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# From Computing with Numbers to Computing with Words: From Manipulation of Measurements to Manipulation of Perceptions

## Abstract

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.

## Keywords

Natural Language Fuzzy Logic Fuzzy Number Possibility Distribution Fuzzy Graph## Preview

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## References and Related Publications

- [1.]H.R. Berenji, Fuzzy Reinforcement Learning and Dynamic Programming, in
*Fuzzy Logic in Artificial Intelligence*,*Proc. IJCAI 93 Workshop*, ed. A.L. Ralescu, Berlin: Springer-Verlag, pp. 1 - 9, 1994.Google Scholar - [2.]
- [3.]P. Bosch, Vagueness, Ambiguity and All the Rest,in
*Sprachstruktur*,*Individuum und Gesselschaft*, eds. M. Van de Velde and W. Vandeweghe, Tubingen: Niemeyer, 1978.Google Scholar - [4.]J. Bowen, R. Lai, and D. Bahler, Fuzzy Semantics and Fuzzy Constraint Networks,
*Proc. of the 1st IEEE Conf. on Fuzzy Systems*, San Francisco, pp. 1009 - 1016, 1992.Google Scholar - [5.]J. Bowen, R. Lai, and D. Bahler,Lexical Imprecision in Fuzzy Constraint Networks,
*Proc. of the National Conf. on Artificial Intelligence*, pp. 616620, 1992.Google Scholar - [6.]M.J. Cresswell,
*Logic and Languages*, London: Methuen, 1973.Google Scholar - [7.]D. Dubois, H. Fargier, and H. Prade, Propagation and Satisfaction of Flexible Constraints, in
*Fuzzy Sets*,*Neural Networks*,*and Soft Computing*, eds. R.R. Yager, L.A. Zadeh, New York: Von Nostrand Reinhold, pp. 166187, 1994.Google Scholar - [8.D. Dubois, H. Fargier, and H. Prade, Possibility Theory in Constraint Satisfaction Problems: Handling Priority, Preference and Uncertainty, to appear in
*Applied Intelligence Journal.*Google Scholar - [9.]D. Dubois, H. Fargier, and H. Prade, The Calculus of Fuzzy Restrictions as a Basis for Flexible Constraint Satisfaction,
*Proc. of the 2nd IEEE Int. Conf. on Fuzzy Systems*, San Francisco, pp. 1131 - 1136, 1993.Google Scholar - [10.]E.C. Freuder and P. Snow, Improved Relaxation and Search Methods for Approximate Constraint Satisfaction with a Maximin Criterion,
*Proc. of the 8th Biennial Conf on the Canadian Society for Computational Studies of Intelligence*, Ontario, pp. 227 - 230, 1990.Google Scholar - [11.]J.A. Goguen, The Logic of Inexact Concepts,
*Synthese*19, pp. 325 - 373, 1969.CrossRefGoogle Scholar - [12.]J.R. Hobbs, Making Computation Sense of Montagues Intensional Logic,
*Artificial Intelligence**9*, pp. 287 - 306, 1978.CrossRefGoogle Scholar - [13.]O. Katai, S. Matsubara, H. Masuichi, M. Ida, et. al., Synergetic Computation for Constraint Satisfaction Problems Involving Continuous and Fuzzy Variables by Using Occam,in
*Transputer/Occam*,*Proc. of the 4th Transputer/Occam Int. Conf*, eds. S. Noguchi and H. Umeo, Amsterdam: IOS Press, pp. 146 - 160, 1992.Google Scholar - [14.]A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, New York: Von Nostrand, 1985.Google Scholar
- [15.]G. Klir and B. Yuan,
*Fuzzy Sets and Fuzzy Logic*, New Jersey: Prentice Hall, 1995.Google Scholar - [16.]K. Lano,
*A Constraint-Based Fuzzy Inference System*, in EPIA 91, 5th Portuguese Conf. on Artificial Intelligence, eds. P. Barahona, L.M. Pereira, and A. Porto, Berlin: Springer-Verlag, pp. 45 - 59, 1991.Google Scholar - [17.]W.A. Lodwick,
