CRN and unary functions Stefano Mazzanti Facoltà di Design e Arti Università Iuav di Venezia Fondamenta delle Terese 2206, 30123 Venezia, Italy email: mazzanti@iuav.it Abstract The set of unary functions of complexity classes satisfying simple con- straints and defined by using concatenation recursion on notation is induc- tively characterized by means of concatenation iteration on notation. In particular, AC 0 ,TC 0 , NC 1 and NC unary functions are then inductively characterized using addition, composition and concatenation iteration on notation. Keywords: concatenation recursion on notation, AC 0 ,TC 0 , NC 1 and NC computable functions. MSC (2000) 03D20, 03D15. The characterization of functions computable with bounded resources is an im- portant issue of computational complexity. One of the main trends comes down from Computability Theory and concerns the inductive definition of complex- ity classes using a set of basic functions and functional operators like function substitution and weak forms of primitive recursion [16, 3]. In classical Computability Theory many efforts have been done to charac- terize the subset of unary (one-argument) functions of primitive and general recursive functions, see [13, 14, 7, 8, 4] and also [5, 6, 9, 10, 15] for recent re- sults. This is motivated by the simple underlying data structure and by the fact that unary functions constitute a monoid with respect to function composition. Moreover, functions can be “simulated” by unary functions, i. e. any function of finite arity can be defined as the composition of a unary function with an encoding of its arguments. On the other hand, for no subclass of the polynomial time computable func- tions a characterization of the unary functions has been given up to now (how- ever, unary linear-space computable functions, unary polynomial-space com- putable functions and unary elementary functions have been characterized as the inductive closure of a finite set of functions with respect to the composition and the bounded primitive iteration operators [11]). In this paper we characterize the set of unary functions of well known com- plexity classes, namely the class AC 0 of functions computable by polysize, con- stant depth boolean circuits, the class TC 0 of functions computable by poly- size, constant depth threshold circuits, the class NC 1 of functions computable by polysize, logarithmic depth boolean circuits and the class NC of functions computable in polylogarithmic time by PRAMs with a polynomial number of 1