• Mathematical foundations of reasoning under uncertainty
• Mathematical theory of neural networks
• Computational complexity
• Machine learning, data mining
• Theory of numbers and its applications

Mathematical foundations of reasoning under uncertainty

• Formal deductive systems of fuzzy logic and their semantics
• Their proof theoretical, computational, and algebraic aspects
• Systematic formalization of fuzzy mathematics
• Applications in fuzzy control and fuzzy data mining

• Systematical analysis of the theory of belief functions (also known as Dempster-Shafer theory)
• New mathematical models and results on randomness quantification and processing alternative to standard probability theory

Mathematical theory of neural networks

Since the work of the first pioneers of artificial neural networks in the former Czechoslovakia, neurocomputing became one of the main research focus of the Institute.

The work of V. Kůrková and J. Šíma is focused on mathematical theory of neural networks and learning, in particular on estimates of network complexity with an emphasis on the role of the dimensionality of data and the choice of the type of computational units. R. Neruda and P. Kudová design, implement and test algorithms based on constructive proof techniques. M. Holeňa explores possibilities of extraction of logical rules from neural networks. Results achieved by the group have been published in in major neurocomputing journals and as chapters in several monographs devoted to neurocomputing and learning. Some results of V. Kůrková also contribute to theory of nonlinear approximation and optimization and have been published in prestigious journals of these fields. V. Kůrková presented several tutorials and invited plenary talks at international conferences. The most prestigious among them were NATO Workshop on the state of the art in learning theory (Leuven, 2002) and Mathematical theory of learning (Barcelona, 2004). J. Šíma has also given invited lectures at international conferences.

Research activities of the members of the team are complemented by organizational efforts: the Institute organized the conference to the 10th anniversary of the journal NNW 2000 with R. Neruda as the conference chair and the 5th International Conference on Neural Networks and Genetic Algorithms ICANNGA 2001 with V. Kůrková as the conference chair. V. Kůrková also organized a special session on Machine Learning and Neural Networks at the 16th COMPSTAT 2004 with participation of several leading scientists of the field. The team is preparing the 18th ICANN 2008 (with V. Kůrková as the program chair and R. Neruda as the organization chair). V. Kůrková is a member of the editorial board of the journal Neural Processing Letters (Kluwer), R. Neruda of the journal Neural Network World (Institute of Computer Science).

At the national scale, the monograph on the theory of neurocomputing by Šíma and Neruda "Teoretické otázky neuronových sítí" (Theoretical aspects of neural nets, published in 1996) became a standard textbook at Czech and Slovak universities. Kůrková and Šíma also contributed by chapters to the series of textbooks "Umĕlá Inteligence", (Artificial Intelligence) published by Academia.

Computational Complexity

During the 1990s, after the move of J. Wiedermann from Slovakia, a relatively small group working in the foundations of computer science, today counting also P. Savický and S. Žák, has been established. In its specialization to computational complexity and machine models the group counts among the strongest in the Czech Republic and thanks to its publication and organizational activities (e.g. organization of an MFCS and ICALP conference, of series of SOFSEM conferences, etc.) has a good reputation also abroad.

Recently, Wiedermann, in cooperation with J. van Leeuwen, Utrecht, opened a new research field investigating so-called interactive evolutionary computing that captures the main features of contemporary computational systems, such as the Internet. These models helped in discovering and characterizing the super-Turing computational potential of related systems, such as biological evolution of organisms, societies of organisms, and communities of agents within artificial life field. Their results have been presented as invited plenary talks at international conferences such as MFCS 2000, FCS 2000, ECAL 2001, and SOFSEM 2001), as well as in journals and in an invited publication Mathematic Unlimited - Year 2001 and Beyond by Springer Publishing Co. which contains papers solicited for the occasion of Year 2000, declared the Year of Mathematics. Savický and Žák have significantly improved several lower bound results in the field of circuit and Boolean complexity. They are regularly invited to e.g. Dagstuhl seminars and have publications in renowned journals in the field.

Machine learning, data mining

Research in this area follows several directions. One of them is the extraction of logical rules from data, nowadays the main stream of data mining. Here, our effort concentrates on further elaboration of the GUHA method, one of the oldest data mining methods, based on a combination of logic and statistics. Another direction of research concentrates on decision trees and decision forests from the point of view of classifier construction. Other approaches to extraction of rules from data are also investigated, in particular rule extraction using artificial neural networks.

Synergy between data mining and soft computing, especially artificial neural networks, fuzzy logic an genetic algorithms is represented by bang project, synergy that reaches from simple combining to deep integration.

Theory of numbers and its applications

The research in number theory covers variety of topics in elementary, combinatorial and analytic number theory, with emphasis to multiplicative number theory (particularly the theory of generalized Beurling integers, a collection of objects having a multiplicative structure but not necessarily an additive one), and to the questions of distribution of sequences and values of arithmetic functions. Š. Porubský is the coauthor of the monograph Distribution of sequences: A sampler  which focuses on the distribution properties of sequences which may be expressed in terms of distribution functions, upper and lower distribution functions, the discrepancy, diaphony, dispersion etc.

Other interests include problems that lie at the interface of number theory with other branches of mathematics and computer science, as algebra, analysis or cryptography.

Since mid of 70's of 20th century Š. Porubský acts as a member of organizing and program committees of the regular Czech and Slovak international conferences on number theory (held every two years) and of the Central European international conferences on cryptology (held every year since 2000).