Automaticity IV : sequences, sets, and diversity

This paper studies the descriptional complexity of (i) sequences over a finite alphabet; and (ii) subsets of N (the natural numbers). If (s (i )) i ~0 is a sequence over a finite alphab et 0394, then we define the k-automaticity of s, Aks(n), to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 ~ i ~ n, takes i expressed in base k as input and computes s(i). We give examples of sequences that have high automaticity in all bases k; for example, we show that the characteristic sequence of the primes has k-automaticity Aks(n) = 03A9(n1/43) for all k ~ 2, thus making quantitative the classical theorem of Minsky and Papert that the set of primes expressed in base 2 is not regular. Research supported in part by a grant from NSERC. Manuscrit regu le 5 avril 1996.