Nondeterministic State Complexity of Positional Addition

TL;DR

This paper analyzes the state complexity of nondeterministic finite automata recognizing the sum of two sets of numbers in base-k positional notation, providing exact bounds and worst-case necessity for automata state counts.

Contribution

It establishes an exact upper bound and proves the necessity of the state count for automata recognizing sums of two number sets in nondeterministic automata.

Findings

The sum of two sets recognized by automata with m and n states can be recognized by an automaton with 2mn+2m+2n+1 states.

This state bound is tight and necessary in the worst case for all bases k≥9.

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.