A superpolynomial lower bound for the size of non-deterministic complement of an unambiguous automaton

TL;DR

This paper proves that converting unambiguous finite automata into automata recognizing their complements can require superpolynomial state increases, disproving a previous conjecture and highlighting complexity differences.

Contribution

It provides the first superpolynomial lower bound for the state complexity of complementing unambiguous automata, challenging existing assumptions.

Findings

Superpolynomial lower bound for complement automata

Languages and complements recognized by sweeping deterministic automata

Disproves the polynomial state increase conjecture

Abstract

Unambiguous non-deterministic finite automata have intermediate expressive power and succinctness between deterministic and non-deterministic automata. It has been conjectured that every unambiguous non-deterministic one-way finite automaton (1UFA) recognizing some language L can be converted into a 1UFA recognizing the complement of the original language L with polynomial increase in the number of states. We disprove this conjecture by presenting a family of 1UFAs on a single-letter alphabet such that recognizing the complements of the corresponding languages requires superpolynomial increase in the number of states even for generic non-deterministic one-way finite automata. We also note that both the languages and their complements can be recognized by sweeping deterministic automata with a linear increase in the number of states.