Learning-Assisted Automated Reasoning with Flyspeck

Abstract

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 % of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

Modelling the way mathematics is actually done

“…Mathematical formalisms take this to an extreme, since they simplify and tightly constrain the ways in which their users can write things down. We note that machine learning methods have been applied to corpora of formalised mathematics to good effect [30,31,33,44]. However, mathematics as it is done by most mathematicians has a very different structure -and there is a lot more of it available, as summarised in The Stack Exchange corpus alone is much larger than the 160000 games in AlphaGo's initial training set [34], but the data looks completely different.…”

Metis-based Paramodulation Tactic for HOL Light

“…The ATP dependencies are obtained as in [KU14]. We evaluate a problem by feeding the conjecture and the dependencies to a proof reconstruction method in HOL Light, such as: If the method can reconstruct the proof within a given time limit, the problem counts as proven.…”

Do Portfolio Solvers Harm?

“…[13], often combined with an SS-portfolio approach. Leading competition versions of solvers for the "main" divisions of the first-order logic theorem proving competition CASC [27] namely E [24], iProver [16] and Vampire [18] are all SS-portfolio solver instances.…”