Step bisimulation is pomset equivalence on a parallel language without explicit internal choice

Abstract

We focus attention on a language ℒ generated by a set of elementary actions under operations of sequential composition, external binary choice, iteration, and non-nested disjoint parallel composition: that is, on an especially simple parallel language without explicit internal, or ‘local’, nondeterminism. ℒ is nondeterministic, despite the restriction to external choice, since a given action may occur in both subterms F and G of the parallel composition F ║ G. We exhibit a single set of equations that axiomatizes both step bisimulation and pomset equivalence on ℒ Given that step bisimulation and pomset equivalence are incomparable on the language ℒ+ obtained from ℒ by relaxing just the constraint on choice, the coincidence of these equivalences on ℒ suggests that the elimination of explicit internal choice can result in simplifications at the semantic level. We reinforce this impression by showing for ℒ that step bisimulation (i) coincides with pomset bisimulation equivalence, (ii) is real-time consistent, and (iii) has step trees as concrete representatives of its equivalence classes. Moreover, we show that none of these results holds for the language ℒ+ Finally, step trace equivalence is proved not to coincide with step bisimulation equivalence on ℒ.