We present an extension of the lambda-calculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good representations of state change. We describe and argue for the usefulness of an extension of LF by features inspired by hybrid logic, which has several benefits. For one, it shows how linear logic features can be decomposed into primitive operations manipulating abstract resource labels. More importantly, it makes it possible to realize a metalogical framework capable of reasoning about stateful deductive systems encoded in the style familiar from prior work with LLF, taking advantage of familiar methodologies used for metatheoretic reasoning in LF.Acknowledgments From the very first computer science course I took at CMU, Frank Pfenning has been an exceptional teacher and mentor. For his patience, breadth of knowledge, and mathematical good taste I am extremely thankful. No less do I owe to the other two major contributors to my programming languages

We show that, given an ordinary lambda-term and a normal resource lambda-term which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambda-term reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine.

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.

: Following J.-L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters --- there is no special constant !. We show here that the well-known result on D, stating that any term which possesses a type in D strongly normalises does not need a new reducibility argument, but is a mere consequence of strong normalization for natural deduction restricted to the conjunction and implication. The proof of strong normalization for natural deduction, and therefore our result, as opposed to reducibility arguments, can be carried out within primitive recursive arithmetic. On the other hand, this enlightens the relation between & and & that G. Pottinger has already wondered about, and can be applied to other situations, like the lambda calculus with multiplicities of G. Boudol. Key-words: Lambda calculus , intersection types , strong normalization. Logic, proof th...

We study the Taylor expansion of lambda-terms in a non-deterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambda-calculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.

We study type-directed encodings of the simply-typed λ-calculus in a session-typed π-calculus. The translations proceed in two steps: standard embeddings of simply-typed λ-calculus in a linear λ-calculus, followed by a standard translation of linear natural deduction to linear sequent calculus. We have shown in prior work how to give a Curry-Howard interpretation of the proofs in the linear sequent calculus as π-calculus processes subject to a session type discipline. We show that the resulting translations induce sharing and copying parallel evaluation strategies for the original λ-terms, thereby providing a new logically motivated explanation for these strategies.

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...

We propose an interpretation of a typed concurrent calculus of objects (conc&) based on the model of Abadi and Cardelli's imperative object calculus. The target of our interpretation is a version of the blue calculus, a variant of the pi-calculus that directly contains the lambda-calculus, with record and first-order types. We show that reduction and type judgements can be derived in a rather simple and natural way, and that our encoding can be extended to "self-types" and synchronisation primitives. We also prove some equational laws on objects.

: 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...