From computing with numbers to computing with words - From manipulation of measurements to manipulation of perceptions
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 that are capable of performing billions of computations per second; we have constructed telescopes that can explore the far reaches of the universe; and we can date the age of rocks that are millions of years old. But alongside the brilliant successes stand conspicuous underachievements and outright failures. We cannot build robots which can move with the agility of animals or humans; we cannot automate driving in heavy traffic; we cannot translate from one language to another at the level of a human interpreter; we cannot create programs which can summarize non-trivial stories; our ability to model the behavior of economic systems leaves much to be desired; and we cannot build machines that can compete with children in the performance of a wide variety of physical and cognitive tasks. It may be argued that underlying the underachievements and failures is the unavailability of a methodology for reasoning and computing with perceptions rather than measurements. An outline of such a methodology – referred to as a computational theory of perceptions – is presented in this paper. The computational theory of perceptions, or CTP for short, is based on the methodology of computing with words (CW). In CTP, words play the role of labels of perceptions and, more generally, perceptions are expressed as propositions in a natural language. CW-based techniques are employed to translate propositions expressed in a natural language into what is called the Generalized Constraint Language (GCL). In this language, the meaning of a proposition is expressed as a generalized constraint, X isr R, where X is the constrained variable, R is the constraining relation and isr is a variable copula in which r is a variable whose value defines the way in which R constrains X. Among the basic types of constraints are: possibilistic, veristic, probabilistic, random set, Pawlak set, fuzzy graph and usuality. The wide variety of constraints in GCL makes GCL a much more expressive language than the language of predicate logic. In CW, the initial and terminal data sets, IDS and TDS, are assumed to consist of propositions expressed in a natural language. These propositions are translated, respectively, into antecedent and consequent constraints. Consequent constraints are derived from antecedent constraints through the use of rules of constraint propagation. The principal constraint propagation rule is the generalized extension principle. The derived constraints are retranslated into a natural language, yielding the terminal data set (TDS). The rules of constraint propagation in CW coincide with the rules of inference in fuzzy logic. A basic problem in CW is that of explicitation of X, R and r in a generalized constraint, X isr R, which represents the meaning of a proposition, p, in a natural language. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers; and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW and CTP. At this juncture, the computational theory of perceptions – which is based on CW – is in its initial stages of development. In time, it may come to play an important role in the conception, design and utilization of information/intelligent systems. The role model for CW and CTP is the human mind.
- Berenji H.R. (1994): Fuzzy reinforcement learning and dynamic programming, In: Fuzzy Logic in Artificial Intelligence (A.L. Ralescu, Ed.). — Proc. IJCAI’93 Workshop, Berlin: Springer-Verlag, pp. 1–9.
- Black M. (1963): Reasoning with loose concepts. — Dialog 2, pp. 1–12.
- Bosch P. (1978): Vagueness, ambiguity and all the rest, In: Sprachstruktur, Individuum und Gesselschaft (M. Van de Velde and W. Vandeweghe, Eds.).—Tubingen: Niemeyer.
- Bowen J., Lai R. and Bahler D. (1992a): Fuzzy semantics and fuzzy constraint networks. — Proc. 1st IEEE Conf. Fuzzy Systems, San Francisco, pp. 1009–1016.
- Bowen J., Lai R. and Bahler D. (1992b): Lexical imprecision in fuzzy constraint networks. — Proc. Nat. Conf. Artificial Intelligence, pp. 616–620.
- Cresswell M.J. (1973): Logic and Languages. — London: Methuen.
- Dubois D., Fargier H. and Prade H. (1993): The calculus of fuzzy restrictions as a basis for flexible constraint satisfaction. — Proc. 2nd IEEE Int. Conf. Fuzzy Systems, San Francisco, pp. 1131–1136.
- Dubois D., Fargier H. and Prade H. (1994): Propagation and satisfaction of flexible constraints, In: Fuzzy Sets, Neural Networks, and Soft Computing (R.R. Yager, L.A. Zadeh, Eds.).—New York: Von Nostrand Reinhold, pp. 166–187.
- Dubois D., Fargier H. and Prade H. (1996): Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty. — Appl. Intell. J., Vol. 6, No. 4, pp. 287–309.
