In [18] we have shown how to construct a 3-layered recurrent neural network that computes the fixed point of the meaning function TP of a given propositional logic program P, which corresponds to the computation of the semantics of P. In this article we consider the first order case. We define a notion of approximation for interpretations and prove that there exists a 3-layered feed forward neural network that approximates the calculation of TP for a given first order acyclic logic program P with an injective level mapping arbitrarily well. Extending the feed forward network by recurrent connections we obtain a recurrent neural network whose iteration approximates the fixed point of TP. This result is proven by taking advantage of the fact that for acyclic logic programs the function TP is a contraction mapping on a complete metric space defined by the interpretations of the program. Mapping this space to the metric space IR with Euclidean distance, a real valued function fP can be defined which corresponds to TP and is continuous as well as a contraction. Consequently it can be approximated by an appropriately chosen class of feed forward neural networks.

One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for first-order normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a wellknown theorem of Funahashi, and necessitate the study of when Funahasi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.

. Inspired by the work of Elman and Wiles [15] and based on the notions developed within the theory of dynamical systems it is shown how a simple recurrent connectionist network with a single hidden unit implements a counter. This result is exemplified by showing that such a network can be used to recognize the context--free language a n b n consisting of strings of some number n of a 's followed by the same number n of b 's for n 250 , whereby the maximum value for n is only restricted by the computing accuracy of the used hardware. 1 Introduction It is a common hypothesis today that the functionality of the brain and nervous system can be understood within the theory of computation (e.g. [11]), it is widely believed that high--level cognitive tasks require the ability to represent structured objects and to execute structure--sensitive processes on top of these objects (e.g. [3,7]), and it seems that --- based on our current data and understanding --- connectionist networks in t...

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.

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

We show that temporal logic and combinations of temporal logics and modal logics of knowledge can be effectively represented in artificial neural networks. We present a Translation Algorithm from temporal rules to neural networks, and show that the networks compute a fixed-point semantics of the rules. We also apply the translation to the muddy children puzzle, which has been used as a testbed for distributed multi-agent systems. We provide a complete solution to the puzzle with the use of simple neural networks, capable of reasoning about time and of knowledge acquisition through inductive learning.

The Connectionist Inductive Learning and Logic Programming System, C-IL 2 P, integrates the symbolic and connectionist paradigms of Artificial Intelligence through neural networks that perform massively parallel Logic Programming and inductive learning from examples and background knowledge. This work presents an extension of C-IL 2 P that allows the implementation of Extended Logic Programs in Neural Networks. This extension makes C-IL 2 P applicable to problems where the background knowledge is represented in a Default Logic. As a case example, we have applied the system for fault diagnosis of a simplified power system generation plant, obtaining good preliminary results.

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.

We argue that the field of neural-symbolic integration is in need of identifying application scenarios for guiding further research. We furthermore argue that ontology learning --- as occuring in the context of semantic technologies --- provides such an application scenario with potential for success and high impact on neural-symbolic integration.