| Grant number: | 05/03648-5 |
| Support Opportunities: | Scholarships in Brazil - Master |
| Start date: | March 01, 2006 |
| End date: | February 29, 2008 |
| Field of knowledge: | Physical Sciences and Mathematics - Computer Science - Computational Mathematics |
| Principal Investigator: | Peter Sussner |
| Grantee: | Rodolfo Miyasaki |
| Host Institution: | Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil |
Abstract By means of this project, we propose to investigate the connections between two seemingly unrelated areas of research: fuzzy relational equations (FREs) and fuzzy morphological associative memories (FMAMs). In contrast to FREs whose theory has been initiated by E. Sanchez almost three decades ago, FMAMs are a very recent development in the area of neuro-fuzzy systems. In fact, a general theory of FMAMs has yet to appear in the literature (we are planning to accomplish this task together with our doctoral student Marcos Valle). We recently introduced a fairly general subclass of FMAMs in the paper "Implicative Fuzzy Morphological Associative Memories" (IFAMs) that will soon appear in the journal "IEEE Transactions on Fuzzy Systems". In this paper, we also describe applications of IFAMs as fuzzy knowledge-based systems, in particular an application in prediction. FREs represent a proven tool in knowledge engineering and have been applied to fuzzy control and fuzzy pattern recognition.In analogy to morphological associative memories (MAMs), IFAMs are simple matrix associative memories that consist of a primal model W and of adual model M. Moreover, the recall phase of IFAMs can also be described using certain matrix vector products, namely max-t and min-s compositions where t denotes a t-norm and s denotes an s-norm. Surprisingly, research papers on FREs do not speak of matrix-vector products although FREs should rather be called systems of FREs. The formulas for FREs are closely related to the recall phase of IFAMs and we observe that the IFAM weight matrix W, respectively M, is chosen so as to coincide with the greatest, respectively the smallest, solution of the corresponding equation. Due to these linkages between FREs and IFAMs/FMAMs, we strongly believe that researchers would benefit from an exchange of ideas between the two areas. For instance, one of the open research problems in the FRE area consists in determining conditions for the solvability of FREs. This problem is closely related to determining conditions for perfect recall using IFAMs. As we have pointed out in our article in the IEEE Trans. on Fuzzy Systems, gray-scale MAMs can be viewed as a special case of IFAMs. We would like to remind the reader that we have already determined conditions for perfect recall using gray-scale MAMs in 1996. However, these conditions are very cumbersome and hard to understand. In a recent conference paper, we presented conditions for perfect recall using binary MAMs. These conditions can be generalized using our results on gray-scale autoassociative morphological memories that are contained in the paper "Gray-Scale Morphological Associative Memories" (accepted for publication in IEEE Trans. on Neural Networks). In this project, we will attempt to generalize these conditions even more to include the class of IFAMs or FMAMs in general and we intend to formulate the corresponding conditions for the solvability of FREs. In the IFAM area, further research is required in order to determine the best type of IFAM for a given application. The (primal) IFAM model depends on the choice of a certain t-norm. For example, the Lukasiewicz IFAM which corresponds to the MAM model empoys the Lukasiewicz t-norm. In our IEEETFS paper, we have tested several types of IFAM in a number of experiments. We observed that the Lukasiewicz IFAM exhibited the best performance among all approaches, including the IFAM models, that were tested in the experiments. However, we do not know what is the bes t IFAM model overall for these and other applications. A method for constructing and optimizing t-norms might provide some means to partially answer this question. Ciaramella et al. construct an infinite number of t-norms from given ones by means of ordinal sums and apply a genetic algorithm in order to choose the best t-norm for a given problem. This approach was also applied to fuzzy relational neural networks. (AU) | |
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