Integrating logical reasoning within deep learning architectures has been a major goal of modern AI systems. In this paper, we propose a new direction toward this goal by introducing a differentiable (smoothed) maximum satisfiability (MAXSAT) solver that can be integrated into the loop of larger deep learning systems. Our (approximate) solver is based upon a fast coordinate descent approach to solving the semidefinite program (SDP) associated with the MAXSAT problem. We show how to analytically differentiate through the solution to this SDP and efficiently solve the associated backward pass. We demonstrate that by integrating this solver into end-to-end learning systems, we can learn the logical structure of challenging problems in a minimally supervised fashion. In particular, we show that we can learn the parity function using single-bit supervision (a traditionally hard task for deep networks) and learn how to play 9x9 Sudoku solely from examples. We also solve a "visual Sudok" problem that maps images of Sudoku puzzles to their associated logical solutions by combining our MAXSAT solver with a traditional convolutional architecture. Our approach thus shows promise in integrating logical structures within deep learning.

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SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver

Today, the dominant paradigm for training neural networks involves minimizing task loss on a large dataset. Using world knowledge to inform a model, and yet retain the ability to perform end-to-end training remains an open question. In this paper, we present a novel framework for introducing declarative knowledge to neural network architectures in order to guide training and prediction. Our framework systematically compiles logical statements into computation graphs that augment a neural network without extra learnable parameters or manual redesign. We evaluate our modeling strategy on three tasks: machine comprehension, natural language inference, and text chunking. Our experiments show that knowledge-augmented networks can strongly improve over baselines, especially in low-data regimes.

Augmenting Neural Networks with First-order Logic

The ability to conduct logical reasoning is a fundamental aspect of intelligent human behavior, and thus an important problem along the way to human-level artificial intelligence. Traditionally, logic-based symbolic methods from the field of knowledge representation and reasoning have been used to equip agents with capabilities that resemble human logical reasoning qualities. More recently, however, there has been an increasing interest in using machine learning rather than logic-based symbolic formalisms to tackle these tasks. In this paper, we employ state-of-the-art methods for training deep neural networks to devise a novel model that is able to learn how to effectively perform logical reasoning in the form of basic ontology reasoning. This is an important and at the same time very natural logical reasoning task, which is why the presented approach is applicable to a plethora of important real-world problems. We present the outcomes of several experiments, which show that our model is able to learn to perform highly accurate ontology reasoning on very large, diverse, and challenging benchmarks. Furthermore, it turned out that the suggested approach suffers much less from different obstacles that prohibit logic-based symbolic reasoning, and, at the same time, is surprisingly plausible from a biological point of view.

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Ontology Reasoning with Deep Neural Networks

One aspect of human commonsense reasoning is the ability to make presumptions about daily experiences, activities and social interactions with others We propose a new commonsense reasoning benchmark where the task is to uncover commonsense presumptions implied by imprecisely stated natural language commands in the form of if-then-because statements For example, in the command "If it snows at night then wake me up early because I don't want to be late for work" the speaker relies on commonsense reasoning of the listener to infer the implicit presumption that it must snow enough to cause traffic slowdowns Such if-then-because commands are particularly important when users instruct conversational agents We release a benchmark data set for this task, collected from humans and annotated with commonsense presumptions We develop a neuro-symbolic theorem prover that extracts multi-hop reasoning chains and apply it to this problem We further develop an interactive conversational framework that evokes commonsense knowledge from humans for completing reasoning chains

Conversational Neuro-Symbolic Commonsense Reasoning

SATNet is an award-winning MAXSAT solver that can be used to infer logical rules and integrated as a differentiable layer in a deep neural network [1]. It had been shown to solve Sudoku puzzles visually from examples of puzzle digit images, and was heralded as an impressive achievement towards the longstanding AI goal of combining pattern recognition with logical reasoning. In this paper, we clarify SATNet’s capabilities by showing that in the absence of intermediate labels that identify individual Sudoku digit images with their logical representations, SATNet completely fails at visual Sudoku (0% test accuracy). More generally, the failure can be pinpointed to its inability to learn to assign symbols to perceptual phenomena, also known as the symbol grounding problem [2], which has long been thought to be a prerequisite for intelligent agents to perform real-world logical reasoning. We propose an MNIST based test as an easy instance of the symbol grounding problem that can serve as a sanity check for differentiable symbolic solvers in general. Naive applications of SATNet on this test lead to performance worse than that of models without logical reasoning capabilities. We report on the causes of SATNet’s failure and how to prevent them.

