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...

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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...

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set. 1 Introduction In this paper, we give a simple proof that certain classes K of algebras have the `finite base property'. This will imply decidability of the universal theory of K, and that any finite algebra in K is representable on a finite set. Examples of such K include the relativized cylindric set algebras in dimension n (Crs n ), polyadic Crs, and the weakly associative relation algebras WA. Most of these results were first established in the paper [ABN2]; the original proofs were substantially longer than the present one. What is the finite base property? Say that we are given a class K of concrete algebras. This is to say that the algebras i...

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.

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig. We point out some consequences in temporal predicate logic and algebraic logic. AMS classification: Primary 03B20; Secondary 03B45, 03C07, 03C13, 03C30, 03G15 Keywords: finite structures, modal logic, modal fragment, packed fragment 1 Introduction Perhaps because beginning students of modal logic are often told that modal logic is more expressive than first-order logic and indeed has some second-order expressive power, or perhaps because they are hoping for something new, it can come as a surprise to them that every modal formula has a `standard translation' into first-order logic. For example, (p !q) is translated to 9y(R(x;y) ^ (P(y) ! 8z(R(y;z) ! Q(z)))): (1) The translation mimics the Kripke semantics for modal logic. Not every first-order formula (with one free variable in the appropriate signature) is the translation of a modal formu...

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.

We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the Region Connection Calculus [33], and we show how to interpret these relations in the collection of regular open sets in the two-dimensional Euclidean plane. 1 Introduction Mereotopology is an area of qualitative spatial reasoning (QSR) which aims to develop formalisms for reasoning about spatial entities [1, 12, 30, 31]. The structures used in mereotopology consist of three parts: 1. A relational (or mereological) part, 2. An algebraic part, 3. A topological part. The algebraic part is often an atomless Boolean algebra, or, more generally, an orthocomplemented lattice, both without smallest element. Due to the presence of the binary relations "part-of" and "contact" in the relational part of mereotopology, composition based reasoning with binary relations has been of interest to the QSR community, and the expressive power, consistency and complexity o...

We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.

We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of non-associative 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.