DeepMath - Deep Sequence Models for Premise Selection DeepMath - Deep Sequence Models for Premise Selection
Paper summary #### Introduction * Automated Theorem Proving (ATP) - Attempting to prove mathematical theorems automatically. * Bottlenecks in ATP: * **Autoformalization** - Semantic or formal parsing of informal proofs. * **Automated Reasoning** - Reasoning about already formalised proofs. * Paper evaluates the effectiveness of neural sequence models for premise selection (related to automated reasoning) without using hand engineered features. * [Link to the paper](https://arxiv.org/abs/1606.04442) #### Premise Selection * Given a large set of premises P, an ATP system A with given resource limits, and a new conjecture C, predict those premises from P that will most likely lead to an automatically constructed proof of C by A #### Dataset * Mizar Mathematical Library (MML) used as the formal corpus. * The premise length varies from 5 to 84299 characters and over 60% if the words occur fewer than 10 times (rare word problem). #### Approach * The model predicts the probability that a given axiom is useful for proving a given conjecture. * Conjecture and axiom sequences are separately embedded into fixed length real vectors, then concatenated and passed to a third network with few fully connected layers and logistic loss. * The two embedding networks and the joint predictor path are trained jointly. ##### Stage 1: Character-level Models * Treat premises on character level using an 80-dimensional one hot encoding vector. * Architectures for embedding: * pure recurrent LSTM and GRU Network * CNN (with max pooling) * Recurrent-convolutional network that shortens the sequence using convolutional layer before feeding it to LSTM. ##### Stage 2: Word-level Models * MML dataset contains both implicit and explicit definitions. * To avoid manually encoding the implicit definitions, the entire statement defining an identifier is embedded and the definition embeddings are used as word level embeddings. * This is better than recursively expanding and embedding the word definition as the definition chains can be very deep. * Once word level embeddings are obtained, the architecture from Character-level models can be reused. #### Experiments ##### Metrics * For each conjecture, the model ranks the possible premises. * Primary metric is the number of conjectures proved from top-k premises. * Average Max Relative Rank (AMMR) is more sophisticated measure based on the motivation that conjectures are easier to prove if all their dependencies occur earlier in ranking. * Since it is very costly to rank all axioms for a conjecture, an approximation is made and a fixed number of random false dependencies are used for evaluating AMMR. ##### Network Training * Asynchronous distributed stochastic gradient descent using Adam optimizer. * Clipped vs Non-clipped Gradients. * Max Sequence length of 2048 for character-level sequences and 500 for word-level sequences. ##### Results * Best Selection Pipeline - Stage 1 character-level CNN which produces word-level embeddings for the next stage. * Best models use simple CNNs followed by max pooling and two-stage definition-based def-CNN outperforms naive word-CNN (where word embeddings are learnt in a single pass).
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DeepMath - Deep Sequence Models for Premise Selection
Alex A. Alemi and Francois Chollet and Geoffrey Irving and Christian Szegedy and Josef Urban
arXiv e-Print archive - 2016 via arXiv
Keywords: cs.AI, cs.LG, cs.LO

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