- Freuder E.C. and Snow P. (1990): Improved relaxation and search methods for approximate constraint satisfaction with a maximin criterion. — Proc. 8th Biennial Conf. Canadian Society for Computational Studies of Intelligence, Ontario, pp. 227–230.
- Goguen J.A. (1969): The logic of inexact concepts. — Synthese 19, pp. 325–373.
- Hobbs J.R. (1978): Making computation sense of Montague’s intensional logic. —Artif. Intell., Vol. 9, pp. 287–306.
- Katai O., Matsubara S., Masuichi H., Ida M., et al. (1992): Synergetic computation for constraint satisfaction problems involving continuous and fuzzy variables by using Occam, In: Transputer/Occam, (S. Noguchi and H. Umeo, Eds.). — Proc. 4th Transputer/Occam Int. Conf., Amsterdam: IOS Press, pp. 146–160.
- Kaufmann A. and Gupta M.M. (1985): Introduction to Fuzzy Arithmetic: Theory and Applications. — New York: Von Nostrand.
- Klir G. and Yuan B. (1995): Fuzzy Sets and Fuzzy Logic.—New Jersey: Prentice Hall.
- Lano K. (1991): A constraint-based fuzzy inference system. — Proc. 5th Portuguese Conf. Artificial Intelligence, EPIA’91 (P. Barahona, L.M. Pereira, and A. Porto, Eds.), Berlin: Springer-Verlag, pp. 45–59.
- Lodwick W.A. (1990): Analysis of structure in fuzzy linear programs. —Fuzzy Sets Syst., Vol. 38, No. 1, pp. 15–26.
- Mamdani E.H. and Gaines B.R., Eds. (1981): Fuzzy Reasoning and its Applications. —London.
- Mares M. (1994): Computation Over Fuzzy Quantities.—Boca Raton: CRC Press.
- Novak V. (1991): Fuzzy logic, fuzzy sets, and natural languages. —Int. J. General Syst., Vol. 20, No. 1, pp. 83–97.
- Novak V., Ramik M., Cerny M. and Nekola J., Eds. (1992): Fuzzy Approach to Reasoning and Decision-Making. — Boston: Kluwer.
- Oshan M.S., Saad O.M. and Hassan A.G. (1995): On the solution of fuzzy multiobjective integer linear programming problems with a parametric study. — Adv. Modell. Anal. A, Vol. 24, No. 2, pp. 49–64.
- Partee B. (1976): Montague Grammar.—New York: Academic Press.
- Pedrycz W. and Gomide F. (1998): Introduction to Fuzzy Sets. — Cambridge: MIT Press, pp. 38–40.
- Qi G. and Friedrich G. (1992): Extending constraint satisfaction problem solving in structural design.—5th Int. Conf. Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, IEA/AIE-92 (F. Belli and F.J. Radermacher, Eds.), Berlin: Springer-Verlag, pp. 341– 350.
- Rasiowa H. and Marek M. (1989): On reaching consensus by groups of intelligent agents, In: Methodologies for Intelligent Systems (Z.W. Ras, Ed.). — Amsterdam: North- Holland, pp. 234–243.
- Rosenfeld A., Hummel R.A. and Zucker S.W. (1976): Scene labeling by relaxation operations.—IEEE Trans. Syst. Man Cybern., Vol. 6, pp. 420–433.
- Sakawa M., Sawada K. and Inuiguchi M. (1995): A fuzzy satisficing method for large-scale linear programming problems with block angular structure.—Europ. J. Oper. Res., Vol. 81, No. 2, pp. 399–409.
- Shafer G. (1976): A Mathematical Theory of Evidence. — Princeton: Princeton University Press.
- Tong S.C. (1994): Interval number and fuzzy number linear programming. — Adv. Modell. Anal. A, Vol. 20, No. 2, pp. 51–56.
- Vallée R. (1995): Cognition et Systeme. — Paris: l’Interdisciplinaire Systeme(s).
- Yager R.R. (1989): Some extensions of constraint propagation of label sets.—Int. J. Approx. Reas., Vol. 3, pp. 417–435.
- Zadeh L.A. (1961): From circuit theory to system theory. — Proc. IRE, Vol. 50, pp. 856–865.
