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Finite equational bases in process algebra: Results and open questions
 Processes, Terms and Cycles: Steps on the Road to Infinity, LNCS 3838
, 2005
"... 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. ..."
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Cited by 28 (19 self)
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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 stateoftheart in axiomatizations for richer process algebras with constructs like sequential and parallel composition. 1
CCS with Hennessy’s merge has no finite equational axiomatization
 Theoretical Computer Science
, 2005
"... 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 equ ..."
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Cited by 19 (16 self)
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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.
On the axiomatizability of priority
 Proceedings of Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006
, 2006
"... 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, groundcomp ..."
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Cited by 14 (7 self)
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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, groundcomplete 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, groundcomplete axiomatization consisting of conditional equations over BCCSP. Sufficient conditions on the priority structure over actions are identified that lead to a finite, groundcomplete axiomatization of bisimulation equivalence using conditional equations. 1
A finite equational base for CCS with left merge and communication merge
 Proceedings of ICALP’06 (part II), volume 4052 of Lecture Notes in Computer Science
, 2006
"... Abstract. Using the left merge and communication merge from ACP, we present an equational base (i.e., a groundcomplete and ωcomplete set of valid equations) for the fragment of CCS without recursion, restriction and relabelling. Our equational base is finite if the set of actions is finite. 1 ..."
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Cited by 10 (6 self)
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Abstract. Using the left merge and communication merge from ACP, we present an equational base (i.e., a groundcomplete and ωcomplete set of valid equations) for the fragment of CCS without recursion, restriction and relabelling. Our equational base is finite if the set of actions is finite. 1
On Cool Congruence Formats for Weak Bisimulations
, 2005
"... In TCS 146, Bard Bloom presented rule formats for four main notions of bisimulation with silent moves. He proved that weak bisimulation equivalence is a congruence for any process algebra defined by WB cool rules, and established similar results for rooted weak bisimulation (Milner’s “observational ..."
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Cited by 6 (0 self)
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In TCS 146, Bard Bloom presented rule formats for four main notions of bisimulation with silent moves. He proved that weak bisimulation equivalence is a congruence for any process algebra defined by WB cool rules, and established similar results for rooted weak bisimulation (Milner’s “observational congruence”), branching bisimulation and rooted branching bisimulation. This study reformulates Bloom’s results in a more accessible form and contributes analogues for (rooted) ηbisimulation and (rooted) delay bisimulation. Moreover, finite equational axiomatisations of rooted weak bisimulation equivalence are provided that are sound and complete for finite processes in any RWB cool process algebra. These require the introduction of auxiliary operators with lookahead, and an extension of Bloom’s formats for this type of operator with lookahead. Finally, a challenge is presented for which Bloom’s formats fall short and further improvement is called for.
Lifting NonFinite Axiomatizability Results to Extensions of Process Algebras
"... This paper presents a general technique for obtaining new results pertaining to the nonfinite 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 reduct ..."
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Cited by 3 (2 self)
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This paper presents a general technique for obtaining new results pertaining to the nonfinite 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 nonfinite axiomatizability of the target language. The proposed technique is applied to obtain a number of new nonfinite axiomatizability theorems in process algebra via reduction to Moller’s celebrated nonfinite 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 nonfinite axiomatizability of CCS modulo bisimilarity.
Is Observational Congruence on µExpressions Axiomatisable in Equational Horn Logic?
, 2007
"... It is well known that bisimulation on µexpressions 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 t ..."
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Cited by 1 (1 self)
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It is well known that bisimulation on µexpressions 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 nonequational sideconditions, 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 prooftheoretically and computationally clean and powerful setting of secondorder 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 higherrank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes.
Is Observational Congruence Axiomatisable in Equational Horn Logic?
"... Abstract. It is well known that bisimulation on µexpressions 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 log ..."
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Abstract. It is well known that bisimulation on µexpressions 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 nonequational sideconditions, 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 prooftheoretically and computationally clean and powerful setting of secondorder 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 higherrank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes. 1
On Finite Alphabets and Infinite Bases
"... Van Glabbeek (1990) presented the linear time – branching time spectrum of behavioral semantics. He studied these semantics in the setting of the basic process algebra BCCSP, and gave finite, sound and groundcomplete, axiomatizations for most of these semantics. Groote (1990) proved for some of van ..."
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Van Glabbeek (1990) presented the linear time – branching time spectrum of behavioral semantics. He studied these semantics in the setting of the basic process algebra BCCSP, and gave finite, sound and groundcomplete, axiomatizations for most of these semantics. Groote (1990) proved for some of van Glabbeek’s axiomatizations that they are ωcomplete, meaning that an equation can be derived if (and only if) all of its closed instantiations can be derived. In this paper we settle the remaining open questions for all the semantics in the linear time – branching time spectrum, either positively by giving a finite sound and groundcomplete axiomatization that is ωcomplete, or negatively by proving that such a finite basis for the equational theory does not exist. We prove that in case of a finite alphabet with at least two actions, failure semantics affords a finite basis, while for ready simulation, completed simulation, simulation, possible worlds, ready trace, failure trace and ready semantics, such a finite basis does not exist. Completed simulation semantics also lacks a finite basis in case of an infinite alphabet of actions.