This paper will give an overview of the various approaches to neurosymbolic integration. Roughly, these can be divided into two strategies: unified strategies aim at attaining neural and symbolic capabilities using neural networks alone, while hybrid strategies combine neural networks with symbolic models such as expert systems, case-based reasoning systems, 2 Chapter 2 and decision trees. These two approaches form the main subtrees of the classification hierarchy depicted in Figure 1. Symbol Proc. Neuronal Unified approach Symbol Proc. hybrids Connectionist Localist Hybrid approach Combined L/D Neurosymbolic integration Functional Chainprocessing Translational Subprocessing hybrids Metaprocessing Distributed Coprocessing Figure 1 Classification of integrated neurosymbolic systems.

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Introduction Research on integrated neural-symbolic systems has made significant progress in the recent past. In particular the understanding of ways to deal with symbolic knowledge within connectionist systems (also called artificial neural networks) has reached a critical mass which enables the community to strive for applicable implementations and use cases. Recent work has covered a great variety of logics used in artificial intelligence and provides a multitude of techniques for dealing with them within the context of artificial neural networks. Already in the pioneering days of computational models of neural cognition, the question was raised how symbolic knowledge can be represented and dealt with within neural networks. The landmark paper [McCulloch and Pitts, 1943] provides fundamental insights how propositional logic can be processed using simple artificial neural networks. Within the following decades, however, the topic did not receive much attention as research in arti

Neural-symbolic systems are hybrid systems that integrate symbolic logic and neural networks. The goal of neural-symbolic integration is to benefit from the combination of features of the symbolic and connectionist paradigms of artificial intelligence. This paper introduces a new neural network architecture based on the idea of fibring logical systems. Fibring allows one to combine di#erent logical systems in a principled way. Fibred neural networks may be composed not only of interconnected neurons but also of other networks, forming a recursive architecture. A fibring function then defines how this recursive architecture must behave by defining how the networks in the ensemble relate to each other, typically by allowing the activation of neurons in one network (A) to influence the change of weights in another network (B). Intuitively, this can be seen as training network B at the same time that one runs network A. We show that, in addition to being universal approximators like standard feedforward networks, fibred neural networks can approximate any polynomial function to any desired degree of accuracy, thus being more expressive than standard feedforward networks.

Graphs of the single-step operator for first-order logic programs --- displayed in the real plane --- exhibit self-similar structures known from topological dynamics, i.e. they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be made mathematically precise. In particular, we give conditions which ensure that those graphs coincide with attractors of suitably chosen iterated function systems, and conditions which allow the approximation of such graphs by iterated function systems or by fractal interpolation. Since iterated function systems can easily be encoded using recurrent radial basis function networks, we eventually obtain connectionist systems which approximate logic programs in the presence of function symbols.

Roughly speaking, adequatness is the property of a theorem proving method to solve simpler problems faster than more difficult ones. Automated inferencing methods are often not adequate as they require thousands of steps to solve problems which humans solve effortlessly, spontaneously, and with remarkable efficiency. L. Shastri and V. Ajjanagadde --- who call this gap the artificial intelligence paradox --- suggest that their connectionist inference system is a first step toward bridging this gap. In this paper we show that their inference method is equivalent to reasoning by reductions in the well-known connection method. In particular, we extend a reduction technique called evaluation of isolated connections such that this technique --- together with other reduction techniques --- solves all problems which can be solved by Shastri and Ajjanagadde's system under the same parallel time and space requirements. Consequently, we obtain a semantics for Shastri and Ajjanagadde's logic. But,...

Intelligent systems based on first-order logic on the one hand, and on artificial neural networks (also called connectionist systems) on the other, differ substantially. It would be very desirable to combine the robust neural networking machinery with symbolic knowledge representation and reasoning paradigms like logic programming in such a way that the strengths of either paradigm will be retained. Current state-of-the-art research, however, fails by far to achieve this ultimate goal. As one of the main obstacles to be overcome we perceive the question how symbolic knowledge can be encoded by means of connectionist systems: Satisfactory answers to this will naturally lead the way to knowledge extraction algorithms and to integrated neural-symbolic systems. 1

The relationship between symbolism and connectionism has been one of the major issues in recent Artificial Intelligence research. An increasing number of researchers from each side have tried to adopt desirable characteristics of the other. A major open question in this field is the extent to which a connectionist architecture can accommodate basic concepts of symbolic inference, such as a dynamic variable binding mechanism and a rule and fact encoding mechanism involving n-ary predicates. One of the current leaders in this area is the connectionist rule-based system proposed by Shastri and Ajjanagadde. We demonstrate that the mechanism for variable binding which they advocate is fundamentally limited and show how a reinterpretation of the primitive components and corresponding modifications of their system can extend the range of inference which can be supported. Our extension hinges on the basic structural modification of the network components and further modifications of the rule a...

Research into the processing of symbolic knowledge by means of connectionist networks aims at systems which combine the declarative nature of logic-based artificial intelligence with the robustness and trainability of artificial neural networks. This endeavour has been addressed quite successfully in the past for propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended beyond propositional logic, it is not obvious at all what neural-symbolic systems should look like such that they are truly connectionist and allow for a declarative reading at the same time. The Core Method – which we present here – aims at such an integration. It is a method for connectionist model generation using recurrent networks with feedforward core. These networks can be trained by standard algorithms to learn symbolic knowledge, and they can be used for reasoning about this knowledge.

The xpoint completion x(P ) of a normal logic program P is a program transformation such that the stable models of P are exactly the models of the Clark completion of x(P ). This is well-known and was studied by Dung and Kanchanasut [15]. The correspondence, however, goes much further: The Gelfond-Lifschitz operator of P coincides with the immediate consequence operator of x(P ), as shown by Wendt [51], and even carries over to standard operators used for characterizing the well-founded and the Kripke-Kleene semantics. We will apply this knowledge to the study of the stable semantics, and this will allow us to almost eortlessly derive new results concerning xed-point and metric-based semantics, and neural-symbolic integration.