Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique

TL;DR

This paper introduces a novel lower bound technique for automata transformations using full automata and alphabet reduction, providing significant bounds for omega-automata complementation problems.

Contribution

It presents a new method for deriving lower bounds on automata complementation complexity, applying it to nondeterministic omega-automata and establishing tight bounds.

Findings

Proves an .76n)^n lower bound for Bhi complementation.

Establishes an ((nk))^n lower bound for generalized Bhi automata.

Demonstrates the technique's applicability to Streett automata.

Abstract

In this paper, we first introduce a lower bound technique for the state complexity of transformations of automata. Namely we suggest first considering the class of full automata in lower bound analysis, and later reducing the size of the large alphabet via alphabet substitutions. Then we apply such technique to the complementation of nondeterministic \omega-automata, and obtain several lower bound results. Particularly, we prove an \omega((0.76n)^n) lower bound for B\"uchi complementation, which also holds for almost every complementation or determinization transformation of nondeterministic omega-automata, and prove an optimal (\omega(nk))^n lower bound for the complementation of generalized B\"uchi automata, which holds for Streett automata as well.