Branching-depth hierarchies

Abstract We study the distinguishing and expressive power of branching temporal logics with bounded nesting depth of path quantifiers. We define the fragments CTL^*_i and CTL_i of CTL ^* and CTL, where at most i nestings of path quantifiers are allowed. We show that for all i ≥ 1, the logic CTL^* _i+1 has more distinguishing and expressive power than CTL ^*_i; thus the branching-depth hierarchy is strict. We describe equivalence relations H_i that capture CTL ^*_i: two states in a Kripke structure are H _i-equivalent iff they agree on exactly all CTL^* _i formulas. While H₁ corresponds to trace equivalence, the limit of the sequence H₁, H₂,... is Milner's bisimulation. These results are not surprising, but they give rise to several interesting observations and problems. In particular, while CTL ^* and CTL have the same distinguishing power, this is not the case for CTL^*_i and CTL_i. We define the branching depth of a structure as the minimal index i for which H _i+1=H_i. The branching depth indicates on the possibility of using bisimulation instead of trace equivalence (and similarly for simulation and trace containment). We show that the problem of finding the branching depth is PSPACE-complete.