R. Milner. Action calculi V: Reflexive Action Calculi. MS., 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is to unify these calculi at a syntactic and operation level. We introduce a symmetric variant which extends the reach of action calculi to cover for example the Fusion calculus [2] and Yoshida s process graphs [3] These symmetric action calculi conservatively extend the reflexive action calculi [4] and have close links to category theory. Background Action calculi and their extensions [4] have constructs for names and nameabstraction in common. The intention is that these naming constructs should represent the naming constructs in other calculi. In addition, each action calculus has a ....

....which extends the reach of action calculi to cover for example the Fusion calculus [2] and Yoshida s process graphs [3] These symmetric action calculi conservatively extend the reflexive action calculi [4] and have close links to category theory. Background Action calculi and their extensions [4] have constructs for names and nameabstraction in common. The intention is that these naming constructs should represent the naming constructs in other calculi. In addition, each action calculus has a number of controls to express the features specific to a particular calculus. For instance, the ....

R. Milner. Action calculi V: Reflexive Action Calculi. MS., 1994.

....present in conventional imperative programming languages. Milner introduced action calculi [Mil96] as a graphical framework within which to explore a foundational understanding of such calculi. We introduce the symmetric action calculi, which conservatively extend the reflexive action calculi [Mil94] and which have di#erent naming primitives. The action calculus framework treats name #x# and name abstraction (x)P as primitive. Name abstraction declares x as a local name which expects an input. The way this calculus expresses the restriction (#x)P is by supplying # Computing Laboratory, ....

....the reflexive action calculi and briefly explain the corresponding graphs. By way of example we give an action calculus corresponding to the synchronous # calculus [MPW92] A more detailed account of action calculi may be found in the introductory paper [Mil96] and of reflexive action calculi in [Mil94]. The categorical models and corresponding type theory presentation is given in [GH97] and a tutorial paper [Gar99a] links the action calculi with other graphical grameworks studied in the literature. The plan is as follows. We construct a set of terms (Definition 2.1) and quotient the terms by ....

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R. Milner. Action calculi V: Reflexive Action Calculi. Manuscript, 1994.

....are the ( 0 translations of the original rules for LAMB. 6 Extensions of Action calculi Milner has introduced two extensions of action calculi: the higher order action calculi [Mil94b] which allow the substitution of actions as well as names for names, and the reflexive action calculi [Mil94a], which in the presence of higher order features gives recursion. We extend closed action calculi to include higher order and reflexive features, and obtain results analogous to those given in Section 4. 6.1 Higher order action calculi Recall the action calculus LAMB given in example 2.8. The ....

....setting: t) x = j xj ( t] x ) ap] x = ap j xj hh k (t)ii = x) h xi Omega id) Delta (hhtii) j xj = k hhap k ii = x)ap; j xj = k We have analogous results to those given in theorems 4.5, 4.7 and 4.8. 6. 2 Reflexive action calculi The reflexive action calculi [Mil94a] are action calculi extended by an additional operator p , called the reflexion operator, which constructs a term p t : m n from t : p Omega m p Omega n. This operator provides a notion of feedback and, together with the higher order features described in the previous section, is ....

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R. Milner. Action calculi V: Reflexive Action Calculi. Manuscript, 1994.

....= p ( q ( p q;p Omega id) Delta a Delta (p p;q ) Omega id) ae 6 m Omega n (s) m ( n (s) Remark 2. 3 Mifsud [Mif96] and Hasegawa [Has97] have shown that the reflexion axioms correspond to the axioms of the trace operator of Joyal, Street and Verity [JSV96] Milner and Jensen [Mil94] have also suggested an additional axiom m (id m ) id , which makes sense at the graphical level, although there are traced symmetric monoidal categories which do not satisfy this axiom. We write BC(K ) to denote the set of basic terms and its associated equational theory BG, specified by ....

....nodes and shows the full power of using names to describe these graphs. In this section, we describe the simple non nesting case and show the connection with sharing graphs. Throughout this paper, we concentrate on action graphs with reflexion, called the reflexive action graphs in the literature [Mil94]; we omit the reflexive adjective. We assume a fixed denumerable set X of names, each of which has a prime arity. We let x; y; z; range over names, and sometimes write x p to indicate that x has prime arity p. Definition 4.1 (Simple Action Terms) The set of simple action terms, ....

R. Milner. Action calculi V: Reflexive Action Calculi. Manuscript, 1994.

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