The Blue Calculus is a direct extension of both the lambda and the pi calculi. In a preliminary work from Gérard Boudol, a simple type system was given that incorporates Curry's type inference for the lambda-calculus. In the present paper we study an implicit polymorphic type system, adapted from the ML typing discipline. Our typing system enjoys subject reduction and principal type properties and we give results on the complexity for the type inference problem. These are interesting results for the blue calculus as a programming notation for higher-order concurrency.

ion and concretization operator from pattern list type to uniform ML list type The abstraction AbsML that maps every pattern list type scheme to a set of standard ML type scheme results from the definition of the concretization and the ordering vP on pattern list types. Property 2. The abstraction and concretization functions (AbsML ; ConcML ) define a lower approximation. ae ` E ) øML list ae ` null(E1) ) bool ae ` E1 ) øML ae ` E2 ) øML list ae ` E1 :: E2 ) øML list ae ` E ) øML list ae ` head(E) ) øML ae ` E1 ) øML list ae ` tail(E1) ) øML list Figure 8: Standard ML list operator behavior For example the abstraction of the pattern list type (ff ! ff)(ff ! num)(!) list of the list x:x :: x:0 :: [] is: (ff ! num) list. With respect to the abstraction and concretization functions (AbsML ; ConcML ), we derive the set of rules describing the behavior of list operators on ML-types--- listed on figure 8---from the set of rules describing the behavior of list operators on patter...