Computing with Words in InformationlIntelligent Systems 1

Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw, Poland E-mail: kacprzyk@ibspan.waw.pl

Vol. 3. A. Geyer-Schulz Fuu.y Rule-Based Expert Systems and Genetic Machine Learning, 2nd ed. 1996 ISBN 3-7908-0964-0

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Vol. 6. E. S. Lee and Q. Zhu Fuu.y and Evidence Reasoning, 1995 ISBN 3-7908-0880-6

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Vol. 10. L.C. Jain (Ed.) Soft Computing Techniques in Knowledge-based Intelligent Engineering Systems, 1997, ISBN 3-7908-1035-5

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Vol. 15. G.J. Klir and M.J. Wiennan Uncertainty-Based Information, 1998 ISBN 3-7908-1073-8

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Vol. 24. E. OrIowska (Ed.) Logic at Work, 1999 ISBN 3-7908-1164-5

continued on page 518

Lotfi A. Zadeh Janusz Kacprzyk (Eds.)

Computing with Words in Information/Intelligent Systems 1 Foundations

With 135 Figures and 22 Tables

Springer-Verlag Berlin Heidelberg GmbH

Prof. Lotti A. Zadeh Berkeley Initiative in Soft Computing (BISC) Computer Science Division and Electronics Research Laboratory Department of Electrical and Electronics Engineering and Computer Science University of California Berkeley, CA 94720-1776 USA E-mail: zadeh@cs.berkeley.edu

Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl

ISBN 978-3-662-11362-2

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Computing with words in information/intelligent systems: with 22 tables I Lotti A. Zadeh, Janusz Kacprzyk (ed.)

1. Foundations. - 1999 (Studies in fuzziness and soft computing; Vol. 33) ISBN 978-3-662-11362-2 ISBN 978-3-7908-1873-4 (eBook) DOI 10.1007/978-3-7908-1873-4

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Foreword

These two volumes consIstmg of Foundations and Applications provide the current status of theoretical and empirical developments in "computing with words".

In philosophy, the twentieth century is said to be the century of language. This is mainly due to Wittgenstein who said:

"The meaning of a word is its use in the language game". "The concept game is a concept with blurred edges".

In the first phrase, "the language game" implies the everyday human activity with language, and in the latter, "game" simply implies an ordinary word. Thus, Wittgenstein precisely stated that a word is fuzzy in real life.

Unfortunately this idea about a word was not accepted in the conventional science. We had to wait for Zadeh's fuzzy sets theory. Remembering Wittgenstein's statement, we should consider, on the one hand, the concept of "computing with words" from a philosophical point of view. It deeply relates to the everyday use of a word in which the meaning of a word is fuzzy in its nature.

On the other hand, "computing with words" can be considered in the perspective of history of computing with language in computer science and also in artificial intelligence. We can go back to the 1950s when an attempt to machine translation started. As we know, this computer scientific adventure to computing with language terminated in 1965 when the US Air Force pronounced machine translation a failure. Researchers continued their activities in computing with language under different titles such as parsing, question-answering or expert systems and the like in the setting of artificial intelligence. There was, however, a strict line between natural language processing and computing, as pointed out by Halliday. Computing was not intelligent in any sense. It was considered just as a tool to realize an intelligent system.

A breakthrough was made in the 1990s by Zadeh's idea of "computing with words". The history of computing with language has now made a revolutionary turn. We have entered the true age af computing with language. Computing itself is now viewed from a perspective of human intelligence. Human cogitation is nothing but "computing with words" as Zadeh points out. Cogitation is essentially connected with recognition. In human recognition, we see the world with words. We articulate the physical world with Wittgenstein's blurred words. According to Zadeh, this articulation is a "fuzzy granulation".

VI

As such, fuzzy logic is a promising tool to playa very important role in intelligent computing. From now on, we will be able to view any computing as "computing with words". This idea would become a main stream to create "an artificial brain".

This volume, Part 1: Foundations, includes introductory papers related to various basic aspects of, and approaches to "computing with words".

I wish to congratulate the editors, Professors Zadeh and Kacprzyk for these volumes for their great success. In particular, I wish to acknowledge Professor Janusz Kacprzyk who has been mainly the driving force behind this project.

Tokyo, March 1999 Michio Sugeno President International Fuzzy Systems Association (IFSA)

What is Computing with Words?

Lotfi A.Zadeh 1

The label "Computing with Words (CW)" lends itself to misinterpretation. There are some who say "What is new? Isn't this what we have been doing all along?" The answer is: No, not really. Then, what is it?

Historically, computing was focused on manipulation of numbers. With the passage of time, the meaning of computing became much more encompassing. Today, computing is a term that is applied to manipulation of a wide variety of data ranging from numbers and symbols to signals, sounds, images, and text. Thus, computing with numbers - as in numerical analysis - is far less prevalent today than processing of a text drawn from a natural language, as in machine translation, spelling correction, interfaces, and summarization. Although such operations do involve manipulation of words, what should be underscored is that they are not representative of what is in the mainstream of computing with words.

The intended meaning of computing with words is reflected in the content of the papers included in this volume. More concretely, the meaning of computing with words is conveyed by examples of problems drawn from its mainstream.

Representative of such problems are the following:

a) solution of a system of linear equations with linguistic rather than numerical coefficients,

b) operations on functions which are defined by fuzzy if-then rules with linguistic values. Example: interpolation or maximization of a function defined by a fuzzy rule set, e.g.:

if X is small then Y is small if X is medium then Y is large if X is large then Y is small

c) computation of linguistic probabilities;

Example:

A box contains balls of various sizes. Most are large and a few are small.

