ZCSZYTY NAUKO'.VE POl.ITECKNIKl ¿LA.SKIF.J Seria; AU TG M A T Y K A z . 61
________ 1981 Nr k b l . 701
Ernest C Z O G A t A , W i to ld PtORYCZ
FUZZY SETS IN INTERACTIVE COMPUTER MA N- MA CH IN E SY 3TEMS
S u m m a r y . The paper deals with some problems of the use of the fuzzy set theory e.g.: linguistic a p p r o x i m a t i o n . approximate rea
soning in designing of an interactive man-machine systems. The ba
sic definir i o n ' of linguistic variables constituting the main feature of this kind of systems has boen presented in details and illustra
ted by means of a numerical example. The general structure of the system has been shown and or.fc of the implementations has been pre
sented as well.
A
1. Introduction
Rece nt ly 30mo new ideas on interactive computer systems have been pu
blished [ o . 13j . The list of hu ma n-oriented software proqrams is extensi
ve (e.g. ELIZA or DEACON [5]'. There is no doubt that they form a usotul tool for solving a wi de class of problems: the designing of complex devices or decision-making in many ill-defined processes.
A n introduction of fuzzy set theory and the posibility theory [?, 8j leads to a general class of interactive systems, wher e a qualitative kind of information can be used and performed. These theories make it possible for us to discuss a new category of models. the so called verbal models
¡J9 . 9, 13]. which may be sy st em at ic al ly analyzed by the use of the fuzzy set theory. It is the aim of this paper to present the use of fuzzy sets in human oriented computer systems, introducing a fuzzy relational equa
tion, a linguistic approximation and the concept of approximate reasoning as a basis of the construction of this kind of systems. At the very befn- ning we have summarized some theoretical results of a fuzzy relational equation, and a linguistic variable.
2. L inguistic variable and linguistic approximation
A linguistic variable, forming the basis of lirguislic algorithms may be defined as follows [ll] .
DEFINI TI ON 1
A linguistic va riable is a system:
< L, T(L). X ,G ,M > '1)
88 E. C z o g a l a , W. Pe drycz
where L, T(L) (or shortly T),3? ,G ,M denote respectively:
L - name of the variable,
T - set of labels of a fuzzy subset of the un iverse of discourse, X - universe of d i s c o u r s e ,
G - syntactic rules, commonly de fined as generative grammar, which de
fine the well -f or me d sentences in T,
M - semantics, which consists of rules by which the meaning of the terms in T can be determined.
If X is a term in T, then its me aning (in the denotational sense) is a subset of X . The pr im or y term in T is a term, whose meaning must be defined a priori. It serves as a basis for the computation of the meaning of the non-primary terms in T.
Modifiers (hedges) such as: v e r y , more or l e s s , slightly less .s l i g h t l y more play a special role in semantics M. T h e y may be defined as follows
[ll] . For a given va lu e of a linguistic vari ab le t e T defined by the use of the membership function:
¿lt : >? — [0,1] (2)
the modifier "m" works as follows:
modifier t *=
e.g.
. a 2
y.e-Q: * = j t t
■1 more or less t - slightly less t ■> ¿1.,.0.75
1 25 slightly more t ” /it '
Another definition of modifiers may be used as well [4], Example 1
Let us illustrate a linguistic variable: pressure, expressing basic e- lements of Def. 1.
L - pressure, \
T ■ ( s m a l l . m i d d l e ,big,more or less .small.very small,very v e r y small... j X - {1 0 ,2 0 ,. .. ,ioo]
G is a grammar:
G - < V , S , P , 6 >
Fuzzy sats In in te ra ct iv e computer.. 89
where
V = | em ai l, mi dd le , b i g , s l i g h t l y , l e s s . v e r y . s l i g h t l y more ,m or e or less|
X ! = | valu e of linguistic v a r i ab le .modifier.prlmaty termj with the following list of productions:
P :
< valu e of linguistic v a r i a b l o : :*<primary t e r m > | < m o d i f i e r > <p ri ma ry term> |<modifler> cv al ue of linguistic variable>
< p r i m a r y term>::= small | middle | big i
< m o d i f i e r > ::>= ve ry | more or less I slightly less I s l ig ht ly more 6 ° < v a l u e of linguistic va ri ab le
Example 2
Ta ki ng into account a linguistic v a r i ab le pr essure as discussed in Ex. 1 and the pr imary set middle de fined by a membership function:
X 10 20 30 40 50 60 70 80 9 0 100
0 .1 .3 .5 1 .5 -.3 .1 0 0
the va lu es of linguistic v a ri ab le s created by mesne of different modi
fiers are given by eq. (4) and illustrated in Fig. 1.
