Weak index versus Borel rank

TL;DR

This paper explores the relationship between weak recognizability and Borel hierarchy in deterministic infinite tree languages, providing an algorithm to minimize weak automata efficiently.

Contribution

It proves the equivalence of Borel and weak index hierarchies for deterministic languages and introduces a quadratic-time algorithm for minimal weak automaton construction.

Findings

Borel and weak index hierarchies coincide for deterministic languages

A quadratic-time algorithm for minimal weak automaton construction is proposed

The algorithm operates within the time complexity of the emptiness problem

Abstract

We investigate weak recognizability of deterministic languages of infinite trees. We prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide. Furthermore, we propose a procedure computing for a deterministic automaton an equivalent minimal index weak automaton with a quadratic number of states. The algorithm works within the time of solving the emptiness problem.