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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 29 (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
Nested Semantics over Finite Trees are Equationally Hard
, 2003
"... 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 ..."
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Cited by 15 (11 self)
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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.
On finite alphabets and infinite bases: From ready pairs to possible worlds
 In Proceedings 7th Conference on Foundations of Software Science and Computation Structures (FOSSACS’04), Barcelona, LNCS 2987
, 2004
"... 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 ..."
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Cited by 9 (7 self)
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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
A finite basis for failure semantics
 In Proceedings 32nd Colloquium on Automata, Languages and Programming (ICALP’05), Lisbon, LNCS 3580
, 2005
"... Abstract. We present a finite ωcomplete axiomatization for the process algebra BCCSP modulo failure semantics, in case of a finite alphabet. This solves an open question by Groote [12]. 1 ..."
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Cited by 9 (7 self)
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Abstract. We present a finite ωcomplete axiomatization for the process algebra BCCSP modulo failure semantics, in case of a finite alphabet. This solves an open question by Groote [12]. 1
A unique decomposition theorem for ordered monoids with applications in process theory
 In Branislav Rovan and Peter Vojtás, editors, Proceedings of MFCS 2003
, 2003
"... Abstract. We prove a unique decomposition theorem for a class of ordered commutative monoids. Then, we use our theorem to establish that every weakly normed process definable in ACP ε with bounded communication can be expressed as the parallel composition of a multiset of weakly normed parallel prim ..."
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Cited by 5 (2 self)
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Abstract. We prove a unique decomposition theorem for a class of ordered commutative monoids. Then, we use our theorem to establish that every weakly normed process definable in ACP ε with bounded communication can be expressed as the parallel composition of a multiset of weakly normed parallel prime processes in exactly one way. 1
On finite alphabets and infinite bases III: Simulation
 Proc. CONCUR’06, LNCS 4137
, 2006
"... 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 fin ..."
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Cited by 5 (1 self)
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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
A Distribution Law for CCS and a New Congruence Result for the Picalculus
 LMCS
"... 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 πcalculus in absence of sum. To our knowledge, this is the only nontrivial subcalc ..."
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Cited by 4 (1 self)
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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 πcalculus in absence of sum. To our knowledge, this is the only nontrivial subcalculus of the πcalculus that includes the full output prefix and for which strong bisimilarity is a congruence.
On finite alphabets and infinite bases II: Completed and ready simulation
 In Proceedings 9th Conference on Foundations of Software Science and Computation Structures (FOSSACS’06), Vienna, LNCS 3921
, 2006
"... Abstract. We prove that the equational theory of the process algebra BCCSP modulo completed simulation equivalence does not have a finite basis. Furthermore, we prove that with a finite alphabet of actions, the equational theory of BCCSP modulo ready simulation equivalence does not have a finite bas ..."
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Cited by 3 (2 self)
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Abstract. We prove that the equational theory of the process algebra BCCSP modulo completed simulation equivalence does not have a finite basis. Furthermore, we prove that with a finite alphabet of actions, the equational theory of BCCSP modulo ready simulation equivalence does not have a finite basis. In contrast, with an infinite alphabet, the latter equational theory does have a finite basis. 1
Decomposition Orders another generalisation of the fundamental theorem of arithmetic
"... We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition orde ..."
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Cited by 1 (0 self)
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We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition iff it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo bisimulation definable in ACP ε with linear communication, with parallel composition as binary operation, has unique decomposition. We also apply our result to establish that the partial commutative monoid associated with a wellfounded commutative residual algebra has unique decomposition. 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.