Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be single-conclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Gore et al., where a complete class of cuts is encapsulated into special "unnest" rules and (3) a cut-free labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus.Comment: In Proceedings CL&C 2010, arXiv:1101.520

The duality of computation

http://www.acm.orgInternational audienceWe present the lambda-bar-mu-mu-tilde-calculus, a syntax for lambda-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call-by-value. This calculus is derived from implicational Gentzen's sequent calculus LK, a key classical logical system in proof theory. Under the Curry-Howard correspondence between proofs and programs, we can see LK, or more precisely a formulation called LK-mu-mu-tilde, as a syntax-directed system of simple types for lambda-bar-mu-mu-tilde-calculus. For lambda-bar-mu-mu-tilde-calculus, choosing a call-by-name or call-by-value discipline for reduction amounts to choosing one of the two possible symmetric orientations of a critical pair. Our analysis leads us to revisit the question of what is a natural syntax for call-by-value functional computation. We define a translation of lambda-mu-calculus into lambda-bar-mu-mu-tilde-calculus and two dual translations back to lambda-calculus, and we recover known CPS translations by composing these translations

Realizability Interpretation and Normalization of Typed Call-by-Need λ\lambda-calculus With Control

We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need \lambda$-$calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need \lambda$$-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice

Call-by-Value Is Dual to Call-by-Name – Reloaded

Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµ-calculus of Parigot (1992). We give translations from the λµ-calculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, translating from λµ to dual and then ‘reloading ’ from dual back into λµ yields a term equal to the original term. Composing the translations with duality on the dual calculus yields an involutive notion of duality on the λµ-calculus. A previous notion of duality on the λµ-calculus has been suggested by Selinger (2001), but it is not involutive. Note This paper uses color to clarify the relation of types and terms, and of source and target calculi. If the URL below is not in blue please download the color version fro