On the power of finite ambiguity in Büchi complementation

Abstract

In this work, we exploit the power of finite ambiguity for the complementation problem of Büchi automata by using reduced run directed acyclic graphs (DAGs) over infinite words, in which each vertex has at most one predecessor; these reduced run DAGs have only a finite number of infinite runs, thus obtaining the finite ambiguity in Büchi complementation. We show how to use this type of reduced run DAGs as a unified tool to optimize both rank-based and slice-based complementation constructions for Büchi automata with a finite degree of ambiguity. As a result, given a Büchi automaton with n states and a finite degree of ambiguity, the number of states in the complementary Büchi automaton constructed by the classical rank-based and slice-based complementation constructions can be improved from 2O(nlog⁡n) and O((3n)n) to O(6n)⊆2O(n) and O(4n), respectively. We further show how to construct such reduced run DAGs for limit deterministic Büchi automata and obtain a specialized complementation algorithm, thus demonstrating the generality of the power of finite ambiguity.