A Completeness Theorem for Nondeterministic Kleene Algebras

A generalization of Kleene Algebras (structures with +·*, 0 and 1 operators) is considered to take into account possible nondeterminism expressed by the + operator. It is shown that essentially the same complete axiomatization of Salomaa is obtained except for the elimination of the distribution P·(Q + R) = P·Q + P·R and the idempotence law P + P = P. The main result is that an algebra obtained from a suitable category of labelled trees plays the same role as the algebra of regular events. The algebraic semantics and the axiomatization are then extended by adding Omega and par operator, and the whole set of laws is used as a touchstone for starting a discussion over the laws for deadlock, termination and divergence proposed for models of concurrent systems.