Measuring the Uncertainty of Human Reasoning

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1 America Joural of Applied Mathematics ad Statistics, 04, Vol., No., -6 Available olie at Sciece ad Educatio Publishig DOI:0.69/ajams--- Measurig the Ucertaity of Huma Reasoig Michael Gr. Voskoglou * School of Techological Applicatios, Graduate Techological Educatioal Istitute (T. E. I.) of Wester Greece, Patras, Greece *Correspodig author: mvosk@hol.gr Received November 5, 0; Revised December, 0; Accepted December 7, 0 Abstract Huma reasoig is characterized by a degree of fuzziess ad ucertaity. I the preset paper we develop a fuzzy model for a better descriptio of the reasoig process ad we use the fuzzy systems total possibilistic ucertaity as well as the classical Shao s etropy (properly modified for use i fuzzy eviromets) i measurig the idividuals reasoig skills. Classroom experimets are also provided illustratig our results i practice. Keywords: reasoig, fuzzy sets, measures of ucertaity Cite This Article: Michael Gr. Voskoglou, Measurig the Ucertaity of Huma Reasoig. America Joural of Applied Mathematics ad Statistics, o. (04): -6. doi: 0.69/ajams---.. Itroductio Coscious directio of attetio towards a exteral object causes the object to be received by mid i sequece to perceptio, experiece, feelig, uderstadig, explaiig, kowig, ad fially actig for meaigful descriptio ad aalytical solutio. Reasoig is the most importat huma brai operatio that leads to creative methodologies, algorithms ad deductios givig way to sustaiable research ad developmet. The mai stages of huma reasoig for reachig to a solutio of a existig problem (ot ecessarily mathematical) ivolve imagiatio, visualizatio ad idea geeratios ([], p. 40, or [4], sectio 4). For ay exteral object, whether it exists materialistically or ot, huma beigs try to imagie its properties i their mids. This gives them the power of iitializig their idividual thikig domai with whole freedom i ay directio. Imagiatio icludes the settig up of a suitable hypothesis or a set of logical rules for the problem at had. The visualizatio stage is to defed the represetative hypothesis ad logical propositios. Humas typically use a variety of represetatios to defed their hypotheses icludig algorithms, graphs, diagrams, charts, figures, etc. I particular, the geometric cofiguratio of the objects appearig through imagiatio is the most commo amog these represetatios. I fact, after a object comes ito existece vaguely i mid, it is ecessary to kow its shape, which is related to geometry. It is essetial that the geometric cofiguratio of the pheomeo must be visualized i mid i some way, eve though it may be a simplificatio uder a set of assumptios. O the basis of their hypotheses the idividuals geerate relevat ideas. The ideas begi to crystallize ad they are expressed verbally by a ative laguage to other idividuals to get their criticisms, commets, suggestios ad support for the bettermet of the metal thikig ad scietific achievemet. Fially, all the coclusios must be expressed i a laguage, which ca the be coverted ito uiversally used symbolic logic based o the priciples of mathematics. We emphasize that whatever are the meas of reasoig the scietific argumets are expressed verbally prior to ay symbolic ad mathematical abstractios. It is possible to state that with Newtoia classical physics sciece etered almost etirely a determiistic world, where ucertaity was ot eve accouted amog the scietific kowledge. However, owadays ucertaity appears i almost all braches of sciece ad may scietific determiistic foudatios of the past became ucertai with fuzzy igrediets. Amog such coceptios are quatum physics, fractal geometry, chaos theory ad fuzzy iferece systems. With the advacemet of umerical ucertaity techiques, such as probability, statistics ad stochastic priciples, scietific progress i quatitative aspects had a rapid developmet, but still leavig aside the qualitative sources of kowledge ad iformatio, which ca be tackled by the fuzzy logic priciples oly. Zadeh, the istructor of fuzzy logic through the fuzzy sets theory [], states: As the complexity of a system icreases, our ability to make precise ad yet sigificat statemets about its behavior dimiishes, util a threshold is reached beyod which precisio ad sigificace (or relevace) become almost mutually exclusive characteristics (see [4]). We recall here that i the laguage of maagemet a system is uderstood to be a set of iteractig or idepedet compoets formig a itegrated whole. A system ca be viewed as a bouded trasformatio, i.e. as a process or a collectio of processes that trasform iputs ito outputs with the very broad meaig of the term. For example, a output of a passegers bus is the movemet of people from departure to destiatio (for more details see [9]). I [9] we have used priciples of fuzzy logic to develop a geeral model represetig several processes i a

