A Fuzzy Inference Model Based on an Uncertainty Forward Propagation Approach

A Fuzzy Inference Model Based on an Uncertainty Forward Propagation Approach Luis M. de Campos and Antonio Gonzfilez Departamento de Ciencias de la Computaci6n e Inteligencia Artificial Universidad de Granada, Spain ABSTRACT The management of uncertainty and imprecision is becoming more and more important in knowledge-based systems. Fuzzy logic provides a systematic basis for representing and inferring with this kind of knowledge. This paper describes an approach for fuzzy inference based on an uncertainty forward propagation method and a change in the granularity of the elements involved. The proposed model is able to handle very general kinds of facts and rules, and it also verifies the most usual properties required by a fuzzy inference model. KEYWORDS: Fuzzy logic, fuzzy inference, uncertainty management, upper and lower probabilities I. INTRODUCTION The facts and/or the rules to be represented in knowledge-based systems may be often uncertain or imprecise. Different models for deductive reasoning are based on mathematical models like Dempster/Shafer's theory of belief functions [1], [2], possibility theory [3], [4], among other alternatives to the standard Bayesian model. In particular, the problem of inference from vague or fuzzy premises [5] will be our main concern. Address correspondence to Antonio Gonz6lez Muhoz, Computer Science and A. L Departmeut, Universidad de Granada, 18071 Granada, Spain. Received May 1, 1992; accepted March 10, 1993. This work was supported by the CICYT under Project TIC92-0665. International Journal of Approximate Reasoning 1993; 9:139-164 © 1993 Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/93/$6.00 139 140 Luis M. de Campos and Antonio Gonz~lez The basic problem may be stated as follows: If X is A then Y is B X is A* then Y is B*. (1) where X and Y are variables on reference sets U 1, U2, respectively. A, B, A* are fuzzy sets of the respective reference sets. These fuzzy sets may be considered as fuzzy information or soft restrictions on each variable. B* is also a fuzzy set, representing a soft restriction of Y obtained from the soft restriction A* on X and the fuzzy rule. This method was initially introduced by Zadeh [6], who proposed the following solution for the basic problem stated in (1) /xn*(s) = sup{/XA*(r ) A (1 -- IxA(r ) + /ZB(S)) } g Thus, U 1 being the antecedent domain and U 2 the consequent domain, the membership function of the predicate B* is given by the projection on U 2 of the intersection of the implication relation /XH, defined by lZl4(r,s) = 1 /x (1tzA(r)+ tz~(S)), and the cylindrical extension on U 1 × U 2 of the membership function /z A *. In contrast to the classical modus ponens, the above rule allows us to use fuzzy predicates A, B, and A*. Moreover, A* is not required to be identical with A. When A = A* and the predicates are crisp, then (1) becomes the classical modus ponens. A more general version of this generalized modus ponens (GMP) can be obtained if we replace the Min operator /x by an alternative t-norm *, and the particular implication relation /zH(r, s) by another tt A _~ B, thus obtaining /z B *(s) = sup{ lz A *(r)*tZA~B(r,s)}. (2) g Several authors have investigated this approach [3], [7], [8]. We are interested in a more general model within which the uncertain knowledge can be included. In this work, we propose a fuzzy inference model based on an uncertainty forward propagation approach [9]. The paper is arranged in nine sections. Section 2 describes the main ideas on which the proposed model is based, and section 3 outlines the propagation approach used to develop a fuzzy inference device. The basic inference model is introduced in section 4, the propagation approach is applied to a simple inference problem. The performance of this model is studied in section 5. Next, in sections 6 and 7, this basic inference model A Fuzzy Inference Model 141 is extended to include uncertainty degrees in the facts and rules, as well as more general kinds of rules. Section 8 includes an example that illustrates the use of the inference model presented, and section 9 comments on the performance and flexibility of our approach, and also points out future lines of research. 