We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a (1/2, 1/3 + $\delta$)-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\ff_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

翻译：我们研究在带计数的不动点逻辑（FPC）中近似独特游戏实例最优值的可行性。形式上，我们证明了针对FPC解释的精确性的下界，这些解释将独特游戏实例（编码为关系结构）映射到有理数，给出可满足约束的近似比例。我们为独特游戏证明了两个新的FPC不可表达性结果：存在一个(1/2, 1/3 + $\delta$)-不可逼近性间隙，以及任意常数因子下的不可逼近性。近期已有研究为少数其他问题建立了类似的FPC不可逼近性结果。我们的构造借鉴了其中一些思想，但包含一项新颖技术。虽然大多数FPC不可表达性结果基于CFI构造的变体，但我们的方法显著不同。我们从具有极大周长的图出发，用$\ff_2$上的随机仿射向量空间标记边，这些空间决定两个结构中的约束。杜普利卡策略涉及在覆盖图中被标记顶点的最小树上维持部分同构。