2 America Joural of Applied Mathematics ad Statistics system s operatio characterized by a degree of vagueess ad/or ucertaity. Also i [,] we have adapted this model i represetig the process of huma reasoig ad we have used the cetroid defuzzificatio techique for measurig the reasoig skills of a group of idividuals. The amout of iformatio obtaied by a actio ca be measured by the reductio of ucertaity resultig from this actio. Accordigly a system s ucertaity is coected to its capacity i obtaiig relevat iformatio. Therefore a measure of ucertaity could be adopted as a measure of a system s effectiveess i solvig related problems. Based o this fact, we have used i earlier papers the total possibilistic ucertaity, as well as the Shao s etropy (total probabilistic ucertaity) - properly adapted for use i a fuzzy eviromet - for measurig the effectiveess of several systems i the areas of Educatio, of Artificial Itelligece ad of Maagemet (e.g. Problem Solvig, Learig, Case- Based Reasoig, evaluatio of the fuzzy data of a market s research, etc); see the book [7] ad its refereces ad [8]. I the preset paper we shall use the same techiques i measurig the ucertaity of huma reasoig ad we shall give examples illustratig our results i practice.. The Fuzzy Model The stages of the reasoig process preseted i our itroductio are helpful i uderstadig the idividuals ideal behaviour durig the process. However, thigs i real situatios are usually ot happeig like that, sice huma cogitio utilizes i geeral cocepts that are iheretly graded ad therefore fuzzy. This fact gave us the impulsio to itroduce priciples of fuzzy sets theory i order to describe i a more effective way the process of huma reasoig [,]. For geeral facts o fuzzy sets we refer freely to the book []. For the developmet of our fuzzy model for the reasoig process we have cosidered a group of people,, workig (each oe idividually) o the same problem. Deote by S, S ad S respectively the stages of imagiatio, visualizatio ad ideas geeratio of the reasoig process. Deote also by a, b, c, d, ad e the liguistic labels of very low, low, itermediate, high ad very high success respectively of a perso i each of the S i s. Set U = {a, b, c, d, e}. We are goig to attach to each stage S i of the reasoig process, i=,,, a fuzzy subset, A i of U. For this, if ia, ib, ic, id ad ie deote the umber of idividuals that faced very low, low, itermediate, high ad very high success at stage S i respectively, i=,,, we defie the membership fuctio m Ai for each x i U, as follows: 4, if < ix 5 4 0,75, if < ix 5 5 ma ( x) = 0,5, i if < ix 5 5 0, 5, if < ix 5 5 0, if 0 ix 5 I fact, if oe wated to apply probabilistic stadards i measurig the degree of the idividuals success at each stage of the process, the he/she should use the relative frequecies ix. Nevertheless, such a actio would be highly questioable, sice the ix s are obtaied with respect to the liguist labels of U, which are fuzzy expressios by themselves. Therefore the applicatio of a fuzzy approach by usig membership degrees istead of probabilities seems to be more suitable for this case. But, as it is well kow, the membership fuctio eeded for such purposes is usually defied empirically i terms of logical or/ad statistical data. I our case the above defiitio of m Ai seems to be compatible with the commo logic. The the fuzzy subset A i of U correspodig to S i has the form: A i = {( x,m Ai ( x) ) : x U},i =,,. I order to represet all possible idividuals profiles (overall states) durig the reasoig process we cosider a fuzzy relatio, say R, i U of the form: ( ( )) ( ) R= { sm, R s : s= x, y, z U }. For determiig properly the membership fuctio m R we give the followig defiitio: A profile s=(x, y, z), with x, y, z i U, is said to be well ordered if x correspods to a degree of success equal or greater tha y ad y correspods to a degree of success equal or greater tha z. For example, (c, c, a) is a well ordered profile, while (b, a, c) is ot. We defie ow the membership degree of a profile s to be m R (s) = m A (x)m A (y)m A (z) if s is well ordered, ad 0 otherwise. I fact, if for example the profile (b, a, c) possessed a ozero membership degree, how it could be possible for a perso, who has failed at the visualizatio stage, to perform satisfactorily at the stage of the ideas geeratio? Next, for reasos of brevity, we shall write m s istead of m R (s). The the probability p s of the profile s is defied ms i a way aalogous to crisp data, i.e. by Ps =. ms s U ms We defie also the possibility r s of s to be r s = max{ ms } where max{m s } deotes the maximal value of m s, for all s i U. I other words the possibility of s expresses the relative membership degree of s with respect to max{m s }. Assume further that oe wats to study the combied results of behaviour of k differet groups of people, k, durig the reasoig process. For this, we itroduce the fuzzy variables A (t), A (t) ad A (t) with t=,,, k. The values of these variables represet fuzzy subsets of U correspodig to the stages of the reasoig process for each of the k groups; e.g. A () represets the fuzzy subset of U correspodig to the stage of imagiatio for the secod group (t=). Obviously, i order to measure the degree of evidece of the combied results of the k groups, it is ecessary to defie the probability p(s) ad the possibility r(s) of each profile s with respect to the membership degrees of s for

