Basic CCS Research Articles

The equational theory of prebisimilarity over basic CCS with divergence

This paper studies the equational theory of prebisimilarity, a bisimulation-based preorder introduced by Hennessy and Milner in the early 1980s, over basic CCS with the divergent process Ω. It is well known that prebisimilarity affords a finite ground-complete axiomatization over this language; this study proves that this ground-complete axiomatization is also complete in the presence of an infinite set of actions. Moreover, in sharp contrast to this positive result, it is shown that prebisimilarity is not finitely based over basic CCS with the divergent process Ω when the set of actions is finite and nonempty.

2-Nested Simulation is not Finitely Equationally Axiomatizable

<p>2-nested simulation was introduced by Groote and Vaandrager<br /> [10] as the coarsest equivalence included in completed trace equivalence<br /> for which the tyft/tyxt format is a congruence format. In the linear<br />time-branching time spectrum of van Glabbeek [8], 2-nested simulation<br />is one of the few equivalences for which no finite equational axiomatization<br /> is presented. In this paper we prove that such an axiomatization<br />does not exist for 2-nested simulation.</p><p>Keywords: Concurrency, process algebra, basic CCS, 2-nested simulation,<br /> equational logic, complete axiomatizations.</p>

A complete equational axiomatization for MPA with string iteration

We study equational axiomatizations of bisimulation equivalence for the language obtained by extending Milner's basic CCS with string iteration. String iteration is a variation on the original binary version of the Kleene star operation p ∗q obtained by restricting the first argument to be a non-empty sequence of atomic actions. We show that, for every positive integer k, bisimulation equivalence over the set of processes in this language with loops of length at most k is finitely axiomatizable, provided that the set of actions is finite. We also offer an infinite equational theory that completely axiomatizes bisimulation equivalence over the whole language. We prove that this result cannot be improved upon by showing that no finite equational axiomatization of bisimulation equivalence over basic CCS with string iteration can exist, unless the set of actions is empty.

A Cook’s Tour of Equational Axiomatizations for Prefix Iteration

Prefix iteration is a variation on the original binary version of the<br />Kleene star operation P*Q, obtained by restricting the first argument to be an atomic action, and yields simple iterative behaviours that can be equationally characterized by means of finite collections of axioms. In this paper, we present axiomatic characterizations for a significant fragment of the notions of equivalence and preorder in van Glabbeek's linear-time/branching-time spectrum over Milner's basic CCS extended with prefix iteration. More precisely, we consider ready simulation, simulation, readiness, trace and language semantics, and provide complete (in)equational axiomatizations for each of these notions over BCCS with prefix iteration. All of the axiom systems we present are finite, if so is the set of atomic actions under consideration.

Axiomatizing Prefix Iteration with Silent Steps

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 by-product, the ω -completeness of the axiomatizations is obtained, as well as their completeness for closed terms.

Axiomatizing Prefix Iteration with Silent Steps

<p>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, eta-congruence, delay congruence and weak congruence. The completeness proofs for eta-, 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 by-product, the omega-completeness of the axiomatizations is obtained as well as their completeness for closed terms.</p><p> </p><p>AMS Subject Classification (1991): 68Q10, 68Q40, 68Q55.<br />CR Subject Classification (1991): D.3.1, F.1.2, F.3.2.<br />Keywords and Phrases: Concurrency, process algebra, basic CCS, prefix iteration, branching bisimulation, eta-bisimulation, delay bisimulation, weak bisimulation, equational logic, complete axiomatizations.</p>

A Complete Equational Axiomatization for MPA with String Iteration

We study equational axiomatizations of bisimulation equivalence for the language obtained by extending Milner's basic CCS with string iteration. String iteration is a variation on the original binary version of the Kleene star operation p*q obtained by restricting the first argument to be a non-empty sequence of atomic actions. We show that, for every positive integer k, bisimulation equivalence over the set of processes in this language with loops of length at most k is finitely axiomatizable. We also offer a countably infinite equational theory that completely axiomatizes bisimulation equivalence over the whole language. We prove that this result cannot be improved upon by showing that no finite equational axiomatization of bisimulation equivalence over basic CCS with string iteration can exist, unless the set of actions is empty.

A Complete Equational Axiomatization for Prefix Iteration with Silent Steps

Fokkink ((1994) Inf. Process. Lett. 52: 333{337) has recently proposed a complete<br />equational axiomatization of strong bisimulation equivalence for MPA_delta^*(A_tau),<br />i.e., the language obtained by extending Milner's basic CCS with prefix iteration.<br />Prefix iteration is a variation on the original binary version of the Kleene star operation<br />p*q obtained by restricting the first argument to be an atomic action. In this<br />paper, we extend Fokkink's results to a setting with the unobservable action by<br />giving a complete equational axiomatization of Milner's observation congruence over<br />MPA_delta^*(A_tau ). <br />The axiomatization is obtained by extending Fokkink's axiom system<br />with two of Milner's standard tau-laws and the following three equations that describe<br />the interplay between the silent nature of tau and prefix iteration:<br />tau . x = tau*x<br />a*(x+tau.y) = a*(x+tau.y + a.y)<br />tau.(a*x) = a*(tau.a*x) .<br />Using a technique due to Groote, we also show that the resulting axiomatization is<br />omega-complete, i.e., complete for equality of open terms.<br />AMS Subject Classification (1991): 68Q40, 68Q42.<br />CR Subject Classification (1991): D.3.1, F.3.2, F.4.2.<br />Keywords and Phrases: Minimal Process Algebra, prefix iteration, equational<br />logic, omega-completeness, observation congruence.

A complete equational axiomatization for prefix iteration

Prefix iteration a * x is added to Minimal Process Algebra (MPA δ , which is a subalgebra of BPA δ equivalent to Milner's basic CCS. We present a finite equational axiomatization for MPA * δ , and 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 show that bisimilar terms have the same normal form.