Complete Axiomatization for Divergent-Sensitive Bisimulations in Basic Process Algebra with Prefix Iteration

We study the divergent-sensitive spectrum of weak bisimulation equivalences in the setting of process algebra. To represent the infinite behavior, we consider the prefix iteration extension of a fragment of Milner’s CCS. The prefix iteration operator is a variant on the binary version of the Kleene star operator obtained by restricting the first argument to be an atomic action and allows us to capture the notion of recursion in a pure algebraic way. We investigate four typical divergent-sensitive weak bisimulation equivalences, namely divergent, stable, completed and divergent stable weak bisimulation equivalences from an axiomatic perspective. A lattice of distinguishing axioms is developed and thus pure equational axiomatizations for these congruences are obtained. A large part of the current paper is devoted to a considerable complicated proof for completeness. This work, to some extent, sheds light on distinct semantics of divergence.