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A quadratic upper bound on the reset thresholds of synchronizing automata containing a transitive permutation group

Yinfeng Zhu

Abstract

For any synchronizing $n$-state deterministic automaton, Černý conjectures the existence of a synchronizing word of length at most $(n-1)^2$. We prove that there exists a synchronizing word of length at most $2n^2 - 7n + 7$ for every synchronizing $n$-state deterministic automaton that satisfies the following two properties: 1. The image of the action of each letter contains at least $n-1$ states; 2. The actions of bijective letters generate a transitive permutation group on the state set.

A quadratic upper bound on the reset thresholds of synchronizing automata containing a transitive permutation group

Abstract

For any synchronizing -state deterministic automaton, Černý conjectures the existence of a synchronizing word of length at most . We prove that there exists a synchronizing word of length at most for every synchronizing -state deterministic automaton that satisfies the following two properties: 1. The image of the action of each letter contains at least states; 2. The actions of bijective letters generate a transitive permutation group on the state set.
Paper Structure (9 sections, 21 theorems, 36 equations)

This paper contains 9 sections, 21 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Synchronizing automata and Černý Conjecture
  3. Automata containing transitive groups and our contribution
  4. Approach and layout
  5. Linear structure
  6. Rystsov digraphs
  7. Digraphs
  8. Automata
  9. Conclusion

Key Result

Theorem 1.2

Let $\mathop{\mathrm{\mathcal{A}}}\nolimits = (Q, \Sigma, \delta) \in \mathop{\mathrm{\mathbb{ST}}}\nolimits$ be an $n$-state automaton. Then $\mathop{\mathrm{\mathsf{rt}}}\nolimits(\mathop{\mathrm{\mathcal{A}}}\nolimits) \le 1+ (n-2)(n - 1 + \mathop{\mathrm{\mathsf{d}}}\nolimits_{\mathop{\mathrm{\m

Theorems & Definitions (39)

  • Conjecture 1.1: Černý-Starke
  • Theorem 1.2: Rystsov
  • Theorem 1.3: Araújo-Cameron-Steinberg
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 29 more