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

TL;DR

This work addresses minimization and learning for deterministic ω-automata in the presence of don't care words, providing a polynomial-time solution for quotient parity optimization under don't cares and a detailed analysis of right-congruence-related structures. It shows that while parity optimization extends efficiently to certain don’t-care settings, minimization for automata with informative right-congruence (IRC) remains NP-hard and may lack unique minimal representatives. The paper establishes that WDBA under trivial right-congruence supports a unique, efficiently computable minimal automaton and extends active learning to incorporate don't cares, yielding polynomial-time learning of D-minimal WDBA. Together, these results clarify where efficient minimization and learning are possible and where intrinsic hardness arises, with potential applications to synthesis and real-time systems relying on ω-regular languages.

Abstract

We study minimization problems for deterministic $ω$-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 Büchi 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 $ω$-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.

Figures (4)

• Figure 1: Illustration of Don't Care Priority Optimization. An optimal parity condition consistent with $(\mathcal{F}_0',\mathcal{F}_1')$ can redefine the priority of $q_0$ to $2$ and thus use one priority less.
• Figure 2: The DBA $\mathcal{A}$ has 3 different corresponding $D$-minimal DBA for the set of don't care words$D:=\Sigma^*b^\omega$.
• Figure 3: The $D'$-minimal DPA $\mathcal{A}_{col}$ and the DPA $\mathcal{A}_G$ for $\mathcal{G}=(\{v_1,v_2,v_3\},\{(v_1,v_2),(v_1,v_3)\})$ recognize $D'$-equivalent $\omega$-languages.
• Figure 4: The situation described in Lemma \ref{['lem:conflict']} with $sx^\omega \in U \backslash D$ and $ty^\omega \not \in U \cup D$.

Theorems & Definitions (20)

• Theorem 3.1
• Lemma 3.2
• Theorem 3.3: BohnL21
• Lemma 3.4
• Definition 4.1: $D$-congruence
• Proposition 4.2
• Definition 4.3: Informative $D$-congruence
• Theorem 4.4
• Corollary 4.5
• Theorem 4.6