Uniform circuits, & Boolean proof nets

Virgile Mogbil (1) , Vincent Rahli (2)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 ULTRA (Useful Logics, Types, Rewriting and their Automation) group

The relationship between Boolean proof nets of multiplicative linear logic (APN) and Boolean circuits has been studied [Ter04] in a non-uniform setting. We refine this results by taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofs-as-programs correspondence between proof nets and deterministic as well as non-deterministic Boolean circuits with a uniform depth-preserving simulation of each other. The Boolean proof nets class m&BN(poly) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the non-deterministic circuit class NNC(poly) is with non-deterministic variables. We obtain uniform-APN = NC and m&BN(poly) = NNC(poly) = NP

Mots clés: linear logic; implicit complexity; Boolean proof net; parallel cut-elimination; Boolean circuits; uniformity; non-determinism

Domaines: Logique en informatique [cs.LO]; Complexité [cs.CC]

Virgile Mogbil, Vincent Rahli. Uniform circuits, & Boolean proof nets. International Symposium of Logical Foundations of Computer Science (LFCS 2007), Jun 2007, New York, United States. pp. 401-421, ⟨10.1007/978-3-540-72734-7⟩. ⟨hal-00122980v2⟩