Abstract
Abstract Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is additionally reversible. This means that the tree shape of a formula including all symbols can be reconstructed from the dense vector representation. We do that by training two decoders: one that extracts the top symbol of the tree and one that extracts embedding vectors of subtrees. The syntactic and semantic logical properties that we aim to preserve include both structural formula properties, applicability of natural deduction steps and even more complex operations like unifiability. We propose datasets that can be used to train these syntactic and semantic properties. We evaluate the viability of the developed encoding across the proposed datasets as well as for the practical theorem proving problem of premise selection in the Mizar corpus.
Highlights
The last two decades saw an emergence of computer systems applied to logic and reasoning
Interactive theorem provers (ITPs) were initially not intended to be used in standard mathematics, subsequent algorithmic developments and modern-day computers allow for a formal approach to major mathematical proofs [Hal08]
We have developed and compared logical formula encodings inspired by the way human mathematicians work
Summary
The last two decades saw an emergence of computer systems applied to logic and reasoning. Two kinds of such computer systems are interactive proof assistant systems [HUW14] and automated theorem proving systems [RV01]. Interactive theorem provers (ITPs) were initially not intended to be used in standard mathematics, subsequent algorithmic developments and modern-day computers allow for a formal approach to major mathematical proofs [Hal08]. Such developments include the proof of Kepler’s conjecture [HAB+17] and the four colour theorem [Gon08]. ITPs are used to formally reason about computer systems, e.g. have been used to develop a formally verified operating system kernel [KAE+10]
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