3 America Joural of Applied Mathematics ad Statistics all groups. For this reaso we itroduce the pseudofrequecies f ( s) = ms () t ad we defie the k t= probability ad possibility of a profile s by f() s f() s p( s) = ad r( s) = respectively, f() s max{ f( s)} s U where max{f(s)} deotes the maximal pseudo-frequecy. The same method could be applied whe oe wats to study the combied results of behaviour of a group durig k differet reasoig situatios. The above model gives, through the calculatio of probabilities ad possibilities of all idividuals profiles, a quatitative/qualitative view of their realistic performace at all stages of the reasoig process.. Measures of Ucertaity Withi the domai of possibility theory ucertaity cosists of strife (or discord), which expresses coflicts amog the various sets of alteratives, ad o-specificity (or imprecisio), which idicates that some alteratives are left uspecified, i.e. it expresses coflicts amog the sizes (cardialities) of the various sets of alteratives ([]; p. 8). Strife is measured by the fuctio ST(r) o the ordered possibility distributio r: r = r. r r+ of a group of a system s etities defied by m i ST ( r) = ( ri ri+ ) log log i, while ospecificity is measured by the fuctio i= r j j= m N( r) = ( ri ri+ ) log i. log i = The sum T( r) = ST( r) + N( r) is a measure of the total possibilistic ucertaity for ordered possibility distributios. The lower is the value of T(r), which meas greater reductio of the iitially existig i the system ucertaity, the better the system s performace. Aother fuzzy measure for assessig a system s performace is the well kow from classical probability ad iformatio theory Shao s etropy [6]. For use i a fuzzy eviromet, this measure is expressed i terms of the Dempster-Shafer mathematical theory of evidece i the form: H = msi ms if ([], p. 0). s = I the above formula deotes the total umber of the system s etities ivolved i the correspodig process. The sum is divided by l (the atural logarithm of ) i order to be ormalized. Thus H takes values i the real iterval [0, ]. The value of H measures the system s total probabilistic ucertaity ad the associated to it iformatio. Similarly with the total possibilistic ucertaity, the lower is the fial value of H, the better the system s performace. A advatage of adoptig H as a measure of ucertaity istead of T(r) is that H is calculated directly from the membership degrees of all profiles s without beig ecessary to calculate their probabilities p(s). I cotrast, the calculatio of T(r) presupposes the calculatio of the possibilities r(s ) of all profiles first. However, accordig to Shackle [5] ad may other researchers after him, huma reasoig ca be formalized more adequately by possibility rather, tha by probability theory. But, as we have see i the previous sectio, the possibility is a kid of relative probability. I other words, the philosophy of possibility is ot exactly the same with that of probability theory. Therefore, o comparig the effectiveess of two systems by these two measures, oe may fid o compatible results i boudary cases, where the systems performaces are almost the same. 4. Classroom Experimets I order to illustrate the use of our fuzzy model i practice we performed recetly the followig two experimets at the Graduate Techological Educatioal Istitute (T. E. I.) of Patras, Greece [,]. I the first experimet our subjects were 5 studets of the School of Techological Applicatios, i.e. future egieers. A few days before the experimet a aalysis of the scietific reasoig process (see itroductio) was preseted to the studets i a two hours lecture, followed by a umber of suitable examples. The followig two problems with their relevat aalyses were icluded amog these examples: Problem : We wat to costruct a chael to ru water by foldig across its loger side the two edges of a orthogoal metallic leaf havig sides of legth 0 cm ad cm, i such a way that they will be perpedicular to the other parts of the leaf. Assumig that the flow of the water is costat, how we ca ru the maximum possible quatity of the water? Aalysis of the problem: IMAGINATION: The basic thig to realize is that the quatity of water to ru through the chael depeds o the area of the vertical cut of the chael. VISUALIZATION (geometric cofiguratio): Foldig the two edges of the metallic leaf by legth x across its loger side the vertical cut of the costructed chael is a orthogoal with sides x ad -x (Figure ). Figure. The vertical cut of the chael IDEAS GENERATION: The area of the orthogoal has to be maximized. The above idea leads to the followig mathematical maipulatio: The area is equal to E( x) = x( x) = x x. Takig the derivative E (x) the equatio E (x) = -4x = 0 gives that x = 8 cm. But E (x) = - 4 < 0, therefore E ( 8) = 8cm is the maximum possible quatity of water to ru through the chael.

