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

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

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

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Abstract. This paper surveys some classic and recent results on the finite axiomatizability of bisimilarity over CCS-like languages. It focuses, in particular, on non-finite axiomatizability results stemming from the semantic interplay between parallel composition and nondeterministic choice. The paper also highlights the role that auxiliary operators, such as Bergstra and Klop’s left and communication merge and Hennessy’s merge operator, play in the search for a finite, equational axiomatization of parallel composition both for classic process algebras and for their real-time extensions. 1 The Problem and its History Process algebras are prototype description languages for reactive systems that arose from the pioneering work of figures like Bergstra, Hoare, Klop and Milner. Well-known examples of such languages are ACP [18], CCS [44], CSP [40] and Meije [13]. These algebraic description languages for processes differ in the basic collection of operators that they offer for building new process descriptions from existing ones. However, since they are designed to allow for the description and analysis of systems of interacting processes, all these languages contain some form of parallel composition (also known as merge) operator allowing one to put two process terms in parallel with one another. These operators usually interleave the behaviours of their arguments, and support some form of synchronization between them. For example, Milner’s CCS offers the binary operator ||, whose intended semantics is described by the following classic rules in the style of Plotkin [49]. x µ → x ′ x | | y µ → x ′ | | y y µ → y ′ x | | y µ → x | | y ′ x α → x ′ , y ¯α → y ′ x | | y τ → x ′ | | y ′ (In the above rules, the symbol µ stands for an action that a process may perform, α and ¯α are two observable actions that may synchronize, and τ is a symbol denoting the result of their synchronization.)

Abstract. We give an axiomatisation of strong bisimilarity on a small fragment of CCS that does not feature the sum operator. This axiomatisation is then used to derive congruence of strong bisimilarity in the finite π-calculus in absence of sum. To our knowledge, this is the only nontrivial subcalculus of the π-calculus that includes the full output prefix and for which strong bisimilarity is a congruence.

It is well known that bisimulation on µ-expressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either the rules are impure by involving non-equational side-conditions, or they are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the proof-theoretically and computationally clean and powerful setting of second-order equational Horn logic. This paper presents a positive and a negative result regarding axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s and Bloom/Ésik’s axiomatisations are incomplete without the congruence rule, and neither system has a complete extension in rank 1 equational axioms. It remains open whether there are higher-rank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes.

Abstract. This paper presents a general technique for obtaining new results pertaining to the non-finite axiomatizability of behavioural (pre)congruences over process algebras from old ones. The proposed technique is based on a variation on the classic idea of reduction mappings. In this setting, such reductions are translations between languages that preserve sound (in)equations and (in)equational proofs over the source language, and reflect families of (in)equations responsible for the non-finite axiomatizability of the target language. The proposed technique is applied to obtain a number of new non-finite axiomatizability theorems in process algebra via reduction to Moller’s celebrated non-finite axiomatizability result for CCS. The limitations of the reduction technique are also studied. In particular, it is shown that prebisimilarity is not finitely based over CCS with the divergent process Ω, but that this result cannot be proved by a reduction to the non-finite axiomatizability of CCS modulo bisimilarity. 1

Abstract This paper presents a general technique for obtaining new results pertain-ing to the non-finite axiomatizability of behavioral semantics over process algebras from old ones. The proposed technique is based on a variation on the classic ideaof reduction mappings. In this setting, such reductions are translations between languages that preserve sound (in)equations and (in)equational proofs over the sourcelanguage, and reflect families of (in)equations responsible for the non-finite axiomatizability of the target language. The proposed technique is applied to obtain a num-ber of new non-finite axiomatizability theorems in process algebra via reduction to Moller's celebrated non-finite axiomatizability result for CCS. The limitations ofthe reduction technique are also studied.

Abstract. It is well known that bisimulation on µ-expressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom / Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either the rules are impure by involving non-equational side-conditions, or they are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the proof-theoretically and computationally clean and powerful setting of second-order equational Horn logic. This paper presents a positive and a negative result regarding axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s and Bloom / Ésik’s axiomatisations are incomplete without the congruence rule, and neither system has a complete extension in rank 1 equational axioms. It remains open whether there are higher-rank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes. 1