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: $\exists$-bisimulation [GL94],
game-bisimulation [HKR97], and $\forall$-bisimulation [LT87].
We show that while game-bisimulation and $\forall$-bisimulation have the same expressiveness hierarchy as tree automata,
$\exists$-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. Beyond the theoretical interest of our results, they are useful when deciding on the type of fairness and the type of fair bisimulation or simulation being used.