# Backward deterministic and weak alternating $\omega$-automata

## TL;DR

This paper introduces a direct method to convert weak alternating omega-automata into backward deterministic automata, improving transformations from nondeterministic B"uchi automata and LTL formulas with optimal results.

## Contribution

It provides the first direct transformation from weak alternating omega-automata to backward deterministic automata, enabling efficient conversions for automata and temporal logic.

## Key findings

- Enables transformation of nondeterministic B"uchi automata into backward deterministic automata
- Achieves optimal backward deterministic automata for LTL formulas
- Uses a novel direct transformation approach

## Abstract

We present a direct transformation of weak alternating $\omega$-automata into equivalent backward deterministic $\omega$-automata and show (1) how it can be used to obtain a transformation of non-deterministic B\"uchi automata into equivalent backward deterministic automata and (2) that it yields optimal equivalent backward deterministic automata when applied to linear-time temporal logic formulas. (1) uses the alternation-free fragment of the linear-time $\mu$-calculus as an intermediate step; (2) is based on the straightforward translation of linear-time temporal logic into weak alternating $\omega$-automata.