Semantically-guided theorem proving for mathematics

Research Project, 2026 – 2030

When mathematicians prove theorems, they make use of semantic information such as the intended interpretation of the functions and the nature of the goal to be proved. Automated theorem provers, however, generally do not. They make deductions rather aimlessly and find proofs by "symbol pushing". This means that they get extremely lost on more complex proof problems.This defect is most clearly visible in equational theorem proving, where the dominant approach, Knuth-Bendix completion, not only makes no use of semantic information but is completely non-goal-directed. Many problems which are easy for humans are impossible for equational theorem provers as a result.In this project we will develop semantics-guided proof procedures for equational logic. The aim is to (1) make use of models, that is intended meanings of the functions in the problem; (2) exploit knowledge of the goal, that is the conjecture to be proved. Models can come from a user, from a database of mathematical objects, or from the proof search itself.We will validate our project by collaborating with mathematicians in loop theory, helping them with their work on the Abelian Inner Mapping conjecture. We will also produce a fully automatic proof of the Robbins conjecture, an open problem solved using a specialised automated theorem prover and much human assistance. We will instead produce a fully automatic proof from a fully general-purpose tool.