Section: New Results

The Meta-Theory of Bisimulation-Up-To

Participants : Kaustuv Chaudhuri, Matteo Cimini, Dale Miller.

The method of proof by bisimulation has proved to be a very successful technique for showing the equivalence of processes. Unfortunately, in process calculi with infinite transition systems, such as in calculi with a replication operator, finding a bisimulation requires exploring an infinite search space, which moreover often tends to have rather intricate and complex structure. One way to combat this complexity—i.e., reduce the size of candidate bisimulation sets—is to identify redundancies among their members and then to replace redundant classes by unique inhabitants. This yields families of bisimulation-up-to proof methods that are parametric over the redundancy relation. For instance, if we consider bisimilarity itself as the redundancy, then we obtain bisimulation up to bisimilarity; with this relation, the singleton set $\left\{\right(!a,!!a\left)\right\}$ is a candidate set for showing that the processes $!a$ and $!!a$ are bisimilar, for example, when the bisimulation set with redundancies is infinite.

Since a priori there is no restriction on such redundancy relations, a key theoretical question is when a bisimulation-up-to relation is sound, i.e., that it is contained in a bisimulation. In the literature there have been a number of techniques proposed for showing soundness, but they often require the use of complex reasoning about lattices of fixed points. In [19] (CPP'15) we show how to use the built-in coinduction facilities of the Abella theorem prover to produce comparatively lightweight proofs of the soundness of many common bisimulation-up-to techniques for CCS and the $\pi$-calculus. A key feature of our approach is that we can use the facilities already provided by the Abella system for reasoning about the binding constructs for the $\pi$-calculus.