On Minimization and Learning of Deterministic $\omega$-Automata in the Presence of Don't Care Words

TL;DR

This paper investigates minimization and learning algorithms for deterministic omega-automata considering don't care words, revealing complexity results and extending active learning methods.

Contribution

It provides efficient minimization techniques for deterministic parity automata with don't care words and analyzes the complexity of minimization in different classes.

Findings

Number of priorities in deterministic parity automata can be minimized efficiently with don't care words.

Minimization for deterministic omega-automata with informative right-congruence is NP-hard.

Extended active learning algorithms to include don't care words with trivial right-congruence.

Abstract

We study minimization problems for deterministic $\omega$-automata in the presence of don't care words. We prove that the number of priorities in deterministic parity automata can be efficiently minimized under an arbitrary set of don't care words. We derive that from a more general result from which one also obtains an efficient minimization algorithm for deterministic parity automata with informative right-congruence (without don't care words). We then analyze languages of don't care words with a trivial right-congruence. For such sets of don't care words it is known that weak deterministic B\"uchi automata (WDBA) have a unique minimal automaton that can be efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give a congruence-based characterization of the corresponding minimal WDBA, and show that the don't care minimization results for WDBA do not extend to deterministic $\omega$-automata with informative right-congruence: for this class there is no unique minimal automaton for a given don't care set with trivial right congruence, and the minimization problem is NP-hard. Finally, we extend an active learning algorithm for WDBA (Maler, Pnueli 1995) to the setting with an additional set of don't care words with trivial right-congruence.