Modeling Fuzzy Rules With Description Logics, Sudhir Agarwal, Pascal Hitzler

Computer Science and Engineering Faculty Publications

In real application scenarios, input data and knowledge is often vague. Likewise, it is often the case that exact reasoning over data is impossible due to complex dependencies between input data and target outputs. For practical applications, however, good approximations often suffice, and efficient calculation of an approximate answer is often preferable over complex processing which may take a long time to come up with an exact answer. Fuzzy logic supports both features by providing fuzzy membership functions and fuzzy IF-THEN rule bases. In this paper, we show how fuzzy membership functions and fuzzy rules can be modeled by means …

Ontology Learning As A Use-Case For Neural-Symbolic Integration, Pascal Hitzler, Sebastian Bader, Artur Garcez

Computer Science and Engineering Faculty Publications

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 occurring in the context of semantic technologies - provides such an application scenario with potential for success and high impact on neural-symbolic integration.

Integrating First-Order Logic Programs And Connectionist Systems - A Constructive Approach, Sebastian Bader, Andreas Witzel, Pascal Hitzler

Computer Science and Engineering Faculty Publications

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. Advances were hindered severely by the lack of concrete algorithms for obtaining the approximating networks which were known to exist: the corresponding proofs are not constructive in that they do not yield concrete methods for building …

Extracting Reduced Logic Programs From Artificial Neural Networks, Jens Lehmann, Sebastian Bader, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Artificial neural networks can be trained to perform excellently in many application areas. While they can learn from raw data to solve sophisticated recognition and analysis problems, the acquired knowledge remains hidden within the network architecture and is not readily accessible for analysis or further use: Trained networks are black boxes. Recent research efforts therefore investigate the possibility to extract symbolic knowledge from trained networks, in order to analyze, validate, and reuse the structural insights gained implicitly during the training process. In this paper, we will study how knowledge in form of propositional logic programs can be obtained in such …

Faster Owl Using Split Programs, Denny Vrandecic, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Knowledge representation and reasoning on the Semantic Web is done by means of ontologies. While the quest for suitable ontology languages is still ongoing, OWL [5] has been established as a core standard. It comes in three flavours, as OWL Full, OWL DL and OWL Lite, where OWL Full contains OWL DL, which in turn contains OWL Lite. The latter two coincide semantically with certain description logics and can thus be considered fragments of first-order predicate logic.

Morphisms In Context, Markus Krotzsch, Guo-Qiang Zhang, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Morphisms constitute a general tool for modelling complex relationships between mathematical objects in a disciplined fashion. In Formal Concept Analysis (FCA), morphisms can be used for the study of structural properties of knowledge represented in formal contexts, with applications to data transformation and merging. In this paper we present a comprehensive treatment of some of the most important morphisms in FCA and their relationships, including dual bonds, scale measures, infomorphisms, and their respective relations to Galois connections. We summarize our results in a concept lattice that cumulates the relationships among the considered morphisms. The purpose of this work is to …

Description Logic Programs: A Practical Choice For The Modelling Of Ontologies, Rudi Studer, York Sure, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Knowledge representation using ontologies constitutes the heart of semantic technologies. Despite successful standardization efforts by the W3C, however, there are still numerous different ontology representation languages being used, and interoperability between them is in general not given. The problem is aggrevated by the fact that current standards lay foundations only and are well-known to be insufficient for the modelling of finer details. Thus, a plethora of extensions of the basic languages is being proposed, rendering the picture of ontology representation languages to be chaotic, to say the least. While semantic technologies start to become applicable and are being applied in …

Integrating Semantic Web Services For Mobile Access, Anupriya Ankolekar, Pascal Hitzler, Holger Lewen, Daniel Oberle, Rudi Studer

Computer Science and Engineering Faculty Publications

We present our work in integrating Semantic Web services for access via mobile devices. We have developed a system, the WebServiceAccessComponent, that transforms a user request for a service on a mobile device, to a Web service request and then selects a matching service from the existing Web services of the Deutsche Telekom, which provide navigational and weather information. In this poster, we present the requirements and design of the WebServiceAccessComponent.

Bridging The Paradigm Gap With Rules For Owl, Jurgen Angele, Boris Motik, Rudi Studer, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Accelerated by the vision of the semantic web, semantic technologies have recently made significant advances. The underlying methods and paradigms are already being transferred to adjacent areas of research in artificial intelligence, knowledge management, and elsewhere. Textbooks explaining the foundations have appeared. Large national and international projects on the topic are under way.

