An algebraic process calculus
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An algebraic process calculus Emmanuel Beffara? Institut de Mathématiques de Luminy CNRS & Université Aix-Marseille II E-mail: Abstract We present an extension of the piI-calculus with formal sums of terms. The study of the properties of this sum reveals that its neutral element can be used to make as- sumptions about the behaviour of the environment of a pro- cess. Furthermore, the formal sum appears as a fundamen- tal construct that can be used to decompose both internal and external choice. From these observations, we derive an enriched calculus that enjoys a confluent reduction which preserves the testing semantics of processes. This system is shown to be strongly normalising for terms without repli- cation, and the study of its normal forms provides a fully abstract trace semantics for testing of piI processes. 1. Introduction The point of this paper is to define a meaningful notion of normalisation for process calculi. Normalisation is not an obvious idea in the context of concurrency because of the non-determinism that is present in most process calculi, and it makes it crucial to distinguish two related notions for term languages: execution, which is a relation that describes the intended dynamics of a term considered as a program in a given model of computation, and evaluation, which is a relation that preserves the computational meaning of terms while simplifying them (in some sense).
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An algebraic process calculus Emmanuel Beffara? Institut de Mathématiques de Luminy CNRS & Université Aix-Marseille II E-mail: Abstract We present an extension of the piI-calculus with formal sums of terms. The study of the properties of this sum reveals that its neutral element can be used to make as- sumptions about the behaviour of the environment of a pro- cess. Furthermore, the formal sum appears as a fundamen- tal construct that can be used to decompose both internal and external choice. From these observations, we derive an enriched calculus that enjoys a confluent reduction which preserves the testing semantics of processes. This system is shown to be strongly normalising for terms without repli- cation, and the study of its normal forms provides a fully abstract trace semantics for testing of piI processes. 1. Introduction The point of this paper is to define a meaningful notion of normalisation for process calculi. Normalisation is not an obvious idea in the context of concurrency because of the non-determinism that is present in most process calculi, and it makes it crucial to distinguish two related notions for term languages: execution, which is a relation that describes the intended dynamics of a term considered as a program in a given model of computation, and evaluation, which is a relation that preserves the computational meaning of terms while simplifying them (in some sense).
- transition system
- transition labelled
- can extend
- standard terms
- equivalence rules
- formal sums
- has no
- both actions
An algebraic process calculus
Emmanuel Be ff ara ∗ Institut de Mathématiques de Luminy CNRS & Université Aix-Marseille II E-mail: beffara@iml.univ-mrs.fr
Abstract We present an extension of the π I-calculus with formal sums of terms. The study of the properties of this sum reveals that its neutral element can be used to make as-sumptions about the behaviour of the environment of a pro-cess. Furthermore, the formal sum appears as a fundamen-tal construct that can be used to decompose both internal and external choice. From these observations, we derive an enriched calculus that enjoys a confluent reduction which preserves the testing semantics of processes. This system is shown to be strongly normalising for terms without repli-cation, and the study of its normal forms provides a fully abstract trace semantics for testing of π I processes.
1. Introduction The point of this paper is to define a meaningful notion of normalisation for process calculi. Normalisation is not an obvious idea in the context of concurrency because of the non-determinism that is present in most process calculi, and it makes it crucial to distinguish two related notions for term languages: execution , which is a relation that describes the intended dynamics of a term considered as a program in a given model of computation, and evaluation , which is a relation that preserves the computational meaning of terms while simplifying them (in some sense). In the λ -calculus, the standard notion of evaluation is β -reduction, which is confluent and strongly normalising for typed terms, while execution refers to particular evaluation strategies, or particular abstract machines. In the π -calculus, the dynamics of terms is usually given either as a labelled transition system or as a reduction relation up to some struc-tural congruence; we refer to the latter form as execution, since it represents the way a process actually runs. A problem in the search for semantics for processes is that there is no related notion of evaluation. For instance, ∗ Work supported by the French ANR project “Choco”.
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consider the most straightforward case of non-determinism: p : = ( ν a )( a . p 1 | a . p 2 | a ¯ ). This process obviously has two pos-sible reductions: p → p 1 and p → p 2 , assuming a does not occur in p 1 or p 2 . We cannot say that the value of p is that of p 1 or p 2 , since this would either lose information about p or imply that p 1 and p 2 have the same value; the most we can say is that the value of p is “either that of p 1 or that of p 2 ”. This cannot be expressed in general in the language of processes, therefore we introduce in the language a formal sum, so that we can formally get the equation p = p 1 + p 2 . This sum has pleasant properties like bilinearity of par-allel composition (i.e. ( p + q ) | r = p | r + q | r ) and linearity of hiding and even prefixes. However, the real gain in ex-pressiveness comes when considering the neutral element of this sum, that we will write 0. This element has to satisfy the equation p | 0 = 0, which is unusual in a process cal-culus. As we will see, it has a meaningful interpretation in terms of testing semantics, where an occurrence of 0 at top level means success. Moreover, a study of the properties of 0 shows that standard actions are not actually linear, since α. 0 , 0, but rather a ffi ne, and we can extract a notion of purely linear action that does satisfy the rule α ˆ . 0 = 0. The introduction of 0 and linear actions provides enough expres-siveness so that we can define for our calculus a system of reduction rules that is normalising and preserves the testing semantics. Contributions After recalling the syntax and operational semantics of the π I-calculus, in section 3 we define the al-gebraic π I-calculus and provide intuition on the meaning of the new constructs. Execution is presented in two equiva-lent forms, namely as a reduction up to structural congru-ence, and as a labelled transition system, and the definition of fair testing is deduced. In section 4, we introduce the system of evaluation rules over processes and we prove its soundness with respect to the fair testing semantics. Exam-ples show that these rules can be used to e ff ectively compute process equivalences. In section 5 we prove that this sys-tem is locally confluent and that it is strongly normalising for terms without replication, and in section 6 we study the