Fig. 1. Pr im ar y set nl ddTe and the va lu es of linguistic vari ab le s created by means of different modifiers
It is in te re st in g to notice a ch ar acteristic feature of modifiers. They can be divided into two groups:
- fuzzification mo di fi er s (m < 1 , e.g. more or less) - co ncentration mo di fi er s (« > 1 , e.g. very)
because the modifiers of the first group increase the degree of fuzziness tpt of pr im ar y seta, i.e.
9C B . C z oca H i . W . redrycz
V t < K t ' (5
and using one of the second group the degree of fuzziness is decreased:
Vx > V » t '6)
wher e :
<Pt = M o x ^ I x j H ' ^ t (x)■ ^ | ) 17.) X€j? xelf
One of the basic problems in approximate reasoning might be stated as follows:
- for a given fuzzy set with a membership function fit "best"
(in the sense of an appriopriate distance; primary sets and modifiers if the number of modifiers is given, i.e. ap proximating a fuzzy set by means of the best fitted value of a linguistic variable.
Let us denote:
{tjJ i” l.l,,,..,m - sst cf membership functions of primary fuzzy sets,
|^jj j “ 1 . 2 k - set of modifiers hedges r. - length of the chain of modifiers
pi - a given membership function So we get the problem:
r T , 7. . b h . .. . h .
Mi n || p - j k || = 1| p - p ^ 01 J °k || (8)
where |j|| denotes the distance.
The solution of the problem presented above needs a great amount of com
putation squal to:
m + m-k + m - k ' k + . ..-nn* k n~* • k =■ m+m-k+m* k“^+. . .+m -kn = m (l+k+k^ + . ..+ k n ) (9)
see Fig. 2.
It. is convenient to use a simpler, suboptimal strategy in order to find the values of the proper fuzzy set and of the modifiers. This has been illustrated in Fig. 3. This method needs the number of calculation to be equal to:
m+m>k + (n-l)-k = m»(ltk) + (n-l)*k tlO)
»
Fuzzy sets in interactive computer. 91
, n modifiers used
Fig. 2. Illu st ra ti on of linguistic ap pr ox im at io n in the case of n m o d i fiers used
Fig. 3. Il lu stration of mo di fi ed l i ng ui st ic 'a pp ro xim at io n in the case of n modi fi er s used
3. Fuzz y relati on al equations
Fuzzy relati on al equations form useful tool for describing. complex, il l- do fi ne d processes, where the re pr es en ta ti on of the process in ca te go
ries of set3 a n d relations is more adeq ua te than in the categories of point3 and functions. Because there is a great number of theoretical and
92 E. Czogala, VV. Pedrycz
^practical co nsiderations on this topic (e.g. [6]) we shall deal with the main problems co nc er ne d with the de sc ri pt io n of a process and some pr o
blems strictly conn ec te d wi th it. We discuss the process described by means fo the fuzzy relati on al equation:
x k+1 = x k°
V R
(11)where X^ , x k+1 are fuzzy sets expressing the state of the process in ti
me mo ments k and (k+l) and R denotes the r e la ti on sh ip existing in the pr ocess (Fig. 4)
v x k+ i
U k : HJ — [0,l] (12)
R : II * Ü? x — [0, l]
where y and HI denote the space of state and control resp ec ti ve ly .E q (ll) is equivalent to the following notation:
■Uy- ( y ) » Maxi Mi n fti, (x), Max(Min(lt (u ) ,w_ (u ,x ,y)))l 1 (13) r X k+ l x€X 1 ^ k ueil r u k ” R J J
G e ne ra l problems concer ni ng this kind of the equation have been discussed in detail (e.g. id en ti fi ca ti on [2], inverse problem (sensitivity) £l ,7]
Eq. 11 is useful in prediction task, so it makes it po ss ib le to be used for a given and constant control to predict future states of the process.
