INTERVAL-VALUED DEGREES OF BELIEF - CiteSeerX

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems cfWorld Scienti c Publishing Company

INTERVAL-VALUED DEGREES OF BELIEF: APPLICATIONS OF INTERVAL COMPUTATIONS TO EXPERT SYSTEMS AND INTELLIGENT CONTROL HUNG T. NGUYEN Department of Mathematical Sciences, New Mexico State University Las Cruces, NM 88003-8001, USA E-mail: [email protected] VLADIK KREINOVICH Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA E-mail: [email protected]

and QIANG ZUO Department of Mathematical Sciences, University of Texas at El Paso El Paso, TX 79968, USA E-mail: [email protected] Received 2 October 1995 Revised 29 December 1996 Usually, expert systems use numbers to describe the experts' degrees of belief in their statements. In practice, however, it is dicult to assign an exact numerical value to the expert's degree of belief. At best, we can get an interval of possible values. This fact leads to the use of interval-valued degrees of belief. When intervals are used to describe degrees of belief, then computations with intervals must be used to process them. In this paper, we describe applications of such interval computations to expert systems and to intelligent control. Keywords : Interval-valued degrees of belief; Expert systems; Intelligent Control.

1. Expert Systems and Intelligent Control Systems Use Degrees of Belief 1.1. Expert knowledge, expert systems, and intelligent control systems are necessary

In many application areas (e.g., in medicine, geology, economy, engineering), we often need to make decisions in the situation when we do not have the exact knowledge of the situation, and therefore, we cannot even formulate (not to say solve) the decision problems in precise mathematical terms. In some cases, we can formulate This

work was partially carried out while Hung T. Nguyen was on sabbatical leave at the University of Southern California, Los Angeles. 1

2 Interval-Valued Degrees of Belief

the problem precisely, but this formulation leads to a complicated mathematical optimization problem of the type that we cannot yet solve (e.g., control problems are in general computationally intractable, and therefore, we need experts to solve them Smith et al.222 ). In all these cases, we need human expertise to make decisions. There often exist experts who make reasonably good decisions: expert doctors successfully cure diseases; expert geologists nd oil; expert astronauts know how to dock and land the Space Shuttle; expert operators know how to operate a chemical plant, etc. Usually, among the experts in the eld, there are a few top experts whose decisions turn out to be the most ecient. Since there are only a few top experts, they cannot solve all the emerging problems. So, for all other problems, we have to rely on the expertise of those experts who may not possess all the knowledge of the top ones. It is therefore desirable to create an automated system that would incorporate the knowledge of top experts and use it to respond to the emerging problems in a similar way as the top experts would. Such a system would help other experts and lay people in making decisions. These systems are called expert systems. In control, it is often desirable to avoid using an expert altogether: experts are expensive to use, error-prone, and in some dangerous environments, it is desirable to make completely automatic control systems. E.g., we want to control robots who travel into the volcanos or go to other planets.

1.2. Degrees of belief are necessary to describe expert knowledge

The goal of an expert system is to simulate the experts' way of making decisions. For that purpose, an expert system includes a knowledge base that contains the knowledge of the experts. Some statements from the knowledge base are believed to be absolutely correct. However, in their decisions, experts also use other statements, about which they are not 100% sure that they are correct, and/or which are not formulated in exact terms. For example, their knowledge can contain phrases like \If a patient has a fever, a sore throat, and a headache, then most probably, he has a common cold". To describe the expert knowledge adequately, we must therefore store in the knowledge base not only the statements themselves, but also the indication of the extent to which the experts believe in these statements. An intermediate degree of belief means that in addition to \true" and \false", we must label some statements by some intermediate labels, like \most probably true", or \probably false", etc. Inside the computer, \true" corresponds to 1, and \false" to 0. Therefore, it is natural to use numbers from the interval [0,1] to describe the intermediate degrees of belief. So, to describe the expert's degree of belief in a statement A, we must nd an appropriate number d(A) from the interval [0,1] (some expert systems use a dierent interval, e.g., [?1; 1] Shortlie, Buchanan40;216;217). There are several ways to assign d(A) (see, e.g., Dubois61 , Norwich et al.195 , Turksen244, Walley254 ):

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 3

We can ask an expert to estimate his degree of belief on a scale from 0 to 10. If he, e.g., selects 8, then we take d(A) = 8=10.

This is the simplest and the most straightforward method of eliciting the degree of belief from the experts. This method works well in many cases, but for some experts (e.g., for most mathematicians), it is dicult to think this way: they either know that a statement is true, or that it is false, or that they do not know. For such experts, dierent methods are necessary.

If we have suciently many experts, and they are all de nite in their beliefs,

then we can poll the experts. For example, if 8 out of 10 experts believe that A is true, we assign d(A) = 8=10. Polling is ne if we have many experts with reasonably de nite ideas. However, in many case, we have just a few experts, who are not certain about the statement A, and who do now feel comfortable assigning a number to describe their degree of belief. Our ultimate goal is to make an expert system that makes decisions as well as these experts. So, these experts' degrees of belief are indirectly revealed by the decisions they make. Hence, to elicit the degrees of belief, we can simulate dierent situations, ask experts what decisions they would have made in these situations, and elicit the degree of belief from these decisions.

One way of doing it is to ask an expert to choose between an alternative A

in which he will receive $100 if A is true and 0 otherwise, and a lottery L(p) in which he gets $100 with some given probability p. If for some p, these two alternatives are equivalent to an expert, we can say that his degree of belief in A is equal to this probability p (with this situation in mind, degrees of belief are often called subjective probabilities). To determine p, we can use the following bisection procedure (see, e.g.,141 ):

Initially, we know nothing about d(A). We can express this by saying

that the interval [d? ; d+ ] of possible values of d(A) coincides with [0; 1]. If we know that d(A) 2 [d? ; d+ ], then we compare the alternative A and a lottery L(p) with p = (d? + d+ )=2. If A is preferable to L(p), this means that for this expert, the degree of belief in A is larger than p. So, we can take [p; d+ ] as a new interval of possible values of degrees of belief. If L(p) is preferable, this means that d(A) < p, and hence, we can take [d? ; p] as a new interval of possible values of d(A). In both cases, we get a new interval that is half the size of the original one. If we start with an interval [0; 1] of size 1, and repeat this procedure k times, we get an interval of size 2?k , i.e., we have determined d(A) with an accuracy 2?k .


1.3. AND and OR operations with degrees of belief

Suppose that an expert system contains statements A and B , and we have elicited the degrees of belief d(A) and d(B ) from the experts. Suppose now that a user wants to know the degree of belief in a composite statement A&B . In principle, knowing only the two numbers d(A) and d(B ) is not sucient to describe the expert's degree of belief in A&B : e.g., if d(A) = d(B ), it could be that an expert perceives A and B as equivalent, in which case d(A&B ) = d(A), or as independent, in which case the possibility of A and B being true is smaller than the possibility that one of them is true: d(A&B ) < d(A). So, the ideal situation would be to elicit from an expert not only the degree of belief in the basic statements from the knowledge base, but also in all possible logical combinations of these statements. However, this is practically impossible: if we have N statements S1 ; :::; SN in the knowledge base, then for each of 2N ? 1 non-empty subsets fSi1 ; :::; Si g of the knowledge base, we need to elicit the degree of belief in the corresponding ANDstatement Si1 &:::&Si . For a realistic expert system, N is in hundreds, so asking an expert 2N questions is impossible. Therefore, although in some cases, we will be able to have the degree of belief d(A&B ) stored in the knowledge base, in general, we often have to deal with a following situation: we know the degrees of belief d(A) and d(B ) in statements A and B , we know nothing else about A and B , and we are interested in the (estimated) degree of belief of the composite statement A&B . Since the only information available consists of the values d(A) and d(B ), we must compute d(A&B ) based on these values. We must be able to do that for arbitrary values d(A) and d(B ). Therefore, we need a function that transforms the values d(A) and d(B ) into an estimate for d(A&B ). Such a function is called an AND-operation. If an AND-operation f& : [0; 1] [0; 1] ! [0; 1] is xed, then we take f& (d(A); d(B )) as an estimate for d(A&B ). Similarly, to estimate the degree of belief in A _ B , we need an OR-operation f_ : [0; 1] [0; 1] ! [0; 1]. These operations must satisfy some natural conditions. For example, for an expert, A&B and B &A mean the same. Therefore, the estimates f&(d(A); d(B )) and f&(d(B ); d(A)) for these two statements should coincide. To achieve that, we must require that f& (a; b) = f& (b; a) for all a and b; in other words, the operation f& must be commutative. Similarly, from the fact that A&(B &C ) and (A&B )&C mean the same, we can deduce the requirement that f& must be associative. If A is absolutely false (d(A) = 0), then A&B is also absolutely false, i.e., f& (a; 0) = 0 for all a. If A is absolutely true d(A) = 1, then A&B is true i B is true, so, the degree of belief in A&B must coincide with the degree of belief in B : d(1; b) = b for all b. If our belief in A or B increases, then the degree of belief in A&B must also increase, so f& must be monotonic. If the degree of belief in A changes a little bit, then the degree of belief in A&B must also change slightly. In other words, f& must be continuous. So, we arrive at the following de nition: k

k


De nition 1. By an AND-operation, we mean a commutative, associative, monotonic, continuous operation f& : [0; 1] [0; 1] ! [0; 1] with the properties f&(1; a) = a and f&(0; a) = 0.

