On the axiomatizability of ready traces, ready simulation, and failure traces (original) (raw)
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Mathematical Structures in Computer Science, 1998
Fokkink and Zantema ((1994) Computer Journal 37:259-267) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA * ). In the light of this positive result on the mathematical tractability of bisimulation equivalence over BPA * , a natural question to ask is whether any other (pre)congruence relation in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence, none of the preorders and equivalences in van Glabbeek's linear time/branching time spectrum, whose discriminating power lies in between that of ready simulation and that of completed traces, has a finite equational axiomatization. This we achieve by exhibiting a family of (in)equivalences that holds in ready simulation semantics, the finest semantics that we consider, whose instances cannot all be proven by means of any finite set of (in)equations that is sound in completed trace semantics, which is the coarsest semantics that is appropriate for the language BPA * . To this end, for every finite collection of (in)equations that are sound in completed trace semantics, we build a model in which some of the (in)equivalences of the family under consideration fail. The construction of the model mimics the one used by Conway ((1971) Regular Algebra and Finite Machines, page 105) in his proof of a result, originally due to Redko, to the effect that infinitely many equations are needed to axiomatize equality of regular expressions.
Finite equational bases in process algebra: Results and open questions
Processes, Terms and Cycles: Steps on the Road to Infinity, 2005
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 state-of-the-art in axiomatizations for richer process algebras with constructs like sequential and parallel composition.
Information Processing Letters, 2011
This note shows that the complete and the ready simulation preorders do not have a finite inequational basis over the language BCCSP when the set of actions is a singleton. Moreover, the equivalences induced by those preorders do not have a finite (in)equational axiomatization either. These results are in contrast with a claim of finite axiomatizability for those semantics in the literature, which was based on the erroneous assumption that they coincide with complete trace semantics in the presence of a singleton set of actions.
Axiomatizing weak ready simulation semantics over BCCSP
2011
This paper is devoted to the study of the (in)equational theory of the largest (pre)congruences over the language BCCSP induced by variations on the classic simulation preorder and equivalence that abstract from internal steps in process behaviours. In particular, the article focuses on the (pre)congruences associated with the weak simulation, the weak complete simulation and the weak ready simulation preorders. For each of these behavioural semantics, results on the (non)existence of finite (ground-)complete (in)equational axiomatizations are given. The axiomatization of those semantics using conditional equations is also discussed in some detail.
Lifting Non-Finite Axiomatizability Results to Extensions of Process Algebras
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 provability over the source language, and reflect families of (in)equations responsible for the non-finite axiomatizability of the target language.
Axiomatizing weak simulation semantics over BCCSP
Theoretical Computer Science, 2013
This paper is devoted to the study of the (in)equational theory of the largest (pre)congruences over the language BCCSP induced by variations on the classic simulation preorder and equivalence that abstract from internal steps in process behaviours. In particular, the article focuses on the (pre)congruences associated with the weak simulation, the weak complete simulation and the weak ready simulation preorders. For each of these behavioural semantics, results on the (non)existence of finite (ground-)complete (in)equational axiomatizations are given. The axiomatization of those semantics using conditional equations is also discussed in some detail.
Basic process algebra with iteration: Completeness of its equational axioms
The Computer Journal, 1994
Bergstra, Bethke and Ponse proposed an axiomatization for Basic Process Algebra extended with (binary) iteration. In this paper, we prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we will set up a term rewriting system, based on the axioms, and prove that this term rewriting system is terminating, and that bisimilar normal forms are syntactically equal modulo commutativity and associativity of the +.
Algebra of communicating processes with abstraction
Theoretical Computer Science, 1985
We present an axiom system ACP, for communicating processes with silent actions ('z-steps'). The system is an extension of ACP, Algebra of Communicating Processes, with Milner's z-laws and an explicit abstraction operator. By means of a model of finite acyclic process graphs for ACP, syntactic properties such as consistency and conservativity over ACP are proved. Furthermore, the Expansion Theorem for ACP is shown to carry over to ACP~. Finally, termination of rewriting terms according to the ACP~ axioms is proved using the method of recursive path orderings.
The Equational Theory of Weak Complete Simulation Semantics over BCCSP
Lecture Notes in Computer Science, 2012
This paper presents a complete account of positive and negative results on the finite axiomatizability of weak complete simulation semantics over the language BCCSP. We offer finite (un)conditional groundcomplete axiomatizations for the weak complete simulation precongruence. In sharp contrast to this positive result, we prove that, in the presence of at least one observable action, the (in)equational theory of the weak complete simulation precongruence over BCCSP does not have a finite (in)equational basis. In fact, the collection of (in)equations in at most one variable that hold in weak complete simulation semantics over BCCSP does not have an (in)equational basis of 'bounded depth', let alone a finite one.
Towards Action-Refinement in Process Algebras
Information and Computation, 1993
ory of concurrency; the symbol ; represents We present a simple process algebra which supports a form of refinement of an action by a process and address the question of a n appropriate equivalence relation for it. The main result of the paper is that an adequate equivalence can be defined in a very intuitive manner and moreover can be axiomatized in much the same way as the standard behavioural equivalences.