Branching-depth hierarchies

Shoham Shamir*, Orna Kupferman, Eli Shamir

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

4 Scopus citations


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 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 CTLi. We define the branching depth of a structure as the minimal index i for which H i+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.

Original languageAmerican English
Pages (from-to)65-78
Number of pages14
JournalElectronic Notes in Theoretical Computer Science
Issue number1
StatePublished - 2003
EventEXPRESS'00, 7th International Workshop on Expressiveness in Concurrency (Satellite Workshop from CONCUR 2000) - , United States
Duration: 21 Aug 200021 Aug 2000


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