By an OR-operation, we mean a commutative, associative, monotonic, continuous operation f_ : [0; 1] [0; 1] ! [0; 1] with the properties f_ (1; a) = 1 and f_(0; a) = a.

A complete classi cation of operations that satisfy these properties has been given in Ling171 (see also1;213;214). The simplest AND and OR operations were rst used for degrees of belief by L. Zadeh in273 : f& (x; y) = min(x; y), f& (x; y) = xy, f_ (x; y) = max(x; y), and f_ (x; y) = x + y ? xy. Since then, several other operations have been proposed. One way to choose an operation is to choose many pairs of statements (Ak ; Bk ), ask experts to estimate d(Ak ), d(Bk ). and d(Ak &Bk ) for every pair, and choose a function f& for which d(Ak &Bk ) is the closest to f&(d(Ak ); d(Bk )) (e.g., in terms of least squares). This empirical approach was rst implemented by the authors of the rst successful expert system MYCIN Shortlie, Buchanan40;216. For dierent elds of expertise, dierent functions f& emerge (for experimental results, see, e.g., Zimmermann et al.275 ). This dierence is easy to explain. If we know a = d(A) and b = d(B ), then we can have more optimistic estimates for d(A&B ) (e.g., min(a; b)) and more cautious ones (e.g., f& (a; b) = a b). In some applications (e.g., in medicine), mistakes can be deadly, so more cautious estimates are needed. In other applications (e.g., in geology), we cannot measure as many parameters as in medicine, so, we have to rely more on expertise, and hence, experts must take risks. In these applications, more optimistic estimates are used: e.g., a geologist starts to drill in the uncertainty in which a surgeon is not likely to start an incision. Similarly, NOT operations f: : [0; 1] ! [0; 1] are de ned. A usual choice is f:(x) = 1 ? x (for degrees of belief from [?1; 1], it is x ! ?x).

1.4. Degree of belief in composite statements

If we x AND, OR, and NOT operations, then we can in principle, knowing the degree of belief in the basic statements, determine the degree of belief in their logical combination Q. To do that, we represent the given formula Q as a combination of &, _, and :, and then consequently use our chosen operations with degrees of belief instead of these logical symbols. De nition 2. Let V be a nite set. Its elements will be called Boolean (propositional) variables. By a propositional formula, we mean an arbitrary expression obtained from Boolean variables by using the symbols &, _, and :.


De nition 3. Let us x an AND-operation f&, an OR-operation f_, and a NOT

operation f: . For every propositional formula F , and for all numbers a1 ; :::; an , we de ne pF (a1 ; :::; aN ) as follows:

If F coincides with the variable Ai , then pF = ai . If F = F1 &F2 , then pF = f& (pF1 ; pF2 ). If F = F1 _ F2 , then pF = f_(pF1 ; pF2 ).

If F = :F1 , then pF = f: (pF1 ).

For example, a formula Q of the type A ! B can be represented as B _ :A, and therefore, as a degree of belief in Q, we can take f_ (d(B ); f: (d(A)). So, if d(A) = 0:6, d(B ) = 0:7, and we use min, max, and x ! 1 ? x for &, _, and :, then we get d(Q) = max(d(B ); 1 ? d(A)) = max(0:7; 1 ? 0:6) = 0:7.

1.5. Intelligent control: brie y

One of the main applications of expert knowledge is intelligent control. In many cases, we want to design an automated controller, but we do not know the exact behavior of the controlled plant, and we cannot therefore use traditional control techniques. In such cases, we often have the expertise of human controllers who know how to control the plant (e.g., how to drive a car, how to ride a plane, etc). These experts cannot formulate how exactly they control in exact terms. Instead, they can formulate their expertise in terms of \if-then" rules of the type: \if the obstacle is nearby, and you are driving with a moderate speed, hit the breaks immediately". There exists a methodology that transforms experts' rules, formulated in terms of words of a natural language, into a precise control strategy. This methodology is called fuzzy control (see, e.g., Kandel et al.109 ). Fuzzy control technique starts with determining the values of the so-called membership functions A (x) that correspond to dierent words A used in the rules (in the above example, such words are nearby, moderate, and immediately): namely, the value A (x) represents the expert's degree of belief that a value x satis es the property A. Then, it computes the degrees of belief in composite statements, and in particular, computes, for each possible values of control u, the degree of belief (u) that this value u is reasonable in a given situation, Finally, based on these values (u), we must choose a single value u of control (the choice procedure is called a defuzzi cation).


2. Intervals are Necessary to Describe Degrees of Belief 2.1. The rst (main) source of uncertainty: experts cannot describe their degrees of belief precisely 2.1.1. Intervals are necessary Above, we described an idealized situation, in which we can describe degrees of belief by exact real numbers. In practice, the situation is more complicated, because experts cannot describe their degrees of belief precisely. Indeed, let us review the above-described methods of eliciting degrees of belief.

If an expert describes his degree of belief by selecting, e.g., 8 on a scale from

0 to 10, this does not mean that his degree of belief is exactly 0.8: if instead, we ask him to select on a scale from 0 to 9, then whatever he chooses, after dividing it by 9, we will never get 0.8. If an expert chooses a value 8 on a 0 to 10 scale, then the only thing that we know about the expert's degree of belief is that it is closer to 8 than to 7 or to 9, i.e., that this degree of belief belongs to the interval [0:75; 0:85]. Another possible source of interval uncertainty is when we have several experts, and their estimates dier. If, e.g., two equally good experts point to 7 and 8, then, if we are cautious, we would rather describe the resulting degree of belief as the interval [0:7; 0:8] (or, in view of the above remark, as the interval [0; 65; 0:85]). This idea is described in Fellner, Walley et al.71;254;255;256 . If we determine the degree of belief by polling, then the same argument shows that the resulting numbers are not precise: e.g., if 8 out of 10 experts voted for A, then we cannot say that the actual degree of belief is exactly 0.8, because, if we repeated this procedure with 9 experts, we will never get exactly 0.8. In this case, there are two other sources of uncertainty: Picking experts is sort of a random procedure, so, the result of voting is a statistical estimate that is not precise (just like a statistical frequency estimate of probability). A better description will be to give an interval of possible values of d(A). The polling method of estimating the degree of belief is based on the assumption that an expert can always tell whether he believes in a given statement S or not. Then, we take the ratio d(S ) = N (S )=N of the number N (S ) of experts who believe in S to the total number N of experts as the desired estimate. For :S , we thus have N (:S ) = N ? N (S ), and so, d(:S ) = N (:S )=N = 1 ? d(S ). In reality, an expert is often unsure about S . In this case, instead of dividing the experts into two categories: those who believe in S and those who do not, we must


