Focusing and proof-nets in linear and non-commutative logic

Abstract

Linear Logic [4] has radsed a lot of interest in computer research, especially because of its resource sensitive nature. One hne of research studies proof construction procedures and their interpretation as computationil models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Focusing", has been described in [2]. Pocusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Pocusing Ccin also be interpreted in the proof-net formalism, where it appecirs, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the CEise of the Non-Commutative logic of [1], and all the computational exploitation of Pocusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case.