一句话：证明椭球法可在带计数不动点逻辑 FPC 中定义，在无穷 FPC 中表达 SDP 精确可行性，并推出图同构的 SoS 层级塌缩到 Sherali-Adams 层级。
本文研究凸优化算法在描述复杂性逻辑中的可定义性。椭球法通过弱分离 oracle 求解凸集的弱可行性与线性优化问题。作者观察到，此前用于说明线性与半定规划可归约到带计数的固定点逻辑（FPC）的方法，实际上适用于任何显式有界的凸集族。由此，他们证明半定规划的精确可行性问题可在无穷 FPC 中表达。作为推论，对图同构问题，Lasserre/平方和（SoS）半定规划松弛层级会塌缩到 Sherali-Adams 线性规划层级，仅损失少量度数。该结果将凸优化、逻辑可定义性与同构问题的代数证明复杂度联系起来。
The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree.