Magic Numbers for Symmetric Difference NFAs

Abstract

Iwama et al [1] showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2ⁿ– α states, for all n ≥ 7 and 5 ≤ α ≤ 2n – 2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata [2]. In the binary case, we show that for any n ≥ 4, there is an n-state ⊕-NFA which needs 2^{n − − 1} + 2^{k − − 1} –1 states, for 2< k ≤ n – 1. In the unary case, we prove the following result for a large practical subclass of unary symmetric difference nondeterministic finite automata: For all n ≥ 2, we show that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent deterministic finite automaton with 2ⁿ – α states, where 0 < α< 2^n − 1. For each n ≥ 2, we quantify such values of α precisely.