Fair equivalence relations

Equivalence between designs is a fundamental notion in verification. The linear and branching approaches to verification induce different notions of equivalence. When the designs are modeled by fair state-transition systems, equivalence in the linear paradigm corresponds to fair trace equivalence, and in the branching paradigm corresponds to fair bisimulation. In this work we study the expressive power of various types of fairness conditions. For the linear paradigm, it is known that the Buchi condition is sufficiently strong (that is, a fair system that uses Rabin or Streett fairness can be translated to an equivalent Buchi system). We show that in the branching paradigm the expressiveness hierarchy depends on the types of fair bisimulation one chooses to use. We consider three types of fair bisimulation studied in the literature: There Exists-bisimulation, game-bisimulation, and For All-bisimulation. We show that while game-bisimulation and For All-bisimulation have the same expressiveness hierarchy as tree automata, There Exists-bisimulation induces a different hierarchy. This hierarchy lies between the hierarchies of word and tree automata, and it collapses at Rabin conditions of index one, and Streett conditions of index two.