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30
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...
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculu ..."
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Cited by 38 (16 self)
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Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
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...
Cutfree Display Calculi for Relation Algebras
, 1997
"... . We extend Belnap's Display Logic to give a cutfree Gentzenstyle calculus for relation algebras. The calculus gives many axiomatic extensions of relation algebras by the addition of further structural rules. It also appears to be the first purely propositional Gentzenstyle calculus for relation ..."
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Cited by 21 (14 self)
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. We extend Belnap's Display Logic to give a cutfree Gentzenstyle calculus for relation algebras. The calculus gives many axiomatic extensions of relation algebras by the addition of further structural rules. It also appears to be the first purely propositional Gentzenstyle calculus for relation algebras. 1 Introduction Given a nonempty set U , the universal relation U \Theta U is the set of all ordered pairs (a; b) where a 2 U and b 2 U . Any subset of U \Theta U is a binary relation over U , and the set of all subsets of U \Theta U is the set of all binary relations over U . Thus any two binary relations R and S are each just a set of ordered pairs, and we can use the settheoretic operations of complement, intersection and union to build other relations. The identity relation is f(a; a) j a 2 Ug while the "relative" analogues of complement, intersection and union are converse (` R) = f(b; a) j (a; b) 2 Rg, composition (R ffi S) = f(a; b) j 9c; (a; c) 2 R and (c; b) 2 Sg and ...
Expressive Power and Complexity in Algebraic Logic
 Journal of Logic and Computation
, 1997
"... Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four ..."
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Cited by 20 (2 self)
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Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tiedin with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The modeltheoretic notion of interpretation is used to generalise known complexity results to a range of other algebraic logics. In particular a number of relation algebras are shown to have intractable network satisfaction problems. 1 Introduction A basic problem in theoretical computing and applied logic is to select and evaluate the ideal formalism to represent and reason about a given application. Many different formalisms are adopted: classical firstorder logic, modal and temporal logics (either...
Representability is not decidable for finite relation algebras
 Trans. Amer. Math. Soc
, 1999
"... Abstract. We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra A is determined by playing a certain two player game G(A) over ‘atomic Anetworks’. It can be shown that the second player in this game has a w ..."
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Cited by 15 (6 self)
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Abstract. We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra A is determined by playing a certain two player game G(A) over ‘atomic Anetworks’. It can be shown that the second player in this game has a winning strategy if and only if A is representable. Let τ be a finite set of square tiles, where each edge of each tile has a colour. Suppose τ includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile T ∈ τ there is a tiling of the plane Z × Z using only tiles from τ in which edge colours of adjacent tiles match and with T placed at (0, 0)? It is not hard to show that this problem is undecidable. From an instance of this tiling problem τ, we construct a finite relation algebra RA(τ) and show that the second player has a winning strategy in G(RA(τ)) if and only if τ is a yesinstance. This reduces the tiling problem to the representation problem and proves the latter’s undecidability. 1.
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.
Container Types Categorically
, 2000
"... A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a ..."
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Cited by 12 (0 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...
An Algebraic Formalization of Fuzzy Relations
 Fuzzy Sets and Systems 101
, 1995
"... This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by G. Schmidt and T. Strohlein. Unlike Boolean ..."
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Cited by 9 (5 self)
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This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by G. Schmidt and T. Strohlein. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semiscalar multiplication. First we present a set of axioms for fuzzy relation algebras and improve the definition of point relations. Then by using relational calculus a representation theorem for such relation algebras is deduced without Tarski rule. Keywords : fuzzy relations, relation algebras, relational calculus, representation theorem. 1 Introduction Since Zadeh's invention the concept of fuzzy sets has been extensively investigated in mathematics, science and engineering. The notion of fuzzy relations is also a basic one in processing fuzzy information in relational structures, see e.g. Pedrycz [9]. Gogue...
Relation Algebras with nDimensional Relational Bases
 Annals of Pure and Applied Logic
, 1999
"... We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , ..."
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Cited by 9 (2 self)
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We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , and use it to give an explicit (hence recursive) equational axiomatisation of RAn , and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete representation. Then we prove that whenever 4 n < l !, RA l is not nitely axiomatisable over RAn . This con rms a conjecture of Maddux. We also prove that Bn is elementary for n = 3; 4 only.