Lower bounds for Choiceless Polynomial Time via Symmetric XOR-circuits

TL;DR

This paper explores the limitations of Choiceless Polynomial Time (CPT) by connecting its expressive power to symmetric XOR-circuits and establishing lower bounds that suggest CPT cannot efficiently solve certain graph isomorphism problems.

Contribution

It introduces CFI-symmetric algorithms, links their capabilities to symmetric XOR-circuits, and proves non-existence of such circuits for hypercubes, advancing understanding of CPT's boundaries.

Findings

CFI-symmetric algorithms can only define the CFI-query if certain symmetric XOR-circuits exist.

Such circuits with strengthened symmetry and fan-in restrictions do not exist for hypercube graphs.

This result nearly separates CPT from polynomial time for the CFI-query.

Abstract

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.