A Semantic Theory for Value–Passing Processes Late Approach Part II: A Behavioural Semantics and Full Abstractness

This is the second of two companion papers on a semantic theory for communicating processes with values based on the late approach. In the first one, [Ing95], we explained the general idea of the late semantic approach. Furthermore weintroduced a general syntax for value-passing process algebra based on the late approach and a general class of denotational models for these languages in the Scott-Strachey style. Then we defined a concrete language, CCSL, which isan extension of the standard CCS with values according to the late approach.We also provided a denotational model for it, which is an instantiation of the general class. This model is a direct extension of the model given by Abramsky[Abr91] to model the pure calculus SCCS. Furthermore we gave an axiomatic semantics by means of a proof system based on inequations and proved its soundness and completeness with respect to the denotational semantics.In this paper we will give a behavioural semantics to the language CCSLin terms of a Plotkin style operational semantics and a bisimulation basedpreorder. Our main aim is to relate the behavioural view of processes we present here to the domain-theoretical one developed in the companion paper [Ing95]. In the Scott-Strachey approach an infinite process is obtained as a chain of finite and possibly partially specified processes. The completely unspecified process is given by the bottom element of the domain. An operational interpretation of this approach is to take divergence into account and give the behaviouralsemantics in terms of a prebisimulation or bisimulation preorder [Hen81,Wal90] rather than by the standard bisimulation equivalence [Par81, Mil83].One of the results in the pure case presented in [Abr91] is that the denotationalmodel given in that reference is fully abstract with respect to the "finitelyobservable" part of the bisimulation preorder but not with respect to the bisimulationpreorder which turns out to be too fine. Intuitively this is due to the algebraicity of the model and the fact that the finite elements in the modelare denotable by syntactically finite terms. The algebraicity implies that thedenotational semantics of a process is completely decided by the semantics ofits syntactically finite approximations, whereas the same can not be said about the bisimulation preorder. In fact we need experiments of an infinite depth to investigate bisimulation while this is not the case for the preorder induced by the model as explained above. An obvious consequence of this observation is that in general, a bisimulation preorder can not be expected to be modeled by an algebraic cpo given that the compact elements are denotable by syntacticallynite elements.In [Hen81] Hennessy defined a term model for SCCS. This model is !-algebraic and fails to be fully abstract with respect to the strong bisimulationpreorder. In the same paper the author introduces the notion of "the finitary part of a relation" and "a finitary relation". The finitary part of a relation R over processes, denoted by RF , is defined bypRF q i 8d:dRp) dRq where d ranges over the set of syntactically finite processes. A relation R isfinitary if RF = R. Intuitively this property may be interpreted as algebraicityat the behavioural level provided that syntactically nite terms are interpretedas compact elements in the denotational model; if a relation is nitary then itis completely decided by the syntactically nite elements.In both [Hen81] and [Abr91] the full abstractness of the respective denotationalsemantics with respect to <F is shown. In [Abr91] it is also shown thatif the language is sort nite and satises a kind of nite branching condition,then <F=< !, where < ! is the strong bisimulation preorder induced by experimentsof nite depth, i.e. the preorder is obtained by iterated application of thefunctional that denes the bisimulation. Note that in general the preorder < isstrictly ner than the preorder < !. However if the transition system is imagenite, i.e. if the number of arcs leading from a xed state and labelled with axed action is nite, then these two preorders coincide.As mentioned above the main aim of this paper is to give a bisimulationbased behavioural semantics for our language CCSL from [Ing95]. To reflect thelate approach the operational semantics will be given in terms of an applicativetransition system, a concept that is a modication of that dened in [Abr90].We generalize the notion of bisimulation [Par81, Mil83] to be applied to applicativetransition systems and introduce a preorder motivated by Abramsky'sapplicative bisimulation [Abr90]. For this purpose we rst introduce the notionof strong applicative prebisimulation and the corresponding strong applicativebisimulation preorder. Following the standard practice this preorder is obtainedas the largest xed point of a suitably dened monotonic functional. We showby an example that this preorder is not nitary in the sense described aboveand is strictly ner than the preorder induced by the model.Next we dene the strong applicative !-bisimulation preorder in the standardway by iterative application of the functional that induces the bisimulationpreorder. This gives as a result a preorder which still is too ne to match thepreorder induced by the denotational model. This will be shown by an example.Intuitively the reason for this is that we still need innite experiments todecide the operational preorder, now because of an innite breadth due to thepossibility of an innite number of values that have to be checked.Then we give a suitable denition of the notion of the \nitary part" ofthe bisimulation preorder to meet the preorder induced by the denotationalmodel. We recall that in [Ing95] we dened the so-called compact terms asthe syntactically nite terms which only use a nite number of values in a nontrivialway. We also showed that these terms correspond exactly to the compactelements in the denotational model in the sense that an element in the modelis compact if and only if it can be denoted by a compact term. This motivatesa denition of the nitary part, <F , of the bisimulation preorder < byp <F q i 8c: c < p ) c < qwhere c ranges over the set of syntactically compact terms. We also deneyet another preorder, <f!, a coarser version of < ! in which we only consider anite number of values at each level in the iterative denition of the preorder.Here it is vital that the set of values is countable and can be enumerated asV al = fv1; v2; g. Thus in the denition of <f1 we only test whether thedening constraints of the preorder hold when the only possible input andoutput value is v1, and in general in the denition of <fn we test the constraintsfor the rst n values only. (Here we would like to point out that this ideaoriginally appears in [HP80].) It turns out that <f! is the nitary part of <in our new sense and that the model is fully abstract with respect to <f!. Wewill prove both these results in this paper using techniques which are similarto those used by Hennessy in the above mentioned reference [Hen81].The structure of the paper is as follows: In Section 2 we give a short survey ofthe result from the companion paper [Ing95] needed in this study. The denitionof the operational semantics and the notion of applicative bisimulation are thesubject of Section 3. Section 4 is devoted to the analysis of the preorder and thedenition of the value-nitary preorder <f!. In Section 5 we give a denition ofthe notion of nitary part of a relation and a nitary relation over processes. Inthe same section we prove that the preorder <f! is nitary and that it coincideswith the nitary part of the preorder < . Finally we prove the soundness andthe completeness of the proof system with respect to the resulting preorder.The full abstractness of the denotational semantics for CCSL, given in [Ing95],then follows from the soundness and the completeness of the proof system withrespect to the denotational semantics. In Section 6 we give some concludingremarks

