Fully abstract models for nondeterministic regular expressions

Fokkink ((1994) Inf. Process. Lett. 52: 333-337) has recently proposed a complete equational axiomatization of strong bisimulation equivalence for MPA * δ (A τ ), i.e., the language obtained by extending Milner's basic CCS with prefix iteration. Prefix iteration is a variation on the original binary version of the Kleene star operation p * q obtained by restricting the first argument to be an atomic action. In this paper, we extend Fokkink's results to a setting with the unobservable action τ by giving a complete equational axiomatization of Milner's observation congruence over MPA * δ (A τ ). The axiomatization is obtained by extending Fokkink's axiom system with two of Milner's standard τ -laws and the following three equations that describe the interplay between the silent nature of τ and prefix iteration:Using a technique due to Groote, we also show that the resulting axiomatization is ω-complete, i.e., complete for equality of open terms.

“…The research literature on process theory has recently witnessed a resurgence of interest in the study of Kleene star-like operations (cf., e.g., the papers [11,3,9,8,23,7]). …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Complete Equational Axiomatization for Prefix Iteration with Silent Steps

Aceto¹,

Ingólfsdóttir²

1995

“…The research literature on process theory has recently witnessed a resurgence of interest in the study of Kleene star-like operations (cf., e.g., the papers [8,17,15,13,32,12,16,3,2]). …”

Section: Introductionmentioning

confidence: 99%

Axiomatizing Prefix Iteration with Silent Steps

Aceto¹,

Fokkink²,

Glabbeek³

et al. 1995

Prefix iteration is a variation on the original binary version of the Kleene star operation P * Q, obtained by restricting the first argument to be an atomic action. The interaction of prefix iteration with silent steps is studied in the setting of Milner's basic CCS. Complete equational axiomatizations are given for four notions of behavioural congruence over basic CCS with prefix iteration, viz. branching congruence, η-congruence, delay congruence and weak congruence. The completeness proofs for η-, delay, and weak congruence are obtained by reduction to the completeness theorem for branching congruence. It is also argued that the use of the completeness result for branching congruence in obtaining the completeness result for weak congruence leads to a considerable simplification with respect to the only direct proof presented in the literature. The preliminaries and the completeness proofs focus on open terms, i.e., terms that may contain process variables. As a byproduct, the ω-completeness of the axiomatizations is obtained as well as their completeness for closed terms.

“…(Cf., e.g., rules like unique fixed-point induction in its various flavours [52,8], the approximation induction principle [11] and ω-induction [39]. ) An alternative, purely algebraic, way of introducing infinite behaviours in process algebras is to augment them with variations on the Kleene star operation familiar from the theory of regular algebra-cf., e.g., the papers [32,9,10,29,24,22]. Some of these studies, notably [10], have investigated the expressive power of variations on standard process description languages in which infinite behaviours are defined by means of Kleene's star operation [44,21] rather than by means of systems of recursion equations.…”

Section: Introductionmentioning

confidence: 99%

A Menagerie of Non-Finitely Based Process Semantics over BPA*: From Ready Simulation Semantics to Completed Traces

Aceto¹,

Fokkink²,

Ingólfsdóttir³

1996

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 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.Our non-finite axiomatizability results apply to the language BPA * over an arbitrary non-empty set of actions. In particular, we show that completed trace equivalence is not finitely based over BPA * even when the set of actions is a singleton. Our proof of this result may be easily adapted to the standard language of regular * BRICS (Basic Research in