24_Maximizing Sets and Fuzzy Markoff Algorithms-1998

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART C: APPLICATIONS AND REVIEWS, VOL. 28, NO. 1, FEBRUARY 1998 9

Maximizing Sets and Fuzzy Markoff AlgorithmsLot A. Zadeh, Life Fellow, IEEE

Abstract A fuzzy algorithm is an ordered set of fuzzy instruc-tions that upon execution yield an approximate solution to a givenproblem.

Two unrelated aspects of fuzzy algorithms are considered inthis paper. The rst is concerned with the problem of maxi-mization of a reward function. It is argued that the conventionalnotion of a maximizing value for a function is not sufcientlyinformative and that a more useful notion is that of a maximizingset. Essentially, a maximizing set serves to provide informationnot only concerning the point or points at which a function ismaximized, but also about the extent to which the values of thereward function approximate to its supremum at other points inits range.

The second is concerned with the formalization of the notionof a fuzzy algorithm. In this connection, the notion of a fuzzyMarkoff algorithm is introduced and illustrated by an example. It

is shown that the generation of strings by a fuzzy algorithm bearsa resemblance to a birth-and-death process and that the executionof the algorithm terminates when no more live strings are left.

I. PREAMBLE

T HIS PAPER is based on a report entitled On FuzzyAlgorithms, [14] or FA for short, which was writtenin 1972 but not submitted for publication at that time.

The report dealt with two issues that have grown in im-portance in the intervening years. The rst issue relates tothe concept of maximization (or minimization). The issuearises because in many real-world problems what matters isnot only the value of , say , at which a function ismaximized, but also the robustness of the maximizing value.This suggests that the concept of a maximizing value bereplaced by the concept of a maximizing fuzzy set or, moresimply, a maximizing set. This is the issue that is addressedin the rst part of FA.

The second part of FA deals with the concept of a fuzzyalgorithm. In an informal way, the concept of a fuzzy algo-rithm was introduced in [13]. One of the main objectives of FAis to formalize the concept of a fuzzy algorithm, employingfor this purpose the concept of a Markoff algorithm, ratherthan that of a Turing machine [1], [2], [11]. An interestingaspect of fuzzy Markoff algorithms relates to the fact that, asin the case of genetic algorithms [6], [9], [10], the result of execution at each stage is a set of strings, rather than a singlestring. However, in the case of fuzzy Markoff algorithms, theset of strings is fuzzy, rather than crisp. Fuzzication of the

Manuscript received June 30, 1994; revised October 22, 1997. This work was supported in part by the Naval Electronic Systems Command ContractN00039-71C-0255.

The author is with the Graduate School and Berkeley Initiative in SoftComputing (BISC), Computer Science Division, Department of ElectricalEngineering and Computer Science, University of California, Berkeley, CA94720-1776 USA (e-mail: [email protected]).

Publisher Item Identier S 1094-6977(98)01523-5.

concept of a genetic algorithm would yield the same result.A natural way of fuzzifying and hence generalizing theconcept of a genetic algorithm is to assume that the tnessfunction is both fuzzy and granular, rather than crisp. Such anassumption would be closer to reality.

The importance of the concept of a fuzzy algorithm stemsfrom the fact that fuzzy algorithms mimic the ways in whichhumans make decisions and act in the presence of imprecision,uncertainty, and partial truth. But what is even more importantis that fuzzy algorithms provide a more realistic way of den-ing many basic concepts that are intrinsically fuzzy, e.g., theconcepts of smoothness, ovalness, stationarity, independence,stability, and causality. This is the issue that was addressed insubsequent papers and especially in [15] and [16], in whichthe basic concepts of a linguistic variable, fuzzy ifthen rule,and fuzzy questionnaire were introduced. In particular, someof the ideas introduced in [16] relate to later theories on roughsets [7] and machine learning [8].

II. INTRODUCTION

Roughly speaking, a fuzzy algorithm [13] is an ordered setof fuzzy instructions that upon execution yield an approximatesolution to a given problem. As in the case of nonfuzzyalgorithms, a fuzzy algorithm is usually expected to be capableof providing an approximate solution to any problem in aspecied class of problems, rather than to a single problem.

Simple examples of fuzzy algorithms that occur in everydayexperience are cooking recipes, instructions for parking a car,instructions for tying a knot, etc. As a more concrete example,consider the following simple control problem. Suppose thatwe wish to transfer a blindfolded subject from an initialposition in a room with no obstacles to a nal position .Furthermore, suppose that the fuzzy instructions are limited tothe following set: 1) turn counterclockwise by approximately

degrees, with being a multiple of, say, 15; 2) take a step;3) take a small step; and 4) take a very small step.

Under these assumptions, a simple fuzzy algorithm for guid-ing from to may be stated as follows. (It is understoodthat after executes an instruction, his or her new position andorientation are observed fuzzily by the experimenter, who thenchooses that instruction from the algorithm that most closelyts the last observation). Basically, the algorithm mimics whata human might do to reach the target.

1) If is facing , then go to 2, else go to 5.2) If is close to , go to 7, else go to 6.3) If is very close to , go to 8, else go to 7.4) If is very very close to , then stop.5) Ask to turn counterclockwise by an amount needed

to make him or her face . Go to 1.

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ZADEH: MAXIMIZING SETS AND FUZZY MARKOFF ALGORITHMS 11

Fig. 1. Maximizing set for a positive-denite reward function.