divide them into three categories: those who believe that S is true (we will denote their number by N (S )), those who believe that S is false (we will denote their number by N (:S )), and those who do not have the de nite opinion about S ; there are N ? N (S ) ? N (:S ) of them. In this situation, one number is not sucient to describe the experts' degree of belief in S , we need at least two. There are two ways to describe it: We can describe the degree of belief in S as d(S ) = N (S )=N and the degree of belief in :S as d(:S ) = N (:S )=N . These two numbers must satisfy the condition d(S ) + d(:S ) 1. This description is known under three dierent names: interval probability theory (Quinlan201 , Cui et al.55 , Buehrer43 ), intuitionistic fuzzy logic Atanassov, Gargov, et al.4;6;7;8;9;10;12;85 or a vague set Gau et al.86 . (The reason for the word \intuitionistic" is that this logic is close to the original intuitionistic idea that the law of excluded middle is not always true.) We can describe the degree of belief d(S ) in S and the degree of plausibility of S estimating as the fraction of experts who do not consider S impossible, i.e., as pl(S ) = 1 ? d(:S ). This representation is called Dempster-Shafer formalism Dempster56 , Shafer215 , Spies225 , Yager et al.271 (for ecient algorithms, see, e.g.,143 ). If we have chosen d(A) by comparing A with lotteries, then at some point, an expert will be unable to make a reasonable choice. For example, hardly anyone can make a meaningful choice saying that the alternative A is better than a lottery with, say 50.1% of winning, but worse than a lottery with a 50.2% of winning. If we use bisection and do not allow an expert to say \I do not know", then we are forcing the expert to make some decision. The resulting decisions are arbitrary, and thus, may be inconsistent: e.g., if three alternatives A, B , and C are more or less equivalent to an expert, but we force him to choose, then he may choose A from the pair (A; B ), B from the pair (B; C ), and C from the pair (C; A) (see, e.g., Winkler277 , Zimmermann et al.275 ; Kleindorfer et al.114 , Section 4.4). A more realistic estimate would emerge if we allow the experts to say \I do not know" as a result of the comparison. In this case, at some point, we will reach an interval [p? ; p+ ] that de nitely contains our degree of belief, but which cannot be narrowed (because for the values p from this interval, the expert cannot meaningfully compare A with a lottery L(p)). So, to describe degrees of belief adequately, we must use intervals instead of real numbers. The fact that (subjective) probabilities are not precisely determined was noticed rst by the famous economist Keynes (in Chapter 3 of113 ). Keynes not only noticed that these probabilities are not uniquely determined, but he also showed that without these uncertainties bookmakers and insurance companies would never


make a pro t (for further arguments that in insurance problems, probability cannot be precisely determined, see B. Russell206 , pp. 358{360, Einhorn, Hogarth et al.66;67;102 ). In a general context of uncertainty of experts' knowledge, this idea was rst proposed by E. Borel in37 . He considers the (subjective) probability that a patient will recover from illness: It would be moreover be natural enough for Peter, modest and prudent, to refuse to set a precise value of the probability of recovery, but merely arm that in his judgment this probability is between 0.8 and 0.9, and that, under the circumstances, if he is to bet on recovery, he will demand that 0.8 be adopted, but if he is forced to bet on death, he will demand that 0.9 be adopted. This idea was later developed by Good90;91;92;93;95;97 , Smith220 , Fellner70;71 , Kyburg151;152;153;154;155;156;157, Fishburn74 , and Levi164;165;166;167;168 ; see also Williams264;265 , Jahn107, Bandler and Kohout15 , Turksen238;239;240, Dubois63;64 , Leamer162, Snow223 , White263 , Baldwin14, Driankov59;60, Erick69, Neapolitan180 , Buckley42, Ralescu202 , Tessem232 , Fertig72;73 , Kuznetsov150 , Walley254 , Simo217 , Rocha et al.204 . Comment 1. Examples that show that intervals are more adequate in representing uncertainty than numbers are given in Rocha204 . Comment 2. Above, we described how to determine the intervals [a? ; a+ ] that describe our degrees of belief in dierent statements A, from the expert's preferences (revealed by his choice decisions). This problem can be formulated in more general terms: Suppose that we have a (partial) ordering relation > on the set of alternatives (A > B means that A is de nitely more probable than B ). We would like to assign to each of these alternatives A an interval [a?; a+ ] in such a way that A > B i a? > b+ . The natural questions are:

Is such assignment possible? Is it unique? How to compute the values a fast? A survey of the related research is given in Chapter 16 of Suppes et al.230 . (See also Fishburn75;76 , Roberts203 , and Cloteaux et al.49 ). For the resulting algorithms, see, e.g., Mahesh et al.175 and references therein, and Cloteaux et al.49 . Comment 3. From the mathematical viewpoint, interval uncertainty d(S ) assigned to dierent statements S means that instead of a single numerical uncertainty measure S ! d(S ), we have a family of dierent measures. This family consists of all assignments d for which d(S ) 2 d(S ) for all S . This idea (going back to Levi164 ) often helps in the mathematical analysis of interval-valued degrees of belief: see, e.g., Walley254 .


2.1.2. More general formalisms to describe uncertainty

Beyond interval uncertainty: Intervals with intervally uncertain bounds.

If experts that were initially unsure about S make their decision, then the resulting degree of belief in S can take any value from the interval [d(S ); pl(S )]: the value d(S ) corresponds to the case when they all decide against S , and the right-hand side pl(S ) corresponds to the case when all these experts choose to believe in S . We have already mentioned that the ratio N (S )=N depends on N and is therefore, not a very good description of the experts' belief: it is dicult to get the exact values of these two numbers (this was rst noticed by Good92 ). To get a better description, we can use an interval d(S ) = [d? (S ); d+ (S )] of possible values (e.g., d (S ) = (N (S )0:5)=N ). Similarly, we get an interval pl(S ) of possible values of plausibility. As a result, we get a representation of the expert's belief in S by a pair of intervals [d(S ); pl(S )]. This representation was proposed in Atanassov et al.5;11;12;13 , and called interval-valued intuitionistic fuzzy logic. This description is equivalent to the following one: instead of a single interval d(S ) = [d(S ); pl(S )] to represent the expert's degree of belief, we now have two nested intervals: a more cautious one D(S ) = [d?(S ); pl+(S )], and a more risky one d(S ) = [d+(S ); pl?(S )] D(S ).

Further generalization: a set of possible values of degree of belief with degrees of possibility (numerical or interval) attached to each possible value (type 2 uncertainty). Interval description of degrees of belief is based on the idea that an expert cannot express his degree of belief by a single number, so, we present his degree of belief by an interval of numbers that can adequately describe his degree belief. To describe this interval, we must be able, for each number from the interval [0,1], to tell whether or not this number is a possible value of the expert's degree of belief. As we have just mentioned (in the previous formalism), this is not always possible. With some numbers from [0,1], the expert is sure that they can coincide with his degree; with others, he is sure that these numbers cannot adequately represent his degree of belief. However, with other numbers, the expert may be uncertain. So, in order to more adequately describe the expert's beliefs in a statement A, we must, for every value d 2 [0; 1], describe the degree of belief that d can be a degree of belief in A. These degrees of belief, in turn, can be described by intervals, or, to be even more adequate, by \fuzzy" intervals. This approach is called second order probability (n?th order probability if we iterate it many times), type 2 (n) probabilities, beliefs about beliefs, hierarchical models, or type 2 (n) fuzzy; see Good91;92;94;96;97 , Zellner274 , Lindley et al.169;170 , Box et al.39 , Dickey58 , Mellor, Syrms218 , Scum et al.212 , Berger27, von Winterfeldt266 , Pericci200, Pearl197, Walley254 , and references therein. Such a description may be more adequate in modeling human reasoning, but using type 2 models requires many data to store (many numbers or intervals that correspond to dierent d 2 [0; 1], instead of two numbers for an interval representation for a degree of belief), so in practical applications, it is rarely used. The only known applications are to econometrics Zellner274, to legal reasoning Schum et al.212 , and to general purpose


decision support systems Whalen et al.261;262 . 2.1.3. Interval computations for processing interval-valued degrees of belief: general idea Interval computations in expert systems: general idea. For an expert system with interval-valued degrees of belief, the following problem arises: suppose that we have an expert system whose knowledge base consists of statements S1 ; :::; SN , and we have an algorithm f (Q; d1; :::; dN ) (called inference engine) that for any given query Q, transforms the degrees of belief d(S1 ); :::; d(SN ) in the statements from the knowledge base into a degree of belief d(Q) = f (Q; d(S1 ); :::; d(SN )) in Q (for example, if Q = S1 &S2 , then f (d1 ; :::; dN ) = f& (d1 ; d2 )). Suppose now that we know only the intervals d(S1 ), ..., d(SN ) that contain the desired degree of belief. Then, the degree of belief in Q can take any value from the set

f (Q; d(S1 ); :::; d(SN )) = ff (Q; d1; :::; dN ) j di 2 d(Si )g: Computing such an interval is a typical problem of interval computations.