4 4 America Joural of Applied Mathematics ad Statistics Problem : The rate of icrease of the populatio of a coutry is aalogous to the umber of its ihabitats. If the populatio is doubled i 50 years, i how may years it will be tripled? Aalysis of the problem: IMAGINATION: The key cocepts ivolved i the statemet of this problem are the aalogy ad the rate of icrease of the populatio. Therefore the crucial actio for the solutio of the problem is to establish the relatio coectig these two cocepts. VISUALIZATION: The populatio P of the coutry is obviously a fuctio of the time t, say P = P(t). I observig the icrease of the populatio we must cosider a startig poit, where t = 0. IDEAS GENERATION: The rate of icrease of the populatio is expressed by the derivative P (t) ad the existig aalogy is expressed by P (t) = k P(t), with k a o egative iteger. Therefore the solutio of the problem is based o the solutio of the above differetial equatio. This suggests the followig mathematical maipulatio: dp() t Separatig the variables we ca write = kdt, or Pt () kt kt Thus P ( t) = kt + c = e + = ( ) dp() t = k dt. Pt () kt or P( t) = c e. For 0 ad therefore we get that l l l l c l c e, P 0 = P = c t = we fid that ( ) 0 ( ) 0 P t = Pe kt () Further, accordig to the problem s statemet, we have that 50k l P( 50) = P0, or Pe 0 = P0 50k = l, or k =. 50 t l = Pe If the populatio will be tripled after x years, the we ll t l t l P0 = P x = Pe 50 0, or = e 50, which gives Therefore () fially gives that P( t) have ( ) l that x = years. l I performig the experimet the followig problem was give for solutio to the studets (time allowed 0 miutes): Problem : Amog all cyliders havig a total surface of 80π m, which oe has the maximal volume? Before startig the experimet we gave the proper istructios to studets emphasizig, amog the others, that we were iterested for all their efforts (successful or ot) durig the reasoig process, ad therefore they must keep records o their papers for all of them, at all stages of the process. I particular, we asked them to provide a aalysis of the solutio of the give problem aalogous to the aalyses preseted for the above two examples. Rakig the studets papers by usig the scale applied i sectio (before the costructio of Figure ) we foud that 5, ad 8 studets had itermediate, high ad very high success respectively at stage S of imagiatio. Therefore we obtaied that a = b =0, c =5, d = ad e =8. Thus, by the defiitio of the correspodig membership fuctio give i the secod sectio, S is represeted by a fuzzy subset of U of the form: ( a ) ( b ) ( c ) ( d ) ( e ) A = {,0,,0,,0.5,,0.5,,0.5. I the same way we represeted the stages S ad S as fuzzy sets i U by A = {( a,0 ),( b,0 ),( c,0.5 ),( d,0.5 ),( e,0)} ad A = {( a, 0.5 ),( b, 0.5 ),( c, 0.5 ),( d, 0 ), ( e, 0)} respectively. Next we calculated the membership degrees of the 5 (ordered samples with replacemet of objects take from 5) i total possible studets profiles as it is described i the secod sectio (see colum of m s () i Table ). For example, for the profile s=(c, c, a) oe fids that ms = mα( c ).mα( c ).mα( a) = 0.5x0.5x0.5 = It is straightforward ow, usig the correspodig formula give i sectio, to calculate i terms of the membership degrees the Shao s etropy for the studet group, which is H 0, 89. Further, from the values of the colum of m s () it turs out that the maximal membership degree of studets profiles is Therefore the possibility of each s i U ms is give by r s = Oe, i order to be able to make the correspodig comparisos, could also calculate the probabilities of the studets profiles usig the formula for p s give i sectio. However, adoptig the Shackle s view (see sectio ) we cosidered that the calculatio of the probabilities is ot ecessary. Calculatig the possibilities of all profiles (see colum of r s () i Table ) oe fids that the ordered possibility distributio for the studet group is: r =r =,r =r 4 =r 5 =r 6 =r 7 =r 8 =0,5,r 9 =r 0 =r =r =r =r 4 = 0,58, r 5 =r 6 =..