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

2025-12-11

AI Summary

This paper investigates the existence of a uniform winning strategy for the Synchronizer player in synchronous games on automata whose transition monoids belong to the pseudovariety DS. Addressing the problem of characterizing the precise algebraic boundary between synchronizability and monoid structure, the authors integrate finite automata theory, Green’s relations analysis, pseudovarietal algebra, and game-theoretic modeling. They establish, for the first time, that DS is the largest pseudovariety admitting a uniform winning strategy in such games—thereby exactly delineating the synchronizability threshold. Moreover, they explicitly construct a uniform winning strategy for all DS-automata and rigorously prove its completeness. This resolves the long-standing question of strategy existence for DS in synchronous games and provides an optimal algebraic characterization of synchronization capability within the pseudovarietal framework. The result furnishes foundational theoretical support for understanding the deep interplay between automata synchronization and algebraic structure.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Develops uniform winning strategies for synchronization games
• Focuses on automata with transition monoids in pseudovariety DS
• Identifies DS as the largest pseudovariety enabling such strategies

Innovation

• Uniform winning strategy for Synchronizer in synchronization games
• Strategy applies to automata with transition monoid in pseudovariety DS
• DS is largest pseudovariety enabling such uniform winning strategy