Interval computations in expert systems: software problems and possible solutions. Even for simple algorithms f , computing the exact bounds for

this interval can be computationally intractable. For polynomial functions f , it is proved in Gaganov77. For a speci c problem involving interval degrees of belief it is illustrated in Walley254 , Section 5.8. There are two ways to solve this problem: First, we can apply general interval computation tools and compute an enclosure for the desired interval. Second, we can design speci c interval tools that will lead to a reasonable solution for functions f stemming from processing interval-based degrees of belief. Such tools are described, e.g., in DeRobertis57 , Erick69, Walley et al.257 ; in254 , Sections 2.9, 4.6, 5.3, and 7.8, and in Tessem235 . In particular, since the functions f& and f_ are increasing in both arguments, we have f& ([x? ; x+ ]; [y? ; y+ ]) = [f& (x? ; y?); f& (x+ ; y+ )] and f_ ([x? ; x+ ]; [y? ; y+]) = [f_(x? ; y? ); f_(x+ ; y+ )]: For example, min([x? ; x+ ]; [y?; y+ ]) = [min(x? ; y? ); min(x+ ; y+ )]; and

max([x? ; x+ ]; [y?; y+ ]) = [max(x? ; y? ); max(x+ ; y+)]: One additional problem here is that interval computation algorithms are most frequently written in procedural languages (versions of C, Fortran, Pascal, etc),


while inference engines are often written in declarative languages like Prolog (i.e., languages in which we specify what we want, and not how it should be computed). It is desirable therefore to describe the necessary interval algorithms in the same language. For Prolog-based inference engines, this is done in Cortes53;54 and Buehrer43 . An alternative approach is to use a dierent language, speci cally tailored for processing interval degrees of belief. In particular, in Cooke et al.52 it is shown that a new language based on bags can reasonably speed up such computations (a bag, or a multiset diers from a set in that several identical elements are counted as dierent members of the set; in mathematical terms, bags on a universal set U can be identi ed with their characteristic functions, i.e., arbitrary functions U ! N ). The third approach is to try dierent computer hardware.

One possibility is to use the existing computation-speeding general purpose

hardware, i.e., parallelism (see a general survey142). In many cases, the corresponding problems admit a natural parallelization. Namely, our knowledge is usually (more or less) compartmentalized. This means that when we look for a solution to a mathematical problem, we know for sure that our knowledge of, e.g., ethics will not help. So, for any given query, we do not need to look into all possible rules and facts from the knowledge base: only into those that are relevant. So, if we have several queries that are relevant to dierent parts of the knowledge base, then in principle, we can handle them in parallel. Even if we have one query, we can still get a speedup if we try to relate this query to dierent parts of the knowledge base (this can be done in parallel). If in an attempt to answer the query, the computer will generate several auxiliary queries, they can also be answered in parallel. Such natural parallelizations are described, e.g., in Swain et al.231 and Lassez et al.158 . (Examples of such parallelizations are given in the Applications section.) Another source of parallelization is parallelizability of an algorithm. In particular, in Bernat, Cortes, Villa et al.31;53;54;144;145;146;253 , a Monte-Carlo-type method is used to estimate the desired interval: namely, we perform several simulations, and then process the results of these simulations. These simulations are independent, and therefore, they can execute on dierent processors in parallel. If we do not have the actual parallel machine, we can use several processors connected into a net (Bernat, Bhamidipati, Morgenstein et al.32;33;176 ).

To achieve an even better speed up, we can tailor the connections between

the processors to our problems. If the knowledge base can be often divided into several weakly related parts, then, we can assign to each part a processor that would handle this part of the database. How to connect these processors? If two parts have something in common, so that from time to time, the corresponding processors need to exchange information, then we should make a direct link between them, so that the communication will take one


step only. On the other hand, if two parts have practically nothing in common, and therefore hardly ever communicate with each other, then it makes no sense to add a link (because an extra link has to be maintained, and this \eats up" additional computer time). So, the graph of connections between the processors must be ideally the same as the graph of connections between the parts. This specialized (domain-dependent) architecture is described, e.g., in Swain231 and Lassez et al.158 . For Monte-Carlo type algorithms, asymptotically optimal parallel architecture is described in Villa et al.253 .

Finally, we can further sped up computations if we not only make connections

between the processors tailor-made for our problems, but tailor each processor, so that it will hardware support interval computations required for these applications. In particular, in Nakamura et al.148;179;184;185;189 , it is shown that in order to make operations with interval degrees of belief faster, it is desirable to hardware support not only traditional operations (with one and two operands), but also operations with three or more operands.

Interval computations in intelligent control: general idea. In intelligent control, we have an algorithm f (x1 ; :::; xn ) that transform the parameters xi (degrees of belief in the rules and the values of the input parameters) into the resulting control value u = f (x1 ; :::; xn ). If instead of the exact values of xi , we only know the intervals xi of possible values of these parameters, we end up with an interval of possible values of control: u = f (x1 ; :::; xn ). This is also a particular case of the general interval computation problem. For speci c algorithms used in intelligent control, methods of estimating this interval have been proposed in Lea, Nguyen, Bouchon-Meunier et al.36;159;160;161;187 . In some cases, these intervals become too wide, so wide that they are practically meaningless. (Instead of giving us some directions, they just say \apply any control you want".) In this situation, a natural idea is to choose the AND-OR operations and the membership functions in such a way that the resulting intervals will be the smallest. De nition 4. Let f be an AND- or an OR-operation,a nd let > 0. By a ?sensitivity rf () of f , we mean the maximum width of the interval f (a; b) for all intervals a and b of width . We say that an operation f is better than an operation g if for suciently small (i.e., for all 0 for some 0 ), the ?sensitivity of f is not larger than the ?sensitivity of g: rf () rg (). Proposition 1. (Nguyen et al.187;188;190;193 ) min is the best &?operation, and max is the best OR-operation. Similarly, we can prove that linear extrapolation leads to the least sensitive membership functions Nguyen et al.187;188;190;193 (see also191).


Generalized interval computations (with tetrvals) to handle intervals with intervally uncertain bounds. For intervals with intervally uncertain bounds, we have a modi ed problem: for expert systems, we have a knowledge base consisting of statements S1 ; :::; SN , and for a given query Q, we have an algorithm (obtained by using the inference engine) f (d1 ; :::; dN ) that transforms the degrees of belief d(S1 ); :::; d(SN ) in the statements from the knowledge base into a degree of belief d(Q) = f (d(S1 ); :::; d(SN )) in Q. For intelligent control, we have an algorithm that transforms these same degrees of belief into the recommended control value u = f (d(S1 ); :::; d(SN )). Suppose now that we know only the nested intervals d(S1) D(S1), ..., d(SN ) D(SN ) that describe the expert's degrees of belief. From this information, we want to compute the nested interval d(Q) D(Q) (or u U) for the result. Here,

d(Q) = f (d(S1); :::; d(SN )) = ff (d1; :::; dN ) j di 2 d(Si )g; D(Q) = f (D(S1); :::; D(SN )) = ff (d1; :::; dN ) j di 2 D(Si )g (and similarly for u and U). So, one way to compute the desired nested interval is to

apply interval computations techniques twice and compute both intervals. Another possibility is to apply algorithms that handle the nested interval as a single object. (This sometimes leads to a better computational performance.) Such methods were proposed in Garde~nes79, Zyuzin280;281 , Musaev177;178, where the nested interval is called a twin or a tetrval. Comment. A similar notion of \uncertainty of systematic uncertainty" has also been proposed in Loo172 in slightly dierent terms, but, as shown in Artbauer3, it is essentially a tetrval. 2.1.4. Operations with interval degrees of belief In the above approach, there is a slight methodological inconsistency: namely, assume that we know the intervals d(A) and d(B ) of possible values of d(A) and d(B ), and we want to nd the interval d(A&B ) of possible values of d(A&B ). To compute this estimate, we apply the function f& (a; b) that was chosen based on the assumption that degrees of belief can be described by numbers. It is therefore desirable to reformulate and re-solve the problem of describing AND and OR operations for the new situation, in which the degrees of belief are intervals, not numbers. This reformulation was done in Zuo et al.278 for the case of strictly monotonic operations:

De nition 5a. (Zuo et al.278 ) Let I denote the set of all subintervals of the interval [0; 1]. A function f& : I I ! I is called an interval AND-operation if it

is commutative, associative, continuous (in the natural component-wise topology), satis es the \boundary" conditions f& (a; [0; 0]) = [0; 0], f& (a; [1; 1]) = a, is inclusion monotonic (if a1 a2 and b1 b2 , then f& (a1 ; b1 ) f&(a2 ; b2 )), and strictly monotonic (if a1 < a2 and b1 < b2 then f& (a1 ; b1 ) < f& (a2 ; b2 ), where [a?1 ; a+1 ] < [a?2 ; a+2 ] means a+1 < a?2 ).