2. OUTLINE OF THE INFERENCE MODEL Fuzzy logic provides a systematic basis for representing and inferring from imprecise knowledge. The main goal of this work is to develop a fuzzy inference model able to handle a wide class of fuzzy facts and rules. The simplest case would include fuzzy propositions such as Rule: if X is low then Y is high Fact: X is very low, but we would like also to include uncertain knowledge and more general kinds of facts and rules, as for example, X is low with certainty degree 0.7, if X is high then Y is very low with certainty degree 0.5, etc. Thus, we are interested in easy-to-implement and computationally efficient models able to do inference with the different kinds of fuzzy propositions (Zadeh [10]), and moreover, verifying the most usual properties required by a fuzzy inference model. The proposed model is based on two main ideas [11]: ° The fuzzy rule "If X is A then Y is B" defines a relation among the elements of the sets U A = {A, -1 A} and U B = {B, -1 B}. • This relation is interpreted as a conditioning, that we represent by means of an uncertainty measure. With respect to the first idea, the models based on the GMP usually assume that the rule "If X is A then Y is B" defines a relation on the cartesian product of the reference sets of X and Y, U 1 x U2. By contrast, we suppose that the level of granularity in this relation is similar to the granularity of the elements involved, that is to say, the rule does not establish how each element of U~ and each element from U 2 are related; it only establishes the relation between the concepts represented by A, ~ A and B, ~ B. On the other hand, the second idea differs from other models that interpret the rule as a material implication (~ A or B). A general input A*, does not usually match any of the antecedent items. Thus, taking into account the above considerations, the fuzzy inference model should translate the information contained in A* to 142 Luis M. de Campos and Antonio Gonz~ilez information about A and -~ A. This translation can be easily done through a compatibility degree between the input and the antecedent of the rule. These degrees (the values al, a 2 in Figure 1) will be interpreted as an uncertainty measure generated by the current input A*, on the set U A. This uncertainty measure will be transferred from the set U A to the set U B through the fuzzy rule by means of an uncertainty propagation model. So far, the answer of the inference model is an uncertainty measure on U B. Finally, by combining the membership functions of B and -1 B, and their uncertainty values (the values /31,/32 in Figure 1), we will obtain a single output B* (see Figure 1). In order to make these general ideas more specific, first we need to choose a formalism to represent uncertainty measures, and a propagation model. The next section is devoted to these topics. 3. THE PROPAGATION MODEL The formalism we will use to represent pieces of uncertain information is by means of a class of fuzzy measures, namely representable measures (also called lower and upper probabilities). This is a very general framework of representation, which includes probabilities, possibilities [4], belief antecedent FUZZY RULE consequent input 1 uncertainty --.... B a, -,B a, uncertainty / propagation model Figure 1. Fuzzy inference model. output , B* A Fuzzy Inference Model 143 functions [1], [2], and Choquet capacities of order two [12] [13] as particular cases. Let us very briefly introduce the concept of lower-upper probabilities: Let P be a family of probability measures on a referential D x. We may associate a pair of lower-upper probabilities with P, (l, u), given by l(A) = inf P(A) VA c_D x PeP u(A) = sup P(A) VA c_D x PeP This defines a pair of ordered fuzzy measures in Sugeno's sense (see [141). Now, let us suppose that we have two variables