Can Nondeterminism Help Complementation?

TL;DR

This paper investigates the relationship between nondeterminism, complementation, and determinization in omega-automata, revealing that for common types, determinization and complementation share the same state complexity, challenging assumptions about their relative difficulty.

Contribution

The paper demonstrates that for all common omega-automata types, determinization and complementation have equivalent state complexities, contrary to the expectation of greater complexity in determinization.

Findings

Determinization and complementation share the same state complexity in omega-automata.

For all common omega-automata, both operations have 2^ heta(n lg n) complexity.

Nondeterminism does not make determinization significantly harder than complementation.

Abstract

Complementation and determinization are two fundamental notions in automata theory. The close relationship between the two has been well observed in the literature. In the case of nondeterministic finite automata on finite words (NFA), complementation and determinization have the same state complexity, namely Theta(2^n) where n is the state size. The same similarity between determinization and complementation was found for Buchi automata, where both operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing question is whether there exists a type of omega-automata whose determinization is considerably harder than its complementation. In this paper, we show that for all common types of omega-automata, the determinization problem has the same state complexity as the corresponding complementation problem at the granularity of 2^\Theta(.).