A Model of Dialog Based on Fuzzy Set Concept

1. INTRODUCTION

In r eal dialog the roles of a speaker and a listener changes alternate ly but in our model those arc fixed for s implic ity. Le t us suppose the following si tuation. The listener acquires a sequence o f information from the speaker by r epeating indirect questions and guesses what the speaker is thinking. In orde r to supple­ ment the obtained information by r easoning or imagination, the lis t ener has a priori knowledge on objects which they talk about. Then listene r eva lua t es thi s information by comparing it with his knowl edge and r epeats this process untill he is convinced that he has understood what the speaker wanted to communicate. Both the informat i on and the knowledge arc so ambiguous that fuzzy se t concept is suitable for the ir mathe ­ matical expr ession.

2. PROCESS OF DIALOG

We first describe the situation o f dialog be­ tween a ·sp.,uker and a listene r by X: a set of objects , A: a se t of a ttributes , AV: sets of values o f each attribute. For exampl e X=(Yama­ quchi , Yoshi naga , Ken, ... }; names of Japanese f cmal<~ singers, A=t Age , Appearance , He ight, .. } AV(Agc)={ Young, Middle , Old, ... }, AV(Appear­ ance)={Beautiful, Normal, ... }, etc . llere it i s assumed that both the speaker and the listene r know the sets X and A. Let x . be the i-th c l ement of X, a. the j-th element 1 of A und avk(aj) the k-th c lc~ent of the se t AV(aj)

which is the set of values of the attribute a . . Then dialog proceeds as follows. J

The speaker chooses x . out of the objects.

The listener: How abo~t the Age of x?

The speaker: She is young.

The listener: ............ .

Here the listener is forbidden to directly ask the name of x. and the speaker is allowed to answer only ifi the form "She is avk(a . )". Generally avk(a.) in the form "She isJavk(a . )" is a fuzzy infotmation on the attribute a . . J Let us assume that this fuzzy informationJis expressed by a fuzzy set. We further assume that the listener has a priori knowledge on the attributes of the objects, e.g., "She, an ob­ ject x, is about 20 years old". This type of fuzzy knowledge can be also expressed by a fuzzy set.

3. EVALUATION AND UNDERSTANDING

Le t F be a fuzzy set associated with a member­ ship function ~ which implies a fuzzy infor­ mation given byFthe speaker as stated in the pr evious section. Further let G. be also a fuzzy set expressing the listenef's fuzzy know­ l edge on an attribute of an object x . . Now we discuss the process of reasoning in which the listener finds, by using a given in­ formation F and a priori knowledges G., what the speaker has chosen out of the objects; where F and G. are of course assumed to have the same unive?se of discourse concerned with an attribute.

In order to s e lect x. out of the objects by comparing F with �.,~we can use truth qualifi­ cation given by zaaeh [1).

where T. : [0, 1) + [0, 1) is a linguistic truth vilue. For example "x . is young (F) is more or less true (T . )" tran~lates into "x . is about 20 years old (~ . )". If T. is false,~then the listener can conclude that ihe speaker is not thinking about the object x. as far as the attribute under consideration i~ concerned. Thus the truth value of "x. is F" with respect to "x. is G. " can be used is a criterion for selecting x~.

According t6 truth qualification rule as is shown in Fig.l, we have

where u denotes a variable in the universe of discourse of F and G . . ~

If llF is one to one mapping, then 11 is given by Ti

where v denotes a variable in the unit interval [0, 1]. In the case that 11 does not have the inverse, llT can be obtainea, for example, as i follows:

That is, T. . ~ mapp~ng ll · termined.F 11 (G . ), the image of G. through In FgeReral T. is not uniquely de­ ~

In this paper the authors define instead of llF(Gi) the following relation RF(Gi) on [0, 1] x [0, 1].

The relation R (G.) is considered as a generali­ zation ofT . . FFi§.2 shows the difference be­ tween 11 (G.t and R (G.). It is easily seen that F ~ F 1 the graph of the membership function of a fuzzy set llF(Gi) is a subset of RF(Gi).

Fig. 2 Difference between IJF (Gi)and RF (Gi) Now our problem is how to evaluate RF(Gi) in comparison of F and G.. To this end l et us de­ fine the following values A, B, C and D with respect to RF(Gi).

where A implies the distance in the sense of max-min from a point (0, 1) to its nearest point in RF(Gi) and B, C, Dare those from (0, 0), (1, 0), (1, 1) respectively as is shown in Fig.3.

Next we introduce a kind of order to the class of R (G.) 's by a function transforming RF(G.) F ~ ~ to a r eal numbe r such that

where A' = 1 -A, C' = 1 - C and a, B, y, o are constants.

We have to take into account that this order preserves the partial order concerned with T. since R (G . ) includes T. in its special case: -y ~ ~

The partial order~ of the class ofT . is de­ fined by T n T 1 = T and T U T 1 T 0 * 1 T 6 T', whe re n is meet* ope ration and U is join* ope­ ration [2].

where t\ stands for min, V for max and l: for union.

Let us take two fuzzy sets F 1 and F 2 and con­ sider the special case where R (G . J and R (G.) Fl ~ F2 ~

are reduced to ordinary functions, more pre­ cisely the graphs of membership functions such as 1J in Eq. ( 2) . T.

Here eonsider the following two equations.

What we should do is to determine a, S, y and 6 in Eq. (6) so that Eq. (7) leads to Eq. (8). The operation in Eq. (7) illustrated in Fig.4 with the values A', B, c' and D.

From the above condition we have a ~ y according t o Fig. 4 (a). From Fig. 4 (b), (c) and (d) follow S ~ 0, y ~ 0 and 6 > a. Finally we have the inequalite s

Conve r se ly, 8q. (8) is obtained from Eq.(7) using Eq. (9).

Now we can introduce an orde r to the c lass of RF(G) 's as follows:

Let us next consider the general case that the information in the attributes of the object x is sequencially obtained. Let F . be the infor­ mation on the j-th attribute andJG .. be a priori knowledge on the j-th attribute of~the objext x .. Let us define ~

Here h(x . ) can be interpreted as the degree of the list~ner's confidence that the speaker is talking about the object x . . Thus the listener can find, according to the~value of h(x.), an x which the speaker might choose out of tfie objects.

4. CONCLUSION

A dialog spoken in natural language is the ex­ change of ambiguous information in many cases. In this paper we develope a mathematical model of a dialog in which one person understands what the partner wants to communicate through ambig­ uous information and knowledge. In our model of dialog, the behaviour of the listener plays the most important role. If this role is played by computer, then in a restricted sence a man can talk with computer in natural language. The key point of this paper is how to evaluate the given fuzzy information though a priori know­ ledge. We have presented a generalized truth value and introduced an order into its class for the evaluation.