*Analysis of Structure in Fuzzy Linear Programs*, Fuzzy Sets and Systems, 38 (1), pp. 15 - 26, 1990.CrossRefGoogle Scholar - [18.]E.H. Mamdani and B.R. Gaines, Eds., Fuzzy Reasoning and its Applications, London, 1981.Google Scholar
- [19.]M. Mares,
*Computation Over Fuzzy Quantities*, Boca Raton: CRC Press, 1994.Google Scholar - [20.]V. Novak,
*Fuzzy Logic, Fuzzy Sets, and Natural Languages*, Int. J. of General Systems 20 (1), pp. 83 - 97, 1991.CrossRefGoogle Scholar - [21.]V. Novak, M. Ramik, M. Cerny and J. Nekola, Eds., Fuzzy Approach to Reasoning and Decision-Making, Boston: Kluwer, 1992.Google Scholar
- [22.]M.S. Oshan, O.M. Saad and A.G. Hassan, On the Solution of Fuzzy Mul- tiobjective Integer Linear Programming Problems with a Parametric Study,
*Advances in Modelling & Analysis A*,*24(2)*, pp. 49 - 64, 1995.Google Scholar - [23.]B. Partee,
*Montague Grammar*, New York: Academic Press, 1976.Google Scholar - [24.]W. Pedrycz and F. Gomide,
*Introduction to Fuzzy Sets*, Cambridge: MIT Press, 1998.Google Scholar - [25.]G. Qi and G. Friedrich, Extending Constraint Satisfaction Problem Solving in Structural Design, in Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 5th Int. Conf, IEA/AIE-92, eds. F. Belli and F.J. Radermacher, Berlin: Springer-Verlag, pp. 341 - 350, 1992.Google Scholar
- [26.]H. Rasiowa and M. Marek, On reaching consensus by groups of intelligent agents, in Z. W. Ras, (ed.):
*Methodologies for Intelligent Systems*, Amsterdam: North-Holland, pp. 234 - 243, 1989.Google Scholar - [27.]A. Rosenfeld, R.A. Hummel and S.W. Zucker, Scene Labeling by Relaxation Operations,
*IEEE Trans. on Systems*,*Man and Cybernetics*,*6*, pp. 420 - 433, 1976.CrossRefGoogle Scholar - [28.]M. Sakawa, K. Sawada, and M. Inuiguchi, A Fuzzy Satisficing Method for Large-Scale Linear Programming Problems with Block Angular Structure,
*European Journal of Operational Research*,*81(2)*, pp. 399 - 409, 1995.Google Scholar - [29.]G. Shafer,
*A Mathematical Theory of Evidence*, Princeton: Princeton University Press, 1976.Google Scholar - [30.]S.C. Tong,
*Interval Number and Fuzzy Number Linear Programming*, Advances in Modelling & Analysis A, 20 (2), pp. 51 - 56, 1994.Google Scholar - [31.]R. Vallee,
*Cognition et Systeme*, Paris: lInterdisciplinaire Systeme(s), 1995.Google Scholar - [32.]R.R. Yager,
*Some extensions of constraint propagation of label sets*Int. J. of Approximate Reasoning, 3, pp. 417 - 435, 1989.CrossRefGoogle Scholar - L.A. Zadeh, From circuit theory to system theory,
*Proc. IRE*,50, pp. 856865, 1961.Google Scholar - [34.]L.A. Zadeh, Fuzzy Sets, Inf. Control, 8, pp. 338 - 353, 1965.CrossRefGoogle Scholar
- [35.]L.A. Zadeh, "
*Probability measures of fuzzy events*," Jour. Math. Analysis and Appl. 23, pp. 421 - 427, 1968.CrossRefGoogle Scholar - [36.]L.A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, J. of Cybernetics 2, pp. 4 - 34, 1972.CrossRefGoogle Scholar
- [37.]L.A. Zadeh, Outline of a New Approach to the Analysis of Complex System and Decision Processes, IEEE Trans. on Systems, Man, and Cybernetics, SMC-3, pp. 28 - 44, 1973.Google Scholar
- [38.]L.A. Zadeh, On the Analysis of Large Scale Systems, Systems Approaches and Environment Problems, ed. H. Gottinger, Gottingen: Vandenhoeck and Ruprecht, pp. 23-37, 1974.Google Scholar
