B\"uchi complementation made tight

TL;DR

This paper introduces a new B"uchi complementation algorithm that nearly matches the theoretical lower bound, significantly improving previous methods and closing the longstanding gap in automata theory.

Contribution

It presents a nearly optimal B"uchi complementation algorithm with complexity close to the known lower bound, advancing the theoretical understanding and practical efficiency.

Findings

Reduces the gap between upper and lower bounds for B"uchi complementation

Achieves complexity within a quadratic factor of the lower bound

Improves previous exponential algorithms to nearly optimal performance

Abstract

The precise complexity of complementing B\"uchi automata is an intriguing and long standing problem. While optimal complementation techniques for finite automata are simple - it suffices to determinize them using a simple subset construction and to dualize the acceptance condition of the resulting automaton - B\"uchi complementation is more involved. Indeed, the construction of an EXPTIME complementation procedure took a quarter of a century from the introduction of B\"uchi automata in the early 60s, and stepwise narrowing the gap between the upper and lower bound to a simple exponent (of (6e)n for B\"uchi automata with n states) took four decades. While the distance between the known upper (O'(0.96 n)n') and lower ('(0.76 n)n') bound on the required number of states has meanwhile been significantly reduced, an exponential factor remains between them. Also, the upper bound on the size of the complement automaton is not linear in the bound of its state space. These gaps are unsatisfactory from a theoretical point of view, but also because B\"uchi complementation is a useful tool in formal verification, in particular for the language containment problem. This paper proposes a B\"uchi complementation algorithm whose complexity meets, modulo a quadratic (O(n2)) factor, the known lower bound for B\"uchi complementation. It thus improves over previous constructions by an exponential factor and concludes the quest for optimal B\"uchi complementation algorithms.