Abstract
We describe several datasets and first experiments with creating conjectures by neural methods. The datasets are based on the Mizar Mathematical Library processed in several forms and the problems extracted from it by the MPTP system and proved by the E prover using the ENIGMA guidance. The conjecturing experiments use the Transformer architecture and in particular its GPT-2 implementation.
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Funding
Funded by the AI4REASON ERC Consolidator grant nr. 649043 and by the Czech project AI&Reasoning CZ.02.1.01/0.0/0.0/15_003/0000466 and the European Regional Development Fund. We thank K. Chvalovský and T. Gauthier for discussions.
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A Additional Data From the Experiments
A Additional Data From the Experiments
Dataset 2 training and loss.
1.1 A.1 XXREAL 1:48 and its GPT-2 predictions
Following are the Mizar premises in the order proposed by GPT-2. The fifth and sixth were not needed for the ATP proof.
1.2 A.2 GROUPP_1:10 and its generalization conjectured by GPT-2
The generalization that avoids finiteness:
We don’t have an ATP proof of the generalization yet. We thank algebraists Michael Kinyon and David Stanovský for confirming that this generalization is provable. Based on this example Stanovský commented that related Mizar theorems can be similarly generalized.
1.3 A.3 SINCOS10:17 and a false conjecture by GPT-2
GPT-2 generated the following conjecture, which is false. Along with another GPT-2 conjecture about the differentiability of sec on the interval, this results in an ATP proof of SINCOS10:17.
1.4 A.4 FUNCTOR1:9 and a GPT-2 conjecture reducing it to FUNCTOR1:7
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Urban, J., Jakubův, J. (2020). First Neural Conjecturing Datasets and Experiments. In: Benzmüller, C., Miller, B. (eds) Intelligent Computer Mathematics. CICM 2020. Lecture Notes in Computer Science(), vol 12236. Springer, Cham. https://doi.org/10.1007/978-3-030-53518-6_24
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