Checking pi-Calculus Structural Congruence is Graph Isomorphism Complete

We show that the problems of checking pi-calculus structural congruence (piSC) and graph isomorphism (GI) are Karp reducible to each other. The reduction from GI to piSC is given explicitly, and the reduction in the opposite direction proceeds by transforming piSC into an instance of the term equality problem (i.e. the problem of deciding equivalence of two terms in the presence of associative and/or commutative operations and commutative variable-binding quantifiers), which is known to be Karp reducible to GI. Our result is robust in the sense that it holds for several variants of structural congruence and some rather restrictive fragments of pi-calculus. Furthermore, we address the question of solving piSC in practice, and describe a number of optimisations exploiting specific features of pi-calculus terms, which allow one to significantly reduce the size of the resulting graphs that have to be checked for isomorphism.