Interval Mathematics for Analysis of Multi -

| The more complex the problem, the more complex the system necessary for solving this problem. For very complex problems, it is no longer possible to design the corresponding system on a single resolution level, it becomes necessary to have multi-level, multiresolutional systems , with multi-level granulation. When analyzing such systems { e.g., when estimating their performance and/or their intelligence { it is reasonable to use the multi-level character of these systems: rst, we analyze the system on the low-resolution level, and then we sharpen the results of the low-resolution analysis by considering higher-resolution representations of the analyzed system. The analysis of the low-resolution level provides us with an approximate value of the desired performance characteristic. In order to make a deenite conclusion, we need to know the accuracy of this approximation. In this paper, we describe interval mathematics { a methodology for estimating such accuracy. The resulting interval approach is also extremely important for tessellating the space of search when searching for optimal control. We overview the corresponding theoretical results, and present several case studies. The more complex the problem, the more complex the system necessary for solving this problem. For very complex problems, it is no longer possible to design the corresponding system on a single resolution level, it becomes necessary to have multi-level systems, with diierent gran-ulations on each level. Granulation methods can be traced to a pioneer paper by L. Zadeh 82]. Multi-level granulation methods, namely, the methodology of multiresolutional search for the optimum solution of a control problem was rst presented by A. Meystel in 42], 43]. These papers contributed to the broad interest in and dissemination of the multiresolutional approach to solving problems of the areas of intelligent control and intelligent systems. Many algorithms based on this methodology were developed since then. The successful practical applications of these algorithms shows that multiresolutional approach is indeed necessary. This empirical conclusion has been supported by many mathematical results; let us name a few recent ones: It has been proven that for general complex (NP-hard) problems, i.e., problems, for which no general feasible algorithm is possible, there always exists an appropriate granu-lation after which the problem becomes easy to solve. The fact that the problem is NP-hard means that there is no general algorithm for automatically nding such a granula-tion, this granulation requires an expert familiar with the particular problem that we are trying to solve 12]. …