Topics for Project or Master Theses

I always have to offer challenging topics for projects and theses related to my past and recent research interests. I give rough sketches for some possible investigations below, but the list is certainly not exhaustive. Please make an appointment with me if you are interested in working under my guidance.

Nonmonotonic Reasoning over Conceptual Structures and Applications

We develop a nonmonotonic reasoning system over conceptual structures coming e.g. from formal concept analysis or description logics. We foresee applications in data- and textmining, and in semantic web technology. Work to be done here includes implementations, the development of application scenarios, and the extension of the theoretical work which so far uses domain-theoretic methodology.

References:

[Hit02] Pascal Hitzler, Contexts, Concepts, and Logic of Domains. Technical Report WV-02-12, Knowledge Representation and Reasoning Group, Department of Computer Science, Dresden University of Technology, 2002.
[Hit03] Pascal Hitzler, Default Reasoning over Domains and Concept Hierarchies. In: Proceedings of the 27th German Conference on Artificial Intelligence, Ulm, Germany, September 2004. Lecture Notes in Artificial Intelligence, to appear.
[Hit0x] Pascal Hitzler, A generalized resolution theorem. Journal of Electrical Engineering 55 (1-2), 2004, Slovak Academy of Sciences.
[HW03] Pascal Hitzler and Matthias Wendt, Formal Concept Analysis and Resolution on Algebraic Domains. In: Aldo de Moor and Bernhard Ganter (Eds.): Using Conceptual Structures - Contributions to ICCS 2003. Shaker Verlag, ISBN 3-8322-1705-3.
[HZ04] Pascal Hitzler and Guo-Qiang Zhang, A cartesian closed category of approximable concept structures. Proceedings of the 12th International Conference on Conceptual Structures, ICCS2004, Huntsville, Alabama, July 2004. Lecture Notes in Computer Science, to appear.

Logic Programs and Artificial Neural Networks

Following the idea of [HKS99], we want to approximate logic programs by artificial neural networks by embedding semantic operators associated with programs into the real numbers. The work we have done so far can be extended in many directions, depending on the interests of the student taking it up. We may also be in need of students doing implementations.

References:

[BH0x] Sebastian Bader and Pascal Hitzler, Logic Programs, Iterated Function Systems, and Recurrent Radial Basis Function Networks. To appear in Journal of Applied Logic.
[Bad03] Sebastian Bader, From Logic Programs to Iterated Function Systems. Master's Thesis, Department of Computer Science, Dresden University of Technology, 2003.
[HHS0x] Pascal Hitzler, Steffen Hölldobler and Anthony K. Seda, Logic Programs and Connectionist Networks. To appear in Journal of Applied Logic.
[HS03] Pascal Hitzler and Anthony K. Seda, Continuity of Semantic Operators in Logic Programming and their Approximation by Artificial Neural Networks. In: Andreas Günter, Rudolf Kruse and Bernd Neumann, KI2003: Advances in Artificial Intelligence. Proceedings of the 26th Annual German Conference on Artificial Intelligence, KI2003, Hamburg, Germany, September 2003. Springer Lecture Notes in Artificial Intelligence Vol. 2821, 2003, pp. 105-119.
[HKS99] Steffen Hölldobler, Yvonne Kalinke and Hans-Peter Störr, Approximating the Semantics of Logic Programs by Recurrent Neural Networks. Applied Intelligence 11, 1999, 45-58.

Characterizing Semantics of Logic Programs

Non-monotonic semantics for logic programs (like the stable and the well-founded semantics) are defined and characterized by very diverse means, including program transformations, monotonic and non-monotonic operators, etc. In [HW02a,HW0x] we have proposed a methodology for obtaining characterizations of many different semantics in a uniform way. This approach needs to be carried over to other semantics and to logic programs under enhanced syntax, including the following.

References:

[Hit03] Pascal Hitzler, Towards a Systematic Account of Different Logic Programming Semantics. In: Andreas Günter, Rudolf Kruse and Bernd Neumann, KI2003: Advances in Artificial Intelligence. Proceedings of the 26th Annual German Conference on Artificial Intelligence, KI2003, Hamburg, Germany, September 2003. Springer Lecture Notes in Artificial Intelligence Vol. 2821, 2003, pp. 355-369.
[HW0x] Pascal Hitzler and Matthias Wendt, A uniform approach to logic programming semantics. To appear in Theory and Practice of Logic Programming.
[HW03] Pascal Hitzler and Matthias Wendt, Characterizing logic programming semantics with level mappings. In: B. Fronhöfer and Steffen Hölldobler, 17. WLP: Workshop Logische Programmierung, December 2002, Dresden, Germany. Technische Berichte der Fakultät Informatik TUD-FI03-03, ISSN 1430-211X, pp. 60-67.
[HW02a] Pascal Hitzler and Matthias Wendt, The Well-Founded Semantics is a Stratified Fitting Semantics. In: M. Jarke, J. Koehler and G. Lakemeyer, Proceedings of the 25th German Conference on Artificial Intelligence (KI2002), Aachen, September 2002. Lecture Notes in Artificial Intelligence 2479, Springer, 2002, 205-221.
[HS04] Pascal Hitzler and Sibylle Schwarz, Level mapping characterizations of selector-generated models for logic programs.Technical Report WV-04-04, Knowledge Representation and Reasoning Group, Department of Computer Science, Dresden University of Technology, 2004.
[Kno03] Matthias Knorr, Level mapping characterizations for quantitative and disjunctive logic programs. Bachelor's Thesis, Department of Computer Science, Dresden University of Technology, 2003.

Generalized metric fixed-point theorems for probabilistic metric spaces

Fixed-point theorems are of fundamental importance in many areas of computer science and mathematics, since many problems can be reduced to finding fixed points of suitable mappings. Many fixed-point theorems are based on order theory, topology, or metric space theory. In [Hit02, Part I], a collection of generalized metric fixed-point theorems which appeared in the literature can be found. On the other hand, metric spaces can be generalized to a probabilistic setting, consequently called probabilistic metric spaces. Work to be done here includes a thorough literature search for fixed-point theorems based on the notion of probabilistic metric spaces. [Zik02] can serve as a starting point, [HP01] gives a thorough introduction. It needs also be investigated to which extend fixed-point theorems for generalized metric spaces can be carried over to (generalized) probabilistic metric spaces.

References:

[Hit02] Pascal Hitzler, Generalized Metrics and Topology in Logic Programming Semantics. PhD thesis, Department of Mathematics, National University of Ireland, University College Cork, 2001.
[HP01] Olga Hadzic and Endre Pap, Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applicaitons vol. 536, Kluwer Academic Publishers, Dordrecht, 2001.
[Zik02] Tatjana Zikic, Multivalued Probabilistic q-Contraction. Journal of Electrical Engineering 53(12/s), 2002, 13-16. Proceedings of the Conference in Applied Mathematics for undergraduate and graduate students, SCAM2002, Bratislava, April 2002, Slovak Academy of Sciences.

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