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.

This paper studies the interplay between functional application and nondeterministic choice in the context of untyped λ-calculus. We introduce an operational semantics which is based on the idea of must preorder, coming from the theory of process algebras. To characterize this relation, we build a model using the classical inverse limit construction, and we prove it fully abstract using a generalization of Böhm trees.

The -calculus with multiplicities is a refinement of the lazy -calculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over the usual -terms. We prove in particular that the observational equivalence induced by contexts with multiplicities coincides with the equality of L'evyLongo trees associated with -terms. This is a consequence of the characterization we give of the corresponding observational precongruence, as an intensional preorder involving j-expansion, namely Ong's lazy Plotkin-Scott-Engeler preorder. 1 Introduction The -calculus with multiplicities was introduced in [5] for the purpose of studying the relationship between the -calculus and Milner's ß-calculus [13]. It is a "resource conscious" refinement of the -calculus, based on the following observation: in an application MN the argument N is infini...

: In this paper we study the semantics of the -calculus induced by Milner's encoding into the ß-calculus. We show that the resulting may testing preorder on -terms coincides with the inclusion of L'evy-Longo trees. To establish this result, we use a refinement of the -calculus where the argument of a function may be of limited availability. In our - calculus with multiplicities, evaluation is deterministic, but it may deadlock, due to the lack of resources. We show that this feature is enough to make the -calculus as discriminating as the ß-calculus. Key-words: functional and concurrent languages, semantics, lambda-calculus (R'esum'e : tsvp) Partially supported by the ESPRIT Basic Research Project 6454 - CONFER. Present address: Universit`a di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy. Unite de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77 -- Telec...