Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

List of Results on the Černý Conjecture and Reset Thresholds for Synchronizing Automata

Mikhail V. Volkov

TL;DR

This living survey collects results on the Černý conjecture and reset thresholds for synchronizing automata up to January 13, 2026. It organizes definitions, methods, and a comprehensive annotated list of results by automata classes, distinguishing cases where the conjecture holds and where quadratic bounds are known. The work emphasizes extensibility-based arguments (top-down compression vs bottom-up extension), reductions via subautomata and quotients, and class-specific structures (e.g., Eulerian, one-cluster, aperiodic, and monoid-based classes). By providing version-stamped updates and cross-referencing historical bounds, it clarifies progress toward (or difficulty in) proving the general bound $\mathfrak{C}(n)=(n-1)^2$ and guides future research directions in synchronizing automata theory.

Abstract

We survey results in the literature that establish the Černý conjecture for various classes of finite automata. We also list classes for which the conjecture remains open, but a quadratic (in the number of states) upper bound on the minimum length of reset words is known. The results presented reflect the state of the art as of January 13, 2026.

List of Results on the Černý Conjecture and Reset Thresholds for Synchronizing Automata

TL;DR

This living survey collects results on the Černý conjecture and reset thresholds for synchronizing automata up to January 13, 2026. It organizes definitions, methods, and a comprehensive annotated list of results by automata classes, distinguishing cases where the conjecture holds and where quadratic bounds are known. The work emphasizes extensibility-based arguments (top-down compression vs bottom-up extension), reductions via subautomata and quotients, and class-specific structures (e.g., Eulerian, one-cluster, aperiodic, and monoid-based classes). By providing version-stamped updates and cross-referencing historical bounds, it clarifies progress toward (or difficulty in) proving the general bound and guides future research directions in synchronizing automata theory.

Abstract

We survey results in the literature that establish the Černý conjecture for various classes of finite automata. We also list classes for which the conjecture remains open, but a quadratic (in the number of states) upper bound on the minimum length of reset words is known. The results presented reflect the state of the art as of January 13, 2026.

Paper Structure

This paper contains 5 sections, 3 theorems, 27 equations, 5 figures.

Table of Contents

  1. Concepts and methods
  2. Basic definitions
  3. Synchronizing automata and the Černý conjecture
  4. Methods
  5. Annotated list of results

Key Result

Proposition 1

Each $\alpha$-extensible DFA with $n>2$ states is synchronizing, and its reset threshold does not exceed $1+\alpha n(n-2)$. In particular, the Černý conjecture holds for $1$-extensible automata.

Figures (5)

  • Figure 1: The automaton $\mathrsfs{C}_n$
  • Figure 2: The one-cluster automaton $\mathrsfs{D}_{10,7}$
  • Figure 3: The automaton $\mathrsfs{R}_n$
  • Figure 4: The automaton $\mathrsfs{V}_n$
  • Figure 5: The subautomaton $\langle C\cup D,\{a,b\}\rangle$ of $\mathrsfs{A}$. Loops labeled $a$ at all states except $e$ are omitted for readability

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2: Volkov:2022
  • Lemma 1: RamirezAlfonsin:2005