Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection

2026-03-30

AI Summary

This work investigates the computational complexity of verifying three-variable identities—such as distributivity—in finite algebraic structures. For the specific problem of testing distributivity between two binary operations, the paper presents the first randomized algorithm running in strongly subcubic time, namely $O(|S|^\omega)$, where $\omega$ denotes the exponent of matrix multiplication. The conditional optimality of this bound is established under conjectures related to triangle detection and 4-term arithmetic progression detection. Furthermore, the study provides a complete complexity classification for a natural class of such identities, yields nearly optimal algorithms for verifying field and ring axioms, and demonstrates that counting triples satisfying distributivity is conditionally strictly harder than merely verifying distributivity itself.

Abstract

We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation $\odot: S\times S\to S$ in optimal time $O(|S|^2)$, they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations $\odot,\oplus: S\times S\to S$. Our results are as follows: * We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time $O(|S|^ω)$, together with a matching conditional lower bound based on the Triangle Detection Hypothesis. * We propose arithmetic progression detection in small universes as a consequential algorithmic challenge: We show that unless we can detect $4$-term arithmetic progressions in a set $X\subseteq\{1,\dots, N\}$ in time $O(N^{2-ε})$, then (a) the 3-uniform 4-hyperclique hypothesis is true, and (b) verifying certain identities requires running time~$|S|^{3-o(1)}$. * A careful combination of our algorithmic and hardness ideas allows us to \emph{fully classify} a natural subclass of identities: Specifically, any 3-variable identity over binary operations in which no side is a subexpression of the other is either: (1) verifiable in randomized time $O(|S|^2)$, (2) verifiable in randomized time $O(|S|^ω)$ with a matching lower bound from triangle detection, or (3) trivially verifiable in time $O(|S|^3)$ with a matching lower bound from hardness of 4-term arithmetic progression detection. * We obtain near-optimal algorithms for verifying whether a given algebraic structure forms a field or ring, and show that \emph{counting} the number of distributive triples is conditionally harder than verifying distributivity.

Problem

• distributivity
• identity verification
• subcubic algorithms
• arithmetic progression detection
• algebraic structure

Innovation

• subcubic algorithm
• distributivity verification
• arithmetic progression detection
• algebraic identity classification
• conditional lower bounds