X and Y that can take on values in the sets D x = {xl, x 2 .... ,x n} and Dy = {Yl, Y2,...,Y,,,}, respectively and a pair of representable measures, ((lx(A), ux(A)), A c_ Dx) , representing our knowledge about the values of X. We also have conditional information about the values of Y, given that we know the true value of X. The problem is to propagate the information from X to Y, through the conditional relationships. So, we want to obtain on Y another pair of lower and upper probabilities representing the knowledge about the value of the variable Y that we can infer from our knowledge about the value of the variable X and the relationships between X and Y. We model the conditional information on Y given some value x~ by also using conditional representable measures (l(B/xi) , u(B/xi)) , B c Dy), x i E D x. In [9] the following general solution for this problem was obtained: to calculate the upper measure of any subset of Dy, we must solve this linear programming problem uy(B) = max ~ u(B/xi)h i i=1 subject to h, <_ ux(A),VA c_ D x (4) x i ~ A Several particular cases of this problem can be directly solved without using any optimization techniques (see [15], [9]). Precisely one of these 144 Luis M. de Campos and Antonio Gonzfilez particular cases will be needed in the fuzzy inference model. Further details about the concrete formulation can be found in the appendix. The following example illustrates how the propagation model works. Example 1: Let us consider two variables X and Y, which stand for the color and the weight of a set of objects, respectively. Suppose that the values these variables can take on are Black (B) and White (W) for X, and Heavy (H) and Light (L) for II. So, the domains for the variables X and Y are D x = {B, W}, Dy = {H, L} respectively. Let us also suppose that we have the following partial information about the color of the objects and about the relationship between color and weight: • 70% of the objects are Black, 10% are White, and 20% can be either Black or White. • 80% of the Black objects are also heavy, and 20% can be either Heavy or Light. • 30% of the White objects are also heavy, 60% are Light, and 10% can be either Heavy or Light. We want information about the weight of the objects in the light of the information about the color, and the weight given that we know the color. These pieces of information can be represented as Dempster-Shafer measures (a particular case of lower and upper probabilities) as follows: Information about the color: ux(B) = 0.9, Ux(W) = 0.3 Ix(B) = 0.7, Ix(W) = 0.1 Conditional information about the weight given the color: • If the color is Black: u(H/B) = 1, l(H/B) = 0.8, • If the color is White: u(H/W) = 0.4, t(H/W) = 0.3, u(L/B) = 0.2 I(L/B) = 0 u(L/W) = 0.7 I(L/W) = 0.6 By applying the propagation model to these measures, we obtain (see the appendix for details) the following measures ly and Uy on Dy: Uy(H) = 0.94, Uy(L) = 0.35 ly(H) = 0.65 ly(L) = 0.06 that correspond to the following partial information about the weight of the objects: 65% of the objects are heavy, 6% are Light and 29%, can be either Heavy or Light. • A Fuzzy Inference Model 145 4. THE FUZZY INFERENCE MODEL In order to develop a fuzzy inference model, we are going to express, within the context of the above propagation model, the following fuzzy rule: IfXisAthenYisB (5) where X and Y are variables on the reference sets U s, U 2, and A and B are fuzzy sets on U 1 and U 2, respectively. Let U A = {A, ~ A} and U B = {B, ~ B} be two fuzzy partitions of U 1 and U 2. The basic idea to develop the inference model consists in replacing U 1 and U 2 by the fuzzy partitions U A and U 8 respectively, and then to consider uncertainty measures on these. The conditional information comes from a semantic interpretation of the rule (5) in the following sense: This rule generates two conditional representable measures on UR, (l(./A),u(./A)), (l(./-.A),u(./-~ A)) defined by l( B/A) = 1 u( B/A) = 1 l( -. B/A) = 0 u( ~ B/A) = 0 (6) l( B/-. A) = 0 u( B/-. A) = 1 I(--1B/ ~ A) =0 u( ~ B/ ~ A) = 1 (7) So, we are interpreting the rule "if X is A then Y is B" as a conditioning (instead of a material implication), that is, if we know that A is true then we can assert that B is also true (this is modelized as the total certainty measure (6)), but if we know that A is false then we cannot infer anything about the truth of B (and it is modelized as the total ignorance measure (7)). Remark As the representable measures we are using are always dual, in the following we will use only the upper measure. The results of the lower measures can be obtained by duality (l(H) = 1 - u(-~ H)). Moreover, we also need an upper probability measure on UA, in order to propagate it on Y through the conditional information (the rule). This measure will be obtained from a matching process between the input A* and each value in UA, that is, between A* and A and between A* and ~ A. For this purpose we will use a particular matching based on a compatibility degree between two fuzzy sets, F and G, through a t-norm *: c(F,G) = sup{ IzF(r)*tzG(r)} (8) r 146 Luis M, de Campos and Antonio Gonzfilez Although we could use any t-norm in (8), we believe the Lukasiewicz t-norm to be adequate because it satisfies the noncontradiction law (a* (i - a) = 0) and thus the compatibility degree between two complementary fuzzy sets is zero: c(F, ~F) = sup{max(/ze(r) + I~F(r) --1,0)} = 0 r Therefore we will use the Lukasiewicz t-norm in (8) because it makes the elements in each fuzzy partition UA and UB incompatible. Remark Although this choice may be controversial, in our context the fuzzy sets A and -7 A, B and ~ B act as the elements of two (crisp) sets U A and UB. Then the use of the Lukasiewicz t-norm guarantees having nonoverlapping dements in these sets. So, we define the upper measure on U A induced by the input A* as ux(A/A* ) == c(A, A*) = sup {max( /zA(r ) + /z A *(r) - 1,0)} r ux(~A/A* ) = c(-~A,A*) = sup {max( /zA *(r ) -- &4(r),0)} (9) r It can be easily proved that c(A, A*) + c(m A, A*) > 1 (10) if we only impose the input A* to be normalized (3r ~ U1/Iza *(r) = l). We will always suppose that A* verifies this property in the rest of the paper. From (10) we may interpret Ux as an upper probability measure. By using the above propagation model and this upper measure on UA, we get an upper measure on U 8. In this case the solution of (4) is very easy (it may be calculated directly without using an optimization technique: we need only to calculate a Choquet integral (see [12], [13]) with respect to the measure Ux; see appendix for details). The solution is uy(B) = 1 ur(-~ B ) =c(-~A,A*) (11) We may consider this upper measure as the first interesting answer of the fuzzy inference: Rule if X is A then Y is B Input X is A* Output Y is B is [1h, 1] Y is -7 B is [0, a], (12) A Fuzzy Inference Model 147 where A = c(--1 A, A*) and "Y is C is a" means a proposition at degree [10]. In our case the uncertainty degree a is an interval [o/inf, O/sup] representing the lower and upper probability respectively. This answer generates certainty values for the results of B and ~ B. Observe that if the compatibility between A* and ~ A is zero then the result is unambiguously B. This only happens if /z A *(r) < tzA(r) 'dr, that is, when the input A* is included in A. When the compatibility between A* and -,A is maximum (3r ~ U1/tz A *(r)= 1 and IZA(r)= 0), for instance if A* = -~ A, then we obtain an uncertainty measure representing the total ignorance, in agreement with the conditional interpretation of the fuzzy rule we have made. The kind of solution given by (12) is similar to that proposed by other authors (see [16]) in the sense that we obtain an uncertainty distribution on the possible answers. In some cases this solution may not be appropriate enough (e.g., in fuzzy control problems). Thus, if we want to obtain an answer B* as output, we could combine the two pieces of information that we have: the upper measure on U B and the membership functions of B and -~ B. The idea in carrying out this combination is to produce the result B* as an expected