Reset Complexity and Completely Reachable Automata with Simple Idempotents

• Published in: Descriptional Complexity of Formal Systems Vol. 13439; pp. 85 - 99
• Main Author: Hoffmann, Stefan
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2022; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: automata with simple idempotents; completely reachable automata; finite automata; set of synchronizing words; sync-maximal automata; synchronization
• ISBN: 9783031132568; 3031132564
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-031-13257-5_7

Every regular ideal language is the set of synchronizing words of some automaton. The reset complexity of a regular ideal language is the size of such an automaton with the minimal number of states. The state complexity is the size of a minimal automaton recognizing a regular language in the usual sense. There exist regular ideal languages whose state complexity is exponentially larger than its reset complexity. We call an automaton sync-maximal, if the reset complexity of the ideal language induced by its set of synchronizing words equals the number of states of the automaton and the gap between the reset complexity and the state complexity of this language is maximal possible. An automaton is completely reachable, if we can map the whole state set to any non-empty subset of states (for synchronizing automata, it is only required that the whole state set can be mapped to a singleton set). We first state a general structural result for sync-maximal automata. This shows that sync-maximal automata are closely related to completely reachable automata. We then investigate automata with simple idempotents and show that for these automata complete reachability and sync-maximality are equivalent. Lastly, we find that for automata with simple idempotents over a binary alphabet, subset reachability problems that are PSPACE-complete in general are solvable in polynomial time.