Abstract
Recent results show that Hoare’s CSP, augmented by one additional operator, can express every operator whose operational semantics are expressible in a new notation and are therefore “CSP-like.” In this paper we show that π-calculus fits into this framework and therefore has CSP semantics. Rather than relying on the machinery of the earlier result we develop a much simpler version from scratch that avoids the extra operator and is sufficient for π-calculus: a much generalised relabelling operator that is expressed in terms of the others. We present a number of different options for the semantics of fresh names, showing how they give semantics that are largely congruent to each other. Finally, we begin the investigation of how these new semantics might be analysed and exploited.
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Notes
- 1.
The uses of relabelling we make later only map visible actions to other visible actions.
- 2.
The only difference is that CCS allows this definition to be used unconditionally, even on under-defined terms such as μp.p. The translations from CCS to CSP in this section are therefore restricted to the case where all recursions add at least one initial action.
- 3.
It is traditional to write such terms using “semantic brackets,” as in \(\mathcal{M}[\![\mathcal{P}]\!]\). We do not follow this convention in this paper because the traditional notation is so similar to CSP renaming P[[R]].
- 4.
These parameters are required because both the common knowledge and availability of fresh names may well have changed at the point of call, and it seems to be correct to evaluate the recursive call in that new world, just as is done by the textual substitution of term rewriting. In the spirit of a pure denotational semantics, fully espousing the traditions of [24], we might well wish to discriminate between Name and the identifiers Ide that denote them: this would allow definitions of input and restriction without syntactic substitution.
- 5.
We can deduce this as follows. Consider the operational semantics of P and Q derived from csp[P]κσ P and csp[Q]κσ Q (without any application of NFN, where σ P and σ Q are the sets assigned to them by the csp[ ⋅]κσ semantics of |). These are finite branching, in the sense that every finite sequence of visible events and τs can only lead to finitely many distinct states. This depends on the fact that no single state of P or Q can have more than a finite number of distinct output \(\overline{x}.y\) actions available, for otherwise | ccs might introduce infinite branching on τ.
The existence of arbitrarily long pairs of finite traces of P and Q such that P|Q, under perhaps different ξs, can give rise to prefixes of a given finite trace s means that csp[P|Q]κσ can itself behave in the same way except that the names of the fresh names extruded might be different. So if we examine that part of the operational semantic tree of csp[P|Q]κσ in which the visible trace is a prefix of s with modifications to the names of extruded fresh names permitted, it is infinite and therefore, by König’s lemma, has an infinite path that necessarily ends (as viewed from the outside) in an infinite sequence of τs. It follows that csp[P|Q]σκ has a divergence that is a prefix of s except that the finitely many extruded names might be different. Since there is certainly a permutation η that maps these names to the ones seen in s, it follows that the corresponding exact prefix of s, and hence s itself from divergence strictness, are divergences of NFN(csp[P|Q]κσ,σ).
- 6.
We say “one” here, because the four options for presenting it are congruent to each other.
- 7.
In such usage it will be necessary to ensure that any CSP used respects the no-selective-input regime required to make channel-based models works.
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Acknowledgements
Peter Welch [26] originally persuaded me to look at mobility in CSP.The fact that I managed to create a mobile parallel operator for him wasone of the main spurs towards looking for a general expressivity result.My initial work on CCS was inspired by the title of [5]. I had usefulconversations with many people about this work, notably Tony Hoare,Michael Goldsmith, Samson Abramsky, Rob van Glabbeek, Gavin Lowe andMichael Mislove. The presentation has benefited greatly from Cliff Jones’sadvice.
The author’s work on this paper was funded by grants from EPSRC and US ONR.
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Roscoe, A.W. (2010). CSP is Expressive Enough for π . In: Roscoe, A., Jones, C., Wood, K. (eds) Reflections on the Work of C.A.R. Hoare. Springer, London. https://doi.org/10.1007/978-1-84882-912-1_16
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