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

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.

We present a fully connectionist system for the learning of first-order logic programs and the generation of corresponding models: Given a program and a set of training examples, we embed the associated semantic operator into a feed-forward network and train the network using the examples. This results in the learning of first-order knowledge while damaged or noisy data is handled gracefully. 1

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

Significant advances have recently been made concerning the integration of symbolic knowledge representation with artificial neural networks (also called connectionist systems). However, while the integration with propositional paradigms has resulted in applicable systems, the case of first-order knowledge representation has so far hardly proceeded beyond theoretical studies which prove the existence of connectionist systems for approximating first-order logic programs up to any chosen precision.

The integration of symbolic and neural-network-based artificial intelligence paradigms constitutes a very challenging area of research. The overall aim is to merge these two very different major approaches to intelligent systems engineering while retaining their respective strengths. For symbolic paradigms that use the syntax of some first-order language this appears to be particularly difficult. In this paper, we will extend on an idea proposed by Garcez and Gabbay (2004) and show how first-order logic programs can be represented by fibred neural networks. The idea is to use a neural network to iterate a global counter n. For each clause C i in the logic program, this counter is combined (fibred) with another neural network, which determines whether C i outputs an atom of level n for a given interpretation I . As a result, the fibred network computes the singlestep operator TP of the logic program, thus capturing the semantics of the program.

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.

Abstract. Bilattice-based annotated logic programs (BAPs) form a very general class of programs which can handle uncertainty and conflicting information. We use BAPs to integrate two alternative paradigms of computation: specifically, we build learning artificial neural networks which can model iterations of the semantic operator associated with each BAP and introduce sound and complete SLD-resolution for this class of programs. Key words: Logic programs, artificial neural networks, SLD-resolution 1

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.

Abstract. In this paper, we develop a theory of the integration of fibring neural networks (a generalization of conventional neural networks) into model-theoretic semantics for logic programming. We present some ideas and results about the approximate computation by fibring neural networks of semantic immediate consequence operators TP and TP, where TP denotes a generalization of TP relative to a many-valued logic analogous to Kleene’s strong logic. We establish a minimalfixed-point semantics for normal logic programs somewhat analogous to the leastfixed-point semantics for definite logic programs. We argue that the class of logic programs for which the approximation by fibring neural networks may be employed to compute minimal fixed points of TP and of TP is the class of normal programs. Our theorems on the approximation of TP and TP for normal programs extend recent results on approximation of these operators for definite programs by conventional neural networks.