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

This paper investigates a probabilistic version of the concurrent constraint programming paradigm (CCP). The aim is to introduce the possibility to formulate so called "randomised algorithms" within the CCP framework. Differently from common approaches in (imperative) high-level programming languages, which rely on some kind of random() function, we introduce randomness in the very definition of the language by means of a probabilistic choice construct. This allows a program to make stochastic moves during its execution. We call the resulting language Probabilistic Concurrent Constraint Programming (PCCP). We present an operational semantics for PCCP by means of a probabilistic transition system such that the execution of a PCCP program may be seen as a stochastic process, i.e. as a random walk on the transition graph. The transition probabilities are given explicitly. This semantics captures a notion of observables which combines results of computations and the probability of those re...

In essence, data mining consists of extracting knowledge from data. This paper proposes a co-evolutionary system for discovering fuzzy classification rules. The system uses two evolutionary algorithms: a genetic programming (GP) algorithm evolving a population of fuzzy rule sets and a simple evolutionary algorithm evolving a population of membership function definitions. The two populations co-evolve, so that the final result of the coevolutionary process is a fuzzy rule set and a set of membership function definitions which are well adapted to each other. In addition, our system also has some innovative ideas with respect to the encoding of GP individuals representing rule sets. The basic idea is that our individual encoding scheme incorporates several syntactical restrictions that facilitate the handling of rule sets in disjunctive normal form. We have also adapted GP operators to better work with the proposed individual encoding scheme.

Fuzzy set theory is based on a `fuzzification' of the predicate 2 (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent in fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets to the framework of GL--monoids that can be understood as a generalization of MV--algebras. Residuation is a basic concept in GL--monoids and many proofs can be formulated in a simple and clear way using residuation instead of concentrating on special properties of the unit interval. 1 Introduction Fuzzy set theory is based on the idea that many non--mathematical properties cannot be described in terms of crisp sets comprising those elements that fulfill a given property. The...

. Numerous formalisms have been proposed for representing and processing uncertainty in expert systems. Several of these formalisms are somewhat ad hoc, in the sense that some of their formulas seem to have been chosen rather arbitrarily. In this paper, we show that random sets provide a natural general framework for describing uncertainty, a framework in which many existing formalisms appear as particular cases. This interpretation of known formalisms (e.g., of fuzzy logic) in terms of random sets enables us to justify many "ad hoc" formulas. In some cases, when several alternative formulas have been proposed, random sets help to choose the best ones (in some reasonable sense). One of the main objectives of expert systems is not only to describe the current state of the world, but also to provide us with reasonable actions. The simplest case is when we have the exact objective function. In this case, random sets can help in choosing the proper method of "fuzzy optimization." As a t...

The material and manufacturing process selection problem is a multi-attribute decision making problem. These decisions are made during the preliminary design stages in an environment characterized by imprecise and uncertain requirements, parameters, and relationships. Material and process selection decisions must occur before design for manufacturing can begin. This paper describes a prototype material and manufacturing process selection system called MAMPS that integrates a formal multi-attribute decision model with a relational database. The decision model enables the representation of the designer’s preferences over the decision factors. A compatibility rating between the product profile requirements and the alternatives stored in the database for each decision criteria is generated using possibility theory. The vector of compatibility ratings are aggregated into a single rating of that alternative’s compatibility. A ranked set of compatible material and manufacturing process alternatives is output by the system. This approach has advantages over existing systems that either do not have a decision module or are not integrated with a database. Keywords: Multi-attribute decision making, material selection, process selection, design for manufacturing, concurrent engineering, possibility theory, fuzzy sets, manufacturing processes.

Personalization tailors a user’s interaction with the Web information space based on information gathered about them. Declarative user information such as manually entered profiles continue to raise privacy concerns and are neither scalable nor flexible in the face of very active dynamic Web sites and changing user trends and interests. One way to deal with this problem is through a complete automated Web personalization system. Such a system can be based on Web usage mining to discover Web usage profiles, followed by a recommendation system that can respond to the users’ individual interests. Significant amounts of error and uncertainty can permeate all the stages of Web personalization. Therefore, we present a fast and intuitive approach to provide Web recommendations using a fuzzy

Measurement methods are a central requirement for the semantic grounding of any mathematical systems theory. Therefore possibility theory, as a branch of General Information Theory (git), requires objective measurement methods to extend its agenda and applications beyond the fuzzy theory from which it emerged. General measuring devices, when defined on intervals of IR, yield empirical random intervals which, when consistent, yield possibility distributions as their plausibilistic traces. These empirical possibility distributions are called possibilistic histograms, and are fuzzy intervals. Their continuous approximations, even for very small sample sizes, yield the the standard fuzzy interval forms commonly used in fuzzy system applications. Keywords: Possibilistic histograms, semiotics, possibilistic measurement, general information theory, fuzzy measures, possibility theory, random sets, random intervals, fuzzy intervals, fuzzy numbers. INTRODUCTION: MEASUREMENT IN POSSIBILISTIC SEM...

Abstract: Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (TrFN). Then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not closed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only approximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applications. We analyze this problem and propose six parameters which define parameterized fuzzy numbers (PFN), of which TFNs are a special case. We provide the methods for performing fuzzy arithmetic and show that the PFN representation is closed under the arithmetic operations. The new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as TFNs. The new method has better accuracy and similar accepted by Fuzzy Sets and Systems: Special Issue on Fuzzy Arithmeticcomputational speed to using TFNs and appears to have benefits when used in engineering applications.

In traditional statistics, we process crisp data -- usually, results of measurements and/or observations. Not all the knowledge comes from measurements and observations. In many real-life situations, in addition to the results of measurements and observations, we have expert estimates, estimates that are often formulated in terms of natural language, like "x is large". Before we analyze how to process these statements, we must be able to translate them in a language that a computer can understand. This translation of expert statements from natural language into a precise language of numbers is one of the main original objectives of fuzzy logic. It is therefore important to extend traditional statistical techniques from processing crisp data to processing fuzzy data. In this paper, we provide an overview of our related research. Keywords: statistical processing, interval data, fuzzy data, random sets, possibility theory, 1