A circuit uniformity sharper than DLogTime

Guillaume Bonfante (1) , Virgile Mogbil (2)

1 CARTE - Theoretical adverse computations, and safety
2 LIPN - Laboratoire d'Informatique de Paris-Nord

We consider a new notion of circuit uniformity based on the concept of rational relations, called Rational-uniformity and denoted Rat. Our goal is to prove it is sharper than DLogTime-uniformity, the notion introduced by Barrington et al. in 1992, denoted DLT, that is: 1) Rational-uniformity implies DLogTime-uniformity, 2) we have NC^0-Rat ⊊ NC^0-DLT ⊊ NC^1-Rat, 3) we have ∀k≥0, NC^k-DLT ⊆ NC^k+1-Rat, 4) Rational-uniformity preserves separation results known with DLogTime-uniformity. In other word, we obtain an interleaved hierarchy: NC^0-Rat ⊊ NC^0-DLT ⊊ NC^1-Rat ⊆ ... ⊆ NC^k-Rat ⊆ NC^k-DLT ⊆ NC^k+1-Rat ⊆ ... ⊆ NC, which implies NC-Rat = NC-DLT . We also prove that Reg ≠ NC^0-Rat. In other words, circuits build by rational relations compute relations not computable by rational relations. Finally, we consider circuits with unbounded fan-in, and we prove the standard result NC^k-Rat ⊆ AC^k-Rat ⊆ NC^k+1-Rat for all k≥0.

Mots clés: boolean circuit; uniformity; complexity theory; NC (Nick's Class); automata

Domaines: Complexité [cs.CC]

Guillaume Bonfante, Virgile Mogbil. A circuit uniformity sharper than DLogTime. 2012. ⟨hal-00701420⟩