Paolo Tranquilli

Bio: Paolo Tranquilli is an academic researcher from École normale supérieure de Lyon. The author has contributed to research in topics: Lambda calculus & Interaction nets. The author has an hindex of 1, co-authored 1 publications receiving 42 citations.

Intuitionistic differential nets and lambda-calculus

Abstract: We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier?s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol?s untyped ?-calculus with resources extended with a linear?nonlinear reduction a la Ehrhard and Regnier?s differential ?-calculus. Such reduction comes in two flavours: baby-step and giant-step s-reduction. The translation, based on Girard?s encoding A?B~!A?B and as such extending the usual one for ?-calculus into proof nets, enjoys bisimulation for giant-step s-reduction. From this result we also derive confluence of both reductions.

Collapsing non-idempotent intersection types

Abstract: We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi.

Solvability in resource lambda-calculus

Abstract: The resource calculus is an extension of the λ-calculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a formal sum. Using the λ-calculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the non-determinism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability.

Linearity, Non-determinism and Solvability

Abstract: We study the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption. Since this calculus is non-deterministic, two different notions of solvability arise, one optimistic (angelical, may) and one pessimistic (demoniac, must). We give a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the must-solvability. Finally, we discuss the open problem of a complete characterization of the must-solvability.

Proof nets and the call-by-value λ-calculus

Abstract: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize an isomorphism between the two systems: every single rewriting step on the calculus maps to a single step on proof nets, and vice-versa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way, and identify a subcalculus that is shown to be as expressive as the full calculus.

Parallel Reduction in Resource Lambda-Calculus

Abstract: We study the resource calculus --- the non-lazy version of Boudol's *** -calculus with resources. In such a calculus arguments may be finitely available and mixed, giving rise to nondeterminism, modelled by a formal sum. We define parallel reduction in resource calculus and we apply, in such a nondeterministic setting, the technique by Tait and Martin-Lof to achieve confluence. Then, slightly generalizing a technique by Takahashi, we obtain a standardization result.