Simon Odense

Simon Odense, Artur d'Avila Garcez

The field of neuro-symbolic AI aims to benefit from the combination of neural networks and symbolic systems. A cornerstone of the field is the translation or encoding of symbolic knowledge into neural networks. Although many neuro-symbolic methods and approaches have been proposed, and with a large increase in recent years, no common definition of encoding exists that can enable a precise, theoretical comparison of neuro-symbolic methods. This paper addresses this problem by introducing a semantic framework for neuro-symbolic AI. We start by providing a formal definition of semantic encoding, specifying the components and conditions under which a knowledge-base can be encoded correctly by a neural network. We then show that many neuro-symbolic approaches are accounted for by this definition. We provide a number of examples and correspondence proofs applying the proposed framework to the neural encoding of various forms of knowledge representation. Many, at first sight disparate, neuro-symbolic methods, are shown to fall within the proposed formalization. This is expected to provide guidance to future neuro-symbolic encodings by placing them in the broader context of semantic encodings of entire families of existing neuro-symbolic systems. The paper hopes to help initiate a discussion around the provision of a theory for neuro-symbolic AI and a semantics for deep learning.

Artur d'Avila Garcez, Simon Odense

Artificial Intelligence (AI) is a powerful new language of science as evidenced by recent Nobel Prizes in chemistry and physics that recognized contributions to AI applied to those areas. Yet, this new language lacks semantics, which makes AI's scientific discoveries unsatisfactory at best. With the purpose of uncovering new facts but also improving our understanding of the world, AI-based science requires formalization through a framework capable of translating insight into comprehensible scientific knowledge. In this paper, we argue that logic offers an adequate framework. In particular, we use logic in a neurosymbolic framework to offer a much needed semantics for deep learning, the neural network-based technology of current AI. Deep learning and neurosymbolic AI lack a general set of conditions to ensure that desirable properties are satisfied. Instead, there is a plethora of encoding and knowledge extraction approaches designed for particular cases. To rectify this, we introduced a framework for semantic encoding, making explicit the mapping between neural networks and logic, and characterizing the common ingredients of the various existing approaches. In this paper, we describe succinctly and exemplify how logical semantics and neural networks are linked through this framework, we review some of the most prominent approaches and techniques developed for neural encoding and knowledge extraction, provide a formal definition of our framework, and discuss some of the difficulties of identifying a semantic encoding in practice in light of analogous problems in the philosophy of mind.

Simon Odense, Artur d'Avila Garcez

Knowledge extraction is used to convert neural networks into symbolic descriptions with the objective of producing more comprehensible learning models. The central challenge is to find an explanation which is more comprehensible than the original model while still representing that model faithfully. The distributed nature of deep networks has led many to believe that the hidden features of a neural network cannot be explained by logical descriptions simple enough to be comprehensible. In this paper, we propose a novel layerwise knowledge extraction method using M-of-N rules which seeks to obtain the best trade-off between the complexity and accuracy of rules describing the hidden features of a deep network. We show empirically that this approach produces rules close to an optimal complexity-error tradeoff. We apply this method to a variety of deep networks and find that in the internal layers we often cannot find rules with a satisfactory complexity and accuracy, suggesting that rule extraction as a general purpose method for explaining the internal logic of a neural network may be impossible. However, we also find that the softmax layer in Convolutional Neural Networks and Autoencoders using either tanh or relu activation functions is highly explainable by rule extraction, with compact rules consisting of as little as 3 units out of 128 often reaching over 99% accuracy. This shows that rule extraction can be a useful component for explaining parts (or modules) of a deep neural network.