A ground-complete axiomatization of finite state processes in process algebra

We consider a generic process algebra of which the standard process algebras ACP, CCS and CSP are subalgebras of reduced expressions. In particular such an algebra is endowed with a recursion operator which computes minimal fixpoint solutions of systems of equations over processes. As model for processes we consider finite-state transition systems modulo Milner's observational congruence and we define an operational semantics for the process algebra. Over such a generic algebra we show the following. We provide a syntactical characterization (allowing as many terms as possible) for the equations involved in recursion operators, which guarantees that transition systems generated by the operational semantics are indeed finite-state. Vice-versa we show that every process admits a specification in terms of such a restricted form of recursion. We then present an axiomatization which is ground-complete over such a restricted signature: an equation can be derived from the axioms between closed terms exactly when the corresponding finite-state transition systems are observationally congruent. Notably, in presenting such an axiomatization, we also show that the two standard axioms of Milner for weakly unguarded recursion can be expressed by using just a single axiom.