- [39.]L.A. Zadeh, Calculus of Fuzzy Restrictions, in
*Fuzzy Sets and Their Applications to Cognitive and Decision Processes*, eds. L.A. Zadeh, K.S. Fu, M. Shimura, New York: Academic Press, pp. 1 - 39, 1975.Google Scholar - [40.]L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part I: Inf. Sci.
*8*, pp. 199-249; Part II:*Inf. Sci*.*8*, pp. 301-357; Part III:*Inf. Sci.**9*, pp. 43 - 80, 1975.Google Scholar - [41.]L.A. Zadeh, A fuzzy-algorithmic approach to the definition of complex or imprecise concepts,
*Int. Jour. Man-Machine Studies**8*, pp. 249 - 291, 1976.CrossRefGoogle Scholar - [42.]L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1, pp. 3 - 28, 1978.CrossRefGoogle Scholar
- [43.]L.A. Zadeh, PRUF - a Meaning Representation Language for Natural Languages,
*Int. J. Man-Machines Studies*, 10, pp. 395 - 460, 1978.CrossRefGoogle Scholar - [44.]L.A. Zadeh, Fuzzy Sets and Information Granularity, in Advances in Fuzzy Set Theory and Applications, eds. M. Gupta, R.Ragade and R. Yager, Amsterdam: North-Holland, pp. 3 - 18, 1979.Google Scholar
- [45.]L.A. Zadeh, A Theory of Approximate Reasoning,
*Machine Intelligence*9, eds. J. Hayes, D. Michie, and L.I. Mikulich, New York: Halstead Press, pp. 149-194, 1979.Google Scholar - [46.]L.A. Zadeh, Test-Score Semantics for Natural Languages and Meaning Representation via PRUF,
*Empirical Semantics*,ed. B. Rieger, W. Germany: Brockmeyer, pp. 281-349. Also*Technical Report Memorandum 246*,AI Center, SRI International, Menlo Park, CA, 1981.Google Scholar - [47.]L.A. Zadeh,
*Test-score semantics for natural languages*, Proc. of the Ninth International Conference on Computational Linguistics*,Prague, pp. 425-430, 1982.*Google Scholar - [48.]L.A. Zadeh,
*Syllogistic reasoning in fuzzy logic and its application to reasoning with dispositions*,Proceedings of the 1984 International Symposium on Multiple-Valued Logic*,Winnipeg, Canada, pp. 148-153, 1984.*Google Scholar - [49.]L.A. Zadeh,
*Outline of a Computational Approach to Meaning and Knowledge Representation Based on a Concept of a Generalized Assignment Statement*, Proc. of the Int. Seminar on Artificial Intelligence and Man-Machine Systems, eds. M. Thoma and A. Wyner, Heidelberg: Springer-Verlag, pp. 198-211, 1986.Google Scholar - [50.]L.A. Zadeh,
*Fuzzy Logic, Neural Networks and Soft Computing*,Communications of the ACM*,37(3), pp. 77-84, 1994.*Google Scholar - [51.]L.A. Zadeh, Fuzzy Logic and the Calculi of Fuzzy Rules and Fuzzy Graphs: A Precis,
*Multiple Valued Logic*1, Gordon and Breach Science Publishers, pp. 1 - 38, 1996.Google Scholar - [52.]L.A. Zadeh,
*Fuzzy Logic = Computing with Words*,EEE Transactions on Fuzzy Systems*,Vol. 4, pp. 103-111, 1996.*Google Scholar - [53.]L.A. Zadeh,
*Toward a Theory of Fuzzy Information Granulation and its Centrality in Human Reasoning and Fuzzy Logic*Fuzzy Sets and Systems 90*,pp. 111-127, 1997.*Google Scholar - [54.]L.A. Zadeh,
*Maximizing Sets and Fuzzy Markoff Algorithms*, IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Reviews, Vol. 28, pp. 9 - 15, 1998.CrossRefGoogle Scholar - [55.]L.A. Zadeh, A New Direction in AI — Toward a Computational Theory of Perceptions,AI Magazine, Vol 22, No. 1, pp. 73 - 84, 2001.Google Scholar