Mathematical conjecture generation using machine intelligence
Challenger Mishra Subhayan Roy Moulik Rahul Sarkar
Conjectures have historically played an important role in the development of
pure mathematics. We propose a systematic approach to finding abstract patterns
in mathematical data, in order to generate conjectures about mathematical
inequalities, using machine intelligence. We focus on strict inequalities of
type f < g and associate them with a vector space. By geometerising this space,
which we refer to as a conjecture space, we prove that this space is isomorphic
to a Banach manifold. We develop a structural understanding of this conjecture
space by studying linear automorphisms of this manifold and show that this
space admits several free group actions. Based on these insights, we propose an
algorithmic pipeline to generate novel conjectures using geometric gradient
descent, where the metric is informed by the invariances of the conjecture
space. As proof of concept, we give a toy algorithm to generate novel
conjectures about the prime counting function and diameters of Cayley graphs of
non-abelian simple groups. We also report private communications with
colleagues in which some conjectures were proved, and highlight that some
conjectures generated using this procedure are still unproven. Finally, we
propose a pipeline of mathematical discovery in this space and highlight the
importance of domain expertise in this pipeline.