Fundamentals of Soft Computing
First, Fuzzy Logic is understood in the sense of a formal calculus. The canonical semantics is the real unit interval, endowed with appropriate operations. A calculus is usually presented in Hilbert-style: a set of axiom schemes together with a set of inference rules. The goal of the research is the development of alternative formalizations of such calculi, the aim being formulations allowing automated theorem proving.
Furthermore, fuzzy formal languages are being investigated. For the first time, fuzzy Petri languages and fuzzy Lindenmayer languages have been investigated. In the case of Chomsky fuzzy languages the main interest is the relationship among fuzzy grammars and the corresponding fuzzy automata. First results suggest that more non-determinismus is needed to accept fuzzy languages than in the crisp case and this would have as consequence a different structure of the (fuzzy) Chomsky hierarchy.
In this area, joint work is being done with the research group of Dr. Héctor Allende from the Technical University Federico Santa María in Valparaíso, Chile. New design algorithms for Ensembles of neural networks are being developed under the point of view of distributed processing. The work is intentionally oriented to be interdisciplinary by including Statistics and Machine Learning aspects in the design of the algorithms. Furthermore, joint work is being pursued with Dr. Igor Aizenberg, from the Texas A&M University, Texarkana, USA for the design of neural networks working in the complex plane, which do not require derivatives to support the training process. Very effective end efficient neural networks have been designed for benchmark problems.
Evolutionary algorithms are very effective to solve search and optimization problems when analytical methods are not available or, if available, they have exponential complexity. Evolutionary algorithms also demand high computational times; however, distributed processing is in this case very attractive, since it is easy to partition these algorithms with the additional positive effect, that in a simple way more diversity may be introduced in the evolutionary process. It is known that for genetic algorithms and for evolutionary strategies an almost linear speed-up has been obtained by distributed processing. The case under present investigation comprises distributed genetic programming on graphs running on a Linux-cluster. As real world benchmark problems, the design of digital circuits for multipliers and for comparators are considered.
In quantum physics, statements of fuzzy nature appear naturally. To every fuzzy set of the domain of some observable, a yes-no test and the corresponding effect in Hilbert space may be associated. The mathematical framework is much more involved than the residuated lattices in fuzzy logic. To formalize reasoning about quantum physical phenomena by means of a logical calculus has remained a challenge even after decades of intensive research in this area.