The Φ-calculus – a Hybrid Extension of the Π-calculus to Embedded Systems

ions apply to concretions more or less exactly as in the π-calculus. Definition 6.3. The application G@C of a φ-abstraction and concretion is defined as follows: ((~ w)P )@ν~z〈 ~y 〉.Q =def ν~z( {~y/~ w}P ‖ Q ) where ~ y and ~ w have the same length, and where names in N must be substituted for names in N , and names in X for names in X . Applications can of course be carried out in the context of an environment E. Example 6.4. For any E (E, (axb).a.[x . = 2].b@〈 cyd 〉.[y . = 3]) = (E, c.[y . = 2].d ‖ [y . = 3] ). We now introduce the environmental restriction operator new xP for x ∈ X . The intent of this operator is to make these variables local to the process P . In the literature, this operator is called “variable hiding”, and sometimes “abstraction”. In φ-calculus usage, however, it is much more akin to the restriction operator νa. Whereas abstractions apply to concretions, this “abstraction” operator has no corresponding concretion. So we define it as an operator from processes to processes. Definition 6.5. For x ∈ X the operator new xP is called the environmental restriction operator. This operator restricts any e-action mentioning the variable x or ẋ (we call such actions x-actions). We allow for α-conversions under the scope of new x, replacing occurrences of x in x-actions by a new variable not occurring free in P . We turn to the rules for commitments in the full φ-calculus. For brevity we do not define reaction rules separately, as is done in Milner. Definition 6.6 (φ-commitments). The commitments of embedded systems (aside from those involving the replication operator) are given in Figure 6.1. In the Sum-e rules, e is an environmental action of the form ψ → (c . = d;F . = G; I . = J) and E is the environment (c, F, I). Further, α ∈ N ∪N , and μ is either an α or an e. Notice the effect of the Env -commitment rule: it is simply to introduce a fresh new variable into P . This variable is local to the scope of new x, as desired. Note: the substitution w/x includes occurrences of the variable x in e-actions under the scope of new x. The flow transition rules for the φ-calculus include those of the φc-calculus. Flow transitions are allowed for processes (and not abstractions) as in Definition 4.13. We need to add a flow transition rule for the environmental restriction operator; this works in analogy with the Res rule in Definition 4.13. Res − x : (E,P [w/x]) t → (E′, P [w/x]) (E, new xP ) t → (E′, new xP ) (w not mentioned in E, not free in P ). The reason for the side conditions can be seen from an example: Example 6.7. Let P = new x[x . = 1] and Q = new y[y . = 1]. Then P and Q are structurally equivalent by α-conversion. Let E be the environment   x : 0 ẋ : 1 TRUE  . If we did not have the 14 WILLIAM C. ROUNDS AND HOSUNG SONG Sum − pi : (E, M + αA+N) α → (E,A) Sum − e : (E,M + e.P +N) e → ((d,G, J), P ) if ψ(c) is true; L −React : (E,P ) w → (E,G) (E,Q) w → (E,C) (E, P ‖ Q ) τ → (E,G@C) R − React : (E,P ) w → (E,C) (E,Q) w → (E,G) (E, P ‖ Q ) τ → (E,G@C) L− par : (E,P ) μ → (E′, A) (E, P ‖ Q ) μ → (E, A ‖ Q ) R − par : (E,Q) μ → (E′, A) (E, P ‖ Q ) μ → (E, P ‖ A ) Res − pi : (E,P ) μ → (E′, A) (E, νwP ) μ → (E, νwA) if μ / ∈ {w, w} Res − x : (E,P ) μ → (E′, Q) (E, new xP ) μ → (E, new xQ) if μ is not an x-action Env : (E, new xP ) τ → (E,P [w/x]) (w ∈ X , w not mentioned in E and not free in P .) Figure 1. Commitment rules for the φ-calculus side conditions, then (E,P ) t → for any t, but (E,Q) cannot flow because y is not defined in the environment. Once again, the new x declaration in P enforces that the local variable x is different from any variable in the environment of P . 6.2. Recursion. The π-calculus rule for recursion is !P ≡ P ‖ !P where ≡ is structural equivalence. We adopt this for the φ-calculus, and we handle commitments just as in the π-calculus with provision for possible environmental changes. Thus when μ is either a π-action or an e-action (E, P ‖ !P ) μ → (E′, Q) (E, !P ) μ → (E′, Q) . The only other thing that we need to add is the flow transition rule. This is simply