Level Mapping Characterizations Of Selector-Generated Models For Logic Programs, Sibylle Schwarz, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Assigning semantics to logic programs via selector generated models (Schwarz 2002/2003) extends several semantics, like the stable, the inflationary, and the stable generated semantics, to programs with arbitrary formulae in rule heads and bodies. We study this approach by means of a unifying framework for characterizing different logic programming semantics using level mappings (Hitzler and Wendt 200x, Hitzler 2003), thereby supporting the claim that this framework is very flexible and applicable to very diversely defined semantics.

The Semantic Web In One Day, York Sure, Pascal Hitzler, Andreas Eberhart, Rudi Studer

Computer Science and Engineering Faculty Publications

To gain momentum, technologies for building private semantic Webs or parts of the World Wide semantic Web must become a commodity and easy to integrate. To determine just how far semantic Web technologies have come, we wanted to create a snapshot of what you could do by applying and assembling existing semantic Web technologies - in one day.

A Uniform Approach To Logic Programming Semantics, Pascal Hitzler, Matthias Wendt

Computer Science and Engineering Faculty Publications

Part of the theory of programming and nonymonotonic reasoning concerns the study of fixed-point semantics for these paradigms. Several different semantics have been proposed during the last two decades, and some have been more successful and acknowledged than others. The rationales behind those various semantics have been manifold, depending on one's point of view, which may be that of a programmer or inspired by commonsense reasoning, and consequently the constructions which lead to these semantics are technically very diverse, and the exact relationships between them have not yet been fully understood. In this paper, we present a conceptually new method, …

Dlp - An Introduction, Denny Vrandecic, Pascal Hitzler, Rudi Studer

Computer Science and Engineering Faculty Publications

DLP - Description Logic Programs - is the name for the common language that is able to integrate knowledge bases described in Description Logic with Logic Programs. In this introduction, we offer a very short overview of DLP, the motivation for it, the benefits it offers and how to use it.

Logic Programs, Iterated Function Systems, And Recurrent Radial Basis Function Networks, Sebastian Bader, Pascal Hitzler

Computer Science and Engineering Faculty Publications

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 …

Continuity Of Semantic Operators In Logic Programming And Their Approximation By Artificial Neural Networks, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

One approach to integrating first-order logic programming and neural network systems employs the approximation of semantic operators by feedforward networks. For this purpose, it is necessary to view these semantic operators as continuous functions on the reals. This can be accomplished by endowing the space of all interpretations of a logic program with topologies obtained from suitable embeddings. We will present such topologies which arise naturally out of the theory of logic programming, discuss continuity issues of several well-known semantic operators, and derive some results concerning the approximation of these operators by feedforward neural networks.

Generalized Metrics And Uniquely Determined Logic Programs, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become non-monotonic when it is introduced, with the result that standard approaches to denotational semantics then become inapplicable. In this paper, we show how generalized metric spaces can be used to obtain fixed-point semantics for several classes of programs relative to the supported model semantics, and investigate relationships between the underlying spaces we employ. Our methods allow the analysis of classes of programs which include the acyclic, locally hierarchical, …

Formal Concept Analysis And Resolution On Algebraic Domains - Preliminary Report, Matthias Wendt, Pascal Hitzler

Computer Science and Engineering Faculty Publications

We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondence 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.

Characterizing Logic Programming Semantics With Level Mappings, Pascal Hitzler, Matthias Wendt

Computer Science and Engineering Faculty Publications

Declarative semantics in logic programming and nonmonotonic reasoning are often defined via fixed points of semantic operators. While many relationships between different semantics known from the literature have been studied, a uniform treatment is still missing. In this paper, we provide uniform operator-free characterizations for some of the most important semantics, more precisely, for the stable, the well-founded, and the Fitting semantics, for the weakly-perfect model semantics, and for the least model semantics for negation-free programs.

The Fixed-Point Theorems Of Priess-Crampe And Ribenboim In Logic Programming, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

Sibylla Priess-Crampe and Paulo Ribenboim recently established a general fixed-point theorem for multivalued mappings defined on generalized ultrametric spaces, and introduced it to the area of logic programming semantics. We discuss, in this context, the applications which have been made so far of this theorem and of its corollaries. In particular, we will relate these results to Scott-Ershov domains, familiar in programming language semantics, and to the generalized metrics of Khamsi, Kreinovich, and Misane which have been applied, by these latter authors, to logic programming. Amongst other things, we will also show that a unified treatment of the fixed-point theory …

Ilp Operators For Propositional Connectionist Networks, Miguel Angel Gutierrez-Naranjo, Pascal Hitzler

Computer Science and Engineering Faculty Publications

No abstract provided.

Towards Nonmonotonic Reasoning On Hierarchical Knowledge, Pascal Hitzler

Computer Science and Engineering Faculty Publications

W.C. Rounds and G.Q. Zhang have recently proposed to study a form of disjunctive logic programming generalized to algebraic domains [RZ01]. This system allows reasoning with information which is hierarchically structured and forms a (suitable) domain. We extend this framework to include reasoning with negative information, i.e. the implicit or explicit absence of bits of information. These investigations will naturally lead to a form of default reasoning which is strongly related to programming with answer sets to stable models, which has recently created much interest amongst artificial intelligence researchers concerned with knowledge representation and reasoning.

