Discovering Infinite Recursive Conjectures Through Genetic Programming

Mathematics is filled with conjectures that involve infinite recursive structure, representing complex structures and underlying deep relationships. Discovering such conjectures is crucial for advancing our understanding of fundamental mathematical principles, as they reveal unexpected patterns and connections across different areas of mathematics. Due to their inherent complexity and infinite nature, these conjectures are challenging to uncover using traditional methods such as manual derivation and numerical calculations. A core task in studying such conjectures is identifying recursive relationships that describe potentially unknown patterns and structures. This task can be framed as a symbolic regression problem, as it involves searching for a suitable mathematical form to represent complex relationships. To address this symbolic regression problem, we propose a gene programming-based algorithm named infinite conjecture explorer (ICE) with a dual-chromosome encoding (DCE) and a two-sided matching operator (TMO). DCE encodes the two sides of a conjectured as separate two chromosomes, providing a clear representation of the underlying structure of an equation. Unlike other encoding methods, DCE improves the efficiency of discovering meaningful conjectures. Since the use of DCE results in two corresponding large and complex search spaces, TMO is designed to efficiently identify expressions that match on both sides of the equation across two spaces. The experimental results show that the proposed ICE is effective in generating promising conjectures with diverse forms.