Least fixpoints as meanings of recursive definitions.

We introduce a calculus which is a direct extension of both the and the π calculi. We give a simple type system for it, that encompasses both Curry's type inference for the -calculus, and Milner's sorting for the π-calculus as particular cases of typing. We observe that the various continuation passing style transformations for -terms, written in our calculus, actually correspond to encodings already given by Milner and others for evaluation strategies of -terms into the π-calculus. Furthermore, the associated sortings correspond to well-known double negation translations on types. Finally we provide an adequate cps transform from our calculus to the π-calculus. This shows that the latter may be regarded as an "assembly language", while our calculus seems to provide a better programming notation for higher-order concurrency.

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We introduce an asynchronous version of Milner's ß-calculus, based on the idea that the messages are elementary processes that can be sent without any sequencing constraint. We show that this simple message passing discipline, together with the restriction construct making a name private for an agent, is enough to encode the synchronous communication of the ß-calculus. 1. Introduction. The purpose of this note is to introduce an asynchronous variant of the ß-calculus of Milner, Parrow and Walker [8]. The ß-calculus is an extension of CCS, based on previous work by Engberg and Nielsen [5], that deals with name passing: in this calculus, agents pass channel names to other agents through named channels. The expressiveness of this link passing discipline has been demonstrated in [8] by a series of examples. Later on, Milner showed in [9] that even a restricted fragment of the original ß-calculus is enough to encode the -calculus. More precisely, he showed that one can mimic in a "mini" ß...

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. This paper presents a formal model of the concurrent graph reduction implementation of non-strict functional programming. This model diers from other models in that: It represents concurrent rather than sequential graph reduction. It represents low-level considerations such as garbage collection. It uses techniques from concurrency theory to simplify the presentation. There are three presentations of this model: An operational semantics based on graph reduction. A denotational semantics in the domain D ' (D !D)? . A program logic and proof system based on coppo types. We can then use abramsky and ong's techniques from the lazy - calculus to show that the denotational semantics is fully abstract for the operational semantics. This proof requires some results about the operational semantics: Since the operational semantics includes garbage collection, reduction is not conuent. We nd a conuent reduction strategy which has the same convergence properties as gr...

We introduce a refinement of the λ-calculus, where the argument of a function is a bag of resources, that is a multiset of terms, whose multiplicities indicate how many copies of them are available. We show that this "λ-calculus with multiplicities" has a natural functionality theory, similar to Coppo and Dezani's intersection type discipline. In our functionality theory the conjunction is managed in a "multiplicative" manner, according to Girard's terminology. We show that this provides an adequate interpretation of the calculus, by establishing that a term is convergent if and only if it has a non-trivial functional character.

In this paper we present a non-deterministic call-by-need (untyped) lambda calculus nd with a constant choice and a let-syntax that models sharing. Our main result is that nd has the nice operational properties of the standard lambda calculus: confluence on sets of expressions, and normal order reduction is sufficient to reach head normal form. Using a strong contextual equivalence we show correctness of several program transformations. In particular of lambdalifting using deterministic maximal free expressions. These results show that nd is a new and also natural combination of non-determinism and lambda-calculus, which has a lot of opportunities for parallel evaluation. An intended application of nd is as a foundation for compiling lazy functional programming languages with I/O based on direct calls. The set of correct program transformations can be rigorously distinguished from non-correct ones. All program transformations are permitted with the slight exception that for transformations like common subexpression elimination and lambda-lifting with maximal free expressions the involved subexpressions have to be deterministic ones.

Plotkin used the models of reduction in order to obtain a semantic characterization of static type inference in the pure -calculus. Here we apply these models to the study of a nondeterministic language, obtaining results analogous to Plotkin's. 1 Introduction The models of reduction are a generalization of the usual syntactic -models for the pure -calculus (see [Plo92] and the references therein). If a term M reduces to a term N then its interpretation in a model of reduction is "smaller than" or equal to the interpretation of N (and not necessarily equal as in -models). Plotkin obtained a series of soundness and completeness results for static type inference with respect to models of reduction. With type inference in mind, it seems natural that M and N be interpreted differently, since it may be possible to infer a type for N but not for M . The study of nondeterministic languages gives rise to an alternative motivation for considering models of reduction. In nondeterministic lang...

We present a natural semantics for the untyped lazy -calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (dened inductively) and divergent behaviour (dened co-inductively). This semantics is equivalent to a small step reduction semantics that corresponds closely to our operational intuitions about McCarthy's amb. We present equivalences for convergent and divergent behaviour based on the natural semantics and prove a Context Lemma for the convergence equivalence. We then give a -theory l 8 , based on the equivalences for convergent and divergent behaviour. Since it is able to distinguish between programs that dier only in their divergent behaviour, the -theory is more discriminating than equational theories based on current domain-theoretic models. It is therefore more suitable for reasoning about functional programs containing McCarthy's amb. Contents 1 Introduction 2 2 Related Work 3 3 ...