A lower bound on the state complexity of transforming two-way nondeterministic finite automata to unambiguous finite automata

Abstract

This paper establishes a lower bound on the number of states necessary in the worst case to simulate an $n$-state two-way nondeterministic finite automaton (2NFA) by a one-way unambiguous finite automaton (UFA). It is proved that for every $n$, there is a language recognized by an $n$-state 2NFA that requires a UFA with at least $\sum_{k=1}^{n} (k - 1)! \cdot k! \cdot \mathrm{stirling2}(n, k) \cdot \mathrm{stirling2}(n+1, k)$ = $\Omega \big( n^{2n+2} / e^{2n} \big)$ states, where $\mathrm{stirling2}(n, k)$ denotes Stirling's numbers of the second kind. This result is proved by estimating the rank of a certain matrix, which is constructed for the universal language for $n$-state 2NFAs, and describes every possible behaviour of these automata during their computation.