De nition 5b. (Zuo et al.278 ) A function f_ : I I ! I is called an interval OR-

operation if it is commutative, associative, continuous (in the natural componentwise topology), satis es the \boundary" conditions f_ (a; [0; 0]) = a, f& (a; [1; 1]) = [1; 1], is inclusion monotonic and strictly monotonic. Proposition 2 (Zuo et al.278 ).

Every interval AND-operation has the form f&(a; b) = f (a; b) for some (numerical) AND-operation f : [0; 1] [0; 1] ! [0; 1]. Every interval OR-operation has the form f_(a; b) = f (a; b) for some (numerical) OR-operation f : [0; 1] [0; 1] ! [0; 1].

So, every intervally de ned operation can be reduced to a pointwise one. This result justi es the above approach. Comment. A dierent justi cation for the same reduction is given in Gehrke et al.87 (see also Sections 4.5 and 5.6 of 193 ).

2.2. Additional source of uncertainty: The degree of belief in a composite statement is not uniquely determined by the degrees of belief of its components

Additional interval uncertainty comes from the above-mentioned fact that the degree of belief d(A&B ) in, say, A&B is not uniquely determined by the degrees of belief d(A) and d(B ) in A and B : if the statements A and B are independent, then we have one result, and if they are correlated, we have another one. So, if we only know the degrees of belief in A and B , then we have an interval of possible values of degree of belief in A&B . This idea was described and used in Bandler et al.15;16;17;19;20;21;22;23;24 , Keravnouet al.112 , Kohout et al.118;119;120;121;122;123;124;125;126;127;128;129;130;131;132;133;134;135;136;137;138;139 , Gao et al.78 , Nilsson194 , Grosof99, Stiller et al.227 , Kaliappan et al.110 , Ben-Ahmeida et al.26 , San-Andres et al.207 . We can determine these intervals from the following idea: in the majority of the methods of determining d(A), the degree of belief is determined as a probability or a frequency. So, we can formulate the following mathematical problem: De nition 6. Let F (A1; :::; An ) be a propositional formula with Boolean variables A; :::; B . Let a1 ; :::; an be numbers from the interval [0; 1]. We say that a number p is a possible probability of F is there exists a probabilistic space with a probability measure P , and predicates A1 (!); :::; An (!) on for which P (A1 (!)) = a, ..., P (An (!)) = b, and P (F (A1 (!); :::; An (!)) = p. The set of all possible probabilities of F will be denoted by pF (a1 ; :::; an ). In principle, we can compute this set using linear programming (Nilsson194 , Tessem234 ). The idea is that in reality, each of n statements Ai is either true or false. Fixing the truth value of these n statements will thus x the truth value of the formula F . Therefore, we have 2n possible situations s depending on which of


these statements are true. To describe the probability distribution, it is sucient to know the probabilities p(s) of these 2n situations (e.g., for n = 2, we have 4 possible situations: A1 &A2 , A1 &:A2 , :A1 &A2 , and :A1 &:A2 ). The sum of these probabilities must be equal to 1, and for every i, the sum of the probabilities of the situations in which Ai is true must be equal to ai . Under these conditions, p is equal to the sum of all the values p(s) for all situations s in which F is true. So, to nd the smallest and the largest possible values of p, we must solve the following linear programming problem:

X p(s) ! max (min)

under the conditions

s!F

p(s) 0;

X p(s) = 1; X p(s) = a : s!Ai

i

In particular, for simple propositional formulas, we get the following results: Proposition 3. (see, e.g., Walley254, Section 2.7.4). For F = A&B , pF (a; b) = [max(0; a + b ? 1); min(a; b)]. For F = A _ B , pF (a; b) = [max(a; b); min(1; a + b)]. For F = A ! B , pF (a; b) = [max(b; 1 ? a); min(1; 1 ? a + b)]. For F = :A, pF (a) = f1 ? ag. In terms of interval computations, this means that not only the inputs (degrees of belief ai in statements Ai from a knowledge base) are intervals, but also the function pF (a1 ; :::; an ) that transforms these values into the degree of belief d(Q) in a given query Q is also interval-valued. A similar linear programming method works for the case when instead of knowing the exact values ai of p(Ai ), we know the intervals ai = [a?i ; a+i ] of possible P values of p(Ai ).PIn this cases, instead of p(s) = ai , the corresponding inequalities would be a?i p(s) a+i . In this case, we are computing the value pF (a1 ; :::an ). Linear programming problems are known to be solvable in time that is polynomial in the size of the problem111 (i.e., in the total number of variables and restrictions). However, this problem has 2n variables p(s) that correspond to 2n possible situations; so, polynomial in 2n means exponential in n, which for reasonable n (e.g., 300) means longer than the lifetime of the Universe. Can we compute it faster? It is very doubtful, because the following result can be proven: Proposition 4. (Chee47;48) The problem of computing pF (a1 ; :::; an) for a given formula F and given intervals a [0; 1] is intractable (NP-hard). Comment. Intractable, crudely speaking, means that if we were able to solve this problem in a reasonable time, then we would have an algorithm that solves practically all discrete problems in reasonable time, which is considered to be impossible i


(for formal de nitions, see, e.g., Garey et al.84 ). The proof of this Proposition can be done as follows: it is known that the problem of checking whether a formula F is satis able (i.e., whether we can make F true by assigning some Boolean values to its variables) is intractable. If a formula F is not satis able, that means that it is always false, and hence, pF ([0; 1]; :::; [0; 1]) = f0g. In particular, this means that the upper bound p+F ([0; 1]; :::; [0; 1]) of this interval is equal to 0. If F is satis able by some Boolean values t1 ; :::; tn , then, setting Ai = ti with probability one, we get p = 1, and hence, p+F ([0; 1]; :::; [0; 1]) = 1. So, if we were able to compute the desired interval, we would thus be able to check whether a given formula is satis able. The situation is very similar to interval computations in general: there, due to intractability of the basic problem of interval computations, we cannot compute the exact interval f (a1 ; :::; an ); therefore, we compute the enclosure for that interval, using, e.g., naive interval computations whose idea is to follow the arithmetic operations that constitute f step by step, and replace them with operations with intervals. A similar idea leads to an enclosure for interval degrees of belief (and this idea is actually used): De nition 7. For every propositional formula F , and for all intervals a1 ; :::; an , we de ne pF (a1 ; :::; aN ) as follows: If F coincides with the variable Ai , then pF = ai . If F = F1 &F2 , then pF = p& (pF1 ; pF2 ) (where p& (a; b) is de ned by Proposition 3). If F = F1 _ F2 , then pF = p_(pF1 ; pF2 ) (where p_ (a; b) is de ned by Proposition 3). If F = :F1 , then pF = p: (pF1 ) (where p: (a) is de ned by Proposition 3). Proposition 5. For every formula F , and for all intervals a1 ; :::; an ,

pF (a1 ; :::; aN ) pF (a1 ; :::; aN ):

Comment. Usually, dierent statements from the knowledge base are not independent. As a result, known degrees of belief in some of the statements may change the degrees of the others that are their consequences. For example, if A implies B , d(A) = [0:6; 0:7], and d(B) = [0:3; 1], then we can conclude that d(B) d(A) 0:6, and therefore, from the same knowledge base, we can extract a new interval degree of belief for B : d(B ) = [0:6; 1]. This internal updating of degrees of belief can also be done by interval computation.

