Computing, in its usual sense, is centered on manipulation of numbers and symbols. In contrast, computing with words, or CW for short, is a methodology in which the objects of computation are words and propositions drawn from a natural language, e.g., small, large, far, heavy, not very likely, the price of gas is low and declining, Berkeley is near San Francisco, it is very unlikely that there will be a significant increase in the price of oil in the near future, etc. Computing with words is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Familiar examples of such tasks are parking a car, driving in heavy traffic, playing golf, riding a bicycle, understanding speech and summarizing a story. Underlying this remarkable capability is the brain’s crucial ability to manipulate perceptions – perceptions of distance, size, weight, color, speed, time, direction, force, number, truth, likelihood and other characteristics of physical and mental objects. Manipulation of perceptions plays a key role in human recognition, decision and execution processes. As a methodology, computing with words provides a foundation for a computational theory of perceptions – a theory which may have an important bearing on how humans make – and machines might make – perception-based rational decisions in an environment of imprecision, uncertainty and partial truth. A basic difference between perceptions and measurements is that, in general, measurements are crisp whereas perceptions are fuzzy. One of the fundamental aims of science has been and continues to be that of progressing from perceptions to measurements. Pursuit of this aim has led to brilliant successes. We have sent men to the moon; we can build computers

A technique to perform design calculations on imprecise representations of parameters has been developed and is presented. The level of imprecision in the description of design elements is typically high in the preliminary phase of engineering design. This imprecision is represented using the fuzzy calculus. Calculations can be performed using this method, to produce (imprecise) performance parameters from imprecise (input) design parameters. The Fuzzy Weighted Average technique is used to perform these calculations. A new metric, called the γ-level measure, is introduced to determine the relative coupling between imprecise inputs and outputs. The background and theory supporting this approach are presented, along with one example. 1.

Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e.,variance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other forms of uncertainty exist in engineering design. Imprecision, or the concept of uncertainty in choice, is one such form. This paper considers real-time techniques for calculating with imprecise parameters. These techniques utilize interval mathematics and the notion of α-cuts from the fuzzy calculus. The extremes or anomalies of the techniques are also investigated, particularly the evaluation of singular or multi-valued functions. It will be shown that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance.

It is a deep-seated tradition in science to view uncertainty as a province of probability theory. The generalized theory of uncertainty (GTU) which is outlined in this paper breaks with this tradition and views uncertainty in a much broader perspective. Uncertainty is an attribute of information. A fundamental premise of GTU is that information, whatever its form, may be represented as what is called a generalized constraint. The concept of a generalized constraint is the centerpiece of GTU. In GTU, a probabilistic constraint is viewed as a special––albeit important––instance of a generalized constraint. A generalized constraint is a constraint of the form X isr R, where X is the constrained variable, R is a constraining relation, generally non-bivalent, and r is an indexing variable which identifies the modality of the constraint, that is, its semantics. The

In many real-life situations, it is very difficult or even impossible to directly measure the quantity y in which we are interested: e.g., we cannot directly measure a distance to a distant galaxy or the amount of oil in a given well. Since we cannot measure such quantities directly, we can measure them indirectly: by first measuring some relating quantities x1 ; : : : ; xn , and then by using the known relation between x i and y to reconstruct the value of the desired quantity y. In practice, it is often very important to estimate the error of the resulting indirect measurement. In this paper, we describe and compare different deterministic and randomized algorithms for solving this problem in the situation when a program for transforming the estimates e x1 ; : : : ; e xn for x i into an estimate for y is only available as a black box (with no source code at hand). We consider this problem in two settings: statistical, when measurements errors \Deltax i = e x i \Gamma x i are inde...

Abstract. Fuzzy control is a very successful way to transform the expert’s knowledge of the type “if the velocity is big and the distance from the object is small, hit the brakes and decelerate as fast as possible ” into an actual control. To apply this transformation one must: 1) choose fuzzy variables corresponding to words like “small”, “big”; 2) choose operations corresponding to “and ” and “or”; 3) choose a method that transforms the resulting fuzzy variable for a into a single value ā. The wrong choice can drastically affect the quality of the resulting control, so the problem of choosing the right procedure is very important. From mathematical viewpoint these choice problems correspond to non-linear optimization and are therefore extremely difficult. We develop a new mathematical formalism (based on group theory) that allows us to solve the problem of optimal choice and thus: 1) explain why the existing choices are really the best (in some situations); 2) explain a rather mysterious fact that the fuzzy control based on the experts’ knowledge is often better than the control by these same experts; 3) give choice recommendations for the cases when traditional choices do not work. Perspectives of space applications will be also discussed.

iii This document and the work it represents was impossible without the support of my wife Ginger. Often one needs non-technical advice to make clear what one is contemplating. Also one always needs a financial supporter. My thesis advisor Erik Antonsson helped focus many of my thoughts. In addition to providing me with technical assistance, he as well provided instruction on the process of conducting academic research, the communication of ideas both orally and written, and the approach to a developing field. I also owe much to my colleagues in the Engineering and Applied Science Division at Caltech. Their comments and advice maintained my comprehension and rigor. Andrew Lewis in particular provided me with invaluable support. Many of the technical proofs were impossible without him. This material and the work it represented were made possible, in part, by a fellowship from the AT&T-Bell Laboratories Ph.D. scholar program, sponsored by the AT&T foundation. Also, the National Science Foundation provided funding under a Presidential Young

Practical scheduling usually has to react to many unpredictable events and uncertainties in the production environment. Although often possible in theory, it is undesirable to reschedule from scratch in such cases. Since the surrounding organization will be prepared for the predicted schedule it is important to change only those features of the schedule that are necessary. We show how on one side fuzzy logic can be used to support the construction of schedules that are robust with respect to changes due to certain types of event. On the other side we show how a reaction can be restricted to a small environment by means of fuzzy constraints and a repair-based problem-solving strategy. We demonstrate the proposed representation and problem-solving method by introducing a scheduling application in a steelmaking plant. We construct a preliminary schedule by taking into account only the most likely duration of operations. This schedule is iteratively "repaired" until some threshold evaluation is found. A repair is found with a local search procedure based on Tabu Search. Finally, we show which events can lead to reactive scheduling and how this is supported by the repair strategy.

This thesis develops the idea of probabilistic arithmetic. The aim is to replace arithmetic operations on numbers with arithmetic operations on random variables. Specifically, we are interested in numerical methods of calculating convolutions of probability distributions. The long-term goal is to be able to handle random problems (such as the determination of the distribution of the roots of random algebraic equations) using algorithms which have been developed for the deterministic case. To this end, in this thesis we survey a number of previously proposed methods for calculating convolutions and representing probability distributions and examine their defects. We develop some new results for some of these methods (the Laguerre transform and the histogram method), but ultimately find them unsuitable. We find that the details on how the ordinary convolution equations are calculated are

This paper deals with simulation of approximate models of dynamic systems. We propose an approach appropriate when the uncertainty intrinsic in some models cannot be reduced by traditional identification techniques, due to the impossibility of gathering experimental data about the system itself. The paper presents a methodology for qualitative modeling and simulation of approximately known systems. The proposed solution is based on the Fuzzy Sets theory, extending the power of traditional numerical-logical methods. We have implemented a fuzzy simulator that integrates a fuzzy, qualitative approach and traditional, quantitative methods. 1. Introduction Simulation can be considered as a part of the process of modeling and forecasting the behavior of a dynamic system. Its task is to reproduce, in the most suitable way, the evolution of a system model in time (Zeigler, 1976). A model is a finite set of formal relations which, in the traditional scientific approach, are mathema...