Symbolic Regression as a Lens for Interestingness: Probabilistic Model Ensembles in Theory Formation

Research Question: Can probabilistic symbolic regression methods (e.g., Bayesian approaches) be adapted to generate, evaluate, and rank mathematical theories by interestingness within automated theory formation frameworks?

Hypothesis: Symbolic regression ensembles naturally generate a diversity of plausible theories; measuring the “surprisingness” or informativeness of ensemble members can provide a principled, information-theoretic interestingness criterion that complements or outperforms heuristic measures.

Experiment Plan: - Integrate a Bayesian symbolic regression engine (Guimerà & Sales-Pardo, 2025) with FERMAT’s theory formation pipeline.

• For each set of definitions/axioms, generate an ensemble of candidate theorems or conjectures, and score them based on measures like model posterior probability, entropy, and divergence from prior expectations.
• Compare the top-ranked ensemble members to those ranked highly by current interestingness heuristics, and evaluate their mathematical value via expert review and downstream utility.
• Investigate whether such probabilistic measures help uncover deeper or more generalizable results in domains like number theory or group theory.

References:

• Guimerà, R., & Sales-Pardo, M. (2025). Bayesian symbolic regression: Automated equation discovery from a physicists' perspective. arXiv.org.
• Pease, A., Colton, S., & Charnley, J. (2013). Automated Theory Formation: The Next Generation.