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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...

These are notes for a short course on relation algebras, finite-dimensional cylindric algebras, and their interconnections, delivered at the Conference on Algebraic Logic, Budapest, Hungary, August 8--14, 1988, sponsored by the the Janos Bolyai Mathematical Society.

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.

Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to deterministic programs. In the second case one is only interested in functional correspondences, since it does not make sense for a pointer to point to two di#erent objects. We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent in the case where the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras. 1

We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to binary relations what boolean algebras are to unary ones. They are used in artificial intelligence, where, for example, the Allen--Koomen temporal planning system checks the consistency of given relations between time intervals. In mathematics, they form a part of algebraic logic. The history of this goes back to the nineteenth century, the early workers including Boole, de Morgan, Peirce, and Schroder; it was studied intensively by Tarski's group (including, at various times, Chin, Givant, Henkin, J'onsson, Lyndon, Maddux, Monk, N'emeti) from around the 1950s, and currently we know of active groups in Amsterdam, Budapest, Rio de Janeiro, South Africa, and the U.S., among other places. Abstract...

The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other non-relational calculi. The graph

ABSTRACT. We investigate in an algebraic setting the question in which logical languages the properties integral, permutational, and rigid of algebras of relations can be expressed. 1.

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 first-order 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) first-order 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, set-theoretical semantics of first-order logic.

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