Finite Equational Bases in Process Algebra: Results and Open Questions

Van Glabbeek (1990) presented the linear time-branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete and omega-complete axiomatizations for them. (An axiomatization E is omega-complete when an equation can be derived from E if, and only if, all its closed instantiations can be derived from E.) Obtaining such axiomatizations in concurrency theory often turns out to be difficult, even in the setting of simple languages like BCCSP. This has raised a host of open questions that have been the subject of intensive research in recent years. Most of these questions have been settled over BCCSP, either positively by giving a finite complete or omega-complete axiomatization, or negatively by proving that such an axiomatization does not exist. Still some open questions remain. This paper reports on these results, and on the state-of-the-art on axiomatizations for richer process algebras, containing constructs like sequential and parallel composition