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...tions” to prove that an axiomatization is ω-complete, and proved for some of the equivalences in the linear time – branching time spectrum that their equational theory in BCCSP has a finite basis. In =-=[3, 5]-=-, a categorization of the equational theories for BCCSP modulo the semantics in the linear time – branching time spectrum is given. For each preorder and equivalence it is studied whether a finite, so...

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...om conditional axiomatizations in a rather uniform fashion. On the other hand, our negative results are shown using proof-theoretic techniques that have their root in, e.g., the seminal journal paper =-=[8]-=-. (The article [3] surveys classic proof techniques for showing non-finite axiomatizability results.) The paper is organized as follows. Section 2 presents the syntax and the operational semantics for...

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...iomatization exists, it is said that there is a finite basis for the equational theory. For concrete semantics, so in the absence of the silent action τ, the existence of finite bases is well-studied =-=[5, 11, 13]-=-, in the context of the process algebra BCCSP, containing the basic process algebraic operators from CCS and CSP. However, for weak semantics, that take into account the τ, hardly anything is known on...

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...red. ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 3 An ω-complete axiomatization enjoys the property that if all closed instances of an equation can be derived from it, then the equation itself can be derived from it. In universal algebra, such an axiomatization is referred to as a basis for the equational theory of the algebra it axiomatizes. Groote [G90] developed a technique of “inverted substitutions” to prove that an axiomatization is ω-complete, and proved for some equivalences in the “linear time – branching time spectrum I” that their equational theory in BCCSP has a finite basis. In [AFIL05, CFLN08], for each preorder and equivalence in the “linear time – branching time spectrum I” it was determined whether a finite, sound, ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With regard to the axiomatizability of weak semantics, relatively little is known compared to concrete semantics. For some semantics in the “linear time – branching time spectrum II” [G93a], a sound and ground-complete axiomatization has been given, in the settin...

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...d in [2, 3, 7, 8, 10, 16]. In the setting of BCCSP, in a seminal journal paper that collects and unifies the results in a series of conference articles, Chen, Fokkink, Luttik and Nain have offered in =-=[17]-=- a definitive classification of the status of the finite basis problem—that is, the problem of determining whether a behavioural equivalence has a finite, complete, equational axiomatization over the ...

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...ase the alphabet of actions is infinite. (An axiomatization E is ω-complete when an equation can be derived from E if, and only if, all of its closed instantiations can be derived from E.) The papers =-=[1,4, 5]-=- offer ⋆ The first and third author were partly supported by the project “The Equational Logic of Parallel Processes” (nr. 060013021) of The Icelandic Research Fund.2 positive and negative results on...