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Assessing SATNet's Ability to Solve the Symbol Grounding Problem

Combining machine learning with logic-based expert systems in order to get the best of both worlds are becoming increasingly popular. However, to what extent machine learning can already learn to reason over rule-based knowledge is still an open problem. In this paper, we explore how symbolic logic, defined as logic programs at a character level, is learned to be represented in a high-dimensional vector space using RNN-based iterative neural networks to perform reasoning. We create a new dataset that defines 12 classes of logic programs exemplifying increased level of complexity of logical reasoning and train the networks in an end-to-end fashion to learn whether a logic program entails a given query. We analyse how learning the inference algorithm gives rise to representations of atoms, literals and rules within logic programs and evaluate against increasing lengths of predicate and constant symbols as well as increasing steps of multi-hop reasoning.

DeepLogic: Towards End-to-End Differentiable Logical Reasoning

Neural networks have been learning complex multi-hop reasoning in various domains. One such formal setting for reasoning, logic, provides a challenging case for neural networks. In this article, we propose a Neural Inference Network (NIN) for learning logical inference over classes of logic programs. Trained in an end-to-end fashion NIN learns representations of normal logic programs, by processing them at a character level, and the reasoning algorithm for checking whether a logic program entails a given query. We define 12 classes of logic programs that exemplify increased level of complexity of the inference process (multi-hop and default reasoning) and show that our NIN passes 10 out of the 12 tasks. We also analyse the learnt representations of logic programs that NIN uses to perform the logical inference.

DeepLogic: End-to-End Logical Reasoning.

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DeepLogic: Towards End-to-End Differentiable Logical Reasoning.

Human reasoning involves recognising common underlying principles across many examples. The by-products of such reasoning are invariants that capture patterns such as "if someone went somewhere then they are there", expressed using variables "someone" and "somewhere" instead of mentioning specific people or places. Humans learn what variables are and how to use them at a young age. This paper explores whether machines can also learn and use variables solely from examples without requiring human pre-engineering. We propose Unification Networks, an end-to-end differentiable neural network approach capable of lifting examples into invariants and using those invariants to solve a given task. The core characteristic of our architecture is soft unification between examples that enables the network to generalise parts of the input into variables, thereby learning invariants. We evaluate our approach on five datasets to demonstrate that learning invariants captures patterns in the data and can improve performance over baselines.

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Learning Invariants through Soft Unification

The task of Video Question Answering (VideoQA) consists in answering natural language questions about a video and serves as a proxy to evaluate the performance of a model in scene sequence understanding. Most methods designed for VideoQA up-to-date are end-to-end deep learning architectures which struggle at complex temporal and causal reasoning and provide limited transparency in reasoning steps. We present the HySTER: a Hybrid Spatio-Temporal Event Reasoner to reason over physical events in videos. Our model leverages the strength of deep learning methods to extract information from video frames with the reasoning capabilities and explainability of symbolic artificial intelligence in an answer set programming framework. We define a method based on general temporal, causal and physics rules which can be transferred across tasks. We apply our model to the CLEVRER dataset and demonstrate state-of-the-art results in question answering accuracy. This work sets the foundations for the incorporation of inductive logic programming in the field of VideoQA.

Nuri Cingillioglu

Papers

DeepLogic: Towards End-to-End Differentiable Logical Reasoning

DeepLogic: End-to-End Logical Reasoning.

DeepLogic: Towards End-to-End Differentiable Logical Reasoning.

Learning Invariants through Soft Unification

HySTER: A Hybrid Spatio-Temporal Event Reasoner.