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. 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...

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 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.

We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...

this paper. each frame of the class.) For example, K is the logic of all n-ary product frames. It is not hard to see that S5 is the logic of all n-ary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cubic universal product S5 -frames. Note that the `i-reduct' F U 1 U n ; R i of F 1 F n is a union of n disjoint copies of F i . Thus, F and F i validate the same formulas, and so L n L 1 L n : There is a strong interaction between the modal operators of product logics. Every n-ary product frame satis es the following two properties, for each pair i 6= j, i; j = 1; : : : ; n: Commutativity : 8x8y8z xR i y ^ yR j z ! 9u (xR j u ^ uR i z) ^ xR j y ^ yR i z ! 9u (xR i u ^ uR j z) Church{Rosser property : 8x8y8z xR i y ^ xR j z ! 9u (yR j u ^ zR i u) This means that the corresponding modal interaction formulas 2 i 2 j p $ 2 j 2 i p and 3 i 2 j p ! 2 j 3 i p belong to every n-dimensional product logic. The geometrically intuitive many-dimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatio-temporal representation and reasoning (see e.g. [33, 34]) or reasoning about the behaviour of multi-agent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexity|even the product of two NP-complete logics can be non-recursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non- nitely axiomatisable [16]

Relation algebras have been used for various kinds of temporal reasoning. Typically the network satisfaction problem turns out to be NP-hard. For the Allen interval algebra it is often convenient to use the propagation algorithm. This algorithm is sound and runs in cubic time but it is not complete. Here we define a series of tractable algorithms that provide approximations to solving the network satisfaction problem for any finite relation algebra. For algebras where all 3-consistent atomic networks are satisfiable, like the Allen interval algebra, we can improve these algorithms so that each algorithm runs in cubic time. These algorithms improve on the Allen propagation algorithm and converge on a complete algorithm. Algebras of relations were studied extensively in the nineteenth century and, together with Frege's logic of quantifiers, form the foundation of modern logic (see [Mad91b, AH91] for some of the history). In the twentieth century algebras of binary relations wer...

We define a finite relation algebra and show that the network satisfaction problem is undecidable for this algebra 1 . Keywords: Network satisfaction problem, relation algebra, undecidability, tiling 1 Notation and Definitions Let A = (A; +; \Gamma; 0; 1; 1 0 ;; ; ) be a relation algebra (see [JT52] for the original axiomatisation or [Mad91] for an introduction to relation algebra). ffl An atom a of A is a minimal, non-zero element. At(A) denotes the set of all atoms of A. A is atomic if for all non-zero a 2 A there exists ff 2 At(A) with ff a. In the following we assume A is atomic. ffl The set of forbidden triples of the atomic relation algebra A is the set of all (ff; fi; fl) 2 3 At(A) such that fl : ff; fi = 0. Forb denotes the set of forbidden triples of A. ffl The set of forbidden triples defines composition in an atomic relation algebra, by a; b = \Sigmaffl 2 At(A) : 9ff; fi 2 At(A); ff a; fi b; &(ff; fi; fl) = 2 Forbg ffl If (ff; fi; fl) is a forbidden triple th...

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-nitely axiomatizable quasivariety and that its equational theory is not nitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrications. A similar result is proved for subreducts of the class of representable sequential algebras. 1 Introduction The aim of this paper is to investigate algebras of relations from the nite axiomatizability point of view. In algebraic logic, the most extensively investigated classes of algebras of relations are the class of (representable) relation algebras and the class of (representable) cylindric algebras, cf. [HMT]. These classes are Boolean algebras equipped with some extra-Boolean operations arising from the nature of relations. In this paper ...

Abstract. The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8-element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator. A binar is a set A with a (total) binary operation ·, and in a partial binar this operation is allowed to be partial. We write x · y ∈ A to indicate that the product of x and y exists. A partial semigroup is an associative partial binar, i.e. for all x, y, z ∈ A, if (x · y) · z ∈ A or x · (y · z) ∈ A, then both terms exist and evaluate to the same element of A. Similarly, a commutative partial binar is a binar such that if x · y ∈ A then x · y = y · x ∈ A. Let (P)(C)Bn and (P)(C)Sg denote the class of all (partial) (commutative) groupoids and all (partial) (commutative) semigroups respectively. For A ∈ PBn the complex algebra of A is defined as Cm(A) = 〈P (A), ∪, ∅, ∩, A, \, ·〉, where X · Y = {x · y | x ∈ X, y ∈ Y and x · y exists} is the complex product of X, Y ∈ Cm(A). Algebras of the form Cm(A) are examples of Boolean algebras with a binary operator, i.e., algebras 〈B, ∨, 0, ∧, 1, ¬, ·〉 such that 〈B, ∨, 0, ∧, 1, ¬ 〉 is a Boolean algebra and · is a binary operation that distributes over finite (including empty) joins in each argument. A Boolean semigroup is a Boolean algebra with an associative binary operator. For a class K of algebras, Cm(K) denotes the class of all complex algebras of K, H(K) is the class of all homomorphic images of K, and V(K) is the variety generated by K, i.e., the smallest equationally defined class that contains K. The aim of this note is to contrast the equational theory of Cm((C)Bn) with that of Cm((C)Sg). It turns out that the former is finitely based while the latter is not.