We propose a deep learning-based approach to the problem of premise selection: selecting mathematical statements relevant for proving a given conjecture. We represent a higher-order logic formula as a graph that is invariant to variable renaming but still fully preserves syntactic and semantic information. We then embed the graph into a vector via a novel embedding method that preserves the information of edge ordering. Our approach achieves state-of-the-art results on the HolStep dataset, improving the classification accuracy from 83 to 90.3 .
Premise selection as a machine learning problem was introduced by @cite_14 , who constructed a corpus of proofs to train a kernelized classifier using bag-of-word features that represent the occurrences of terms in a vocabulary. Deep learning techniques were first applied to premise selection in the DeepMath work by @cite_36 , who applied recurrent networks and convolutional to formulas represented as textual sequences, and showed that deep learning approaches can achieve competitive results against baselines using hand-engineered features. Serving the needs for large datasets for training deep models, @cite_31 introduced the HolStep dataset that consists of 2M statements and 10K conjectures, an order of magnitude larger than the DeepMath dataset @cite_36 .
@cite_36 @cite_31 @cite_14
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The success rates of automated theorem provers in large theories highly depend on the choice of given facts. Premise selection is the task of choosing a subset of given facts, which is most likely to lead to a successful automated deduction proof of a given conjecture. Premise selection can be viewed as a multi-label classification problem, where machine learning from related proofs turns out to currently be the most successful method. Random forests are a machine learning technique known to perform especially well on large datasets. In this paper, we evaluate random forest algorithms for premise selection. To deal with the specifics of automated reasoning, we propose a number of extensions to random forests, such as incremental learning, multi-path querying, depth weighting, feature IDF inverse document frequency, and integration of secondary classifiers in the tree leaves. With these extensions, we improve on the k-nearest neighbour algorithm both in terms of prediction quality and ATP performance. Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. This work develops learning-based premise selection in two ways. First, a fine-grained dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed, extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50 improvement on the benchmark over the state-of-the-art Vampire SInE system for automated reasoning in large theories. We study the effectiveness of neural sequence models for premise selection in automated theorem proving, a key bottleneck for progress in formalized mathematics. We propose a two stage approach for this task that yields good results for the premise selection task on the Mizar corpus while avoiding the hand-engineered features of existing state-of-the-art models. To our knowledge, this is the first time deep learning has been applied theorem proving on a large scale. The main contribution of this paper is a simple semi-supervised pipeline that only uses the original training set without collecting extra data. It is challenging in 1) how to obtain more training data only from the training set and 2) how to use the newly generated data. In this work, the generative adversarial network (GAN) is used to generate unlabeled samples. We propose the label smoothing regularization for outliers (LSRO). This method assigns a uniform label distribution to the unlabeled images, which regularizes the supervised model and improves the baseline. We verify the proposed method on a practical problem: person re-identification (re-ID). This task aims to retrieve a query person from other cameras. We adopt the deep convolutional generative adversarial network (DCGAN) for sample generation, and a baseline convolutional neural network (CNN) for representation learning. Experiments show that adding the GAN-generated data effectively improves the discriminative ability of learned CNN embeddings. On three large-scale datasets, Market-1501, CUHK03 and DukeMTMC-reID, we obtain +4.37 , +1.6 and +2.46 improvement in rank-1 precision over the baseline CNN, respectively. We additionally apply the proposed method to fine-grained bird recognition and achieve a +0.6 improvement over a strong baseline. The code is available at this https URL