Representing the Adverb Very in Fuzzy Set Theory

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1 9 Representing the Adverb Very in Fuzzy Set Theory Martine De Cock Applied Mathematics and Computer Science, University of Gent Krijgslaan 28 (S9), B-9 Gent, Belgium Abstract. We recall the concept of a linguistic variable and the representation of its values (i.e. linguistic terms) by means of fuzzy sets. In this framework adverbs are represented by fuzzy modifiers, i.e. operators acting on these fuzzy sets. We investigate two important classes of popular fuzzy modifiers. Grounding on results from psycholinguistic research, we discuss their pro s and contra s for representing the adverb very. 9. Linguistic variables 9.. The concept of a linguistic variable The concept of a linguistic variable was introduced by Zadeh in the 7 s [ZAD975]. While the values of a numerical variable (often used in classical mathematics, physics, economics,...) are numbers, the values of a linguistic variable are linguistic terms. E.g. the numerical variable size has values 5 cm 2, cm 2,... while the linguistic variable Size has values large, small, very large, rather small, not very large and not small,... The set consisting of all possible values of a linguistic variable is called the term set. Perhaps the most important computational beauty of the concept of a linguistic variable is the fact that the elements of its term set show a specific structure. Starting from one, two, or more base terms (e.g. large, small), every other term can be constructed using the following scheme : 223 Proceedings of the ESSLLI Student Session 999 Amalia Todirascu (editor). Chapter 9, Copyright c 999, Martine De Cock.

2 Representing the Adverb Very in Fuzzy Set Theory: Martine De Cock /224 large close to Figure 9.: Fuzzy sets for large and close to 5 <conjunction> := and ; <disjunction> := or ; <modifier> := very rather more or less extremely quite sort of really slightly...; <term> := <base term> <term> <conjunction> <term> <term> <disjunction> <term> not <term> <modifier> <term>; 9..2 Representing the meaning of a term by a fuzzy set The meaning of each term can be represented by a fuzzy set. If X is a universe of discourse, then a fuzzy set A in X is characterized by its membership function A : X [, ] x A(x), x X The membership function of A maps every object x of the universe X onto a degree of membership, i.e. the degree to which x belongs to the fuzzy set A. F(X) is the set of all fuzzy sets in X. E.g. if X =[, 2] then the terms large and close to 5 could be represented by the fuzzy sets in fig. 9.. For simplicity we will make no notational distinction between a fuzzy set A, its membership function A and the term A represented by that fuzzy set. Once the membership function for the base terms is known, the membership function of all other terms can be deduced. Representation of conjunction and disjunction The conjunctor and and the disjunctor or canbemodelledusingmin and max respectively. If A and B are two fuzzy sets representing the meaning of two linguistic terms

3 225\ ESSLLI Student Session 999 then the fuzzy sets for A andb andfor AorB canbederivedinthe following way : ( x X)((A andb)(x) =(A min B)(x) =min(a(x),b(x))) ( x X)((A orb)(x) =(A max B)(x) =max(a(x),b(x))) Representation of linguistic modifiers In this paper however we will focus on the deduction of the membership function of a modified term from the membership function of that term. With a linguistic modifier, i.e. an adverb in natural language, a fuzzy modifier m is associated, i.e. a F(X) F(X) mapping m : F(X) F(X) A m(a), A F(X) So a fuzzy modifier is an operator acting on a fuzzy set, transforming a fuzzy set into another one in the same universe. Inclusion can be defined for fuzzy sets A F(X), B F(X) inthe following way : A B ( x X)(A(x) B(x)) So the fuzzy set A is a subset of the fuzzy set B iff for every object x the degree to which x belongs to A doesnot exceed the degree to which x belongs to B. Now we can distinguish two important subclasses of fuzzy modifiers : m is expansive ( A F(X))(A m(a)) m is restrictive ( A F(X))(m(A) A) A restrictive modifier leads to a decrease in the degrees of membership; it pushes the original membership function down. Bouchon and Jia [BOU992] call this reinforcement, Lakoff[LAK973] talks about intensifiers. Restrictive modifiers are thus often associated with intensifying adverbs such as very, extremely, highly,... An expansive modifier on the other hand leads to an increase of the degrees of membership; it lifts the original membership function up. Bouchon and Jia[BOU992] call this weakening, Lakoff[LAK973] talks about deintensifiers. Expansive modifiers are thus often associated with deintensifying adverbs such as more or less, rather, a bit,... In this paper we will focus on the representation of one particular adverb, namely very. We will among other things explain that it can t always be represented by a restrictive fuzzy modifier.

