Learning Interestingness in Automated Mathematical Theory Formation
📝 Abstract
We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce $\emph{FERMAT} $, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through $\emph{FERMAT} $: automatically scoring the $\emph{interestingness}$ of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines. We open-source the $\emph{FERMAT}$ environment at this URL(https://github.com/trishullab/Fermat) .
💡 Analysis
We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce $\emph{FERMAT} $, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through $\emph{FERMAT} $: automatically scoring the $\emph{interestingness}$ of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines. We open-source the $\emph{FERMAT}$ environment at this URL(https://github.com/trishullab/Fermat) .
📄 Content
AI researchers have dreamed of building an “automated mathematician” since the 1950s [29]. Such a system would allow human mathematicians to harness the vast processing capacity of computers to discover entirely new areas of mathematics [42]. An emerging body of work seeks to realize this dream using the tools of modern machine learning. In particular, the AI community has developed a wide range of systems that can prove formal theorems [13,47] and search for programs discovering mathematical constructions [31,40].
However, a key limitation of much of this research is that it is focused on solving predefined problems. Mathematicians develop theories through an open-ended process of defining new concepts, studying their properties, making conjectures, and proving or finding counterexamples. While some work [35] has offered systems that construct new problems in addition to solving them, there is currently no framework that supports the full theory-formation process, including, for example, the synthesis of new definitions in addition to problems.
A central challenge in this open-ended process is guiding the search. The space of possible definitions and conjectures is combinatorially vast, and most paths lead to trivial or dull mathematics. Human mathematicians navigate this space using a nuanced, intuitive sense of “interestingness” -a judgment of scientific potential that directs their focus. An explicit formulation of this concept has long been debated, with different perspectives valuing properties such as the surprising connection between disparate fields [36], depth and generality [22], or its unexpected real-world applicability [49].
In this paper, we take two key steps towards addressing these challenges. First, we provide a reinforcement learning (RL) framework, called FERMAT (Figure 1), which can be used to design and evaluate new algorithms for automatic theory formation. The system generalizes the early symbolic computing-prover system HR [8], which used a system of production rules to generate new concepts and conjectures, either symbolically or from explicit examples, and proof mechanisms for resolving conjectures. We model these symbolic steps as the actions of a Markov Decision Process (MDP), and the mathematical knowledge available at a given point during exploration as an MDP state. This formulation opens up numerous RL problems relevant to theory formation.
Our second contribution is a solution to a particular algorithmic problem in FERMAT: learning an interestingness heuristic for selecting mathematical concepts to develop. To form a theory, one must navigate a combinatorial search space of mathematical objects, most objects in which are not meaningful or worthy of study. Prior works were attentive to this problem, but required hard-coded measures to formalize the concept of interestingness [8,26]. In contrast, we frame the discovery of this heuristic as a learning problem. We specifically learn interestingness measures as programmatic representations, as this makes them interpretable and allows us to analyze what may contribute to fruitful discovery. To this end, we develop an LLM-driven method, called EvoAbstract, for learning the intrinsic value of mathematical objects in the context of the current theory. EvoAbstract is an evolutionary program synthesis algorithm that extends the FunSearch [40] approach with a form of abstraction learning, allowing for interpretable abstractions to be discovered during function search. We experimentally show that EvoAbstract can automatically synthesize interestingness measures that lead to significant improvements in discovering concepts in elementary number theory and finite fields over hard-coded baselines.
2 Problem Formulation and Motivation
Figure 1: A high-level description of FERMAT, our environment for mathematical theory formation. At any given time, the current theory (state) is represented as a knowledge graph consisting of the mathematical definitions, conjectures, and theorems discovered so far. At each step, the policy π inputs the current state and selects an action to apply, updating the theory with additional information. The action space allows the production of new definitions, conjectures, and proofs of theorems.
To rigorously study automated mathematical theory formation using reinforcement learning, we first formalize the process as an MDP (S, A, T , R). This framework allows us to model the sequential nature of mathematical discovery, where an agent iteratively expands a body of knowledge by making choices about definitions, conjectures, and proof attempts. Let M denote the universe of all well-formed mathematical entities. The components of this MDP are defined as follows:
• Mathematical State Space (S): A state S ∈ S represents the current state of mathematical knowledge, represented as a directed knowledge graph G = (V, E), where:
• V ⊆ M is the set of mathematical entities, categorized into definitions D, conjecture
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