If and U are given, so using Ma x - M i n composition we get:
V i s X. o U o R m
k X. o G
k \
X k+2 9 X. k+l ° U ° R = X. . « G - X .oG oG
k+l k (14)
X k+p - X k ° H ° ■ • • o G - X. » G P k
where p denotes the horizon of prediction.
Fuzzy 3ets In Interactive computer. 93
An ot he r important task is the one-step local control of a process. Let
where p H (*,*) stands for the Ha mming di st an ce and of > 0 . The pr oblem of control is stated as follows: For a given state find optimal i.e. mi
nimize the qu ality index given by (15). The method of computing is as follows
1. Compute
T h e co ns id er at io ns given above are used in the cons tr uc ti on of an inter
active system.
4. Ge ne ra l structure of an interactive computer m a n - ma ch in e system
In this section we present the structure of an in te ra ct iv e ma n- ma ch in e system, which can be used in the case of pr oc es si ng a qu al it at iv e form of information. The main blocks of 't he system are de picted in the Fig. 5 Let us briefly discuss some of them.
us assume that the optimal state X _ and optimal control U . are qi- r opt r opt 3 ven in the form of membership functions:
r (xi } ' ^ U
/V
opt opt J
The qu ality index takes the form:
Q “ Ï H ^ o p t ' W ♦ * ? N * U k'U o p t ) (15)
“ ? H (XoP f V V R) + ^ H (CJ 'U o p t ) (16) J -1 . 2 ... Of > 0
where
Uj : UJ ~ [ 0 , l ]
with the me mb er sh ip function:
(17)
2. Put as a fuzzy set with the member sh ip function equal to:
¿XU k <'U i') “ 1 “
Q ( G i ) Max Q(ff,) l<i<p
(18)
94 E. Czogała, W. Pedrycz
Subrouti.no defining univer
ses of d isco u rse an d membership function of basic
fuxxy sets
Settin g th e basic valu es
Input n u m erical Cor lin g u istic) inform.
humeri-\
ca l in fe r motion^.
T ra n sfo rm a tio n of In f o r m a tio n
Subroutine Syntax analysis
N u m e rica l of p red ic
a lg o r ith m s lion or control I S T A R T ~~)
Transform ation of infor
m a tio n .
Fig. 5. Ge neral st ru ct ur e o^f an interactive na n- m a c h i n e system
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Fuzzy sets in Interactive computer.. 95
1 .5 .3 .2 .1 0
4 .6 1 .4 .3 .1
3 .4 .7 1 .4 .3
1 .3 .5 1 .3 .1
0 , . 3 .4 .8 1 .9
0 .2 .3 .7 .9 1
1 .9 .8 .7 .5 .3
1 .8. .5 .2 .1 0
9 •1 .9 .5 .4 .3
9 .9 1 .7 .3 .1
9 .9 1 .9 .4 .2
1 .2 .5 .6 1 .5
3 .5 .9 1 .8 .7
4 .5 .8 1 .9 .7
3 .5 1 .6 .3 .2
7 .8 1 .4 .2 0
9 .9 1 .8 .5 .2
9 1 .5 .2 0 0
3 .7 .8 .9 1 1
5 .8 .9 1 .2 .2
6 .9 1 .9 .4 .4
5 .8 1 .6 .3 .1
4 .5 .9 1 .3 .1
4 .7 1 .3 .1 .1
1 .4 .3 .2 0 0
8 1 .5 .3 .1 .1
4 .5 1 .2 .2 .1
3 .8 1 .3 0 0
7 1 .8 .5 0 0
5 .9 1 .1 0 0
Fig. 7. Fuzzy relation R describing the process
96 E. Czogała, W. Pe drycz
Setting of the basic valuas - In this block the primary fuzzy set9 and mo
difiers are defined and the spaces and HI are created.
