A Complete Axiomatization for Observational Congruence of Prioritized Finite-State Behaviors
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
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Milner''s complete proof system for observational congruence is crucially based on the possibility to equate $\tau$ divergent expressions to non-divergent ones by means of the axiom $\ms{rec} X. (\tau.X + E) = \ms{rec} X. \tau. E$. This possibility of ``escaping'''' $\tau$ divergence is known as fairness. In the presence of a notion of priority, where e.g. actions of type $\delta$ have a lower priority than $\tau$ actions, this axiom is no longer sound because a $\delta$ action performable by $E$ is pre-empted in the left-hand term but not in the right-hand term. It has previously been claimed that introducing such a notion of priority in process algebras could cause observational congruence to become unfair, i.e. make it not always possible to escape $\tau$ divergence. In fact we show that it is always possible to escape $\tau$ divergence even in the presence of priority by simply replacing $E$ in the axiom above by a term $\ms{pri}(E)$ where all initial $\delta$ actions have been pre-empted. This technique allows us to provide a complete axiomatization for Milner''s observational congruence over finite-state terms of a process algebra with priority and recursion.