Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform winning strategies for the synchronization games on subclasses of finite automata

Henning Fernau, Carolina Haase, Stefan Hoffmann, Mikhail Volkov

TL;DR

The paper analyzes synchronization games on DFAs through the lens of algebraic structure, focusing on the DS pseudovariety of finite monoids. It proves that every synchronizing DS-automaton admits a uniform winning strategy, with the proof leveraging a semilattice decomposition of the transition monoid into semigroups nilpotent over their kernels. It also shows that DS is the largest pseudovariety enjoying this uniform strategy property, by exhibiting a counterexample outside DS, and connects the findings to known automata families (definite, commutative, weakly acyclic). The work further discusses decidability, bounds on strategy length, and robustness under a modified adversarial rule, and outlines open questions for efficiency and broader generalizations.

Abstract

The pseudovariety $\mathbf{DS}$ consists of all finite monoids whose regular $D$-classes form subsemigroups. We exhibit a uniform winning strategy for Synchronizer in the synchronization game on every synchronizing automaton whose transition monoid lies in $\mathbf{DS}$, and we prove that $\mathbf{DS}$ is the largest pseudovariety with this property.

Uniform winning strategies for the synchronization games on subclasses of finite automata

TL;DR

The paper analyzes synchronization games on DFAs through the lens of algebraic structure, focusing on the DS pseudovariety of finite monoids. It proves that every synchronizing DS-automaton admits a uniform winning strategy, with the proof leveraging a semilattice decomposition of the transition monoid into semigroups nilpotent over their kernels. It also shows that DS is the largest pseudovariety enjoying this uniform strategy property, by exhibiting a counterexample outside DS, and connects the findings to known automata families (definite, commutative, weakly acyclic). The work further discusses decidability, bounds on strategy length, and robustness under a modified adversarial rule, and outlines open questions for efficiency and broader generalizations.

Abstract

The pseudovariety consists of all finite monoids whose regular -classes form subsemigroups. We exhibit a uniform winning strategy for Synchronizer in the synchronization game on every synchronizing automaton whose transition monoid lies in , and we prove that is the largest pseudovariety with this property.

Paper Structure

This paper contains 18 sections, 7 theorems, 11 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Game rules and presentation
  3. Game rules
  4. On board designs
  5. Algebraic preliminaries
  6. Transition monoids and synchronization
  7. Structure of monoids in DS
  8. A characterization of DS as a pseudovariety
  9. Uniform winning strategy in synchronization games on DS-automata
  10. Uniform winning strategies: definition and examples
  11. Uniform winning strategy within DS
  12. Corollaries
  13. No uniform winning strategy beyond DS
  14. Open questions and a generalization
  15. Decidability and complexity
  16. ...and 3 more sections

Key Result

Lemma 1

Alice has a winning strategy in the synchronization game on a DFA if and only if she has a winning strategy for the task of synchronizing every 2-element subset.

Figures (5)

  • Figure 1: Designing boards for the synchronization game: two different proposals
  • Figure 2: The automaton $\mathrsfs{B}_2$ with the transition monoid isomorphic to $B_2^1$
  • Figure 3: The automaton $\mathrsfs{B}'_2$
  • Figure 4: The automaton $\mathrsfs{C}_n$ (above) and its duplication (below), with $q_0=0$ and letter $b$ selected
  • Figure 5: The automaton $\mathrsfs{E}$

Theorems & Definitions (14)

  • Lemma 1: (FomMarVol2013)
  • Definition 1
  • Lemma 2
  • proof
  • Definition 2
  • Lemma 3
  • Lemma 4: Mar81
  • Definition 3
  • Theorem 5
  • proof
  • ...and 4 more