’, can be defined for arbitrary relation algebras. It will be shown here that the constraint satisfiability problem is NPcomplete for almost all compass and interval algebras.
Algebraic Polymodal Logic: A Survey
 LOGIC JOURNAL OF THE IGPL
, 2000
"... This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with ..."
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Cited by 2 (0 self)
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This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with
Relation algebras and their application in qualitative spatial reasoning
, 2005
"... www.cosc.brocku.ca Relation algebras and their application in temporal and spatial reasoning ..."
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www.cosc.brocku.ca Relation algebras and their application in temporal and spatial reasoning
Representable sequential algebras and observation spaces
"... Abstract. We define the concepts of representable and abstract sequential Qalgebra, which are generalizations of the (relational) Qalgebras in [10]. Just as in that paper, we then prove that the two concepts coincide. In the following section we recall the concept of observation space and note tha ..."
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Abstract. We define the concepts of representable and abstract sequential Qalgebra, which are generalizations of the (relational) Qalgebras in [10]. Just as in that paper, we then prove that the two concepts coincide. In the following section we recall the concept of observation space and note that all complex algebras of observation spaces are representable sequential algebras. Finally we give an uncountable family of representable sequential algebras that generate distinct minimal varieties (i.e. covers of the variety of oneelement algebras). 1