Towards Verified Lazy Implementation of Concurrent Value-Passing Languages (Abstract)

n/

Characteristic Formulae: From Automata to Logic

This paper discusses the classic notion of characteristic formulae for processes using variations on Hennessy-Milner logic as the underlying logical specification language. It is shown how to characterize logically (states of) finite labelled transition systems modulo bisimilarity using a single formula in Hennessy-Milner logic with recursion. Moreover, characteristic formulae for timed automata with respect to timed bisimilarity and the faster-than preorder of Moller and Tofts are offered in terms of the logic L_nu of Laroussinie, Larsen and Weise

A Fully Abstract Denotational Model for Observational Congruence

Denotational Model for Observational Congruence Anna Ing olfsd ottir Andrea Schalk BRICS Report Series RS-95-40 ISSN 0909-0878 August 1995 Copyright c fl 1995, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark Telephone:+45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through WWW and anonymous FTP: http://www.brics.aau.dk/BRICS/ ftp ftp.brics.aau.dk (cd pub/BRICS) A Fully Abstract Denotational Model for Observational Congruence Anna Ing'olfsd'ottir BRICS Dep.of Maths and Computer Science ..

Testing Hennessy-Milner Logic with Recursion

This study offers a characterization of the collection of propertiesexpressible in Hennessy-Milner Logic (HML) with recursion that can be testedusing finite LTSs. In addition to actions used to probe the behaviour of thetested system, the LTSs that we use as tests will be able to perform a distinguished action nok to signal their dissatisfaction during the interaction with the tested process. A process s passes the test T iff T does not perform the action nok when it interacts with s. A test T tests for a property phi in HML with recursion iff it is passed by exactly the states that satisfy phi. The paper gives an expressive completeness result offering a characterization of the collection of properties in HML with recursion that are testable in the above sense

A Characterization of Finitary Bisimulation

Following a paradigm put forward by Milner and Plotkin, a primary criterion to judge the appropriateness of denotational models for programming and specification languages is that they be in agreement with operational intuition about program behaviour. Of the "good t" criteria for such models that have beendiscussed in the literature, the most desirable one is that of full abstraction.Intuitively, a fully abstract denotational model is guaranteed to relate exactly all those programs that are operationally indistinguishable with respect to some chosen notion of observation. Because of its prominent role in process theory, bisimulation [12] has been a natural yardstick to assess the appropriateness of denotational models for several process description languages. In particular, when proving full abstractionresults for denotational semantics based on the Scott-Strachey approach for CCS-like languages, several preorders based on bisimulation have been considered; see, e.g., [6, 3, 4]. In this paper, we shall study one such bisimulationbasedpreorder whose connections with domain-theoretic models are by now well understood, viz. the prebisimulation preorder . investigated in, e.g., [6, 3]. Intuitively, p < q holds of processes p and q if p and q can simulate each other'sbehaviour, but at times the behaviour of p may be less specified than that of q. A common problem in relating denotational semantics for process descriptionlanguages, based on Scott's theory of domains or on the theory of algebraic semantics, with behavioural semantics based on bisimulation is that the chosen behavioural theory is, in general, too concrete. The reason for this phenomenon is that two programs are related by a standard denotational interpretation if, in some precise sense, they afford the same finite observations. On the other hand, bisimulation can make distinctions between the behaviours of two processesbased on infinite observations. (Cf. the seminal study [1] for a detailed analysis of this phenomenon.) To overcome this mismatch between the denotationaland the behavioural theory, all the aforementioned full abstraction results are obtained with respect to the so-called finitely observable, or finitary, part of bisimulation. The finitary bisimulation is defined on any labelled transition system thus: