Call-by-Value and Call-by-Name Dual Calculi with Inductive and Coinductive Types

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it has the duality of inductive and coinductive types, that is, the duality of terms and coterms for inductive and coinductive types, and the duality of their reduction rules. Its strong normalization is also proved, which is shown by translating it into a second-order dual calculus. The strong normalization of the second-order dual calculus is proved by translating it into the second-order symmetric lambda calculus. This paper then introduces a call-by-value system and a call-by-name system of the dual calculus with inductive and coinductive types, and shows the duality of call-by-value and call-by-name, their Church-Rosser properties, and their strong normalization. Their strong normalization is proved by translating them into the non-deterministic dual calculus with inductive and coinductive types.Comment: The conference version of this paper has appeared in RTA 200

An Approach to Call-by-Name Delimited Continuations

International audienceWe show that a variant of Parigot's λμ-calculus, originally due to de Groote and proved to satisfy Böhm's theorem by Saurin, is canonically interpretable as a call-by-name calculus of delim- ited control. This observation is expressed using Ariola et al's call-by-value calculus of delimited control, an extension of λμ-calculus with delimited control known to be equationally equivalent to Danvy and Filinski's calculus with shift and reset. Our main result then is that de Groote and Saurin's variant of λμ-calculus is equivalent to a canonical call-by-name variant of Ariola et al's calculus. The rest of the paper is devoted to a comparative study of the call-by-name and call-by-value variants of Ariola et al's calculus, covering in particular the questions of simple typing, operational semantics, and continuation-passing-style semantics. Finally, we discuss the relevance of Ariola et al's calculus as a uniform framework for representing different calculi of delimited continuations, including "lazy" variants such as Sabry's shift and lazy reset calculus

The First-Order Hypothetical Logic of Proofs

The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Proofs and Refutations for Intuitionistic and Second-Order Logic

The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results

Stream Associative Nets and Lambda-mu-calculus

$\Lambda\mu$-calculus has been built as an untyped extension of Parigot's $\lambda\mu$-calculus in order to recover Böhm theorem which was known to fail in $\lambda\mu$-calculus. An essential computational feature of $\Lambda\mu$-calculus for separation to hold is the unrestricted use of abstractions over continuations that provides the calculus with a construction of streams. Based on the Curry-Howard paradigm Laurent has defined a translation of $\Lambda\mu$-calculus in polarized proof-nets. Unfortunately, this translation cannot be immediately extended to $\Lambda\mu$-calculus: the type system on which it is based freezes \Lm-calculus's stream mechanism. We introduce \emph{stream associative nets (SANE)}, a notion of nets which is between Laurent's polarized proof-nets and the usual linear logic proof-nets. SANE have two kinds of \lpar (hence of \ltens), one is linear while the other one allows free structural rules (as in polarized proof-nets). We prove confluence for SANE and give a reduction preserving encoding of $\Lambda\mu$-calculus in SANE, based on a new type system introduced by the second author. It turns out that the stream mechanism at work in $\Lambda\mu$-calculus can be explained by the associativity of the two different kinds of \lpar of SANE. At last, we achieve a Böhm theorem for SANE. This result follows Girard's program to put into the fore the separation as a key property of logic