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, 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. AMS Subject Classification (1991): 68Q10, 68Q40, 68Q55.CR Subject Classification (1991): D.3.1, F.1.2, F.3.2.Keywords and Phrases: Concurrency, process algebra, basic CCS, prefix iteration, branching bisimulation, eta-bisimulation, delay bisimulation, weak bisimulation, equational logic, complete axiomatizations

Axiomatizing Flat Iteration

Flat 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 sum of atomic actions. It generalizes prefix iteration, in which the first argument is a single action. Complete finite equational axiomatizations are given for five notions of bisimulation congruence over basic CCS with flat iteration, viz. strong congruence, branching congruence, eta-congruence, delay congruence and weak congruence. Such axiomatizations were already known for prefix iteration and are known not to exist for general iteration. The use of flat iteration has two main advantages over prefix iteration: 1.The current axiomatizations generalize to full CCS, whereas the prefix iteration approach does not allow an elimination theorem for an asynchronous parallel composition operator. 2.The greater expressiveness of flat iteration allows for much shorter completeness proofs. In the setting of prefix iteration, the most convenient way to obtain the completeness theorems for eta-, delay, and weak congruence was by reduction to the completeness theorem for branching congruence. In the case of weak congruence this turned out to be much simpler than the only direct proof found. In the setting of flat iteration on the other hand, the completeness theorems for delay and weak (but not eta-) congruence can equally well be obtained by reduction to the one for strong congruence, without using branching congruence as an intermediate step. Moreover, the completeness results for prefix iteration can be retrieved from those for flat iteration, thus obtaining a second indirect approach for proving completeness for delay and weak congruence in the setting of prefix iteration.Comment: 15 pages. LaTeX 2.09. Filename: flat.tex.gz. On A4 paper print with: dvips -t a4 -O -2.15cm,-2.22cm -x 1225 flat. For US letter with: dvips -t letter -O -0.73in,-1.27in -x 1225 flat. More info at http://theory.stanford.edu/~rvg/abstracts.html#3

Axiomatizing Maximal Progress and Discrete Time

Milner's complete proof system for observational congruence is crucially based on the possibility to equate $\tau$ divergent expressions to non-divergent ones by means of the axiom $recX. (\tau.X + E) = recX. \tau. E$. In the presence of a notion of priority, where, e.g., actions of type $\delta$ have a lower priority than silent $\tau$ actions, this axiom is no longer sound. Such a form of priority is, however, common in timed process algebra, where, due to the interpretation of $\delta$ as a time delay, it naturally arises from the maximal progress assumption. We here present our solution, based on introducing an auxiliary operator $pri(E)$ defining a "priority scope", to the long time open problem of axiomatizing priority using standard observational congruence: we provide a complete axiomatization for a basic process algebra with priority and (unguarded) recursion. We also show that, when the setting is extended by considering static operators of a discrete time calculus, an axiomatization that is complete over (a characterization of) finite-state terms can be developed by re-using techniques devised in the context of a cooperation with Prof. Jos Baeten

A Cook’s Tour of Equational Axiomatizations for Prefix Iteration

Prefix iteration is a variation on the original binary version of theKleene 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

Equational Axioms for Probabilistic Bisimilarity (Preliminary Report)

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (omega-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity. Hence probabilistic bisimilarity (over finite-state agents) has an equational axiomatization relative to iteration algebras

Canonical Solutions to Recursive Equations and Completeness of Equational Axiomatisations

In this paper we prove completeness of four axiomatisations for finite-state behaviours with respect to behavioural equivalences at various ?-abstract levels: branching congruence, delay congruence, ?-congruence, and weak congruence. Instead of merging guarded recursive equations, which was the approach originally used by Robin Milner and has since become the standard strategy for proving completeness results of this kind, in this work we take a new approach by solving guarded recursive equations with canonical solutions which are those with the fewest reachable states. The new strategy allows uniform treatment of the axiomatisations with respect to different behavioural equivalences

Specification and Automated Verification of Real-Time Behaviour —A Case Study

In this paper we sketch a method for specification and automaticverification of real-time software properties. The method combinesthe IEC 848 norm and the recent specification techniques TCCS (TimedCalculus of Communicating Systems) and TML (Timed Modal Logic) - supported by an automatic verification tool, Epsilon. The methodis illustrated by modelling a small real-life steam generator example andsubsequent automated analysis of its properties.Keywords: Control system analysis; formal specification; formal verification; real-time systems; standards

On the Axiomatisation of Branching Bisimulation Congruence over CCS

In this paper we investigate the equational theory of (the restriction, relabelling, and recursion free fragment of) CCS modulo rooted branching bisimilarity, which is a classic, bisimulation-based notion of equivalence that abstracts from internal computational steps in process behaviour. Firstly, we show that CCS is not finitely based modulo the considered congruence. As a key step of independent interest in the proof of that negative result, we prove that each CCS process has a unique parallel decomposition into indecomposable processes modulo branching bisimilarity. As a second main contribution, we show that, when the set of actions is finite, rooted branching bisimilarity has a finite equational basis over CCS enriched with the left merge and communication merge operators from ACP

Topological Aspects of Traces

This paper is a little mathematical study of some models of concurrency. The most elementary one is the concept of an independence structure, which is nothing but a set L with a binary, irreflexive and symmetric relation on it, the independence relation. This leads to the notion of a trace: a string of elements of L, modulo the equivalence generated by swapping adjacent, independent elements of the string. There are two aspects of finite traces: they form an order, hence a topology; on the other hand they form a monoid, a quotient of the free monoid on L. Unfortunately, these two points of view are hard to bring together, since the monoid structure can never be continuous or even order-preserving. It is therefore not surprising that many papers on trace theory consist of two, disjoint, parts. In this paper I concentrate on the order-theoretic and topological aspects