Results 1  10
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14
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...
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...
Complete Representations in Algebraic Logic
 JOURNAL OF SYMBOLIC LOGIC
"... A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary. ..."
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Cited by 19 (8 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 neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
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.
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.
On the Search for a Finitizable Algebraization of First Order Logic
, 2000
"... We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of thi ..."
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Cited by 9 (1 self)
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We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 (a concrete version of which is given by Theorems 2.8 and 4.2), while its logical form is Corollary 5.2. For first order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case  instead of finitely axiomatizing the corresponding class of all representable algebras  we finitely axiomatize only the equational theory of that class. See Subsection 6.1, especially Remark 6.6 there. The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also "Further main results" in the
Relation Algebras from Cylindric Algebras, I
, 1999
"... We characterise the class SRaCAn of subalgebras of relation algebra reducts of ndimensional cylindric algebras (for finite n 5) by the notion of a `hyperbasis', analogous to the cylindric basis of Maddux, and by relativised representations. A corollary is that SRaCAn = SRa(CAn " Crs n ) = SRa(CAn ..."
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Cited by 8 (6 self)
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We characterise the class SRaCAn of subalgebras of relation algebra reducts of ndimensional cylindric algebras (for finite n 5) by the notion of a `hyperbasis', analogous to the cylindric basis of Maddux, and by relativised representations. A corollary is that SRaCAn = SRa(CAn " Crs n ) = SRa(CAn " Gn ). We outline a gametheoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of SRaCAn .
Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory
, 1998
"... We confirm a conjecture about neat embeddings of cylindric algebras made in 1969 by J. D. Monk, confirm a later conjecture by Maddux about relation algebras obtained from cylindric algebras, and solve a problem raised by Tarski and Givant. These results in algebraic logic have the following conseque ..."
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Cited by 6 (2 self)
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We confirm a conjecture about neat embeddings of cylindric algebras made in 1969 by J. D. Monk, confirm a later conjecture by Maddux about relation algebras obtained from cylindric algebras, and solve a problem raised by Tarski and Givant. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal ff 3 there is a logically valid sentence ', in a firstorder language with equality and exactly one nonlogical binary relation symbol, such that ' contains only 3 variables (each of which can occur arbitrarily many times), ' has a proof containing exactly ff+1 variables, but no proof containing only ff variables. 1 Introduction and summary Cylindric algebras are an algebraisation of the theory of ffary relations. For the special case of binary relations, there is an alternative approach to algebraisation, using relation algebras. This paper is concerned with the connection between the two approaches and the consequences for ffvariable p...
NonRepresentable Algebras of Relations
, 1997
"... this dissertation. More precisely, we are referring to what is called the orthodox version of these logics in these works ..."
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Cited by 4 (0 self)
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this dissertation. More precisely, we are referring to what is called the orthodox version of these logics in these works
Provability with Finitely Many Variables
"... For every finite n 4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); 'n has a proof in firs ..."
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Cited by 3 (1 self)
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For every finite n 4 there is a logically valid sentence 'n with the following properties: 'n contains only 3 variables (each of which occurs many times); 'n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); 'n has a proof in firstorder logic with equality that contains exactly n variables, but no proof containing only n \Gamma 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) firstorder binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that 'n has a proof with only n variables. To show that 'n has no proof with only n \Gamma 1 variables we use alternative semantics in place of the usual, standard, settheoretical semantics of firstorder logic.