Top-down Automated Theorem Proving (Notes for Sir Timothy)

TL;DR

This paper proposes a top-down approach to automated theorem proving that mimics human mathematical reasoning by focusing on domain concepts and human-like proofs, contrasting with traditional bottom-up formal methods.

Contribution

It introduces a novel top-down ATP methodology that emphasizes human-like proofs and domain concepts, differing from the conventional bottom-up formalization approach.

Findings

Highlights the importance of human-like proof structures

Suggests coding domain concepts for more effective ATP

Contrasts top-down and bottom-up approaches

Abstract

We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code all relevant domain concepts. These concepts encode a large portion of the knowledge in any domain. Furthermore, researchers should write programs that produce proofs of the kind that human mathematicians write (and publish); this means proofs that might sometimes have mistakes; and this means making inferences that are sometimes invalid. This approach is meant to contrast with the historically dominant "bottom up" approach: coding fundamental types (typically sets), axioms and rules for (valid) inference, and building up from this foundation to the theorems of mathematical practice and to their outstanding questions. It is an important fact that the actual proofs that mathematicians publish in math journals do not look like the formalized proofs of Russell & Whitehead's Principia Mathematica (or modern computer systems like Lean that automate some of this formalization). We believe some "lack of rigor" (in mathematical practice) is human-like, and can and should be leveraged for ATP.