State Complexity of the Set of Synchronizing Words for Circular Automata and Automata over Binary Alphabets

Hoffmann, Stefan

doi:10.1007/978-3-030-68195-1_25

Cited by 9 publications

(3 citation statements)

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“…This is a stronger notion than being synchronizing by stipulating that, starting from the whole state set, not only some singleton set is reachable, but every non-empty subset of states is reachable by some word. In fact, this property was also previously observed for many classes of synchronizing automata [15,25,32,33].…”

Section: Introductionsupporting

confidence: 75%

“…The set of synchronizing words of an n-state automaton is a regular language and can be recognized by an automaton of size 2 n ´n [13,32,45]. A property shared by most families of slowly synchronizing automata is that for them, every automaton for the set of synchronizing words needs exponentially many states [25,32,33]. Note that, of course, by taking an automaton and adjoining a letter mapping every state to a single state, as the extremal property of the set of synchronizing words is preserved by adding letters, automata whose sets of synchronizing words have exponential state complexity in the number of states are not necessarily slowly synchronizing.…”

Section: Introductionmentioning

confidence: 99%

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Sync-Maximal Permutation Groups Equal Primitive Permutation Groups

Hoffmann¹

2021

Preprint

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The set of synchronizing words of a given n-state automaton forms a regular language recognizable by an automaton with 2 n ´n states. The size of a recognizing automaton for the set of synchronizing words is linked to computational problems related to synchronization and to the length of synchronizing words. Hence, it is natural to investigate synchronizing automata extremal with this property, i.e., such that the minimal deterministic automaton for the set of synchronizing words has 2 n ´n states. The sync-maximal permutation groups have been introduced in [S. Hoffmann, Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words, LATA 2021] by stipulating that an associated automaton to the group and a non-permutation has this extremal property. The definition is in analogy with the synchronizing groups and analog to a characterization of primitivity obtained in the mentioned work. The precise relation to other classes of groups was mentioned as an open problem. Here, we solve this open problem by showing that the sync-maximal groups are precisely the primitive groups. Our result gives a new characterization of the primitive groups. Lastly, we explore an alternative and stronger definition than sync-maximality.

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Section: Introductionsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Sync-Maximal Permutation Groups Equal Primitive Permutation Groups

Hoffmann¹

2021

Preprint

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“…For example, it can be verified that any DFA that recognizes the language Sync C 4 , where C 4 is the synchronizing DFA with four states presented in Example 1.1, has at least 12 states. Among the publications in this area we should mention [72], [88], [92], [93], [119]- [121], [139], and [146]. Now we turn back to the computational complexity of synchronizability recognition.…”

Section: Algorithmic and Complexity Issuesmentioning

confidence: 99%