# Kolmogorov-Arnold Classifier Systems as Universal Approximators

arXiv: https://arxiv.org/abs/2609.37958 · HTML: https://mlsanity.com/paper/2609.37958/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 77 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [A Literature Survey of Benchmark Functions For Global Optimization Problems](https://mlsanity.com/paper/1308.4008/) (2013)
- [TKAN: Temporal Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.07344/) (2024)
- [F-KANs: Federated Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.20100/) (2024)
- [Autoencoding with a Classifier System](https://mlsanity.com/paper/1910.10579/) (2019)
- [X-KAN: Optimizing Local Kolmogorov-Arnold Networks via Evolutionary Rule-Based Machine Learning](https://mlsanity.com/paper/2505.14273/) (2025)
- [Adapting Rule Representation With Four-Parameter Beta Distribution for Learning Classifier Systems](https://mlsanity.com/paper/2506.03602/) (2025)

## Abstract (arXiv)

As the input dimension n grows, rule-based machine learning, such as Learning Classifier Systems (LCSs), faces a fundamental scalability bottleneck for function approximation: both rule count and parameter count grow exponentially with n. Traditional LCSs partition the n-dimensional input space directly, requiring O(mn) rules for adequate coverage, where m is the per-variable resolution. This article breaks from this paradigm by reorganizing rules dimension-wise, guided by the Kolmogorov-Arnold representation theorem: any continuous n-dimensional function can be expressed as a finite superposition of one-dimensional functions. The proposed Kolmogorov-Arnold Classifier System (KACS) decomposes the target function into one-dimensional subproblems and assigns a dedicated ruleset to each, reducing the worst-case rule count from O(mn) to O(mn2) and replacing n-dimensional local models with one-dimensional models requiring only two parameters per rule, independent of n. We also provide the first constructive proof that an LCS, namely KACS, is a universal approximator for continuous functions on compact domains. Evaluated against a direct n-dimensional input space partitioning approach under otherwise identical conditions, KACS achieves competitive accuracy in many settings while using only 2% to 40% of the parameters. Our implementation is available at https://github.com/YNU-NakataLab/KACS.