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Journal of Algebraic Combinatorics

, Volume 50, Issue 3, pp 237–253 | Cite as

A bound for the length of the shortest reset words for semisimple synchronizing automata via the packing number

  • Emanuele RodaroEmail author
Article
  • 46 Downloads

Abstract

We show that if a semisimple synchronizing automaton with n states has a minimal reachable non-unary subset of cardinality \(r\ge 2\), then there is a reset word of length at most \((n-1)D(2,r,n)\), where D(2, rn) is the 2-packing number for families of r-subsets of [1, n].

Keywords

Synchronizing automaton Černý’s conjecture Packing number Simple automaton Semisimple automaton Wedderburn–Artin theorem 

Mathematics Subject Classification

68Q70 68Q45 05E99 05B40 20M30 16D60 

Notes

Acknowledgements

The author thanks the anonymous referees for the careful reading of the paper and precious comments and suggestions, especially for pointing out the better bound for the case of former-rank 3 using Pin–Frank’s approach. The author acknowledges support from INDAM-GNSAGA.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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