Efficient Polynomial Identity Testing Over Nonassociative Algebras

2025-09-14

AI Summary

This work addresses the Polynomial Identity Testing (PIT) problem in non-associative polynomial algebras. Motivated by the longstanding lack of efficient algorithms for PIT in this setting, we establish the first Amitsur–Levitzki-type theorem for non-associative algebras and construct canonical algebraic models free of low-degree identities. Leveraging these structural results, we design a randomized polynomial-time black-box PIT algorithm and a deterministic polynomial-time white-box PIT algorithm. Furthermore, we construct a quasipolynomial-size hitting set for constant-depth non-associative arithmetic circuits. Our results break fundamental complexity barriers for PIT in the non-associative regime, resolving several long-standing open problems. They also introduce a novel computational framework for modeling non-associative structures within algebraic complexity theory, thereby advancing the theoretical foundations of non-associative computation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson (2010) and Fijalkow, Lagarde, Ohlmann, and Serre (2021) from the identity testing perspective. Our main results are the following: (1) We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. (2) On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity. (3) In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.

Problem

• Designing efficient polynomial identity testing algorithms
• Addressing nonassociative polynomial algebra challenges
• Providing deterministic and randomized testing methods

Innovation

• First efficient polynomial identity testing algorithms
• Constructing nonassociative algebras without low degree identities
• Quasipolynomial-size hitting set for small depth circuits