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## The Theory of Action calculi

In Confer-1, Action structures have been proposed as an algebra for both the syntax and the semantics of interactive computation. Action calculi are action structures of a concrete nature, with added structure, and which represent a wide variety of calculi of computation - functional, sequential, and concurrent. Each action in an action calculus is represented as an assembly of molecules; the syntactic binding of names is the means by which molecules are bound together. One action calculus differs from another only in its generators, called controls. The work on action calculi has been pursued this year in Cambridge in the following directions:

• Labelled transition systems and bisimulation congruences The semantics of a process calculus, several of which have been introduced during CONFER, can often be most intuitively given as a reduction relation, defined by a structural congruence and reduction axioms. The action calculus framework embraces a rather wide class of such definitions. A central problem is then that of deriving, from such a definition of reduction, a labelled transition semantics for which bisimulation is a satisfactory congruence. Considerable progress has been made, e.g. by Jensen [Jen98] for a form of graph rewriting approximating to a graphical presentation of action calculi, and by Sewell for special cases of term rewriting. Work on more general cases is being undertaken by Gardner, Jensen, Leifer, Milner and Sewell.
• Expressiveness A uniform framework for the operational semantics of calculi allows comparisons of expressiveness to be made. Using natural homomorphisms between control structures (the models of action calculi), it is possible to characterise important properties of calculi such as mobility of channels. Leifer has shown that the -calculus is mobile in this sense, whilst many other calculi such as CCS, the -calculus, and Petri nets, are not. This work, extending the work of Mifsud at Edinburgh, gives a precise meaning to the intuitive sense in which the -calculus is more expressive than, say, CCS.
• Gardner has introduced a type-theoretic description of action calculi [GH97] which develops her previous work on a name-free account of action calculi and gives a logical interpretation based on the general binding operators studied by Aczel. With her Ph.D student Hasegawa, at Edinburgh, she has shown that the type-theoretic account of higher-order action calculi corresponds to Moggi's commutative computational -calculus. With Gordon Plotkin and Andrew Barber, also at Edinburgh, they have introduced a further extension of action calculi, called the linear action calculi [BGHP97]. Linear action calculi extend the higher-order case, and the associated type theory corresponds to the linear type theories studied by Barber and Nick Benton of Cambridge. The conservative extension results follow from looking at the corresponding categorical models.
• Examples Several example action calculi have been considered, including a functional programming language with mutable store and exceptions, the ambient calculus of Abadi and Cardelli, and distributed process calculi.
• Higher-order reflexive action calculi The first of these examples required two previously separate enrichments of AC, namely, the higher-order and the reflexive extensions (the latter in a simplified form). The proof that an AC combining the two is well-defined reduces to a proof of strong normalisation for the higher-order beta-reduction of the molecular forms corresponding to this enriched AC. Leifer completed this proof and, in the process of doing so, developed several new techniques for handling molecular forms, including the formalisation of manipulations involving reflexive substitutions.

[[BGHP97]] Andrew Barber, Philippa Gardner, Masahito Hasegawa, and Gordon Plotkin.
From action calculi to linear logic,.
In Proceedings of CSL 97, Aarhus, 1997.

[[Gar98]] Philippa Gardner.
Closed action calculi.
Theoretical Computer Science, 1998.
To appear.

[[GH97]] Philippa Gardner and Masahito Hasegawa.
Types and models in higher-order action calculi.
In Proceedings of TACS 97, Sendai, Japan, 1997.

[[Jen98]] Ole Høgh Jensen.
PhD thesis, University of Cambridge.
Forthcoming, 1998.

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1/10/1998