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

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and well-foundedness; second, an algebraic reconstruction of propositional Hoare logic.Comment: 40 page

CCS with Hennessy's merge has no finite-equational axiomatization

Abstract This paper confirms a conjecture of Bergstra and Klop¿s from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner¿s Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy¿s merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired. 2000 MATHEMATICS SUBJECT CLASSIFICATION: 08A70, 03B45, 03C05, 68Q10, 68Q45, 68Q55, 68Q70. CR SUBJECT CLASSIFICATION (1991): D.3.1, F.1.1, F.1.2, F.3.2, F.3.4, F.4.1. KEYWORDS AND PHRASES: Concurrency, process algebra, CCS, bisimulation, Hennessy¿s merge, left merge, communication merge, parallel composition, equational logic, complete axiomatizations, non-finitely based algebras