Abstract. 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 axiomatizations for them. 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 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 in axiomatizations for richer process algebras with constructs like sequential and parallel composition. 1

This paper confirms a conjecture of Bergstra and Klop’s from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner’s Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy’s merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired.

We provide an answer to an open question, posed by van Glabbeek [4], regarding the axiomatizability of ready trace semantics. We prove that

This paper studies nested simulation and nested trace semantics over the language BCCSP, a basic formalism to express finite process behaviour. It is shown that none of these semantics affords finite (in)equational axiomatizations over BCCSP. In particular, for each of the nested semantics studied in this paper, the collection of sound, closed (in)equations over a singleton action set is not finitely based.

Abstract. This paper studies the equational theory of bisimulation equivalence over the process algebra BCCSP extended with the priority operator of Baeten, Bergstra and Klop. It is proven that, in the presence of an infinite set of actions, bisimulation equivalence has no finite, sound, ground-complete equational axiomatization over that language. This negative result applies even if the syntax is extended with an arbitrary collection of auxiliary operators, and motivates the study of axiomatizations using conditional equations. In the presence of an infinite set of actions, it is shown that, in general, bisimulation equivalence has no finite, sound, ground-complete axiomatization consisting of conditional equations over BCCSP. Sufficient conditions on the priority structure over actions are identified that lead to a finite, ground-complete axiomatization of bisimulation equivalence using conditional equations. 1

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Abstract. This paper contributes to the study of the equational theory of the semantics in van Glabbeek’s linear time- branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respectively, ground-complete) equational axiomatization of any behavioral congruence lying between ready simulation equivalence and partial traces equivalence from a complete (respectively, ground-complete) inequational axiomatization of its underlying precongruence—that is, of the precongruence whose kernel is the equivalence. The algorithm preserves finiteness of the axiomatization when the set of actions is finite. 1

Fokkink ((1994) Inf. Process. Lett. 52: 333--337) has recently proposed a complete equational axiomatization of strong bisimulation equivalence for MPA i.e., the language obtained by extending Milner's basic CCS with prefix iteration. p q obtained by restricting the first argument to be an atomic action. In this paper, we extend Fokkink's results to a setting with the unobservable action by giving a complete equational axiomatization of Milner's observation congruence over with two of Milner's standard -laws and the following three equations that describe the interplay between the silent nature of and prefix iteration: Using a technique due to Groote, we also show that the resulting axiomatization is !-complete, i.e., complete for equality of open terms.

Abstract. We present a finite ω-complete axiomatization for the process algebra BCCSP modulo failure semantics, in case of a finite alphabet. This solves an open question by Groote [12]. 1

Abstract. We prove that if a finite alphabet of actions contains at least two elements, then the equational theory for the process algebra BCCSP modulo any semantics no coarser than readiness equivalence and no finer than possible worlds equivalence does not have a finite basis. This semantic range includes ready trace equivalence. 1