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.

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. 3 De nition An (abstract) relation algebra A (RA) is a tuple (A; _; ?; 0; 1; ; ;^; Id) which satises the following axioms, essentially due to Tarski. For all a; b; c 2 A, 1. (A; _; ?; 0; 1) is a Boolean algebra (1 is the universal element) so we can introduce^; as the usual abbreviations 2. ; is an associative binary operator on A 3. (a^)^= a 4. Id; a = a; Id = a 5. a; (b _ c) = a; b _ a; c 6. (a _ b)^= a^_ b7 . (a ? b)^= a^? b8 . (a; b)^= b^; a9 . (a; b)^c^= 0 , (b; c)^a^= 0 triangle axiom]. An atom of A is a minimal non-zero element under the ordering . The set of all atoms of A is denoted At(A).Note that ? here is a unary operation, but we can de ne the binary operation a?b def = ?(b_?a). In a PRA the operation ? is interpreted as complement relative to the top element 1. These axioms are clearly sound over PRA, but they turn out not to be complete Lyn50] | there are (even nite) relation algebras which are not isomorphic to any proper relation algebra. Let us de ne a representation (X; D) of A to be an isomorphism X from A to some proper relation algebra P with domain D i.e. a bijection from the elements of A to the binary relations in P over D and X must respect all the operations. For any representation, it is always the case that X(1) is an equivalence relation over D. This follows from the equations Id 1; 1^= 1 and 1; 1 = 1 which are consequences of axioms 1 to 9. If X(1) = D D then call (X; D) a square representation. Lyndon's result shows that not every relation algebra is representable. It has been shown Mon64] that no nite set of axioms can be sound and complete over PRA.Let (X; D); (Y; E) be representations of a relation algebra A. A base-isomorphism h : (X; D) ! (Y; E) is a bijection from D to E preserving the relations i.e. for all d; d 0 2 D, for all a 2 A, (d; d 0 ) 2 X(a) if and only if (h(d); h(d 0 )) 2 Y (a).Notation As is standard in model theory (though perhaps not in algebraic logic) we reduce notational clutter by letting the same symbol X stand for the map, the domain of a representation and the name of the representation itself. Thus x 2 X means that x is a point in the domain of the representation and for a 2 A, X(a) is the binary relation corresponding to a. These di erent uses are to be interpreted according to their context. If x is an n-tuple of elements from the represen...

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 characterized 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. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-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 two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.Other instances of this approach are looked at, and include the step by step method.

No abstract

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.