Showing papers by "Christian Szegedy published in 2020"

Graph Representations for Higher-Order Logic and Theorem Proving

Abstract: This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significant challenge for deep learning. Higher-order logic is highly expressive and, even though it is well-structured with a clearly defined grammar and semantics, there still remains no well-established method to convert formulas into graph-based representations. In this paper, we consider several graphical representations of higher-order logic and evaluate them against the HOList benchmark for higher-order theorem proving.

Mathematical Reasoning via Self-supervised Skip-tree Training

Abstract: We examine whether self-supervised language modeling applied to mathematical formulas enables logical reasoning. We suggest several logical reasoning tasks that can be used to evaluate language models trained on formal mathematical statements, such as type inference, suggesting missing assumptions and completing equalities. To train language models for formal mathematics, we propose a novel skip-tree task. We find that models trained on the skip-tree task show surprisingly strong mathematical reasoning abilities, and outperform models trained on standard skip-sequence tasks. We also analyze the models' ability to formulate new conjectures by measuring how often the predictions are provable and useful in other proofs.

A Promising Path Towards Autoformalization and General Artificial Intelligence

Abstract: An autoformalization system is an AI that learns to read natural language content and to turn it into an abstract, machine verifiable formalization, ideally by bootstrapping from unlabeled training data with minimum human interaction. This is a difficult task in general, one that would require strong automated reasoning and automated natural language processing capabilities. In this paper, it is argued that autoformalization is a promising path for systems to learn sophisticated, general purpose reasoning in all domains of mathematics and computer science. This could have far reaching implications not just for mathematical research, but also for software synthesis. Here I provide the outline for a realistic path towards those goals and give a survey of recent results that support the feasibility of this direction.

Mathematical Reasoning in Latent Space

Abstract: We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can compress this information by embedding the formula in a vector space, such that the vector associated with a statement can be used to predict whether a statement can be rewritten by other theorems. Predicting the embedding of a formula generated by some rewrite rule is naturally viewed as approximate reasoning in the latent space. In order to measure the effectiveness of this reasoning, we perform approximate deduction sequences in the latent space and use the resulting embedding to inform the semantic features of the corresponding formal statement (which is obtained by performing the corresponding rewrite sequence using real formulas). Our experiments show that graph neural networks can make non-trivial predictions about the rewrite-success of statements, even when they propagate predicted latent representations for several steps. Since our corpus of mathematical formulas includes a wide variety of mathematical disciplines, this experiment is a strong indicator for the feasibility of deduction in latent space in general.

Language Modeling for Formal Mathematics

Abstract: We examine whether language modeling applied to mathematical formulas enables logical reasoning. We suggest several logical reasoning tasks that can be used to evaluate language models trained on formal mathematical statements, such as type inference, suggesting missing assumptions and completing equalities. To train language models for formal mathematics, we propose a novel skip-tree task, which outperforms standard language modeling tasks on our reasoning benchmarks. We also analyze the models' ability to formulate new conjectures by measuring how often the predictions that do not fit the ground truth or any training data turn out to be true and useful statements.

Text Embeddings for Retrieval from a Large Knowledge Base

Abstract: Text embedding representing natural language documents in a semantic vector space can be used for document retrieval using nearest neighbor lookup In order to study the feasibility of neural models specialized for retrieval in a semantically meaningful way, we suggest the use of the Stanford Question Answering Dataset (SQuAD) in an open-domain question answering context, where the first task is to find paragraphs useful for answering a given question First, we compare the quality of various text-embedding methods on the performance of retrieval and give an extensive empirical comparison on the performance of various non-augmented base embedding with, and without IDF weighting Our main results are that by training deep residual neural models, specifically for retrieval purposes, can yield significant gains when it is used to augment existing embeddings We also establish that deeper models are superior to this task The best base baseline embeddings augmented by our learned neural approach improves the top-1 paragraph recall of the system by \(14\%\)