Answer by Alan Bundy for Proofs that require fundamentally new ways of thinking

2011-01-19T11:37:16Z

I work in automated theorem proving. I certainly agree, in principle, that there are no proofs that are inherently beyond the ability of a computer to solve, but I also think that there are fundamental methodological problems in addressing the problem as posed.

The problem is to come up with a solution that would not be regarded as 'cheating', i.e., somehow building the solution into the automated prover to start with. New proof methods can be captured by what are called 'tactics', i.e., programs that guide a prover through a proof. Clearly, it would not be satisfactory to analyse the original proof, extract a tactic from it (even a generic one) that captures the novel proof structure and then demonstrate that the enhanced prover could now 'discover' the novel proof. Rather, we want the prover to invent the new tactic itself, perhaps by some analysis of the conjecture to be proved, and then apply it. So we need an automated prover that learns. But anticipating what kind of tactic we want to be learnt may well influence the design of the learning mechanism. We've now kicked the 'cheating' problem up a level.

Methodologically, what we want is a large class of problems of this form. Some we can use for development of the learning mechanism, and some we can use to test it. Success on a previously unseen test set would demonstrate the generality of the learning mechanism and, hence, the absence of cheating. Unfortunately, these challenges are usually posed as 'can a prover prove this theorem' rather than 'can it solve this wide range of theorems, each requiring a different form of novelty. Clearly, this latter form of the question is hugely challenging and we're unlikely to see if solved in the foreseeable future.

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Answer by Alan Bundy for Proofs that require fundamentally new ways of thinking