Choiceless Polynomial Time (CPT) is one of the few remaining candidate logics for capturing PTIME. In this paper, we make progress towards separating CPT from polynomial time by firstly establishing a connection between the expressive power of CPT and the existence of certain symmetric circuit families, and secondly, proving lower bounds against these circuits. We focus on the isomorphism problem of unordered Cai-F\"urer-Immerman-graphs (the CFI-query) as a potential candidate for separating CPT from P. Results by Dawar, Richerby and Rossman, and subsequently by Pakusa, Schalth\"ofer and Selman show that the CFI-query is CPT-definable on linearly ordered and preordered base graphs with small colour classes. We define a class of CPT-algorithms, that we call "CFI-symmetric algorithms", which generalises all the known ones, and show that such algorithms can only define the CFI-query on a given class of base graphs if there exists a family of symmetric XOR-circuits with certain properties. These properties include that the circuits have the same symmetries as the base graphs, are of polynomial size, and satisfy certain fan-in restrictions. Then we prove that such circuits with slightly strengthened requirements (i.e. stronger symmetry and fan-in and fan-out restrictions) do not exist for the n-dimensional hypercubes as base graphs. This almost separates the CFI-symmetric algorithms from polynomial time - up to the gap that remains between the circuits whose existence we can currently disprove and the circuits whose existence is necessary for the definability of the CFI-query by a CFI-symmetric algorithm.

翻译：无选择的聚合时间(CPT) 是捕捉 PTIME 的少数剩下的候选逻辑之一 。 在本文中, 我们取得进展, 将CPT 与多元时间分开, 首先在CPT 的表达力和某些对称电路家族的存在之间建立联系, 其次, 证明这些电路的界限较低 。 我们集中关注未经排序的 Cai- F\\" ur- Imerman- graphy- graphy- logy (CFI- commer- commer- compress) 的无足轻重的逻辑问题 。 由 Dawar 、 Richerby 和 Rossman 和 之后的 Pakusa、 Schalth\ “ ofer 和 Selman 一起的结果 。 我们的 CFIFI 和 Slormayalalal 的解算性在一定的直径解式直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直线和直径直径直径直径直径直径直径直径直的直径直径直的直等等级。