An Isomorphism between a Fragment of Sequent Calculus and an Extension of Natural Deduction

Abstract

Variants of Herbelin's λ-calculus, here collectively named Herbelin calculi, have proved useful both in foundational studies and as internal languages for the efficient representation of λ-terms. An obvious requirement of both these two kinds of applications is a clear understanding of the relationship between cut-elimination in Herbelin calculi and normalisation in the λ-calculus. However, this understanding is not complete so far. Our previous work showed that λ is isomorphic to a Herbelin calculus, here named λP, only admitting cuts that are both left-and right-permuted. In this paper we consider a generalisation λPh admitting any kind of right-permuted cut. We show that there is a natural deduction system λN h which conservatively extends λ and is isomorphic to λPh. The idea is to build in the natural deduction system a distinction between applicative term and application, together with a distinction between head and tail application. This is suggested by examining how natural deduction proofs are mapped to sequent calculus derivations according to a translation due to Prawitz. In addition to β, λN h includes a reduction rule that mirrors left permutation of cuts, but without performing any append of lists/spines.

The λ-Calculus and the Unity of Structural Proof Theory

“…Strong cut-elimination for λ Gtz is proved via an interpretation into the calculus of "delayed substitutions" λs of [10]; strong normalisation for λ Nat follows by isomorphism. These results constitute, for the logic under analysis here, considerable improvements over [16,1,27,8].…”

“…With these mappings one proves that normalisation is a "homomorphic" image of cut-elimination [28,24]. 8 The typing system of λ Nat clarifies the puzzling relation between cut and substitution. Consider rule Cut in λ Gtz and rule Subst in λ Nat .…”

“…As compared to [16], we covered full sequent calculus, where the constraints on left introduction that define Herbelin's fragment LJT are dropped. Already [1] and [7,8] considered implicitly Herbelin's programme. The improvement over these works is that the spine calculus of [1], when restricted to the logic of this paper, and the sequent systems in [7,8] are all fragments of Herbelin's LJT and, therefore, are under the restrictions already mentioned.…”

“…Indeed, the λ-calculus is isomorphic (even at level of reduction) to the structure V [u 1 , ..., u m ] (where V is a value, i.e. a variable or abstraction) [6,7], and only an extension of the λ-calculus, equipped with a distinction between applicative term and application, is isomorphic to the structure t[u 1 , ..., u m ] [7,8].…”

“…The developments in [6][7][8] only comprise, in the sequent calculus side, fragments of LJT . But LJT is a permutation-free fragment, where only a restricted form of left introduction is available and where the computational meaning of permutation (so typical of sequent calculus) is absent.…”