Bisimilarity, Hypersets, and Stable Partitioning: a Survey

Since Hopcroft proposed his celebrated $n \log n$ algorithm for minimizing states in a finite automaton, the race for efficient partition refinement methods has inspired much research in algorithmics. In parallel, the notion of bisimulation has gained ground in theoretical investigations not less than in applications, till it even pervaded the axioms of a variant Zermelo-Fraenkel set theory. As is well-known, the coarsest stable partitioning problem and the determination of bisimilarity (i.e., the largest partition stable relative to finitely many dyadic relations) are two faces of the same coin. While there is a tendency to refer these topics to varying frameworks, we will contend that the set-theoretic view not only offers a clear conceptual background (provided stability is referred to a non-well-founded membership), but is leading to new insights on the algorithmic complexity issues.