Abstract
This work explores the application of deep learning, a machine learning technique that uses deep neural networks (DNN) in its core, to an automated theorem proving (ATP) problem. To this end, we construct a statistical model which quantifies the likelihood that a proof is indeed a correct one of a given proposition. Based on this model, we give a proof-synthesis procedure that searches for a proof in the order of the likelihood. This procedure uses an estimator of the likelihood of an inference rule being applied at each step of a proof. As an implementation of the estimator, we propose a proposition-to-proof architecture, which is a DNN tailored to the automated proof synthesis problem. To empirically demonstrate its usefulness, we apply our model to synthesize proofs of the minimal propositional logic. We train the proposition-to-proof model using a training dataset of proposition–proof pairs. The evaluation against a benchmark set shows the very high accuracy and an improvement to the recent work of neural proof synthesis.
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- 1.
Following the convention of neural network research, we use the word “model” both for a statistical model and for a trained DNN.
- 2.
We can annotate \(\textsf {typeof}( M )\) by applying a standard type-inference algorithm for the simply typed lambda calculus to \( M \).
- 3.
It is not clear from Mou et al. [25] how they express an AST as a binary tree in their implementation.
- 4.
Epoch is the unit that means how many times the dataset is scanned during the training.
- 5.
See the full version [30] for the detail. The low accuracy of these rules does not influence the overall accuracy much because the validation dataset does not contain many applications of them either.
- 6.
The source code is at http://www2.disco.unimib.it/fiorino/pitp.html; accessed on 26/8/2018.
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Acknowledgments
We are grateful to Akifumi Imanishi, Kensuke Kojima, and Takayuki Muranushi for the discussion and the contribution to the earlier version of the present work. Kohei Suenaga is supported by JST PRESTO Grant Number JPMJPR15E5, Japan. Taro Sekiyama is supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST.
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Sekiyama, T., Suenaga, K. (2018). Automated Proof Synthesis for the Minimal Propositional Logic with Deep Neural Networks. In: Ryu, S. (eds) Programming Languages and Systems. APLAS 2018. Lecture Notes in Computer Science(), vol 11275. Springer, Cham. https://doi.org/10.1007/978-3-030-02768-1_17
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