Generalized Morphisms: a Tool for Evaluation of Adequacy of Logic Connectives Used in Symbolic and Soft Computing

I. The need for comparative evaluation of performance of different systems of fuzzy connectives in soft computing applications This paper continues our work on developing new methods and tools for comparative assessment of the adequacy of diierent many-valued implication operators and other fuzzy connectives. To deal with the problem of selecting the optimal fuzzy logic connectives, new methods for mutual comparison (meta-analysis) of relational structures are required. In this paper, we describe one of the methods we have developed that serves this purpose. It is well known for example from control engineering applications that diierent fuzzy implication operators are suitable for getting the best performance in diierent control tasks. This is also the case in relational data analysis, where diierent many-valued base logics used in fuzzy computations over the same set of data might yield diierent fuzzy preorders or equivalences. This may innuence the results of relational computations employed in structural analysis of scientiic and commercial data 19] as well as the performance of symbolic computing systems such as automated reasoning systems 16],,17]. The clear and generally accepted conclusion is that the choice of the optimal fuzzy connectives strongly depends on a particular domain of application. The fact that such studies should be done is recognized in control enginnering and fuzzy symbolic computing application. It is not only the performance but also the correct-ness of computations that depends on adequate choices. The methods adumbrated in this paper can be used not only in the above mentioned domains but also for evaluating of neural network and genetic algorithms based clas-siiers. Such systems should provide correct classiication and results should be veriiable considering these as black boxes. The Hasse diagram techniques for testing classiica-tion hierarchies described here can be used for this purpose. II. Methods for analysis of relational structures based on BK-products of fuzzy relations Fuzzy relational methods can be used to discover meaningful structures implicit in scientiic data or business data. In particular, fuzzy relational products and algorithms for computing of fuzzy closures and interiors, when applied to the real world scientiic, management or engineering data 1 make it possible to discover and further analyze structural relationships between individual items of data, which can not always be adequately analyzed by the methods based on conventional statistics. The fuzzy relational approach that could be used for this sort of analysis of data has to provide an adequate theory of fuzzy relations, …