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Deep Network Guided Proof Search

Sarah Loos
Geoffrey Irving
Christian Szegedy
Cezary Kaliszyk
LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, EasyChair (2017), pp. 85-105


Deep learning techniques lie at the heart of several significant AI advances and break- throughs in the past years including object recognition and detection, image captioning, machine translation, speech recognition and synthesis, and playing the game of go. Automated first-order theorem provers can aid in the formalization and verification of mathematical theorems and play a crucial role in program analysis, theory reasoning, security, interpolation, and system verification. Here we suggest deep learning based guidance to the proof process of E Prover. We train and compare several deep neural network models on the traces of existing ATP proofs of Mizar statements and use them to select processed clauses during proof search. We give experimental evidence that with a hybrid, two-phase approach, deep learning based guidance can significantly reduce the average number of proof search steps while increasing the number of theorems proved. Using a few proof guidance strategies that leverage deep neural networks, we have found first-order proofs of 7.18% of the first-order logic translations of the Mizar Mathematical Library theorems that did not previously have ATP generated proofs. This increases the ratio of statements in the corpus with ATP generated proofs from 56% to 59%.