Axiomatizing Maximal Progress and Discrete Time

Milner's complete proof system for observational congruence is crucially based on the possibility to equate $τ$ divergent expressions to non-divergent ones by means of the axiom $recX. (τ.X + E) = recX. τ. E$. In the presence of a notion of priority, where, e.g., actions of type $δ$ have a lower priority than silent $τ$ actions, this axiom is no longer sound. Such a form of priority is, however, common in timed process algebra, where, due to the interpretation of $δ$ as a time delay, it naturally arises from the maximal progress assumption. We here present our solution, based on introducing an auxiliary operator $pri(E)$ defining a "priority scope", to the long time open problem of axiomatizing priority using standard observational congruence: we provide a complete axiomatization for a basic process algebra with priority and (unguarded) recursion. We also show that, when the setting is extended by considering static operators of a discrete time calculus, an axiomatization that is complete over (a characterization of) finite-state terms can be developed by re-using techniques devised in the context of a cooperation with Prof. Jos Baeten.