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16
On Finite Alphabets and Infinite Bases
, 2007
"... 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.
On the Axiomatizability of Impossible Futures: Preorder versus Equivalence
, 2008
"... bisimulation We investigate the (in)equational theory of impossible futures semantics over the process algebra BCCSP. We prove that no finite, sound axiomatization for BCCSP modulo impossible futures equivalence is groundcomplete. By contrast, we present a finite, sound, groundcomplete axiomatizat ..."
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bisimulation We investigate the (in)equational theory of impossible futures semantics over the process algebra BCCSP. We prove that no finite, sound axiomatization for BCCSP modulo impossible futures equivalence is groundcomplete. By contrast, we present a finite, sound, groundcomplete axiomatization for BCCSP modulo impossible futures preorder. If the alphabet of actions is infinite, then this axiomatization is shown to be ωcomplete. If the alphabet is finite, we prove that the inequational theory of BCCSP modulo impossible futures preorder lacks such a finite basis. We also derive nonfinite axiomatizability results for nested impossible futures semantics. completed simulation simulation 2nested simulation ready simulation possible worlds ready traces failure traces readies failures completed traces possible futures impossible futures traces 1
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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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.
Ready to Preorder: an Algebraic and General Proof
"... There have been quite a few proposals for behavioural equivalences for concurrent processes, and many of them are presented in Van Glabbeek’s linear timebranching time spectrum. Since their original definitions are based on rather different ideas, proving general properties of them all would seem to ..."
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There have been quite a few proposals for behavioural equivalences for concurrent processes, and many of them are presented in Van Glabbeek’s linear timebranching time spectrum. Since their original definitions are based on rather different ideas, proving general properties of them all would seem to require a casebycase study. However, the use of their axiomatizations allows a uniform treatment that might produce general proofs of those properties. Recently Aceto, Fokkink and Ingólfsdóttir have presented a very interesting result: For any process preorder coarser than the ready simulation in the linear timebranching time spectrum they show how to get an axiomatization of the induced equivalence. Unfortunately, their proof is not uniform and requires a casebycase analysis. Following the alternative approach suggested above, in this paper we present a much simpler (and algebraic) proof of that result which, in addition, is more general and totally uniform, so that it does not need to consider one by one the different semantics in the spectrum.
Equational Characterization of CovariantContravariant Simulation and Conformance Simulation Semantics
"... Covariantcontravariant simulation and conformance simulation generalize plain simulation and try to capture the fact that it is not always the case that ”the larger the number of behaviors, the better”. We have previously studied their logical characterizations and in this paper we present the axio ..."
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Covariantcontravariant simulation and conformance simulation generalize plain simulation and try to capture the fact that it is not always the case that ”the larger the number of behaviors, the better”. We have previously studied their logical characterizations and in this paper we present the axiomatizations of the preorders defined by the new simulation relations and their induced equivalences. The interest of our results lies in the fact that the axiomatizations help us to know the new simulations better, understanding in particular the role of the contravariant characteristics and their interplay with the covariant ones; moreover, the axiomatizations provide us with a powerful tool to (algebraically) prove results of the corresponding semantics. But we also consider our results interesting from a metatheoretical point of view: the fact that the covariantcontravariant simulation equivalence is indeed ground axiomatizable when there is no action that exhibits both a covariant and a contravariant behaviour, but becomes nonaxiomatizable whenever we have together actions of that kind and either covariant or contravariant actions, offers us a new subtle example of the narrow border separating axiomatizable and nonaxiomatizable semantics. We expect that by studying these examples we will be able to develop a general theory separating axiomatizable and nonaxiomatizable semantics. 1 Introduction and some related work
Axiomatizing Weak Ready Simulation Semantics over BCCSP
"... Ready simulation has proven to be one of the most significant semantics in process theory. It is at the heart of a number of general results that pave the way to a comprehensive understanding of the spectrum of process semantics. Since its original definition by Bloom, Istrail and Meyer in 1995, sev ..."
