Synchronous linear constraint system games

TL;DR

This paper explores the algebraic structures underlying synchronous linear constraint system games, revealing their equivalence to graph isomorphism games and unifying different algebraic approaches.

Contribution

It shows that the game algebra is a quotient of the solution group's algebra, establishing a fundamental connection between these algebraic objects.

Findings

Game algebra is a quotient of the solution group algebra

Linear constraint system games are equivalent to graph isomorphism games

Algebraic objects encode the existence of perfect strategies

Abstract

Synchronous linear constraint system games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated to these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here we show that these objects are essentially the same, i.e., that the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that linear constraint system games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.