Semantic Operators And Fixed-Point Theory In Logic Programming, Anthony K. Seda, Pascal Hitzler

Computer Science and Engineering Faculty Publications

We consider rather general operators mapping valuations to (sets of) valuations in the context of the semantics of logic programming languages. This notion generalizes several of the standard operators encountered in this subject and is inspired by earlier work of M.C. Fitting. The fixed points of such operators play a fundamental role in logic programming semantics by providing standard models of logic programs and also in determining the computability properties of these standard models. We discuss some of our recent work employing topological ideas, in conjunction with order theory, to establish methods by which one can find the fixed points …

A "Converse" Of The Banach Contraction Mapping Theorem, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

We prove a type of converse of the Banach contraction mapping theorem for metric spaces: if X is a T₁ topological space and f: X -> X is a function with the unique fixed point a such that fⁿ(x) converges to a for each x is a member of X, then there exists a distance function d on X such that f is a contraction on the complete ultrametric space (X,d) with contractivity factor 1/2. We explore properties of the resulting space (X,d).

Unique Supported-Model Classes Of Logic Programs, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

We study classes of programs, herein called unique supported-model classes, with the property that each program in the class has a unique supported model. Elsewhere, the authors examined these classes from the point of view of operators defined relative to certain three-valued logics. In this paper, we complement our earlier results by considering how unique supported-model classes fit into the framework given by various classes of programs in several well-known approaches to semantics.

Kontraktionssatze Auf Verallgemeinerten Metrischen Raumen, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Classes Of Logic Programs Which Possess Unique Supported Models, Anthony K. Seda, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Logic programming is concerned with the use of logic as a programming language. The main manifestation of this computing paradigm is in the various versions of Prolog which are now available, in which computation is viewed as deduction from sets of Horn clauses, although there is also growing interest in the related form known as answer set programming, see [10]. The reference [1] contains a good survey of the growth of logic programming over the last twenty-five years both as a stand-alone programming language and as a software component of large information systems. One advantage a logic program P has …

A New Fixed-Point Theorem For Logic Programming Semantics, Anthony K. Seda, Pascal Hitzler

Computer Science and Engineering Faculty Publications

We present a new fixed-point theorem akin to the Banach contraction mapping theorem, but in the context of a novel notion of generalized metric space, and show how it can be applied to analyse the denotational semantics of certain logic programs. The theorem is obtained by generalizing a theorem of Priess-Crampe and Ribenboim, which grew out of applications within valuation theory, but is also inspired by a theorem of S.G. Matthews which grew out of applications to conventional programming language semantics. The class of programs to which we apply our theorem was defined previously by us in terms of operators …

Characterizations Of Classes Of Programs By Three-Valued Operators, Anthony K. Seda, Pascal Hitzler

Computer Science and Engineering Faculty Publications

Several important classes of normal logic programs, including the classes of acyclic, acceptable, and locally hierarchical programs, have the property that every program in the class has a unique two-valued supported model. In this paper, we call such classes unique supported model classes. We analyse and characterize these classes by means of operators on three-valued logics. Our studies will motivate the definition of a larger unique supported model class which we call the class of Phi-accessible programs. Finally, we show that the class of Phi -accessible programs is computationally adequate in that every partial recursive function can be implemented by …

Multivalued Mappings, Fixed-Point Theorems And Disjunctive Databases, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

In this paper, we discuss the semantics of disjunctive programs and databases and show how multivalued mappings and their fixed points arise naturally within this context. A number of fixed-point theorems for multivalued mappings are considered, some of which are already known and some of which are new. The notion of a normal derivative of a disjunctive program is introduced. Normal derivatives are normal logic programs which are determined by the disjunctive program. Thus, the well-known single-step operator associated with a normal derivative is single-valued, and its fixed points can be found by well-established means. It is shown how fixed …

Strictly Level-Decreasing Logic Programs, Pascal Hitzler, Anthony K. Seda

Computer Science and Engineering Faculty Publications

We study strictly level-decreasing logic programs (sld-programs) as defined earlier by the present authors. It will be seen that sld-programs, unlike most other classes of logic programs, have both a highly intuitive declarative semantics, given as a unique supported model, and are computationally adequate in the sense that every partial recursive function can be represented by some sld-program P. Allowing for a safe use of cuts, an interpreter based on SLDNF-resolution, as implemented for example in standard Prolog systems, is shown to be sound and complete with respect to this class of programs. Furthermore, we study connections between topological …