Non-deterministic Boolean Proof Nets

16 pagesInternational audienceWe introduce Non-deterministic Boolean proof nets to study the correspondence with Boolean circuits, a parallel model of computation. We extend the cut elimination of Non-deterministic Multiplicative Linear logic to a parallel procedure in proof nets. With the restriction of proof nets to Boolean types, we prove that the cut-elimination procedure corresponds to Non-deterministic Boolean circuit evaluation and reciprocally. We obtain implicit characterization of the complexity classes NP and NC (the efficiently parallelizable functions)

Substructural Logic for Orientable and Non-Orientable Surfaces

4 pp., extended abstractWe present a generalization of Permutative logic (PL) which is a non-commutative variant of Linear logic suggested by some topological investigations on the geometry of linear proofs. The original logical status based on a variety-presentation framework is simplified by extending the notion of q-permutation to the one of pq-permutation. Whereas PL is limited to orientable structures, we characterize the whole range of topological surfaces, orientable as well as non-orientable. The system we obtain is a surface calculus that enjoys both cut elimination and focussing properties and comes with a natural phase semantics whenever explicit context is considered

LINK: a Proof Environment based on Proof nets

Colloque avec actes et comité de lecture. internationale.International audienceLINK is a proof environment including proof nets-based provers for multiplicative linear logics: mixed linear logic, or recently called non-commutative logic (MNL), commutative linear logic (MLL) and non-commutative (or cyclic) linear logic (MCyLL). Its main characteristic is the provability analysis through automatic proof nets construction that is a powerful alternative to deal with resource management in proof search. These provers can be also seen as implementations of new connection methods for these linear logic fragments