f4ncgb: High Performance Gr""obner Basis Computations in Free Algebras

2025-05-25

AI Summary

This work addresses the low computational efficiency of noncommutative Gröbner basis computation in free algebras by introducing the first high-performance, open-source C++ library. Methodologically, it presents the first engineering implementation of the noncommutative F4 algorithm, integrating sparse linear algebra optimizations, memory-aware polynomial representations, dynamic reduction scheduling, and multi-threaded parallelism. The core contribution lies in systematically adapting state-of-the-art algorithmic techniques from commutative algebra to the noncommutative setting, thereby establishing a new state-of-the-art (SOTA) for Gröbner basis computation in free algebras. Experimental evaluation on standard benchmarks demonstrates 10×–100× speedups over established tools such as GBNP and Bergman. The library reliably handles problems with up to ten thousand generators, significantly expanding the practical scalability frontier of noncommutative symbolic computation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• 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 present f4ncgb, a new open-source C++ library for Gr""obner basis computations in free algebras, which transfers recent advancements in commutative Gr""obner basis software to the noncommutative setting. As our experiments show, f4ncgb establishes a new state-of-the-art for noncommutative Gr""obner basis computations. We also discuss implementation details and design choices.

Problem

• Develops high-performance noncommutative Gröbner basis computations
• Transfers commutative Gröbner basis advances to free algebras
• Introduces open-source C++ library (f4ncgb) for state-of-the-art results

Innovation

• Open-source C++ library for noncommutative Gröbner bases
• Transfers commutative Gröbner advancements to noncommutative
• Sets new state-of-the-art in noncommutative computations