value of B and ~ B, weighted by their upper measures, through a fuzzy integral. In the above process, we obtained the answer ur(.) from Ux(.) and the conditional relation A ~ B as an average of u(./. ) weighted by the measure u x. In a similar way, we could obtain B* (ix B *) from ur(.) and {/z~(s), tz_, B(s)} because we interpret the membership function of a fuzzy set C, tZc(S), as the conditional possibility (a particular case of upper probability measure) of s given that C is true: tzc(s)= rr(s/C). Then again we have an upper measure uy(.) on U B and two conditional upper measures 7r(s/B) =/zs(s) and rr(s/~ B) =/z B(s) = 1 -/zB(s). The model of forward propagation, when it is applied to this case, produces (using again the Choquet integral) the following membership function for the output B*: tZB(S) if i~B(S) >_ 1/2 /ZB*(S ) = U({S}) ---- ~/ZR(S)( 1 _ 2A) + A if /Z~(S) _< 1/2 or equivalently ~B*(s) = (1 - A)~8(s ) + Amax(~8(s),l - ~B(s)), (13) where h = c(~ A, A*). This method has the following drawback: let us suppose that h = uv(~ B) = 1, that is, ~ B is as credible as B. This happens, for instance, 154 Luis M. de Campos and Antonio Gonz~lez include in this model are described by X is A is [a,/3] Xis -~A is [1 -/3,1 - a] (24) where a,/3 ~ [0, 1], a _</3, and the intervals [.,. ] represent lower and upper probabilities. When we take /3 = 1 we obtain (22), and from a = 1 (and then /3 = 1, too) we get the true facts included in category 1. In order to simplify the notation, from now on we will represent (24) only by writing the first expression X is A is [ a,/3 ] (25) because it is obvious that once we know (25) then we know (24) too. In this way, the inference problem for uncertain facts can be written as If X is A then Y is B X is A* is [a,/3] (26) The way in which we solved the basic inference model given by (12) may be applied again, we only need to get a measure u x on UA. To do that we could transfer to U A the measure we have on UA *, which is, Ix(A*) = ,~, ux( A*) =/3 lx(~A*) =1- /3, Ux(-TA*) = l - a (27) This transference will be made through a fictitious, uncertain rule, such as if X is A* then X is A where the conditional measures that define this rule are the compatibilities among elements of UA and U A *, u(A/A*) = c(A, A*) u(A/~A*) =c(A,-~A*) u( ~ A/A*) = c( ~ A, A*) u( ~ A/-~ A*) = c( -1 A, -1 A*) (28) Then, by chaining this rule with the real rule, we obtain a measure on UB, or equivalently, a fuzzy output B*. Next, we are going to develop this process. From the measure (27), the conditional measures (28) and the propaga- A Fuzzy Inference Model 155 tion model, we get the If u( A /A* ) < If u( A /A* ) > If u(-1 A/A*) < If u( ~ A/A*) > following measure on UA: u( A/ ~ A*) ~ ux( A ) = au( A/A*) +(1a)u(A/~A*) u( A/ ~ A*) ~ Ux( A ) = ~u( A/A*) +(1 - ¢I)u(A/-~A*) u( -7 A~-1 A*) ~ ux( -~ A) = au( -7 A/A*) +(1 - c~)u(~ A/~ A*) u( ~ A~ ~ A*) ~ Ux( -7 A) = ¢lu( ~ A/A*) +(1 -/3)u(~ A/~ A*) Finally, by propagating (29) to U B through the rule we obtain = B(s) • 3' with [ ac(~ A, A*) + (1 - a)c(~ A, ~ A*) 3" = ~c(~ A, A*) + (1 fl)c(~ A, -7 A*) (29) if c( ~ A, A*) _< c( A, A*) if c(~ A, A*) >_ c( A, A*) (30) This last equation can be easily simplified in many cases. Since c(-~A,-~A*) = 1 when 3ro/tzA(r o) = /ZA*(r 0) = 0 (for instance, this condition is obviously verified for fuzzy quantities with a bounded support set), (30) is written in this case as 3'= ac(~A,A*)+ 1-a = c(~ A, A*) @ (1 - a), and therefore the resultant output is t%*(s) = ~B(S) @ (h @ (1 - a)) (31) with A = c(-7 A, A*). By considering this last value of 3, = h @ (1a), several particular cases of uncertain facts can be studied: • If a = 1 then /~B * (S) = tZB(S) @ h. That is, when the fact does not present any uncertainty, the basic inference model given by (17) is obtained again. • If a = 0 then tz B * (s) = /ZB(S) @ 1 = 1. So, the ignorance of the facts makes every value of the consequent domain completely possible. • If A*_A then h =0 and /z B*(s)= I~B(S)@(1-- a). In this case the uncertainty of the conclusion only comes from the uncertainty of the fact. • If A* c_ --7 A then h = 1 and /z 8*(s) = /zs(s) @ 1 = 1. In this way, facts completely opposed to the antecedent, even being uncertain, give rise to ignorance of the consequent. 