Executable Functional Abstractions: Inferring Generative Programs for Advanced Math Problems

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from reinforcement learning (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for mathematical reasoning as problem generators for stress-testing models. However, prior work has been limited to automatically constructing abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced mathematics problems by developing EFAGen, which operationalizes the task of automatically inferring an EFA for a given seed problem and solution as a program synthesis task. We first formalize the properties of any valid EFA as executable unit tests. Using execution feedback from the unit tests, we search over candidate programs sampled from a LLM to find EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. We then apply the tests as a reward signal, training LLMs to become better writers of EFAs. We show that EFAs inferred by EFAGen are faithful to the seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across diverse sources of competition-level math problems. Finally, we show uses of model-written EFAs e.g., finding harder/easier problem variants, as well as data generation.