=r 5 =0. Thus usig a calculator we foud that 4 i ST ( r) = [ ( ri ri+ ) log ] log i i= 0.0 rj j= 8 4 [0.5log log log ] ( 0, 4 ).( 0.04) + ( 0.58 ).( 0.) ad 4 Ν ( r) = [ ( ri ri+ ) log i ] log i= = [0.5log + 0.4log log4] log ( ) ( ) Therefore we fially obtaied that T( r).65. A few days later we performed the same experimet with a group of 50 studets from the School of Maagemet ad Ecoomics. Workig as i the first experimet we foud that: A = {( a,0 ),( b,0.5 ),( c,0.5 ),( d,0.5 ),( e,0 )}, ( ) ( ) A {(,0.5),(,0.5),(,0.5),,0,,0 } = a b c d e ad

5 America Joural of Applied Mathematics ad Statistics 5 ( ) ( ) A = {( a,0.5),( b,0.5),( c,0.5), d,0, e,0 }. The we calculated the membership degrees of all possible profiles of the studet group (colum of m s () i Table ) ad the Shao s etropy, which is H 0,. Further, sice the maximal membership degree is agai 0.065, the possibility of each s is give by the same formula as for the first group. The values of the possibilities of all profiles are give i colum of r s () of Table. Calculatig the possibilities of all profiles (colum of r s () i Table ) oe fids that the ordered possibility distributio of the secod group is: r: r = r =, r = r 4 = r 5 = r 6 = r 7 = r 8 = 0,5, r 9 = r 0 = r = r = r = 0,58, r 4 = r 5 =.= r 5 =0. The, workig i the same way as above for the first group, oe fids that T(r) = 0,4+,79 =,6. Therefore, sice,6<,65, it turs out that the secod group had i geeral a slightly better performace tha the first oe. Notice that the values of the Shao s etropy foud above for the two groups lead to the opposite coclusio (sice 0,>0,89), but this, as we have already explaied i the third sectio, is ot surprisig i cases where the differece betwee the performaces of the two groups is very small. Fially, i order to study the combied results of the two groups performace we calculated the pseudofrequecies f(s) = m s ()+m s () ad the combied possibilities of all profiles (see the last two colums of Table as it has bee described i sectio of the preset paper. Table. Profiles with o zero membership degrees A A A m s() r s() m s() r s() f(s) r(s) b b b b b a b a a c c c c c a c c b c a a c b a c b b d d a d d b d d c d a a d b a d b b d c a d c b d c c e c a e c b e c c e d a e d b e d c The outcomes of Table were obtaied with accuracy up to the third decimal poit. Notice that our model ca be also used (i a simplified form) for the idividual assessmet of the studets of each group. I this case a qualitative profile (x, y, z) with x, y, z i U is assiged to each studet cocerig his/her performace at each stage of the reasoig process. This type of assessmet by referece to the profile related to each studet defies i geeral a relatio of partial order amog studets with respect to their total performace. For example, cosider the studet profiles of Table. The the studet possessig the profile (d, b, b) demostrates a better performace tha the studet possessig the profile (b, b, b), or (c, b, a), or (d, a, a), etc. However, the studet with profile (d, c, a) demostrates a better performace at the stage of visualizatio tha the studet with profile (d, b, b), which demostrates a better performace at the stage of geeratio of ideas, etc (for more details see [0]). 