Fig. 2. Maximizing set for a nondenite reward function.

Case 3: A reward function is negative-denite if it isnonpositive for all in , that is, . In this case, (2)is applied to a translate of , expressed by

(5)

yielding the denition

(6)

Taken together, (2), (4), and (6) constitute a general def-inition of the maximizing set for a reward function that isbounded both from above and from below.

The following properties of the maximizing set are imme-diate consequences of the above denition.

1) The maximizing set for is unique.2) The maximizing set for is invariant under linear

scaling.In other words

(7)

where is any real constant.3) The maximizing set for is not invariant under trans-

lation.

Fig. 3. Maximizing set for a negative-denite reward function.

Thus, if and constant , then in general

More specically, for any xed , the dependence of on is of the form

where and are constants. This implies that, asthe magnitude of increases, the membership functiontends to unity.

4) For , the maximizing set is convex if and onlyif the reward function is quasiconcave.By denition, a fuzzy subset of (linear vectorspace of -tuples of real numbers) is convex [12] if all

of its level sets are convex, that is, if the sets

(8)

are convex in . Equivalently, is convex if satises

(9)

for all in and all in .Also by denition, a function on is quasiconcaveif all of the level sets

are convex for all nite in .Now the denition of in terms of implies that if

is quasiconcave, so is . This in turn implies the convexityof . Thus, if is quasiconcave, then is convex andvice-versa.

Example: The maximizing sets for the reward functionsshown in Figs. 13 are convex.

Comment: It should be observed that a concave rewardfunction can be transformed into a quasiconcave function thatis bounded both from above and from below by an order-preserving transformation.


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ZADEH: MAXIMIZING SETS AND FUZZY MARKOFF ALGORITHMS 15

the dead in a cemetery with (32) representing the addition of the newly deceased to that population.

As in a birth-and-death process, the population of livestrings can grow explosively if the productions aresuch that each execution of results in signicantly morebirths than deaths. This rather interesting aspect of fuzzyMarkoff algorithms is not present in conventional Markoff algorithms.

In the foregoing discussion, we have restricted ourselvesto formulating what appears to be a natural extension of thenotion of a Markoff algorithm. Exploration of the propertiesof such algorithms will be pursued in subsequent papers onthis subject.

REFERENCES

[1] A. T. Berztiss, Data Structures . New York: McGraw-Hill, 1971.[2] S. S. L. Chang and L. A. Zadeh, Fuzzy mapping and control, IEEE

Trans. Syst., Man, Cybern. , vol. SMC-2, pp. 3034, 1972.[3] Z. Chi, H. Yan, and T. Pham, Fuzzy Algorithms: With Applications to Im-

age Processing and Pattern Recognition . Singapore: World Scientic,1996.

[4] H. Galler and A. Perlis, A View of Programming Languages . Reading,MA: Addison-Wesley, 1970.

[5] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Ma-chine Learning . Reading, MA: Addison-Wesley, 1989.

[6] C. Z. Janikow, A genetic algorithm method for optimizing fuzzydecision trees, Inf. Sci. , vol. 90, pp. 122, 1995.

[7] R. Pawlak, Rough SetsTheoretical Aspects of Reasoning About Data .Boston, MA: Kluwer, 1990.

[8] J. R. Quinlan, Programs for Machine Learning . San Mateo, CA:Morgan Kaufmann, 1993.

[9] E. Sanchez, T. Shibata, and L. A. Zadeh, Genetic Algorithms and Fuzzy Logic Systems Soft Computing Perspectives . Singapore: WorldScientic, 1997.

[10] S. K. Pal and P. P. Wang, Genetic Algorithms for Pattern Recognition .Boca Raton, FL: CRC, 1996.

[11] E. Santos, Fuzzy algorithms, Inf. Control , vol. 17, pp. 326339, 1970.[12] L. A. Zadeh, Fuzzy sets, Inf. Control , vol. 8, pp. 338353, 1965.[13] , Fuzzy algorithms, Inf. Control , vol. 12. pp. 99102, 1968.[14] , On fuzzy algorithms, , Elec. Res. Lab., Univ. California,

Berkeley, Memo. ERL-M325, 1972.[15] , Outline of a new approach to the analysis of complex systems

and decision process, IEEE Trans. Syst., Man, Cybern. , vol. SMC-3,pp. 2844, 1973.

[16] , A fuzzy-algorithmic approach to the denition of complex orimprecise concepts, Int. J. Man-Mach. Stud. , vol. 8, pp. 249291, 1976.

[17] , Fuzzy logic, neural networks and soft computing, Commun. ACM , vol. 37, no. 3, pp. 7784, 1994.

[18] , Soft computing and fuzzy logic, IEEE Software , vol. 11, nos.16, pp. 4856, 1994.

[19] , The roles of soft computing and fuzzy logic in the conception,design and deployment of intelligent systems, Br. Telecom J. , vol. 14,no. 4, 1996.

[20] , Toward a theory of fuzzy information granulation and itscentrality in human reasoning and fuzzy logic, Fuzzy Sets Syst. , vol.90, pp. 111128, 1997.

Lot A. Zadeh (S45A47M47SM56F58LF87), for a photographand biography, see this issue, p. 8.

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24_Maximizing Sets and Fuzzy Markoff Algorithms-1998