Tau laws for pi calculus

The paper investigates the non-symbolic algebraic semantics of the weak bisimulation congruenceson finite pi processes. The weak bisimulation congruences are studied both in the absence and inthe presence of the mismatch operator. Some interesting phenomena about the open congruences arerevealed. Several new tau laws are discovered and their relationship is discussed. The contributionsof the paper are mainly as follows: 1. It is proved that Milner’s three tau laws fail to lift a complete system for the strong opencongruence to a complete system for the weak open congruence in the absence of both themismatch operator and the restriction operator. A fourth tau law is proposed to deal with thematch operator under the prefix operation. It is shown that for this calculus a complete systemfor the strong open congruence extended with all the four tau laws is complete for the weakopen congruence. 2. It is verified that the four tau laws are also enough for the weak open congruence of the pi calculuswithout the mismatch operator. Two complete systems are given, one using distinctions andthe other using a schematic law for the restriction operator. 3. It is pointed out that the standard definition of the weak open congruence gives rise to a badequivalence relation in the presence of the mismatch operator. Two alternatives are proposed.These are the late open congruence and the early open congruence. Their difference is similarto that between the weak late congruence and the weak early congruence. Complete axiomaticsystems for the two weak open congruences are given.