A behavioural theory for a π-calculus with preorders

We study the behavioural theory of piP, a pi-calculus featuring restriction as the only binder. In contrast with calculi such as Fusions and Chi, reduction in piP generates a preorder on names rather than an equivalence relation. We present two characterisations of barbed congruence in piP: the fi rst is based on a compositional LTS, and the second is an axiomatisation. The results in this paper bring out basic properties of piP, mostly related to the interplay between the restriction operator and the preorder on name

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

Compositional Reasoning for Explicit Resource Management in Channel-Based Concurrency

We define a pi-calculus variant with a costed semantics where channels are treated as resources that must explicitly be allocated before they are used and can be deallocated when no longer required. We use a substructural type system tracking permission transfer to construct coinductive proof techniques for comparing behaviour and resource usage efficiency of concurrent processes. We establish full abstraction results between our coinductive definitions and a contextual behavioural preorder describing a notion of process efficiency w.r.t. its management of resources. We also justify these definitions and respective proof techniques through numerous examples and a case study comparing two concurrent implementations of an extensible buffer.Comment: 51 pages, 7 figure

Coinduction up to in a fibrational setting

Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity

On the Representation of McCarthy's amb in the π-calculus

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Unique Solutions of Contractions, CCS, and their HOL Formalisation

The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's "unique solution of equations" technique. The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions. (The formalisation consists of about 20,000 lines of proof scripts in Standard ML.) Some refinements of the theory itself are obtained. In particular we remove the constraints on summation, which must be weakly-guarded, by moving to rooted contraction, that is, the coarsest precongruence contained in the contraction preorder.Comment: In Proceedings EXPRESS/SOS 2018, arXiv:1808.0807

True concurrency can be traced

In this paper sets of labelled partial orders are employed as fundamental mathematical entities for modelling nondeterministic and concurrent processes thereby obtaining so-called noninterleaving semantics. Based on closures of sets of labelled partial orders, a simple algebraic language with refinement is given denotational models fully abstract w.r.t. corresponding behaviourally motivated equivalences

Name-passing calculi: from fusions to preorders and types

This is the appendix of the paper "Name-passing calculi: from fusions to preorders and types" (D Hirschkoff, JM. Madiot, D. Sangiorgi), to appear in LICS'2013