A circuit uniformity sharper than DLogTime

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. [3], denoted DLT , that is: 1) Rational-uniformity implies DLogTime-uniformity, 2) we have NCRat NCDLT ( NCRat, 3) we have ∀k > 0, NCDLT ⊆ NC Rat , 4) Rationaluniformity preserves separation results known with DLogTime-uniformity. In other word, we obtain an interleaved hierarchy: NCRat ( NCDLT ( NCRat ⊆ · · · ⊆ NCRat ⊆ NCDLT ⊆ NC Rat ⊆ · · · ⊆ NC which implies NCRat = NCDLT . We also prove that Reg 6= NCRat. In other words, circuit build by rational relations compute relations not computable by rational relations. Finally, we consider circuit with unbounded fan-in, and we prove the standard result NCRat ⊆ ACRat ⊆ NC Rat for all k > 0.