- Zadeh L.A. (1965): Fuzzy sets. — Inf. Control, Vol. 8, pp. 338– 353.
- Zadeh L.A. (1968): Probability measures of fuzzy events. — J. Math. Anal. Appl., Vol. 23, pp. 421–427.
- Zadeh L.A. (1972): A fuzzy-set-theoretic interpretation of linguistic hedges. — J. Cybern., Vol. 2, pp. 4–34.
- Zadeh L.A. (1973): Outline of a new approach to the analysis of complex system and decision processes. — IEEE Trans. Syst. Man Cybern., Vol. SMC–3, pp. 28–44.
- Zadeh L.A. (1974): On the analysis of large scale systems. — Systems Approaches and Environment Problems (H. Gottinger, Ed.), Gottingen: Vandenhoeck and Ruprecht, pp. 23–37.
- Zadeh L.A. (1975a): Calculus of fuzzy restrictions, In: Fuzzy Sets and Their Applications to Cognitive and Decision Processes, (L.A. Zadeh, K.S. Fu, M. Shimura, Eds.). — New York: Academic Press, pp. 1–39.
- Zadeh L.A. (1975b): The concept of a linguistic variable and its application to approximate reasoning. — Part I: Inf. Sci., Vol. 8, pp. 199–249; Part II: Inf. Sci., Vol. 8, pp. 301–357; Part III: Inf. Sci., Vol. 9, pp. 43–80.
- Zadeh L.A. (1976): A fuzzy-algorithmic approach to the definition of complex or imprecise concepts. — Int. J. Man- Mach. Stud., Vol. 8, pp. 249–291.
- Zadeh L.A. (1978a): Fuzzy sets as a basis for a theory of possibility. — Fuzzy Sets Syst., Vol. 1, pp. 3–28.
- Zadeh L.A. (1978b): PRUF – A meaning representation language for natural languages. — Int. J. Man-Mach. Stud., Vol. 10, pp. 395–460.
- Zadeh L.A. (1979a): Fuzzy sets and information granularity, In: Advances in Fuzzy Set Theory and Applications (M. Gupta, R.Ragade and R. Yager, Eds.). — Amsterdam: North-Holland, pp. 3–18.
- Zadeh L.A. (1979b): A theory of approximate reasoning. — Mach. Intell., Vol. 9 (J. Hayes, D. Michie and L.I. Mikulich, Eds.), New York: Halstead Press, pp. 149–194.
- Zadeh L.A. (1981): Test-score semantics for natural languages and meaning representation via PRUF. — Empirical Semantics (B. Rieger, W. Germany, Eds.), Brockmeyer, pp. 281–349. Also Technical Report Memorandum 246, AI Center, SRI International, Menlo Park, CA.
- Zadeh L.A. (1982): Test-score semantics for natural languages. —Proc. 9-th Int. Conf. Computational Linguistics, Prague, pp. 425–430.
- Zadeh L.A. (1984): Syllogistic reasoning in fuzzy logic and its application to reasoning with dispositions. — Proc. Int. Symp. Multiple-Valued Logic, Winnipeg, Canada, pp. 148–153.
- Zadeh L.A. (1986): Outline of a computational approach to meaning and knowledge representation based on a concept of a generalized assignment statement. — Proc. Int. Seminar on Artificial Intelligence and Man-Machine Systems (M. Thoma and A. Wyner, Eds.), Heidelberg: Springer- Verlag, pp. 198–211.
- Zadeh L.A. (1994): Fuzzy logic, neural networks and soft computing. —Comm. ACM, Vol. 37, No. 3, pp. 77–84.
- Zadeh L.A. (1996a): 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.
- Zadeh L.A. (1996b): Fuzzy logic = Computing with words. — IEEE Trans. Fuzzy Syst., Vol. 4, pp. 103–111.
- Zadeh L.A. (1997): Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. — Fuzzy Sets Syst., Vol. 90, pp. 111–127.
- Zadeh L.A. (1998): Maximizing Sets and Fuzzy Markoff Algorithms. — IEEE Trans. Syst. Man Cybern., Part C: Appli. Rev., Vol. 28, pp. 9–15.