Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words

Hoffmann, Stefan

doi:10.1007/978-3-030-68195-1_24

Cited by 11 publications

(12 citation statements)

References 17 publications

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“…In [22], the following characterization of the primitive permutation groups, connecting them to completely reachable automata, was shown.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In the same work [22], in analogy with Theorem 2 and 3, the sync-maximal permutation groups were introduced. Definition 4.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In [22] it was shown that every sync-maximal permutation groups is primitive. The main result of the present work is that the converse implication holds true.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In [22,Lemma 3.1] it was shown that distinguishability of all sets reduces to distinguishablity of the 2-subsets. Formulated without reference to automata, this gives the next result.…”

Section: Proposition 6 ([22]mentioning

confidence: 99%

“…It has been proven in [22] that a permutation group of degree n is primitive if and only if in the transformation monoid generated by the group and an arbitrary non-permutation with an image of size n ´1, there exists, for every non-empty subset of the permutation domain, an element mapping the whole permutation domain to this subset, or said differently that an associated automaton is completely reachable. In the same paper [22] the sync-maximal permutation groups were introduced by stipulating that, for an associated n-state automaton, the smallest automaton for the set of synchronizing words has size 2 n ´n. It was shown that the sync-maximal permutation groups are contained between the 2-homogeneous and the primitive permutation groups, and it was posed as an open problem if they are properly contained between them, and if so, what the precise relation to other permutation groups is.…”

Section: Introductionmentioning

confidence: 99%

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Sync-Maximal Permutation Groups Equal Primitive Permutation Groups

Hoffmann¹

2021

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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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“…In [22], the following characterization of the primitive permutation groups, connecting them to completely reachable automata, was shown.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In the same work [22], in analogy with Theorem 2 and 3, the sync-maximal permutation groups were introduced. Definition 4.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In [22] it was shown that every sync-maximal permutation groups is primitive. The main result of the present work is that the converse implication holds true.…”

Section: Hence a Transformation Monoid M ď T N Contains A Constant Ma...mentioning

confidence: 99%

“…In [22,Lemma 3.1] it was shown that distinguishability of all sets reduces to distinguishablity of the 2-subsets. Formulated without reference to automata, this gives the next result.…”

Section: Proposition 6 ([22]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations