Proof nets for unit-free multiplicative-additive linear logic

Abstract

A cornerstone of the theory of proof nets for unit-free multiplicative linear logic (MLL) is the abstract representation of cut-free proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof. Thus the problem of finding a satisfactory notion of proof net for unit-free multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory.

Uniform Circuits, & Boolean Proof Nets

“…A more general work can be done using the additive fragment expressiveness fully: a garbage is no more needed but the cut-elimination is no more confluent. Other approaches are interesting like the additivesà la Hughes and van Glabbeek [HvG03,HvG05]: we believe to realize a stronger seed-up.…”

“…Our encoding and the defined Boolean proof nets satisfy this property, so we restrict our attention to this setting. Strong normalization and confluence of the additive proof nets have been studied in various directions [Tor03], in the polarized fragment of LL [LdF04] or recently with a set of linkings on a formula [HvG03,HvG05]. We have not yet fully explored this last approach which seems to give another kind of speed-up.…”

Canonical Sequent Proofs via Multi-Focusing

“…MLL − proof nets are certainly the most concise canonical structures for this fragment. There are candidates to extend MLL − proof nets to broader fragments (MLL with units [14], MALL [12] or MELL) but they are not as satisfactory as for MLL − . The analysis we just made could be carried to MALL proof nets as introduced by Hughes and van Glabbeek [12] for the appropriate extension of definition 17 (in particular to take into account the fact that with MALL proof nets there is not only one linking but a set of linkings corresponding to the additive slices of the proof net).…”

“…We show that maximally multi-focused sequent proofs (modulo the weak "iso-polar" equivalence) are in one-to-one correspondence with MLL proof nets [9]: we show how to uniquely associate a maximally multi-focused proof to an MLL proof net. We also discuss proof nets in MALL without units [10,12] and for other fragments of linear logic: maximal multi-focusing proofs should also be applicable in various other richer logics where the nature of proof nets is less well developed or satisfying, such as linear logic with units and exponentials. This paper is organized as follows: in Sec.…”

From Proof Nets to the Free *-Autonomous Category

“…This should not be very hard, given the work done in [HvG03]. The true challenge is to include also the additive units.…”