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 CTLi 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 Hi that capture CTL∗i: two states in a Kripke structure are Hi-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 CTLi. We define the branching depth of a structure as the minimal index i for which Hi+1=Hi. 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.