Showing papers by "Christian Szegedy published in 2019"
Proceedings Article•
24 May 201980 citations
Posted Content•
05 Apr 2019
TL;DR: This work provides an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment and presents a deep reinforcement learning driven automated theorem provers, DeepHOL, with strong initial results on this benchmark.
Abstract: We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.
29 citations
Posted Content•
TL;DR: In this paper, the first use of graph neural networks (GNNs) for higher-order proof search was presented and demonstrated that GNNs can improve upon state-of-the-art results in this domain.
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.
28 citations
Posted Content•
TL;DR: The 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, a strong indicator for the feasibility of deduction in latent space in general.
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.
19 citations
Posted Content•
TL;DR: In this article, the authors present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic called HOL Light, which can be used as a reinforcement learning environment.
Abstract: We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.
16 citations
Patent•
09 Aug 2019
TL;DR: In this paper, the authors present a method and a system for image processing using deep neural networks, where the deep neural network comprises a plurality of sub-networks, wherein the subnetworks are arranged in a sequence from lowest to highest.
Abstract: The invention relates to a method and a system for processing images. Methods, systems, and apparatus, including computer programs encoded on computer storage media, for image processing using deep neural networks, are provided. One of the methods includes receiving data characterizing an input image; processing the data characterizing the input image using a deep neural network to generate an alternative representation of the input image, wherein the deep neural network comprises a plurality of subnetworks, wherein the subnetworks are arranged in a sequence from lowest to highest, and whereinprocessing the data characterizing the input image using the deep neural network comprises processing the data through each of the subnetworks in the sequence; and processing the alternative representation of the input image through an output layer to generate an output from the input image.