We show that it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. As a corollary, there is a boolean constraint system (BCS) game for which the value of the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy does not attain the commuting operator value at any finite level. Our contribution involves establishing a computable mapping from Turing machines to BCS nonlocal games in which the halting property of the machine is encoded as a decision problem for the commuting operator value of the game. Our techniques are algebraic and distinct from those used to establish MIP*=RE.

翻译：我们证明，判断非局域博弈的对易算子值是否严格大于1/2是不可判定的。作为推论，存在一个布尔约束系统（BCS）博弈，其Navascués-Pironio-Acín（NPA）层级在任何有限层级上的值都无法达到该博弈的对易算子值。我们的贡献在于构建了一个从图灵机到BCS非局域博弈的可计算映射，其中图灵机的停机性质被编码为博弈对易算子值的判定问题。我们的技术方法是代数性的，与证明MIP*=RE所采用的技术有本质区别。