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16
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 98 (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 ..."
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Cited by 83 (9 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 41 (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.
Decompositions in quantum logic
 Transactions of the AMS
, 1996
"... In 1996, Harding showed that the binary decompositions of any algebraic, relational, or topological structure X form an orthomodular poset Fact X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompo ..."
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Cited by 17 (9 self)
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In 1996, Harding showed that the binary decompositions of any algebraic, relational, or topological structure X form an orthomodular poset Fact X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompositions in Fact X is compatible if and only if all the binary decompositions in S can be built from a common nary decomposition of X. This characterization of compatibility is used to show that for any algebraic, relational, or topological structure X, the orthomodular poset Fact X is regular. Special cases of this result include the known facts that the orthomodular posets of splitting subspaces of an inner product space are regular, and that the orthomodular posets constructed from the idempotents of a ring are regular. This result also establishes the regularity of the orthomodular posets that Mushtari constructs from bounded modular lattices, the orthomodular posets one constructs from the subgroups of a group, and the orthomodular posets one constructs from a normed group with operators. Moreover, all these orthomodular posets are regular for the same reason. The characterization of compatibility is also used to show that for any structure X, the finite Boolean subalgebras of Fact X correspond to finitary direct product decompositions of the structure X. For algebraic and relational structures X, this result is extended to show that the Boolean subalgebras of Fact X correspond to representations of the structure X as the global sections of a sheaf of structures over a Boolean space. The above results can be given a physical interpretation as well. Assume that the true or false questions 4 of a quantum mechanical system correspond to binary direct product decompositions of the state space of the system, as is the case with the usual von Neumann interpretation of quantum mechanics. Suppose S is a subset of 4. Then a necessary and sufficient condition that all questions in S can be answered simultaneously is that any two questions in S can be answered simultaneously. Thus, regularity in quantum mechanics follows from the assumption that questions correspond to decompositions. 1.
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
"... Abstract 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. 1Introduction We say that a relation algebra is small if it has no more ..."
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Cited by 6 (2 self)
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Abstract 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. 1Introduction We say that a relation algebra is small if it has no more than eight elements. A relation algebra is a Boolean algebra with additional operators, so every small relation algebra has cardinality 1, 2, 4, or 8. There are eighteen isomorphism types of small relation algebras. One of the types contains oneelement algebras, thirteen of them contain simple algebras, and the remaining four contain direct products of simple relation algebras. Asimple or oneelement relation algebra A is representable if it is isomorphic to a subalgebra of ReU, for some set U, where ReU = 〈 ReU, ∪, ∼, , −1 〉, IdU is the relation algebra of all binary relations on the set U. A representation of A on U is an isomorphism that embeds A into ReU. Direct products of representable relation algebras are also representable. It has long been known that every simple small relation algebra is representable. Therefore all small relation algebras are representable.
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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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