On the base-dependence of sets of numbers recognizable by finite automata

Abstract

It is known that the set of powers of two is recognizable by a finite automaton if the notational base used for representing numbers is itself a power of two but is unrecognizable in all other bases. On the other hand, the set of multiples of two is recognizable no matter what the notational base. It is shown that the latter situation is the exception, the former the rule: the only sets recognizable independently of base are those which are ultimately periodic; others, if recognizable at all, are recognizable only in bases which are powers of some fixed positive integer. © 1969 Swets & Zeitlinger B.V.