Transformation of information - Input information is transformed into nu
merical values by means of defined fuzzy sets and modifiers. Output in
formation can be numerical in character or form the value of a linguistic variable. In the second case the subroutine of linguistic approximation is used. The algorithms of prediction and control have been presented in the previu3 section. A computer programme based on the ge ne ra l' sc he me gi
ven in Fig. 5-has been realised using FORTRAN 1900 (Fig. 6). Spaces and U consist of 6 and 5 elements:
* ■ { X 1 ,X2 ’* 3 ,X4 '*5 ,X6 } { u i . u2. u3. u4.u's ) ID
and relation R describing the process is presented in Fig. 7. Defini
tions of basic sets and modifiers (hedges) are given in Fig. 8. Some re
sults obtained by means of this programme are presented in Fig. 9.
/ D E F I N E D B A S I C S E T S A N D H E D G E S _ S H A U
XI 1 , 0 0 0 X 2 0 , 6 0 0 X 3 0 . 4 0 0 X 4 0 , 2 0 0 X 5 0, 0 0 0 X 6 ■ 0 , 0 0 0
M I D D L E XI 0 . 0 0 0
33 V M
X 4 0 , 3 0 0 X 5 0. 0 0 0 X 6 0 , 0 0 0
DIG XI Ô . Q 0 Ô
■*2 0.200 X 3 0 . 4 0 0
* 4 0 . 6 0 0
3! ? : S 88
V E R Y j - - - 2 , 0 0 0
M Q k E O R L E S S ---- 0 . 5 0 0
Fig. 8. Definition of b B B i c sets end modifiers (hedges)
PH E 01 CT ION
INITIAL VALUES S T Ä T i “ o f t h e P R O C E S S
*\
x a o . * o o o
x5 O.UQO
X 6 0 . 0 0 0
_ L I N G U I S T I C A P P « 0 X I M A T I U H c o n t r o l o r t h e p r o c e s s
U 1 0 . 0 0 0
Si ?:S88 ul 8:888
P R E D I C T I O N
S T E P O F P R E D I C T I O N ^ 1 q X 2 > 0 1 3 0 0 X 3 O . V O O
X 6It J:8a
L I N G U I S T I C Ü ? P K O X ] M A TiUn S T E P o f P R E D I C T I O N 2
X I c . b o o , x z O . b O O X 3 1 . 0 Q 0 X A O . H O O X 5 0 . 3 0 0
L I N G U I S T I C A P P R O X I H A T I U N S T E P O f P R E D I C T I O N 3
X I 0 . 7 0 0
11 ?:888
X A O . b O O X 5
X 6
0 . b O O 0 . 7 0 0
L I N G U I S T I C A P P R O X I M A T I O N S T E P O f P R E D I C T I O N A
x1 o . b o o
ii ?:§8o
X A 0 . b O O X 5 O . b O O X 6 0 . 7 0 0
L I N G U I S T I C A P P R O X I H A T I U N S T E P O F P R E D I C T I O N 5
X I O . b O O
X A o . b o o X 5 O . b O O X 6 0 , 7 0 0
L I N G U I S T I C A P P R O X I M A T I O N , S T E P O F P R E D I C T I O N 6
X I O . b O O
ii ?:888
X A O . b O O X 3 O . b O O , X 6 . 0 . 7 0 0 , L I N G U I S T I C A P P R O X I M A T I O N S T E P O f P R E D I C T I O N
f:«9b o o
Si
Si r s s .
X A O . b O O x 5 o . b o o X 6 0 . 7 0 0 ,
L I N G U I S T I C A P P R O X I H A T I U N C O N T R O L
O P T I M A L S T A T E 0 . 1 0 0 0 . 2 0 0 0 . 3 0 0 O . V O O X 5 1 . 0 0 0 X 6 1 ^ 0 0 0 o p t i m a l c & n t r o l u i 1 . 0 0 0
Si
8:588
U A 0 . 0 0 0
IJ5 0*000
S T A T E O F t H E P R O C E S S
M O R E O R L E S S
M O R E O R L E S S S M A L L
M O R E O R L E S S B I G
!
