Strong normalization property for second order linear logic

Abstract

International audienceThe paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard's original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard's proof-nets, and we apply to sps Gandy's method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard's reducibility candidates

Confluence of Pure Differential Nets with Promotion

“…As is evident in [10], this is the first step in proving strong normalization in the typed case 2 . Furtherly, as can be deducted from [9], this can be the ground for new work on calculi with explicit substitutions: whether by extending some results to untyped calculi; or by considering explicit substitutions for nondeterministic calculi akin to Boudol's λ-calculus with resources (see [6]).…”

“…One can already see two big differences with respect to LL and the work done with it in [10]: firstly, sums may arise even without the "logical" step of dereliction on box; moreover, the codereliction on box rule, which reduces a commutative cut, changes the possible cuts on all other cuts of the box. These problems prevent an immediate adaptation of the measures used in [10]. The nets in Figure 5 are examples already known in LL showing issues about correctness and types.…”

“…In [15], Danos proved the counterpart of the finite development theorem for MELL, and Pagani and Tortora de Falco did the same for the whole of second order LL in [10]. In this setting the actual definition of what a "new" redex is gets more technical.…”

“…Our technique, reminiscent of the work done on LL in [10], uses a finite development theorem used to prove a strong confluence property of a suitable notion of parallel reduction.…”

Compressing Polarized Boxes

“…So even local confluence is nontrivial and requires a new technique. Unfortunately, the state of the art in terms of proof of strong normalization for proof nets relies on local confluence [51], [58].…”