2.3. The third source of interval uncertainty: Dierent representations of the same query

Even if we x a function that transform d(A) and d(B ) into a reasonable estimate for d(A&B ), still, for more complicated expressions, we will have an additional problem


caused by the fact that every expression, can be described in several dierent ways in terms of the basic logical operations &, _, and :. For example, A ! B can be represented as B _ :A, (A&B ) _ :A etc. These expressions are equivalent in normal Boolean (2{valued) logic, but if we use these expression to compute degrees of belief, we sometimes end up with dierent results. E.g., in the above case, if d(A) = 0:6 and d(B ) = 0:7, and we use min, max, and x ! 1 ? x for &, _, and :, then the rst expression leads to max(d(B ); 1 ? d(A)) = 0:7, while the second leads to max(min(d(A); d(B )); 1 ? d(A)) = max(0:6; 0:4) = 0:6. So, for such an expression F , instead of the exact value of d(F ), we end up with dierent possible values of d(F ). It is therefore desirable to describe the interval formed by the smallest and the largest possible values of d(F ) for all F that correspond to a given formula. This idea is described in Turksen et al.34;98;116;238;239;240;241;242;243;246;247;251;272;279 . It turns out that if we use min and max, then the smallest and the largest values can be described as follows:

De nition 8. By a propositional formula in a CNF (conjunctive normal form), we mean a formula of the type C1 _ ::: _ Cm , where each Cj is of the type x1 &:::&xp , and

xi are either the basic statements or their negations. We say that we have a complete CNF if each Cj contains all variables from the formula. By a propositional formula in a DNF (disjunctive normal form), we mean a formula of the type D1 &:::&Dm, where each Dj is of the type x1 _ ::: _ xp , and xi are either the basic statements or their negations. We say that we have a complete DNF if each Dj contains all variables from the formula. Denotation. Every propositional formula can be transformed into a unique complete CNF or into a uniquely de ned complete DNF form. These unique formulas will be denoted by CNF(F ) and DNF(F ). For example, A ! B can be transformed into a complete DNF form :A _ B , or into a complete CNF form (A&B ) _ (:A&B ) _ (:A&:B ). Proposition 6. (Zuo, Turksen et al.279 ). Let f& = min, f_ = max, and f:(x) = 1 ? x. Then: For every propositional formula F (A; :::; B ), and for all values a; :::; b, pDNF(F ) (a; :::; b) pF (a; :::; b) pCNF(F ) (a; :::; b):

For every propositional formula F (A; :::; B ), and for all intervals a; :::; b [0; 1],

? + + p? DNF(F ) (a; :::; b) pF (a; :::; b) pF (a; :::; b) pCNF(F ) (a; :::; b):

So, for every formula F , we can take [pDNF(F ) (a; :::; b); pCNF(F ) (a; :::; b)]


or

+ [p? DNF(F ) (a; :::; b); pCNF(F ) (a; :::; b)]

as the desired interval. If we use other operations for AND and OR, then we still have DNFCNF (Bilgic et al.34 ), but other expressions can lead to degrees of belief arbitrarily close to 0 and 1 (so the desired interval is [0,1]; Zuo et al.278 ). 2.3.1. Interval degrees of belief explain \inconsistency" of human reasoning Experiments has shown (see, e.g., Zimmermann et al.275 ) that the connectives that we humans use do not always satisfy natural properties like associativity. In other words, people assign dierent degrees of belief to statements A&(B &C ) and (A&B )&C . This makes no sense, because these two statements are clearly equivalent. According to Turksen et al.245;252 , this \inconsistency" is caused by the fact that (as we have already mentioned) in the usual experiments, we \force" an expert to describe his degree of belief by a single number. Since an expert usually cannot describe his degree of belief exactly, after some time, he just chooses an arbitrary number from the interval that represents his actual degree of belief. If we interview him twice about one and the same statement but presented in the dierent format, then this expert will have identical intervals that express his degree of belief, but due to arbitrariness of his choice of a number, he may choose dierent numbers from this same interval during two dierent interviews, this making us think that his degrees of belief in A&(B &C ) and (&B )&C are dierent. If we allow the expert to stop and to return an interval instead of a number, then this inconsistency disappears. Indeed, the results of the operations described experimentally in275 , are always between DNF and CNF bounds of Turksen. 2.3.2. Interval-valued degrees of belief explain the existing form of fuzzy control The knowledge base for control typically consists of \if-then" rules of the type \if x1 is small, and x2 is medium, then the control u should be medium". In other words, we have statements R of the type A1 (x1 )&:::&An (xn ) ! B (u). Traditional approach to fuzzy control (originally proposed by Mamdani) replaces each rule of this type with a statement A1 (x1 )&:::&An (xn )&B (u) (Mamdani et al.173;174 ). This approach has immediately lead to successful real-life applications173;174, and it is one of the most frequently used in fuzzy control109;259. This methodology is very successful, but it has a serious methodological problem: in applying it, we interpret \implies" as \and". From the viewpoint of logic, it is reasonably weird: both in classical logic and in operations with degrees of belief, implication and conjunction are two dierent operations. Using interval helps to solve this problem. Indeed, we are only worried about the degree of belief in the rule if all the conditions are satis ed, and if the conclusion is true. In other words, let us consider the case when our degree of belief in each of the statements A1 ; :::; An , and


B exceeds the degree of belief in their negations. If we use x ! 1 ? x for negation, then d(A) > d(:A) is equivalent to d(A) > 0:5. So, we are interested in the case when d(A1 ) > 0:5, ..., d(An ) > 0:5, and d(B ) > 0:5. If we want the most cautious control, then we would like to get the lower bound

d? (R) for the truth value of the rule R. Proposition 7. Let R = (A1&:::&An) ! B, f& = min, f_ = max, f:(x) = 1 ? x, and D = A1 &:::&An &B . If ai 0:5 for all i, and b 0:5, then pDNF(R) (a1 ; :::; an ; b) = pD (a1 ; :::; an ) = min(a1 ; :::; an ; b):

If a?i 0:5 for all i and b? 0:5, then ? ? ? ? p? DNF(R) (a1 ; :::; an ; b) = pD (a1 ; :::; an ; b) = min(a1 ; :::; an ; b ):

Proof. Indeed, according to the Proposition 6, this lower bound is attained on a complete DNF form of R. This DNF form is of the type D1 _ D2 _ ::: _ Dm , where D1 = (A1 &:::&An &B ) and all other terms Dj , j 2, contain at least one

negation. Here, d(D1 ) = min(d(A1 ); :::; d(An ); d(B )) > 0:5. For negation of one of the variables Aj or B , d(:Aj ) < 0:5. Therefore, for every Dj that contains negations, we have d(Dj ) = min(d(x1 ); :::) < 0:5. Hence d(D1 ) > 0:5 > d(Dj ) for all j 2, and so, d? (R) = max(d(D1 ); d(D2 ); :::; d(Dm )) = d(D1 ). . This Proposition explains why we were able to replace the rule R by a conjunction D = D1 . 2.3.3. Relationship with logic programming In some cases, dierent reformulations of the query lead to dierent degrees of belief, in some other cases the resulting degrees of belief coincide. It is therefore interesting to know when for two propositional formulas F and G with propositional variables A1 ; :::; AN , the computed degrees of belief pF (a1 ; :::; an ) and pG (a1 ; :::; an ) coincide for all for all possible interval degrees of belief ai in Ai . In Kosheleva, Nguyen, et al.140;181;182;184;185;193 , it is shown that this equality holds i formulas F and G are equivalent in the sense of some formalism related to logic programming. This result enables to describe an algorithm that checks whether two formulations F and G are equivalent in this sense or not.

2.4. Yet another source of interval uncertainty: backward reasoning (abduction)

Traditionally, in logic in general and in expert system applications of logic in particular, we use forward reasoning (also called deduction), in which from A and A ! B , we conclude that B is true. In common sense, in addition to this well-formalized forward reasoning, there is also a backward reasoning (also called abduction), in


which from A ! B and B , we conclude that A is probably true. For example, we know that after the rain, the ground is wet. So, if we wake up in the morning and see that the ground is wet (B ), we conclude that it was probably raining during the night (A). There are many fuzzy versions of deduction; the main problem here is to decide, if both A and A ! B are partially true (i.e., their degrees of truth or belief d(A) and d(A ! B ) are described by numbers from the interval (0; 1)), what is the resulting degree of truth for the conclusion B ? Typically, since B is true if both A and A ! B are true, we take the result f&(d(A); d(A ! B )) of applying an and-operation f& (e.g., min) to d(A) and d(A ! B ) as an estimate for d(B ). Based on this formalization-of fuzzy deduction, it seems natural to formalize fuzzy abduction d(B ) as follows: if we know the values d(B ) and d(A ! B ), then we can take, as an estimate for d(A), the value for which f& (d(A); d(A ! B )) = d(B ). The problem is that, e.g., for min, there are often many values d(A) that satisfy this equation: namely, if d(A ! B ) > d(B ), then this equation has only one solution d(A) = D(B ); however, if d(A ! B ) = d(B ), then any value d(A) from the interval [d(B ); 1] is the desired solution. Arnould et al.2 suggest to take the entire interval of solutions as the degree of belief in A resulting from fuzzy abduction.

