The field of neurosymbolic 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 neurosymbolic 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 neurosymbolic methods. This paper addresses this problem by introducing a semantic framework for neurosymbolic 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 neurosymbolic 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, neurosymbolic methods, are shown to fall within the proposed formalization. This is expected to provide guidance to future neurosymbolic encodings by placing them in the broader context of semantic encodings of entire families of existing neurosymbolic systems. The paper hopes to help initiate a discussion around the provision of a theory for neurosymbolic AI and a semantics for deep learning.

The field of neurosymbolic 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 neurosymbolic 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 neurosymbolic methods. This paper addresses this problem by introducing a semantic framework for neurosymbolic 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 neurosymbolic 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, neurosymbolic methods, are shown to fall within the proposed formalization. This is expected to provide guidance to future neurosymbolic encodings by placing them in the broader context of semantic encodings of entire families of existing neurosymbolic systems. The paper hopes to help initiate a discussion around the provision of a theory for neurosymbolic 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.

The Restricted Boltzmann Machine (RBM) has proved to be a powerful tool in machine learning, both on its own and as the building block for Deep Belief Networks (multi-layer generative graphical models). The RBM and Deep Belief Network have been shown to be universal approximators for probability distributions on binary vectors. In this paper we prove several similar universal approximation results for two variations of the Restricted Boltzmann Machine with time dependence, the Temporal Restricted Boltzmann Machine (TRBM) and the Recurrent Temporal Restricted Boltzmann Machine (RTRBM). We show that the TRBM is a universal approximator for Markov chains and generalize the theorem to sequences with longer time dependence. We then prove that the RTRBM is a universal approximator for stochastic processes with finite time dependence. We conclude with a discussion on efficiency and how the constructions developed could explain some previous experimental results.