Tau laws for pi calculus

The paper investigates the nonsymbolic algebraic semantics of the weak bisimulation congruences on finite pi processes. The weak bisimulation congruences are studied both in the absence and in the presence of the mismatch operator. Some interesting phenomena about the open congruences are revealed. Several new tau laws are discovered and their relationship is discussed. The contributions of 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 open congruence to a complete system for the weak open congruence in the absence of both the mismatch operator and the restriction operator. A fourth tau law is proposed to deal with the match operator under the prefix operation. It is shown that for this calculus a complete system for the strong open congruence extended with all the four tau laws is complete for the weak open congruence. 2. It is verified that the four tau laws are also enough for the weak open congruence of the pi calculus without the mismatch operator. A complete system using distinctions is given. 3. It is pointed out that the standard definition of the weak open congruence gives rise to a bad equivalence 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 similar to that between the weak late congruence and the weak early congruence. Complete axiomatic systems for the two weak open congruences are given.