Axiomatizability of representable domain algebras

Abstract

The family of domain algebras provide an elegant formal system for automated reasoning about programme verification. Their primary models are algebras of relations, viz. representable domain algebras. We prove that, even for the minimal signature consisting of the domain and composition operations, the class of representable domain algebras is not finitely axiomatizable. Then we show similar results for extended similarity types of domain algebras.

Domain and range for angelic and demonic compositions

“…Completeness of these axioms seems more elusive. Work involving the present authors showed that no finite system of axioms is sufficient to capture the full first order theory of the algebra of relations under composition with domain and/or range (amongst other operations) [20,25]. This fact is just one of a swathe of negative results relating to the theory of binary relations.…”

“…Thus axiomatising the representation classes may require different methods. Note that the classes R(;, D, R) and R(;, D, R, +) of algebras of binary relations in fact have no finite axiomatisation [20,25]. On the other hand, the class of partial maps under composition, domain and range has a finite axiomatisation [34].…”

Monoids with tests and the algebra of possibly non-halting programs

“…We now demonstrate that no system of equational axioms can replace law (18) in either the definition of twisted agreeable monoids with tests, or restriction monoids with tests. The arguments are slightly complicated by the fact that the systems we are considering are two-sorted, with the test sort embedded as a subset of the other elements.…”

“…Many authors have investigated algebraic foundations for facets of the theory of computer programs: amongst others we have in mind are sum-ordered (partial) semirings (Manes and Benson [29]), dynamic algebras (Pratt and others, see [34]), Kleene algebras with tests (KAT) (Kozen [25]), Kleene algebra with domain (KAD; Desharnais, Möller and Struth [9]; see also Hirsch and Mikulás [18] and Desharnais, Jipsen and Struth [8]), modal semirings (Möller and Struth [32]), refinement algebras (von Wright [40]), correctness algebras (Guttmann [16]), as well as the authors' own contributions such as modal restriction semigroups [22]. Approaches based on the full Tarski algebra of relations are detailed in Maddux [27].…”

Ordered domain algebras

“…Domain algebras provide an elegant, one-sorted formalism for automated reasoning about program and system verification, see [DS08a,DS08b] and [HM11] for details and further motivation. The algebraic behaviour of domain algebras have been investigated, e.g.…”