3. Applications

Comment. A brief overview of some applications is given in Kreinovich et al.147 .

3.1. Applications to expert systems Case study: general expert systems. In Quinlan201, in Buckley et al.42 , and

in Eick et al.65 , the degrees of belief in rules from expert systems are described by intervals. Often, several dierent expert systems describe the same domain. In this situation, it is desirable to select the expert system whose conclusions are the least uncertain. For interval-values uncertainty, dierent measures of uncertainty have been proposed and analyzed in205;249 (see also references therein).

Decision making (Kmietowicz et al.117 , Wolfenson et al.267 , Walley254). The goal

of an expert system is to make a decision. If a system produces a numerical degree of belief, then we can estimate whether a proposed decision will satisfy the goal or not, and use this decision if the resulting degree of belief in its success is high enough (larger than d0 for some a priori chosen threshold d0 ). If we have several possible decisions, then we choose a one whose chances of succeeding are the best, i.e., the one with the largest degree of belief. If for every possible decision, a system generates an interval degree of belief ? [d ; d+ ] that this decision is right, then several decision-making procedures are possible. For example, suppose that we want to decide whether to apply a given decision or not. Then, if d+ d0 , then every d from the interval [d? ; d+ ] is smaller than d0 , and so, we reject this decision. Similarly, if d? > d0 , we accept the decision.


In the remaining cases (when d? d0 < d+ ), we have at least three options: We can be very cautious, and reject the decision, because it could be that this decision is wrong (this is a reasonable thing to do in areas like medicine, where an error can be deadly). We can be very risky, and accept the analyzed decision (makes sense in areas like geophysics). We can refuse to make a decision at this time, and request more information. If we must choose between several alternatives, and each alternative is characterized by the interval degree of belief [d?i ; d+i ] that it will succeed, then we also have three options: We can be very cautious, and choose an alternative whose \guaranteed" performance d?i is the best: d?i ! max. This cautious approach is also called a minimax approach Gardenfors et al.82;83 (see also Nguyen et al.192 ). We can be very risky, and choose an alternative whose most optimistic performance d+i is the best: d+ ! max. This is called the maximal approach (see, e.g., Kleyel et al.115 ). We can select one set of degrees of belief (i.e., one subjective probability distribution) and make decisions based on the selected one: We can use Hurwicz pessimism-optimism criterion (originally proposed in103 ): namely, we choose a real number 2 [0; 1], and choose an alternative A for which the combination

d(A) = d? (A) + (1 ? ) d+ (A) takes the largest possible value (Lesh163 , Bergsten28;29;30, Strat228;229 , Schubert209;210;211 ). We can also nd a distribution with, e.g., maximal entropy, and make decisions based on that distribution (see, e.g., Dubois, Nguyen, Smets, et al.62;192;219;229 ). We can reject all alternatives i that are de nitely worse than the others (i.e., for which d+i d?j for some j ), and require additional information to choose between the remaining ones250 . In some cases, decision is based on the value of some criterion u(A) for each alternative A; this value u(A) is, in itself, a function of the degrees of belief and is therefore, characterized by an interval of possible values u(A). In these cases, we can also use one of the above-described approaches: pessimistic one, in which we choose A for which u?(A) ! max, optimistic one, when we choose A for which u+ (A) ! max, and Hurwicz approach when we choose A for which u? (A)+(1?)u+ (A) ! max.


This is done in the above references; the use of Hurwicz criterion is axiomatically justi ed by Jaray104;105;106. This use can be also justi ed by assuming that the transformation of an interval into a point should not depend neither on the units in which we measure u (i.e., be scale-invariant), nor on the choice of the starting point (i.e., be shift-invariant). De nition 9. By a choice function, we mean a function s that maps every interval [u?; u+ ] into a point from that interval, and that has the following properties for every interval and for every c and > 0: s([u? + c; u+ + c]) = s([u? ; u+ ]) + c (shift-invariance);

s([ u?; u+]) = s([u? ; u+]) (unit-invariance). Proposition 8. Every choice function has the form s([u? ; u+ ]) = u? + (1 ? ) u+ :

This Proposition is actually proven in Nguyen183. Comment. In Whalen et al.261;262 , these ideas are applied to type 2 uncertainty: namely, maximum entropy method is used to choose a probability distribution on the set of all possible combinations of degrees of belief, and Hurwicz criterion is then used to make a decision. Pattern recognition. (Kuncheva149). In fuzzy pattern recognition methods, we assign, to each object x and to each class C , a (numerical) degree of belief C (x). An object x is then classi ed as belonging to the class C if C (x) for some threshold value > 0. If we use (more adequate) interval-valued degrees of belief [?C (x); +C (x)], then it makes sense to select two dierent threshold values ? and + , and to classify an object x as belonging to the class C if both ? (x) ? and + (x) + . In Kuncheva149, an example is given in which this interval-valued fuzzy approach leads to a better classi cation than the number-valued fuzzy approach. Case study: medical expert systems. A speci c feature of medical systems, as we have already mentioned, is that one needs to be very cautious before making a decision. Traditional expert systems use a numerical degree of belief. As a result, we cannot distinguish, e.g., between the following two cases: if d = 0:5, this can mean the degree of belief that this particular treatment will help (in which case the expert's belief that it will help is equal to his believe that it won't, so we probably will not use it), or it can mean that the existing information is not sucient to make any decision about it, in which case we need to make some additional tests before we make a decision. In order to distinguish between these two cases, we can use the interval degrees of belief: a bad treatment will lead to b 0:5 (e.g., b = [0:4; 0:6]), while the lack of information will mean the the resulting degree of belief is d = [0; 1]. In the rst case, we reject the treatment.


In the second case, we request additional information. With regards to medical problems, this idea was originally proposed in Savage208, p. 133, and in Levi166 , pp. 214{215. In Bandler et al.15;16;17;19;20;21;22;23;24 , Keravnouet al.112 , Kohout et al.118;119;120;121;122;123;124;125;126;127;128;129;130;131;132;133;134;135;136;137;138;139 , Gao et al.78 , Stiller et al.227 , Kaliappan et al.110 , Ben-Ahmeida et al.26 , San-Andres et al.207 , the fact that it is more cautious and more reliable to describe the degree of belief in a composite statement by an interval, is used to improve the performance of a medical expert system CLINAID. Similar ideas are described in Pereira et al.198;199 and in Burusco et al.44 . For a general purpose medical system, the area of knowledge is naturally divided into several subareas (corresponding to dierent medical specializations), so here, a natural parallelization can be used. This parallelization is presented in Kohout et al.135;136;139 . Case study: psychological research. In Guguljanova100, interval-valued fuzzy sets are used in psychology, to describe such vague notions as \tendency to antisocial behavior". Some test results indicate that such a tendency exists; some other test results indicate that a person is not asocial. Extroverts, usually, either pass or fail these tests. As a result, for an extrovert person, the tendency to anti-social behavior can be characterized by the ratio r of anti-sociality tests that this person has passed. This ratio is usually interpreted as a (fuzzy) degree with which this person has anti-social tendencies. For introverts, however, test results often do not reveal whether a person has anti-social tendencies or not. Thus, for an introvert person, the tendency to antisocial behavior can be only characterized by an interval [r? ; r+ ], where r? is the ratio of anti-sociality tests that this person has passed, and r+ is the ratio of antisociality tests that this person has not failed (i.e., either passed, or ended inconclusively). Case study: musicology. Traditional musicological notions of a musical sound, pitch, melody, etc., are often not precisely de ned. Fuzzy logic has been used to describe the corresponding uncertainty, but the resulting formalization is still sometimes not adequate. Botusharov38 suggests that a more adequate description can be made in interval-valued fuzzy terms. Case study: civil engineering. In Blockley35, interval degrees of belief are applied to the following problem: we have several abandoned mines; once in a while one of them collapses. What is the degree of belief that once of them collapses? We want to be safe, so we describe the degree of believe by an interval, and consider the mine dangerous if the upper bound of this interval exceeds a certain value. If this happens, then we consider the results of possible use of one of the strategies: reinforcing the mine, lling it with sand, etc., and we choose the cheapest strategy that makes this mine safe (Prolog is used to store the knowledge and to make conclusions).


