Non deterministic Linear logic: application to Boolean circuits

Virgile Mogbil (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

Following the Curry-Howard isomorphism but for parallel model of computation, we study proof nets of the non-deterministic multiplicative Linear logic, i.e. with an explicit rule to sum up (as done in [Mau03]). We define nmBN (k(n)) the uniform families of multiplicative Boolean proof nets with O(k(n)) amount of explicit nondeterminism. For k(n) respectively polynomial and constant, we obtain a Curry-Howard characterization of the complexity class respectively NP and NC (the efficiently parallelizable functions). If k(n) is polylogarithmic then we characterize the class NNC (polylog), that is, NC with polylogarithmic amount of non-deterministic variables. The depth of a proof net being defined to be the maximal logical depth of cut formulas in it, the cut-elimination corresponds to Boolean circuit evaluation and reciprocally.

Mots clés: Implicit computational complexity; Linear logic; Nondeterministic Boolean circuits; Nondeterministic Boolean proof nets; Parallel cut-elimination

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

Virgile Mogbil. Non deterministic Linear logic: application to Boolean circuits. 2007. ⟨hal-00122981v2⟩