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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.

The past decade has amply demonstrated the remarkable functionality that can be realized by learning complex input/output relationships. Algorithmically, one of the most important and opaque relationships is that between a problem's structure and an effective solution method. Here, we quantitatively connect the structure of a planning problem to the performance of a given sampling-based motion planning (SBMP) algorithm. We demonstrate that the geometric relationships of motion planning problems can be well captured by graph neural networks (GNNs) to predict SBMP runtime. By using an algorithm portfolio we show that GNN predictions of runtime on particular problems can be leveraged to accelerate online motion planning in both navigation and manipulation tasks. Moreover, the problem-to-runtime map can be inverted to identify subproblems easier to solve by particular SBMPs. We provide a motivating example of how this knowledge may be used to improve integrated task and motion planning on simulated examples. These successes rely on the relational structure of GNNs to capture scalable generalization from low-dimensional navigation tasks to high degree-of-freedom manipulation tasks in 3d environments., Comment: 8 pages; To appear in proceedings of 2022 IROS, Kyoto Japan

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.