Abstract-In the traditional fuzzy logic, as truth values, we take all real numbers from the interval [0,1]. In some situations, this set is not fully adequate for describing expert uncertainty, so a more general set is needed. From the mathematical viewpoint, a natural extension of real numbers is the set of complex numbers. Complex-valued fuzzy sets have indeed been successfully used in applications of fuzzy techniques. This practical success leaves us with a puzzling question: why complex-valued degree of belief, degrees which do not seem to have a direct intuitive meaning, have been so successful? In this paper, we use latest results from theory of computation to explain this puzzle. Namely, we show that the possibility to extend to complex numbers is a necessary condition for fuzzy-related computations to be feasible. This computational result also explains why complex numbers are so efficiently used beyond fuzzy, in physics, in control, etc. I. FORMULATION OF THE PROBLEMFrom the mathematical viewpoint, complex-valued fuzzy sets are natural. In classical (2-valued) logic, every statement is either true or false. In the computer, "true" is usually represented as 1, and "false" as 0. As a result, in the 2-valued logic, the set of possible truth values is a 2-element set {0, 1}.The traditional 2-valued logic is well equipped to represent:• situations when we are completely confident that a given statement is true and • situations when we are completely confident that a given statement is false. However, the traditional logic cannot adequately represent intermediate situations, when we only have some degree of confidence that a statement is true. To describe such intermediate situations, L. Zadeh invented fuzzy logic; see, e.g., [7], [11], [13]. In the original version of fuzzy logic, the set of possible truth values is an interval [0, 1]; this is still the most widely used set of possible truth values.Fuzzy logic has been successful in many applications. However, in some applications, the [0, 1]-based fuzzy logic is itself not fully adequate. For example, the [0, 1]-based logic assumes that we can describe an expert's intermediate degree of confidence by an exact number from the interval [0, 1]. Reallife experts often cannot meaningfully distinguish between nearby numbers: it is difficult to meaningfully distinguish between degree of confidence 0.78 and 0.79. So, a more adequate description is needed.One of the main objectives of studying fuzzy knowledge is to use the resulting models in computer-based systems. Because of this objective, researchers are looking for well-developed models, models which use well-established mathematical constructions. Thus, we are looking for wellestablished mathematical constructions that generalize the set of all real numbers from the interval [0, 1].In mathematics, one of the natural generalizations of real numbers are complex numbers. Not surprisingly, complexvalued generalizations of fuzzy sets have been proposed and used; see, e.g., From the intuitive viewpoint, complex-valu...
Fuzzy sets have been originally introduced as generalizations of crisp sets, and this is how they are usually considered. From the mathematical viewpoint, the problem with this approach is that most notions allow many different generalizations, so every time we try to generalize some notions to fuzzy sets, we have numerous alternatives. In this paper, we show that fuzzy sets can be alternatively viewed as limits of crisp sets. As a result, for some notions, we can come up with a unique generalization -as the limit of the results of applying this notion to the corresponding crisp sets.
Abstract-An expert opinion describes his or her opinion about a quantity by using imprecise ("fuzzy") words from a natural language, such as "small", "medium", "large", etc. Each of these words provides a rather crude description of the corresponding quantity. A natural way to refine this description is to assign degrees to which the observed quantity fits each of the selected words. For example, an expert can say that the value is reasonable small, but to some extent it is medium. In this refined description, we represent each quantity by a tuple of the corresponding degrees.Once we have such a tuple-based information about several quantities x1, . . . , xm, and we know that another quantity y is related to xi by a known relation y = f (x1, . . . , xm), it is desirable to come up with a resulting tuple-based description of y. In this paper, we describe why a seemingly natural idea for computing such a tuple does not work, and we show how to modify this idea so that it can be used.
Scheduling of Grid workflows has been a prevalent research area as it is the mechanism that helps improving the efficiency of the Grid workflow execution. The most common goal of scheduling workflow in the Grid is to minimize execution makespan and many algorithms have been proposed to address this issue. However, most of the algorithms so far usually focus and are implemented based on a single principle and act according to that principle. Only few algorithms consider adaptive scheduling process that acts differently in different situations.In this paper, we propose an adaptive Grid workflow scheduling that reacts to the presence of bottlenecks and different execution context. With this adaptive approach, the proposed algorithm can achieve better overall makespan. Simulations of parameter sweep workflows with single instance and multiple instances executed in parallel are used to evaluate the algorithm against four existing algorithms.
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