4 Representing the Adverb Very in Fuzzy Set Theory: Martine De Cock /226 large very large close to 5 very close to Figure 9.2: Effect of a powering modifier on large and close to Two popular classes of fuzzy modifiers 9.2. Powering modifiers In the early 7 s Zadeh[ZAD972] introduced a class of fuzzy powering modifiers that has become very popular. For α [, + [, P α is a F(X) F(X) mapping with for A F(X) P α (A) : X [, ] x (A(x)) α, x X If α thenp α is a restrictive modifier, i.e. ( A F(X))(P α (A) A). Nowadays the fuzzy modifier P 2 is still a very popular choice for representing the adverb very (see e.g. [BAB998], [YASMUK998]). Fig. 9.2 shows the effect of P 2 on the fuzzy sets large and close to Shifting modifiers Another type of fuzzy modifiers, the shifting modifiers, was already informally suggested by Lakoff[LAK973] in the 7 s. A.o. Hellendoorn [HEL99] and Bouchon[BOU993] used it in a more formal manner. Since the shifting is an operation on objects of the universe (and not an operation on their degree of membership like the powering is), it s only applicable for fuzzy sets in a universe equipped with such an operation. For X = R, i.e. the set of real numbers, and α R we define the shifting modifier S α with for A F(R) S α (A) : R [, ] x A(x α), x R

5 227\ ESSLLI Student Session 999 large very large close to 5 AB Figure 9.3: Effect of shifting modifiers on (a) large and (b) close to Shortcomings of the fuzzy modifiers as a representation for very 9.3. Behaviour of S α with regard to restrictiveness Unlike with the powering modifiers, we can t distinguish a set of values for α for which S α is restrictive. Let A F(R) then the following properties hold : If A is increasing and α thens α (A) A If A is decreasing and α thens α (A) A If A is increasing, then very A could be represented by e.g. S. If A is decreasing, then very A could be represented by e.g. S. But in general, there is no α so that ( A F(X)) (S α (A) A). Fig. 9.3a shows how we can generate the membership function for very large from the increasing membership function for large using S. The membership function for close to 5 however is neither increasing or decreasing; we cannot use a shifting modifier S α to generate very close to 5. Fig. 9.3b illustrates the effect of S and S on close to 5 (A = S (close to 5), B = S (close to 5)) Behaviour of P 2 with respect to kernel and support For A F(X) : We recall that for A F(R) kernel A = {x x X A(x) =} support A = {x x X A(x) > } A is increasing ( (x, y) R 2 )(x y A(x) A(y)) A is decreasing ( (x, y) R 2 )(x y A(x) A(y))

6 Representing the Adverb Very in Fuzzy Set Theory: Martine De Cock /228 Probably the best-known shortcoming of the powering modifiers is the fact that they keep the kernel and the support ([KER993] p. 35, [HEL99] p. 38, [LAK973] p. 488, [DES988] p. 64). For α [, + [ : ( A F(X))(kernel(P α (A)) = kernel A support(p α (A)) = support A) If we choose to represent very by P 2, then every square x that is certainly large (x kernel(large)) isalsovery large (x kernel(very large)). In other words in this representation there can t be any squares that are considered to be large to degree, but very large to a lower degree. Every square that is large to degree, is automatically considered to be very large to degree as well. In some real life situations however a square with a size of 7.5cm e.g. can be considered large to degree, but not very large to the same degree. (Compare with : some people consider a man of 75 years certainly old (75 kernel(old)), but not certainly very old (75 / kernel(very old).) Furthermore : every square x that isn t very large to any degree (x / support(very large)), can t be considered to be large in any degree either (x / support(large)). These properties are clearly counterintuitive. It s worth mentioning that the shifting modifiers do not have this shortcoming Two interpretations of very In the mid 7 s, Hersh and Caramazza[HECA976] did some psycholinguistic experiments on the accuracy of P 2 as a representation for very (and the usefulness of fuzzy sets to model linguistic terms in general). One of their most interesting findings was that there are two different interpretations of the adverb very. Vanden Eynde[VDEYN] also did research on this subject and came to a similar conclusion. The two interpretations are. The inclusive interpretation (fig. 9.4a) : the fuzzy set very large is included in the fuzzy set large. Semantical entailment is clearly respected : ( x X)(x is very large x is large). 2. The non inclusive interpretation (fig. 9.4b) : the fuzzy set very large isn t included in large, neither is large in very large. large and very large denote two different (overlapping) categories. This doesn t mean that the person interpreting large and very large doesn t respect semantical entailment, but in this case he is following Grice s maxime of conversation [GRI978] : Make your speech contribution as informative as required. When a listener hears that xislarge, he assumes that x isn t very large, because in the latter case the speaker would have used the more informative utterance x is very large. Solarge + > not very large. We would like to remark that the difference between the two interpretations of large and very large doesn t only have consequences for the choice

