On the Power of Unambiguity in B\"uchi Complementation

TL;DR

This paper leverages unambiguity in B"uchi automata to optimize complementation constructions, reducing state complexity for automata with finite ambiguity by employing reduced run DAGs.

Contribution

It introduces a unified approach using reduced run DAGs to improve both rank-based and slice-based B"uchi automata complementation methods for automata with finite ambiguity.

Findings

State complexity improved to 2^{O(n)} for rank-based complementation.

State complexity improved to O(4^n) for slice-based complementation.

Applicable to automata with finite degree of ambiguity.

Abstract

In this work, we exploit the power of \emph{unambiguity} for the complementation problem of B\"uchi automata by utilizing reduced run directed acyclic graphs (DAGs) over infinite words, in which each vertex has at most one predecessor. We then show how to use this type of reduced run DAGs as a \emph{unified tool} to optimize \emph{both} rank-based and slice-based complementation constructions for B\"uchi automata with a finite degree of ambiguity. As a result, given a B\"uchi automaton with $n$ states and a finite degree of ambiguity, the number of states in the complementary B\"uchi automaton constructed by the classical rank-based and slice-based complementation constructions can be improved, respectively, to $2^{O(n)}$ from $2^{O(n\log n)}$ and to $O(4^n)$ from $O((3n)^n)$.