Nondeterministic State Complexity of Positional Addition

Abstract

Consider nondeterministic finite automata recognizing base-k positional notation of numbers. Assume that numbers are read starting from their least significant digits. It is proved that if two sets of numbers S and T are represented by nondeterministic automata of m and n states, respectively, then their sum {s+t | s in S, t in T} is represented by a nondeterministic automaton with 2mn+2m+2n+1 states. Moreover, this number of states is necessary in the worst case for all k>=9.