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

: Given a general measure (nite or innite), we develop possibility and necessity measures as upper and lower estimators of . We provide a method for constructing such fuzzy measures and show that the measure can be approximated with arbitrary closeness using fuzzy measures constructed this way. Using the extension principle, these consistent possibility and necessity measures are used to produce possibility and necessity measures on the range space of a measurable function which are consistent with the measure on the range space induced by the measurable function. This induced measure can be approximated with arbitrary closeness by extending consistent possibility and necessity measures constructed on the domain space. Keywords: Possibility Theory, Non-additive Measures, Measure Theory 1

It is shown that possibility distributions can be formulated within the context of probability theory and that membership values of fuzzy set theory can be interpreted as cumulative probabilities. The basic functions and operations of possibility theory are interpreted within this setting. The probabilistic information that can be derived from possibility distributions is examined. This leads to two functionals that provide estimates for the expected value of a random variable, the expected average of a single possibility distribution and the estimated expectation that requires two special possibility distributions to compute. Secondly, the space of fuzzy numbers is examined. It is shown that this space can be partitioned into a vector space and that the expected average functional motivates a norm on this space. It is shown that for most applications, Cauchy sequences converge in this space. Thirdly, applications of this theory to problems in optimization are examined. The concept of ...