Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the well-known call-by-value and call-by-name CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages.Comment: In Proceedings DCM 2011, arXiv:1207.682

Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit substitutions

International audienceThe lambda-bar-mu-mu-tilde-calculus, defined by Curien and Herbelin, is a variant of the lambda-mu-calculus that exhibits symmetries such as terms/contexts and call-by-name/call-by-value. Since it is a symmetric, and hence a non-deterministic calculus, usual proof techniques of normalization needs some adjustments to work in this setting. Here we prove the strong normalization (SN) of simply typed lambda-bar-mu-mu-tilde-calculus with explicit substitutions. For that purpose, we first prove SN of simply typed lambda-bar-mu-mu-tilde-calculus (by a variant of the reducibility technique from Barbanera and Berardi), then we formalize a proof technique of SN via PSN (preservation of strong normalization), and we prove PSN by the perpetuality technique, as formalized by Bonelli

Towards a canonical classical natural deduction system

Preprint submitted to Elsevier, 6 July 2012This paper studies a new classical natural deduction system, presented as a typed calculus named lambda-mu- let. It is designed to be isomorphic to Curien and Herbelin's lambda-mu-mu~-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's lambda-mu -calculus with the idea of "coercion calculus" due to Cervesato and Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus and the mentioned isomorphism Theta offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of "read-back" of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the lambda-mu-mu~-calculus, that explains co-terms and cuts as, respectively, contexts and hole- filling instructions. lambda-mu-let is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and provides the "read-back" map and the formalized semantics, based on the precise notions of context and "hole-expression" provided by lambda-mu-let. We use "read-back" to achieve a precise connection with Parigot's lambda-mu , and to derive lambda-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics , when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in lambda-mu-mu~ ("co-context"), with which one can give a new de nition of eta-reduction

Towards a canonical classical natural deduction system

This paper studies a new classical natural deduction system, presented as a typed calculus named \lml. It is designed to be isomorphic to Curien-Herbelin's calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's $\lambda\mu$-calculus with the idea of ``coercion calculus'' due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. \lml is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of Curien-Herbelin's calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by \lml. The third problem is the lack of a robust process of ``read-back'' into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of $\lambda$-calculi for call-by-value. An isomorphic counterpart to the $Q$-subsystem of Curien-Herbelin's-calculus is derived, obtaining a new $\lambda$-calculus for call-by-value, combining control and let-expressions.Fundação para a Ciência e a Tecnologia (FCT

Control Reduction Theories: the Benefit of Structural Substitution

L'article contient une annexe historique par Matthias Felleisen sur la génèse des opérateurs de contrôle à l'université d'Indiana à la fin des années 80.International audienceThe historical design of the call-by-value theory of control relies on the reification of evaluation contexts as regular functions and on the use of ordinary term application for jumping to a continuation. To the contrary, the lambda-C-tp control calculus, developed by the authors, distinguishes between jumps and terms. This alternative calculus, which derives from Parigot's lambda-mu-calculus, works by direct "structural substitution" of evaluation contexts. We review and revisit the legacy theories of control and argue that lambda-C-tp provides an observationally equivalent but smoother theory. In an additional note contributed by Matthias Felleisen, we review the story of the birth of control calculi during the mid to late eighties at Indiana University

The discriminating power of multiplicities in the λ\lambda-calculus

The $\lambda$-calculus with multiplicities is a refinement of the lazy $\lambda$-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 $\lambda$-terms. We prove in particular that the observational equivalence induced by contexts with multiplicities coincides with the equality of Lévy-Longo trees associated with $\lambda$-terms. This is a consequence of the characterization we give of the corresponding observational precongruence, as an intensional preorder involving $\eta$-expansion, namely Ong's lazy Plotkin-Scott-Engeler preorder

The call-by-value λ\lambda-calculus, the SECD machine, and the π\pi-calculus

We present an encoding of the call-by-value $\lambda$-calculus into the $\pi$-calculus, alternative to the well-known Milner's encodings. We show that our encoding is barbed congruent (under typed contexts) to Milner's "light" encoding, and that it takes two $\pi$-steps to mimic a beta-reduction for normalizing terms. We describe a translation of Plotkin's SECD machine into the $\pi$-calculus, and show that there is an operational correspondence between a SECD machine and its encoding. Equipped with a notion of a state-based machine and two kinds of correspondences between them, we compare the encodings of the call-by-value $\lambda$-calculus and the SECD machine into the $\pi$-calculu