Neural disjunctive normal form: Vertically integrating logic with deep learning for classification

Inspired by the limitations of pure deep learning and symbolic logic-based models, in this thesis we consider a specific type of neuro-symbolic integration called vertical integration to bridge logic reasoning and deep learning and address their limitations. The motivation of vertical integration is to combine perception and reasoning as two separate stages of computation, while still being able to utilize simple and efficient end-to-end learning. It uses a perceptive deep neural network (DNN) to learn abstract concepts from raw sensory data and uses a symbolic model that operates on these abstract concepts to make interpretable predictions. As a preliminary step towards this direction, we tackle the task of binary classification and propose the Neural Disjunctive Normal Form (Neural DNF). Specifically, we utilize a per- ceptive DNN module to extract features from data, then after binarization (0 or 1), feed them into a Disjunctive Normal Form (DNF) module to perform logical rule-based classi- fication. We introduce the BOAT algorithm to optimize these two normally-incompatible modules in an end-to-end manner. Compared to standard DNF, Neural DNF can handle prediction tasks from raw sensory data (such as images) thanks to the neurally-extracted concepts. Compared to standard DNN, Neural DNF offers improved interpretability via an explicit symbolic representation while being able to achieve comparable accuracy despite the reduction of model flexibility, and is particularly suited for certain classification tasks that require some logical composition. Our experiments show that BOAT can optimize Neural DNF in an end-to-end manner, i.e. jointly learn the logical rules and concepts from scratch, and that in certain cases the rules and the meanings of concepts are aligned with human understanding. We view Neural DNF as an important first step towards more sophisticated vertical inte- gration models, which use symbolic models of more powerful rule languages for advanced prediction and algorithmic tasks, beyond using DNF (propositional logic) for classification tasks. The BOAT algorithm introduced in this thesis can potentially be applied to such advanced hybrid models.