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

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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

Abstract. This paper studies the (in)equational theory of simulation preorder and equivalence over the process algebra BCCSP. We prove that in the presence of a finite alphabet with at least two actions, the (in)equational theory of BCCSP modulo simulation preorder or equivalence does not have a finite basis. In contrast, in the presence of an alphabet that is infinite or a singleton, the equational theory for simulation equivalence does have a finite basis. 1

Abstract. In a functional calculus, the so called ω-rule states that if two terms P and Q applied to any closed term N return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the λβ-calculus the ω-rule does not hold, even when the η-rule (weak extensionality) is added to the calculus. A long-standing problem of H. Barendregt (1975) concerns the determination of the logical power of the ω-rule when added to the λβ-calculus. In this paper we solve the problem, by showing that the resulting theory is Π 1 1-Complete.

Vol. 5 (2:6) 2009, pp. 1–21 www.lmcs-online.org