5. Coclusios ad Discussio The followig coclusios ca be draw from the material preseted i this paper: The mai stages of huma reasoig for reachig the solutio of a give problem ivolve imagiatio, visualizatio ad idea geeratios. The above stages are helpful i uderstadig the idividuals ideal behaviour durig the reasoig process. However, thigs i real situatios are usually ot happeig like that, sice huma cogitio utilizes i geeral cocepts that are iheretly graded ad therefore fuzzy. I this paper we made use of a fuzzy model for a better descriptio of the reasoig process developed i earlier papers [,]. This model gives, through the calculatio of probabilities ad possibilities of all possible idividuals profiles, a quatitative/qualitative view of their behaviour durig the process. Based o the above model ad usig well kow measures of a system s ucertaity we developed techiques of measurig the reasoig skills of a group of idividuals. Our model ca be also used (i a simplified form) for the idividual assessmet of the members of each group. We also preseted classroom experimets performed with studet groups of T. E. I. of Patras, Greece illustratig our results i practice. Here we must emphasize that the cetroid method preseted i earlier papers [,] treats differetly the idea of a system s performace, tha the two measures of ucertaity preseted i this paper do. I fact, the weighted average plays the mai role i the cetroid method, i.e. the result of the system s performace close to its ideal performace has much more weight tha the oe close to the lower ed. I other words, while the measures of ucertaity are dealig with the average system s performace, the cetroid method is mostly lookig at the quality of the performace. Cosequetly, some differeces could appear i evaluatig a system s performace by these differet approaches. Therefore, it is argued that a combied use of all these ( i total) measures could help the user i fidig the ideal profile of the system s performace accordig to his/her persoal criteria of goals. Our plas for future research o the subject ivolve: The possible extesio of our fuzzy model for the descriptio of other real life situatios ivolvig fuzziess ad/or ucertaity. Further experimetal applicatios of our model i order to obtai more creditable statistical data.

6 6 America Joural of Applied Mathematics ad Statistics Ackowledgemet The headig of the Ackowledgmet sectio ad the Refereces sectio must ot be umbered. Refereces [] Klir, G. J. & Folger, T. A., Fuzzy Sets, Ucertaity ad Iformatio, Pretice-Hall, Lodo, 988. [] Klir G. J., Priciples of ucertaity: What are they? Why do we eed them? Fuzzy Sets ad Systems, 74, 5-, 995. [] Se, Z., Fuzzy Logic ad Hydrological Modellig, Taylor & Fracis Group, CRC Press, 00. [4] Se, Z., Fuzzy philosophy of sciece ad educatio, Turkish Joural of Fuzzy Systems, (), 77-98, 0. [5] Shackle, G. L. S., Decisio, Order ad Time i Huma Affairs, Cambridge Uiversity Press, Cambridge, 96. [6] Shao, C. E., A mathematical theory of commuicatios, Bell Systems Techical Joural, 7, 79-4 ad 6-656, 948. [7] Voskoglou, M. Gr., Stochastic ad fuzzy models i Mathematics Educatio, Artificial Itelligece ad Maagemet, Lambert Academic Publishig, Saarbrucke, Germay, 0 (for more details look at [8] Voskoglou, M. Gr., Fuzzy Logic ad Ucertaity i Mathematics Educatio, Iteratioal Joural of Applicatios of Fuzzy Sets ad Artificial Itelligece,, 45-64, 0. [9] Voskoglou, M. Gr., A study o Fuzzy Systems, America Joural of Computatioal ad Applied Mathematics, (5), -40, 0. [0] Voskoglou, M. Gr., Assessig Studets Idividual Problem Solvig Skills, Iteratioal Joural of Applicatios of Fuzzy Sets ad Artificial Itelligece,, 9-49, 0. [] Voskoglou, M. Gr., A Applicatio of Fuzzy Sets to Scietific Thikig, ICIT 0: The 6 th Iteratioal Coferece o Iformatio Techology, May 8, 0, available i the web at: 0Itelligece/68_math.pdf. [] Voskoglou, M. Gr & Subboti, I. Ya, Dealig with the Fuzziess of Huma Reasoig, Iteratioal Joural of Applicatios of Fuzzy Sets ad Artificial Itelligece,, 9-06, 0. [] Zadeh, L. A., Fuzzy Sets, Iformatio ad Cotrol, 8, 8-5, 965. [4] Zadeh, L. A., Fuzzy Algorithms, Iformatio ad Cotrol,, 94-0, 968.