7 229\ ESSLLI Student Session 999 large very large large very large Figure 9.4: (a) Inclusive interpretation (b) non inclusive interpretation of a fuzzy modifier to model very (restrictive or not), but also influences the shape of the membership function for large. It is clear that Zadeh s restrictive P 2 can only be used to model very in the inclusive interpretation. Finally we would like to mention that Hersh and Caramazza s research showed another shortcoming of Zadeh s P 2 in the inclusive interpretation. While P 2 increases the slope of an increasing membership function, the slopes of the functions for tall and very tall resulting from the experiment were approximately equal. In fact Hersh and Caramazza decided that the membership function for tall could be shifted to the right to obtain a pretty good approximation for the membership function for very tall Choosing the best representation The two kinds of fuzzy modifiers described above, i.e. the powering and the shifting modifiers, are easy operators from a mathematical point of view. However they both have their own advantages and disadvantages from a linguistic perspective. Neither one of them is the best in general, but we can give some guidelines on how to choose the most suitable in a specific situation. We assume that in a non inclusive interpretation the fuzzy sets have membership functions that can be approximated by a Gaussian curve like in fig. 9.4b, while in an inclusive interpretation they can have any shape. Furthermore we keep in mind that P 2 has the disadvantage of keeping the kernel and the support, so we ll avoid the use of P 2 whenever possible. For X a universe and A F(X) we can use the following guidelines to decide which fuzzy modifier we will use to represent very :

8 Representing the Adverb Very in Fuzzy Set Theory: Martine De Cock /23 start no yes inclusive interpretation? X equipped with shifting? S α yes no no X equipped with shifting? P 2? yes no A increasing or decreasing? P 2 yes S α Conclusion and future work The adverb very can be modelled in a lot of situations using powering or shifting modifiers. The choice between those two depends mainly on the interpretation (inclusive or non inclusive) and the kind of universe (equipped with a shifting operation or not). Some cases however can not be handled (cfr. a non inclusive interpretation in a universe not equipped with a shifting operation). Furthermore, even in situations where very can be represented using P 2, this fuzzy modifier has the great disadvantage of keeping the kernel and the support. These observations encourage us to search for another kind of fuzzy modifier. Among other things we would like that this new type of fuzzy modifier : Doesn t keep the kernel and the support in every case. Is applicable in all kinds of universes (not only in those equipped with a special operation, like shifting). Provides a framework in which both the inclusive and the non inclusive interpretation can be modelled in a similar mathematical way. Acknowledgement The author would like to thank the Fund for Scientific Research Flanders (FWO) for funding the research reported on in this paper.

9 Bibliography [BAB998] R. BABU SKA, Fuzzy Modeling for Control, International Series in Intelligent Technologies, Kluwer Academic Publishers, (998) [BOU992] B. BOUCHON-MEUNIER, Y. JIA, Linguistic Modifiers and Imprecise Categories, International Journal of Intelligent Systems, 7(992), [BOU993] B. BOUCHON-MEUNIER, La Logique Floue, Que sais-je?, 272, Paris, (993) [DES988] S. DESPRÉS, Un apport àlaconceptiondesystèmes à base de connaissances : les opérations de déduction floues, Thèse de doctorat de l Université Pierre et Marie Curie (Paris VI), (988) [GRI978] H. P. GRICE, Further notes on logic and conversation, P. Cole (ed.), Syntax and Semantics 9 : Pragmatics, New York : Academic Press 978, 3-28 [HEL99] H. HELLENDOORN, Reasoning with fuzzy logic, Ph. D. thesis, T.U. Delft, 99 [HECA976] H.M. HERSH, A. CARAMAZZA, A Fuzzy Set Approach to Modifiers and Vagueness in Natural Language, Journal of Experimental Psychology : General, 5(976),nr. 3, [KER993] E. E. KERRE, Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications, Communication and Cognition, Gent, (993) [LAK973] G. LAKOFF, Hedges : a Study in Meaning Criteria and the Logic of Fuzzy Concepts, Journal of Philosophical Logic, 2(973), [VDEYN] C. VANDEN EYNDE, A very difficult problem : modelling modification using VERY. A semantic pragmatic approach., (in Dutch) FKFO project, private communication 23 Proceedings of the ESSLLI Student Session 999 Amalia Todirascu (editor). Chapter 9, Copyright c 999, Martine De Cock.

10 Representing the Adverb Very in Fuzzy Set Theory: Martine De Cock /232 [YASMUK998] H. YASUI, M. MUKAIDONO, Fuzzy Prolog based on Lukasiewicz implication, Logic Programming and Soft Computing, edited by T.P. Martin, F. Arcelli Fontana; Research Studies Press Ltd., (998) [ZAD972] L. A. ZADEH, A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges, Journal of Cybernetics, 2,3(972), 4-34 [ZAD975] L. A. ZADEH, The Concept of a Linguistic Variable and its Application to Approximate Reasoning I, II, III, Information Sciences, 8(975), , 3-357, 9(975), 43-8