Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-F\"urer-Immerman graphs (the CFI-query). We study the expressive power of CPT with respect to this problem and develop a partial characterisation of solvable instances in terms of properties of symmetric XOR-circuits over the CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if: For each graph $G$, there exists an XOR-circuit $C$, whose input gates are labelled with edges of $G$, such that $C$ is sufficiently symmetric with respect to the automorphisms of $G$ and satisfies certain other circuit properties. We also give a sufficient condition for CFI being solvable in CPT and develop a new CPT-algorithm for the CFI-query. It takes as input structures which contain, along with the CFI-graph, an XOR-circuit with suitable properties. The strongest known CPT-algorithm for this problem can solve instances equipped with a preorder with colour classes of logarithmic size. Our result implicitly extends this to preorders with colour classes of polylogarithmic size (plus some unordered additional structure). Finally, our work provides new insights regarding a much more general problem: The existence of a solution to an unordered linear equation system $A \cdot x = b$ over a finite field is CPT-definable if the matrix $A$ has at most logarithmic rank (with respect to the size of the structure that encodes the equation system). This is another example that separates CPT from fixed-point logic with counting.

翻译：无选择的聚合时间(CPT) 是目前捕捉 PTIME (即,它包含在 PTIME 中,而且尚未与它分离) 的唯一候选逻辑。 PTIME 中一个不为 CPT 所定义的决定问题的一个突出例子就是未排序的 Cai-F\"urer-Immerman 图形( CFI-query) 上的无序问题。 我们研究 CPT 对这一问题的表达力, 并开发了在 CFI 中对齐的 XOR 电路路段特性方面最强烈的可溶性实例。 CFI- due- cluel 直径: CFIFI- delioral 结构中, CPT 直径的 CPT 直径为直径直径直径, 仅当下列情况下, CPT 才会被确定为 CPT : $ 的每张图$G, 它的输入门被贴在 $G 的边缘上贴上, 因此, $CFIPT 和某些新电路系的直径直径的直径的直径的直径系统, 我们的直径的直径的直径直径的直径的直径直径的直径的直径的直径, 我们的直方的直的直方的直径的直径向的直方的直方的直路路路路的直径向的直径提供了一个直方的直方的直方的直方的直径的直方的直方的直方的直径方的直方的直方的直径。