Bases for AC 0 and other complexity classes Stefano Mazzanti Dipartimento di Culture del Progetto Università Iuav di Venezia Fondamenta delle Terese 2206, 30123 Venezia, Italy email: mazzanti@iuav.it Abstract Function complexity classes are defined as the substitution closure of finite function sets by improving a method of elimination of concatenation recursion from function algebras. Consequently, the set of AC 0 functions and other canonical complexity classes are defined as the substitution closure of a finite function set. Keywords: concatenation recursion on notation, substitution basis. MSC (2000) 03D20, 03D15. Function substitution is a very natural operation used in recursion-theoretic characterizations of computable functions. A finite function set F is a substitution basis for a function class C (and C is the substitution closure of F ) when C can be defined using only the functions in F , the projection functions and the substitution operator. The search for substitution bases started with a question of Grzegorczyk about the existence of a basis for the class of Kalmár’s elementary functions. Over the years, several authors discovered more and more simple bases, see [9] for a full historical account. Moreover, partial recursive functions, Grzegorczyk classes E n for n ≥ 2 [8, 9, 12] and polynomial time computable functions [11, 7] have a substitution basis. Recently, the set {x + y,x ˙ −y,x ∧ y, ⌊x/y⌋ , 2 |x| 2 } where x ∧ y is the bitwise and of x and y, has been shown to be a basis for the class TC 0 of functions computable by polysize, constant depth threshold circuits [13]. 1 In a previous paper inspired by [13], we showed that a function class closed with respect to substitution and concatenation recursion on notation (CRN) admits a substitution basis, provided that it contains integer division [10]. By applying this result to Clote-Takeuti characterizations of TC 0 and NC 1 [4], we obtained the above mentioned basis for TC 0 and a new basis for NC 1 because, according to [5], division is in TC 0 . In this paper, we improve the techniques and results of [10]. The existence of a basis for a function class does not need the division requirement anymore and a basis for AC 0 is introduced. The results of this paper are obtained by refining the method of elimination of CRN introduced in [10], based on the use of a functional operator, denoted 1 Names AC 0 ,TC 0 ,NC 1 are usually intended to denote language classes. However, in this paper they will always denote function classes, since no misunderstanding is possible. Moreover we consider only Dlogtime-uniform circuit families. 1