On Minimization and Learning of Deterministic $Ï$-Automata in the Presence of Don't Care Words
For researchers working on automata theory and formal verification, this paper provides both efficient algorithms and hardness results for minimization with don't care words, clarifying the boundaries of tractability.
The paper studies minimization of deterministic ω-automata with don't care words, showing that priority minimization for deterministic parity automata is efficient under arbitrary don't care sets, but minimization for automata with informative right-congruence under trivial don't care sets is NP-hard. It also extends active learning for weak deterministic Büchi automata to handle don't care words.
We study minimization problems for deterministic $Ï$-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 $Ï$-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.