1 Professor in the Graduate School and Director, Berkeley Initiative in Soft Computing, University of Cali fomi a, Berkeley, CA 94720-1776

VIII

What is the probability that a ball drawn at random is neither large nor small?

d) syllogistic reasoning;

Example:

Most young men are healthy Robert is young What is the probability that Robert is healthy?

e) Dispositional reasoning:

Example:

Slimness is attractive Cindy is slim What can be said about Cindy's attractiveness?

What we see is that problems of this type do not lend themselves to solution by conventional methods drawn from logic, probability theory and numerical analysis. They do fall within the province of Computing with Words (CW).

It is helpful to distinguish between two types of problems in computing with words:

• level 1 problems (CWl) are those in which members are replaced with words, as in examples (a), (b), (c), and (d),

• level 2 problems (CW2) in which the premises are propositions drawn from natural language whose meaning cannot be simply defined, as in example (e).

There are four principal rationales for computing with words:

1. The don't know rationale

Values of variables are not known with sufficient precision to justify the use of numbers.

2. The don't need rationale

Precision is not needed.

3. The can't solve rationale

A problem cannot be solved or a task cannot be performed through the use of numbers.

4. The can't define rationale

A concept is too complex for numerical definition. Examples: causality, relevance, summary.

IX

Since the objects of computation in computing with words are propositions drawn from a natural language, it is necessary to have in computing with words a means of defining meaning in a way that makes semantics amenable to computation. In computing with words, this is accomplished through the use of constraint-centered semantics of natural languages (CSNL). In this semantics, the meaning of a proposition is represented as a generalized constraint on an implicit variable. The constraint-centered semantics of natural languages has a much higher expressive power than conventional meaning-representation systems based on predicate logic.

The high expressive power of the constraint-centered semantics of natural languages makes it possible to use computing with words as a foundation for what may be called the computational theory of perceptions (CTP). This theory suggests many new applications for the basic methodology of computing with words.

In essence, the computational theory of perceptions is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Everyday examples of such tasks are: parking a car, driving in heavy traffic, playing golf, balancing a pole and summarizing a story. Underlying this capability is the brain's ability to manipulate perceptions - perceptions of time, distance, speed, force, direction, color, likelihood, intent and truth, among others.

An essential difference between measurements and perception is that, in general, measurements are crisp whereas perceptions are fuzzy. In the computational theory of perceptions, the point of departure is a description of perceptions as propositions expressed in a natural language as, e.g., Robert is very honest, overeating causes obesity, it is very unlikely that there will be a significant decline in the price of oil in the near future, etc. Once perceptions are described as propositions, the machinery of computing with words may be employed to reason with them. This, in essence, is the key idea underlying the computational theory of perceptions.

It is a long-standing tradition in science to accord more respect to numbers than to words. But it is becoming increasingly clear that the remarkable human ability to manipulate words and perceptions without any measurements and any computations plays a key role in human intelligence. In the final analysis, the role model for computing with words and the computational theory of perceptions is the human mind.

Berkeley, February, 1999. Lotfi A. Zadeh

Contents

Foreword M.Sugeno

What is Computing with Words? L.A. Zadeh

1. INTRODUCTORY SECTIONS

Fuzzy Logic = Computing with Words L.A. Zadeh

Performing Approximate Reasoning with Words? D. Dubois, L. Foulloy, S. Galichet and H. Prade

Approximate Reasoning as a Basis for Computing with Words R.R. Yager

What is Intelligent Measurement? L. Reznik

2. COMPUTING WITH WORDS: LINGUISTIC ASPECTS

Semiotics and Computational Linguistics. On Semiotic Cognitive

v

vii

3

24

50

78

Information Processing 93 B.B. Rieger

Words about Uncertainty: Analogies and Contexts 119 M. J. Smithson

Virtual Meaning: Problems of Interpretation in the Social Sciences 136 V. Dimitrov and B. Hodge

Towards Fixing Some 'Fuzzy' Catchwords: A Terminological Primer 154 H.Toth

XII

3. COMPUTING WITH WORDS AND INFORMATION GRANULARITY

Granular Computing: Fuzzy Logic and Rough Sets T.Y. Lin

Towards an Adaptive Calculus of Granules L. Polkowski and A. Skowron

Semantics and Modelling of Flexible Time Indications R. De Caluwe, F. Devos, P. Maesfranckx, G. De Tn! and B. Van der Cruyssen

4. COMPUTING WITH WORDS: LOGICAL ASPECTS

Towards Fuzzy Logic W. Ostasiewicz

Fuzzy Negation E. Ferri, A. Kandel and G. Langholz

Triangular Operations, Negations, and Scalar Cardinality of a Fuzzy Set

M. Wygralak

Fuzzy Implications in Approximate Reasoning E. Czogala and J. L~ski

On the Semantics of Fuzzy Linguistic Quantifiers H. Thiele

Evaluating Linguistic Expressions and Functional Fuzzy Theories in Fuzzy Logic

V. Novak and I. Perfilieva

5. COMPUTING WITH WORDS: NUMERICAL ASPECTS

Calculation over Verbal Quantities M. Mares and R. Mesiar

183

201

229

259

297

326

342

358

383

409

XIII

Aggregation of Linguistic Information Based on a Symbolic Approach 428 M. Delgado, F. Herrera, E. Herrera-Viedma, J.L. Verdegay and M.A. Vila

6. GENERAL FRAMEWORKS FOR COMPUTING WITH WORDS

Object Networks: A Computational Framework to Compute with Words 443 R.R. Gudwin and F.A.C. Gomide

Study on a Frameworkfor Solving Ill-defined Problems Using Patterns and Symbols 479

Y. Hattori and T. Furuhashi

From Expert Words Directly to Numerical Simulations: Group-Theoretic Approach to Computing with Words in InformationlIntelligent Systems 495

V. Kreinovich, B. Penn and S. Starks