M O R E O R L E S S
M O R E O R lESjS * B I G
M O R E O R L E S S
M O R E O R L E S S
XI
X 2 X 3 X A X 5 X 6 C O N T R O L
0 . 3 0 0 0 . b O O
o: - 2
288
0.909
U A U 5
f:88o
0 . 3 7 9
0.000
Pig. 9. E x a m p l e s of r e su lt s o b t a i n e d b y me a n s of p r o g r a m m e
Fuzzy sets in Interactive computer. 97
5. Concluding remarks
The concept of the fuzzy set theory forms a formal frame of expressing vague, ambiguous and ill-defined concepts. A human-oriented computer sy
stem,,using the methods of these theories e.g. fuzzy relational equations, linguistic approximation, po ss ib il it y theory can operate on a qualitative kind of information giving results of linguistic or numerical character, which are used in many areas, where the human factor forms an Important element of the system. Otherwise, such a system can form a proper tool for linguistic simulation and for the modelling of complex, not well-de
fined systems.
R EFERENCES
[1] Czogała E . , Pedrycz W. : Computer Fault Diagnostics by the use of Mul tivalued logic and Fuzzy Set Theory, II International Conf. on Fault Tolerant Systems Diegnositcs, Brno 1979.
[2] Czogała E. , Pedrycz W. : An Approach to the Identification of Fuzzy Systems, Ar ch iw um A u to ma ty ki i Telemechaniki 1, 1980.
[3] Go ug en O . A . : Concept representation in Natural and Artificial Langua
ges: Axioms, Extensions, and Applications, Int. Journal Man-Machine Studies, 6, 1974.
[4] Lakoff G. , Hedges: A study in Me an in g Criteria and the Logic of Fuz
zy concepts, 0. Philos. Logic, 2, 1973.
[5] Ma rt in 0.: Dialog człowieka z maszynę c y f r o w ę , W N T , Wa rszawa 1976.
[6] Sanchez E. : Resolution of Composite Relation Equations, Inf and Con
trol, 1, 1976.
[ 7 ] Terano T. , Ts uk am ot o Y. et al: Diagnostics of En gi ne Tr ouble by Fuz
zy Logic, VII Triennal Congress IFAC, Vol. 3, Helsinki 1978.
[8] llmano M. , Mizu mo to M. , Tanaka K. : FSDS System: a Fuzzy Set Manipula
tion System, Information Sciences, 14, 1978.
[9] Wenstp F . : Deductive verbal models of organizations. Int. Journal Ma n- Ma ch in e Studies, 8, 1976.
£10] Mac Vi ca r-Whelan P.J. : Fuzzy Sets for Ma n- Machine Interaction, Int.
Jo urnal M a n - Ma ch in e Studies, 8, 1976.
[llj Zadeh L.A. : The Concept of a Linguistic Va riable and Its A p p l i c a t i o n to Ap pr oximate Reasoning, A m e r ic an Elsevier Publ. Comp. N e w York 1973 [l2] Zadeh L.A. : Fuzzy Sets as a Basis for a T h e o r y of Possibility, Fuzzy
Sets and Systems, 1, 1978.
[133 Zadeh L.A. : PRUF-Meaning Re pr es en ta ti on Language For Natural Lan
guages, Int , Journal Man- Ma ch in e Studies, 10, 1978.
Złożono w red. 21 .02.80 r. Recenzent
W formie ostatecznej 15.05.80 r. Doc. dr hab. inż. Andrzej Tylikow3ki
98 E. Czogała, W. Pedrycz
ZBIORY ROZMYTE W INTERAKTYWNYCH SYSTEMACH KOMPUTEROWYCH
S t r e s z c z e n i e
W pracy rozpatrzono niektóre problemy dotyczęce zastosowania teorii zbiorów rozmytych (np. aproksymacji lingwistycznej, rozumowania przybli
żonego) w projektowaniu interaktywnych sy stemów komputerowych. S z cz eg ół o
wo przedstawiono definicję i interpretację zmiennej lingwistycznej graję- cej iBtotnę rolę w tego rodzaju systemach. Jak również podano przykład numeryczny. Praca zawiera ogólnę strukturę systemu wraz z Jednę z m o ż
liwych implementacji.
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