ABSTRACT. Refining and extending previous work by Retoré (1994), we develop a systematic approach to intersection types via natural deduction. We show how a step of beta reduction can be seen as performing, at the level of typing derivations, Prawitz reductions in parallel. Then we derive as immediate consequences of Subject reduction the main theorems about normalization for intersection types: for system D, strong normalization, for system DΩ, the leftmost reduction termination for terms typable without Ω.

ABSTRACT. We investigate the possibility of a semantic account of the execution time (i.e. the number of beta v-steps leading to the normal form, if any) for the shuffling calculus, an extension of Plotkin's untyped call-by-value lambda-calculus. For this purpose, we use a linear logic based denotational model that can be seen as a non-idempotent intersection type system: relational semantics. Our investigation is inspired by similar ones for linear logic proof-nets and untyped call-by-name lambda-calculus. We first prove a qualitative result: a (possibly open) term is normalizable for weak reduction (which does not reduce under abstractions) if and only if its interpretation is not empty. We then show that the size of type derivations can be used to measure the execution time. Finally, we show that, differently from the case of linear logic and call-by-name lambda-calculus, the quantitative information enclosed in type derivations does not lift to types (i.e. to the interpretation of terms). In order to get a genuine semantic measure of execution time in a call-by-value setting, we conjecture that a refinement of its syntax and operational semantics is needed.

ABSTRACT. We study the question of extending the BCD intersection type system with additional type constructors. On the typing side, we focus on adding the usual rules for product types. On the subtyping side, we consider a generic way of defining a subtyping relation on families of types which include intersection types. We find back the BCD subtyping relation by considering the particular case where the type constructors are intersection, omega and arrow. We obtain an extension of BCD subtyping to product types as another instance. We show how the preservation of typing by both reduction and expansion is satisfied in all the considered cases. Our approach takes benefits from a ``sub-formula property'' of the proposed presentation of the subtyping relation.

ABSTRACT. A new proof of strong normalization for simple type assignment for λ-calculus is obtained, through a translation from this system to a system of uniform intersection types, which is equivalent to it as typability power and whose strong normalization property can be easily proved by induction on derivation.

ABSTRACT. We show (1) For each strongly normalizable lambda term M, with beta eta normal form N, there exists an intersection type A such that in BCD (without top element) we have |-M:A and N is the unique beta-eta normal term s.t.|- N:A. A similar result holds for finite sets of strongly normalizable terms. (2) For each intersection type A, if the set of all closed terms M such that in BCD (without top element) |-M:A is infinite, then when closed under beta-eta conversion this set forms an adequate numeral system for untyped lambda calculus.In particular, all these terms are generated from a single 0 by the application of a successor S, S(...(S0)...) and by beta-eta conversion.

ABSTRACT. Gradual typing integrates dynamic and static types. Since its introduction, it has been successfully applied to several extensions of simple types, including subtyping, parametric polymorphism and substructural types. This work studies its application to intersection type systems. We introduce a new approach to define a gradual version of the original intersection type system of Coppo and Dezani. We then present a new operational semantics for terms typed with static and dynamic intersection types, which enables dynamic type casts and identifies the causes for type errors in a framework inspired by the blame calculus. Finally we prove several properties of our system including a correctness criteria and soundness of the extension of the original gradual type system.