Choiceless Polynomial Time, Symmetric Circuits and Cai-F\"urer-Immerman Graphs

TL;DR

This paper explores the limits of Choiceless Polynomial Time (CPT) logic in solving the graph isomorphism problem for Cai-F"urer-Immerman graphs, developing new algorithms and characterizations based on symmetric XOR-circuits.

Contribution

It provides a partial characterization of CPT-definability for the CFI-graph isomorphism problem using symmetric XOR-circuits and introduces a new CPT-algorithm for this problem.

Findings

CPT-definability depends on the existence of symmetric XOR-circuits for each graph.

A new CPT-algorithm can solve instances with certain preorders and color class sizes.

CPT can define solutions to linear equations with low-rank matrices over finite fields.

Abstract

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.