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

Curry-Howard for sequent calculus at last!

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction.(undefined

The polarized λ-calculus

A natural deduction system isomorphic to the focused sequent calculus for polarized intuitionistic logic is proposed. The system comes with a language of proof-terms, named polarized λ-calculus, whose reduction rules express simultaneously a normalization procedure and the isomorphic copy of the cut-elimination procedure pertaining to the focused sequent calculus. Noteworthy features of this natural deduction system are: how the polarity of a connective determines the style of its elimination rule; the existence of a proof-search strategy which is equivalent to focusing in the sequent calculus; the highlydisciplined organization of the syntax - even atoms have introduction, elimination and normalization rules. The polarized λ-calculus is a programming formalism close to call-by-push-value, but justified by its proof-theoretical pedigree.This research was financed by Portuguese Funds through FCT Fundac¸ao para a Ci ˜ encia ˆ e a Tecnologia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio