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

TL;DR

This paper proves a significant lower bound on the number of states needed for unambiguous finite automata to simulate two-way nondeterministic finite automata, highlighting fundamental complexity limitations.

Contribution

It introduces a new lower bound on state complexity for converting 2NFAs to UFAs, using matrix rank estimation related to Stirling numbers.

Findings

Lower bound of (n^{2n+2}/e^{2n}) states for UFA simulation

Estimation of matrix rank for universal language behaviors

Demonstration of fundamental complexity limits in automata conversion

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.