Recent work by Atserias and Dawar (J. Log. Comp 2019) and Tucker-Foltz (LMCS 2024) has established undefinability results in fixed-point logic with counting (FPC) corresponding to many classical complexity results from the hardness of approximation. In this line of work, NP-hardness results are turned into unconditional FPC undefinability results. We extend this work by showing the FPC undefinability of any constant factor approximation of weighted 2-to-2 games, based on the NP-hardness results of Khot, Minzer and Safra. Our result shows that the completely satisfiable 2-to-2 games are not FPC-separable from those that are not epsilon-satisfiable, for arbitrarily small epsilon. The perfect completeness of our inseparability is an improvement on the complexity result, as the NP-hardness of such a separation is still only conjectured. This perfect completeness enables us to show the FPC undefinability of other problems whose NP-hardness is conjectured. In particular, we are able to show that no FPC formula can separate the 3-colourable graphs from those that are not t-colourable, for any constant t.

翻译：Atserias与Dawar（J. Log. Comp 2019）以及Tucker-Foltz（LMCS 2024）近期的研究，已在带计数的不动点逻辑（FPC）中建立了与近似算法复杂性理论中诸多经典结果相对应的不可定义性结论。在这类研究中，NP困难性结果被转化为无条件的FPC不可定义性结果。我们基于Khot、Minzer与Safra的NP困难性结论，通过证明带权2-对-2博弈的任意常数因子近似在FPC中不可定义，进一步拓展了该研究方向。我们的结果表明：对于任意小的ε，完全可满足的2-对-2博弈与那些不可ε-满足的博弈在FPC中是不可区分的。该不可区分性所具有的完美完备性是对原复杂性结果的改进，因为此类分离问题的NP困难性目前仍仅为猜想。这种完美完备性使我们能够证明其他NP困难性尚属猜想的问题在FPC中同样不可定义。特别地，我们证明了对于任意常数t，不存在FPC公式能将3-可着色图与不可t-着色图进行区分。