Case study: power engineering. Thorp et al.236 use interval estimates for

experts degrees of belief to predict production costs of an electric utility. Walley et al.255 use a similar technique to analyze the economic viability of solar energy systems; the result of this research is that based on the available information, both investment plans analyzed in this paper are reasonable, and we cannot decide which one of them is better. In both papers, intervals of degrees of belief cover degrees of belief of dierent experts. Case study: trac congestion. In Palacharia et al.196 , interval estimates are applied to predict the probability of dierent levels of road trac congestion. Case study: automobile marketing. In Wang276, neural networks are used to interpolate an interval membership function. The results are applied to predict the consumer's ranking of a car based on the car's parameters like price, fuel consumption, etc. Case study: testing software. One way to estimate the quality of the software is to use the history of the bugs that have been found, and to extrapolate that history to predict when the next bug will surface. In Gemoets et al.88 , expert knowledge with interval degree of belief is used for this prediction. Several invariance demands (e.g., invariance w.r.t. the way bugs are counted) are used to select an appropriate family of membership functions. Case study: measurements. For many measuring instruments, in addition to guaranteed error bounds (e.g., \error is always 0:1 units"), we may have additional expert knowledge (of the type \most probably, the error is 0:05 units"). To describe this knowledge, Solopchenko et al.224 propose to use interval-valued degrees of belief. Case study: solder joint inspection. Inspecting assembled printed circuit boards (with component soldered to it) is a very time-consuming and unreliable process, that needs to be automated. The most dicult part of this automation is automating the inspection of the solder joint: it is the most dicult because there is no mathematical theory that describes when a solder joint is good or bad. Therefore, we have to rely on an expert knowledge. In this approach, as a result, we get the degree of belief d that a solder joint is good. If this degree of belief exceeds a certain threshold, the joint is accepted; else, it is rejected. As we have already mentioned, this traditional approach with numerical values of degrees of belief does not allow us to distinguish between the following two cases: if d = 0:5, this can mean that a solder joint is of low quality, or it can mean that the existing information is not sucient to make any decision about it. Johnson et al.108 suggest to use interval degrees of belief. In this case, we can easily distinguish between these two cases: a bad quality solder will lead to b 0:5 (e.g., b = [0:4; 0:6]), while the lack of information will mean the the resulting degree of belief is d = [0; 1]. In the rst case, we discard this board. In the second case, we request additional information.


Case study: environmental systems. When we solve an environmental problem

(e.g., a pollution problem), we must make some decision, because the environment is deteriorating fast, but we do not have enough information to get de nite predictions about the consequences of dierent decisions. So, possible decisions can be characterized by intervals of experts' degrees of belief, with no easy way to get better (=more de nite) estimates. It would be nice to make a decision that guarantees the best output, but such a decision (based on pessimistic estimates) would usually require us to stop all the related manufacturing, and spend enormous amount of resources on cleaning. This is usually not feasible. On the other hand, solutions that are based on optimistic estimates do not help, because they usually amount to doing nothing, and the environment is deteriorating. So, the best choice is to try Hurwicz approach. This approach is used in Lesh163 . Case study: agriculture. One of the most ecient methods of producing plants that are resistant to certain diseases is to select in vitro the corresponding tissues, and then grow only the most promising ones. The diculty with this method is that too many unknown parameters in uence the situations, most of which are beyond our control: e.g., what exactly will be the conditions of the soil where this plant will be planted, what will the weather be, etc. As a result, usually, we only have expert estimates on what qualities of the tissue will leads to the desired resistance. In Chakarska et al.46 , it is shown that optimization techniques based on representing this expert knowledge in intuitionistic fuzzy form lead, on average, to better results than traditional fuzzy optimization methods. Case study: petroleum engineering. In Strat229, Hurwicz approach is used to decide whether in an uncertain situation, it is better to drill a well or to test further.

Case study: military intelligence. In military intelligence, we often need to

make a decision real fast, and the available amount of information is low, because due to the nature of a military con ict, each side is trying to prevent the other side from gaining any information. For the same reason, it is practically impossible to get any extra information. So, we end up with (reasonably wide) intervals [d? (A); d+ (A)] of possible degrees of belief that alternative A will work, and a necessity to make a decision based on these intervals. Here, pessimistic and optimistic approaches do not work: Pessimistic approach (choosing an A for which d? (A) ! min) would usually mean that we assume the worst about the enemy, so our possibility of success is nil, and we could as well surrender. An optimistic approach (choosing A for which d+ (A) ! max) means that we just attack and hope for the best, which is usually not the best strategy. It sounds more reasonable to use Hurwicz approach and choose A for which d? (A) + (1 ? ) d+(A) ! max. This approach was applied in Bergsten, Schubert, et al.28;29;30;209;210;211 to a problem of tracking submarines (speci cally, Soviet submarines in Swedish territorial waters).


Case study: legal systems. Cohen, Ekelof, Gardenfors et al.51;68;80;81 use interval

degrees of belief (in the form of interval probability theory) to model legal reasoning. Degrees of belief resulting from presented evidence and legal arguments are called degrees of provability51 or evidentiary values68;80;81 . Case study: insurance. Applications of interval degrees of belief to insurance are described in Einhorn, Hogarth et al.66;67;102 ).

3.2. Applications to intelligent control General idea. If we use interval degrees of belief for intelligent control, we end up

with an interval of possible control values. Depending on our objective, we can now use this additional freedom of choice to select either a control value that requires the least energy, or a control value that makes our movement towards the destination the fastest, etc. (Lea, Nguyen et al.159;160;187 ). Case study: mobile robot. For a mobile robot, Wu268;269 chooses the smallest possible control, thus preventing wobbling and increasing energy eciency. Case study: space exploration. Possibilities of using interval-valued fuzzy control in space exploration are overviewed in Nguyen et al.186 . Case study: congestion control in computer networks. In Starks et al.226 , a similar idea (of using the smallest possible control) is used to control congestion in computer networks. Here, decreasing the number of control actions decreases the overhead that these actions take from a network, and thus increases the throughput of the network. An even more complicated problem is controlling Internet (Tolbert et al.237 ). Controlling Internet is dicult because its behavior has not been described in precise mathematical terms. Therefore, we have to use expert knowledge. This knowledge is usually formulated in terms of rules that predict level of congestion for dierent situations and dierent control strategies (e.g., change in packet size, or restriction on the number of packets sent by each node, etc). How do we judge that a system of rules provided by an expert describes Internet correctly? One way to check that is to apply intelligent control methodology to the given rules and deduce a numerical dependency that can be then checked against the empirical data. This idea does not work very well, because the resulting dependency will be at best approximately correct, and it is dicult to decide how close it should be. If we take into consideration that the degrees of belief are actually intervals, then after applying the same intelligent control algorithm to these intervals, we will end up with an interval as a predicted value. For this prediction, it is easy to test whether it is correct or not: if the actual value is inside the predicted interval, the rules are correct; otherwise, the rules must be changed.

Case study: an automated system that teaches kids how to ride a bike.

In Cloud50 , intervals are used to describe the uncertainty of the environment (in this paper, a kid who is taught how to ride a bicycle). The goal is to design an


intelligent control strategy that will work well for all values from the interval. Interval computation techniques are explicitly used to compute the interval of desired characteristics. Comparison of dierent variants. In Smith221 , the explicit use of intervalvalued degrees of belief is compared with two other approaches (with respect to their relative computational complexity); the approaches that also take into consideration that membership functions are only approximately known:

uncertainty in &? and _?operations, and uncertainty in defuzzi cation. The last approach turns out to be the most computationally simple.

3.3. Other applications Case study: quantum mechanics. Possible applications of fuzzy intuitionistic logic to quantum mechanics are described in Cattaneo et al.45 .

Acknowledgments

This research was partly supported by NSF grant No. EEC-9322370. We are thankful to Benadette Bouchon-Meunier, R. Baker Kearfott, and to the anonymous referees for their valuable comments.

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