The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of path-consistency plays a central role. Algorithms for path-consistency can be implemented on matrices of relations and on matrices of elements from a relation algebra. We give an example of a 4-by-4 matrix of infinite relations on which no iterative local path-consistency 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...

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense 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 pair-dense 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.

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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 2-4, 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 CCR-8814921 ...

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 first-order 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 non-trivial intervals over the rational numbers. Algorithms are given for quantifer-elimination, consistency checking, and satisfaction of arbitrary first-order formulas in the Interval Calculus. A natural countable model of the calculus is presented, the TUS , in which clock- and calendar-time may be represented in a straightforward way. Allen an...

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...

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 tied-in with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The model-theoretic 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 first-order logic, modal and temporal logics (either...

We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 ! n ! !. Completely analogous statement holds for the case n !. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...

Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper or...

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 A-networks’. 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 yes-instance. This reduces the tiling problem to the representation problem and proves the latter’s undecidability. 1.