Towards a Systematic Account of Different Semantics for Logic Programs

In [Hitzler and Wendt 2002, 2005], a new methodology has been proposed which allows to derive uniform characterizations of different declarative semantics for logic programs with negation. One result from this work is that the well-founded semantics can formally be understood as a stratified version of the Fitting (or Kripke-Kleene) semantics. The constructions leading to this result, however, show a certain asymmetry which is not readily understood. We will study this situation here with the result that we will obtain a coherent picture of relations between different semantics for normal logic programs.Comment: 20 page

Dimensions of Neural-symbolic Integration - A Structured Survey

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. We present a comprehensive survey of the field of neural-symbolic integration, including a new classification of system according to their architectures and abilities.Comment: 28 page

Towards a unified theory of logic programming semantics: Level mapping characterizations of selector generated models

Currently, the variety of expressive extensions and different semantics created for logic programs with negation is diverse and heterogeneous, and there is a lack of comprehensive comparative studies which map out the multitude of perspectives in a uniform way. Most recently, however, new methodologies have been proposed which allow one to derive uniform characterizations of different declarative semantics for logic programs with negation. In this paper, we study the relationship between two of these approaches, namely the level mapping characterizations due to [Hitzler and Wendt 2005], and the selector generated models due to [Schwarz 2004]. We will show that the latter can be captured by means of the former, thereby supporting the claim that level mappings provide a very flexible framework which is applicable to very diversely defined semantics.Comment: 17 page

Formal Concept Analysis and Resolution in Algebraic Domains

We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos. We show how to identify the notion of formal concept in the domain theoretic setting. In particular, we show that a special instance of the resolution rule from the domain logic coincides with the concept closure operator from formal concept analysis. The results shed light on the use of contexts and domains for knowledge representation and reasoning purposes.Comment: 14 pages. We have rewritten the old version according to the suggestions of some referees. The results are the same. The presentation is completely differen

The Integration of Connectionism and First-Order Knowledge Representation and Reasoning as a Challenge for Artificial Intelligence

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.Comment: In Proceedings of INFORMATION'2004, Tokyo, Japan, to appear. 12 page

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page