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Ready simulation has proven to be one of the most significant semantics in process theory. It is at the heart of a number of general results that pave the way to a comprehensive understanding of the spectrum of process semantics. Since its original definition by Bloom, Istrail and Meyer in 1995, several authors have proposed generalizations of ready simulation to deal with internal actions. However, a thorough study of the (non)existence of finite (in)equational bases for weak ready simulation semantics is still missing in the literature. This paper presents a complete account of positive and negative results on the axiomatizability of weak ready simulation semantics over the language BCCSP. In addition, this study offers a thorough analysis of the axiomatizability properties of weak simulation semantics.
The Saga of Finite Equational Bases over BCCSP
"... Abstract. Van Glabbeek (1990) presented the “linear time – branching time spectrum”, a lattice of behavioral semantics over labeled transition systems ordered by inclusion. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete axiomatizations ..."
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Abstract. Van Glabbeek (1990) presented the “linear time – branching time spectrum”, a lattice of behavioral semantics over labeled transition systems ordered by inclusion. 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, which had raised a host of open questions that were the subject of intensive research in recent years. All of these questions are settled over BCCSP now, either positively by giving a finite complete axiomatization, or negatively by proving that such an axiomatization does not exist. This essay reports on these results. 1
On Finite Bases for Weak Semantics:Failures versus Impossible Futures
"... Abstract. We provide a finite basis for the (in)equational theory of the processalgebra BCCS modulo the weak failures preorder and equivalence. We also give positive and negative results regarding the axiomatizability of BCCS moduloweak impossible futures semantics. 1 Introduction Labeled transition ..."
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Abstract. We provide a finite basis for the (in)equational theory of the processalgebra BCCS modulo the weak failures preorder and equivalence. We also give positive and negative results regarding the axiomatizability of BCCS moduloweak impossible futures semantics. 1 Introduction Labeled transition systems constitute a widely used model of concurrent computation.They model processes by explicitly describing their states and their transitions from state to state, together with the actions that produce these transitions. Several notions ofbehavioral semantics have been proposed, with the aim to identify those states that afford the same observations [9, 11]. For equational reasoning about processes, one needsto find an axiomatization that is sound and groundcomplete modulo the semantics under consideration, meaning that all equivalent closed terms can be equated. Ideally,such an axiomatization is also!complete, meaning that all equivalent open terms canbe equated. If such a finite axiomatization exists, it is said that there is a finite basis for
Simulations upto and Canonical Preorders (extended abstract) Abstract
, 2007
"... In this paper we define simulations upto a preorder and show how we can use them to provide a coinductive, simulationlike, characterization of semantic preorders for processes. The result applies to a wide class of preorders, in particular to all semantic preorders coarser than the ready simulatio ..."
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In this paper we define simulations upto a preorder and show how we can use them to provide a coinductive, simulationlike, characterization of semantic preorders for processes. The result applies to a wide class of preorders, in particular to all semantic preorders coarser than the ready simulation preorder in the linear timebranching time spectrum. An interesting but unexpected result is that, when built from an equivalence relation, the simulation upto is a canonical preorder whose kernel is the given equivalence relation. These canonical preorders have several nice properties, the main being that since all of them are defined in a homogeneous way, their properties can be proved in a generic way. In particular, we present an axiomatic characterization of each of these canonical preorders, that is obtained just by adding a single axiom to the axiomatization of the original equivalence relation. This gives us an alternative axiomatization for every axiomatizable preorder in the linear timebranching time spectrum, whose correctness and completeness can be proved once and for all.
On Finite Bases for Weak Semantics: Failures versus Impossible Futures
"... We provide a finite basis for the (in)equational theory of the process algebra BCCS modulo the weak failures preorder and equivalence. We also give positive and negative results regarding the axiomatizability of BCCS modulo weak impossible futures semantics. ..."
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We provide a finite basis for the (in)equational theory of the process algebra BCCS modulo the weak failures preorder and equivalence. We also give positive and negative results regarding the axiomatizability of BCCS modulo weak impossible futures semantics.