156 Luis M. de Campos and Antonio Gonzfilez The equation (31) shows how the output B* does not depend on the upper probability/3. Curiously, we decided to represent the uncertainty of a fact in the formalism of the representable measures, but the inference model only uses the lower probability, and therefore to represent uncertain facts it suffices to use (21) and (22). That is to say, the possibilistic interpretation of (21) is enough for the purpose of this inference model. Uncertain rules An uncertain rule can be written as If X is A then Y is B is [a,/3] (32) In a similar way as we interpreted the uncertainty in facts by means of lower and upper probabilities, we are going to interpret the uncertain rule (32) as the following conditional lower and upper measures l( B/A) = a u( B/A) =/3 l(-~ B/A) = 1 - /3 u(--~ B/A) = 1 - a (33) I(B/-~ A) = 0 u(B/-~ A) = 1 1( ~ B/-~ A) = 0 u( ~ B/-~ A) = 1 (34) Here, by propagating the measure (9) through these conditional measures we obtain u),(B) =/3(1 - c( ~ A, A*)) + c( -7 A, A*) = fl ~ c(-nA,A*) =/3~h Uy(-7 B) = (1 - ~)(1 - c(~ A, A*)) + c(~ A,A*) = (1 - • c( A, A*) = (1 - • (35) where A = c(~ A, A*). The output B* produced by combining (35) and the membership function of B and -7 B is /zn*(s) = (~B(s) • h ~9 (1 - a)) - (~B(s)(1 - fl)(1 - h)) (36) This equation defines an unnormalized fuzzy set for/3 4: 1. The lack of normalization does not create any problems when B* is a final output of the inference model. On the contrary, when the output may be used as input for another rule, unnormalization blocks the way of B* through the second rule. To prevent this kind of problem, we impose the condition /3 = 1 to the rules. Thus, we are restricting the kind of uncertainty in the rules to the possibilistic interpretation, that is, If X is A then Y is B is a (37) A Fuzzy Inference Model 157 with l( B/A) = a u( B/A) = 1 l(-~ B/A) = 0 u(-~ B/A) = 1 - a (38) and preserving (34). Therefore, when the system uses uncertain rules, the output generated is /,,*(s) = g~(s) ¢ A • (1 - a) (39) Note how uncertainty in rules and facts (at the same degree) produces the same output. This coincidence shows how the system uniformly manages the uncertainty throughout their different components. When uncertain facts and rules appear together, that is, when we have a rule "if X is A then Y is B is c~" and a fact "X is A* is a '", then the output B* is /z,*(s) =/,,(s) ¢ A ¢ (1 - (~) ¢ (1 - a') (40) where A = c( --, A, A*). 7. MANAGEMENT OF CONJUNCTIONS AND DISJUNCTIONS IN RULES The kind of rules contained in a real knowledge base are not always as simple as the basic rule (1). In this section, we extend the inference model to include conjunctions and disjunctions in rules. Conjunctions in premises Let us consider the following kind of fuzzy inference IfX 1 isA landX 2isA zand ... andX nisA nthenYisB X 1 is A~' and X 2 is A'~ and ... and Xn is A* (41) Yis B* where X i are variables on reference sets U/, Y is a variable on reference set V, A i are fuzzy sets on U/, and B is a fuzzy set on V. A* are the inputs and B* is the output of the inference model. The expression (41) can be rewritten in the following way If X is A then Y is B X is A* (42) where X = (X1, X 2 ..... Xn), A = (A 1, A z, . .. , A,) and A* = (A T , A~ ..... A* ). 158 Luis M. de Campos and Antonio Gonz~ilez Thus, by replacing (41) by (42) we get a similar situation to the previous one, and the only problem is to define a measure on UA and a conditional measure to represent this rule. For this purpose, we need to define the compatibility between the conjunction of Ai,'s and the conjunction of A*'s. The greater all the compatibilities between each A i and A* (c(A i, A*)) are, the greater the compatibility between the vectors A and A* (c(A, A*)) is, because these vectors denote a conjunction of facts. For the same reason, as soon as an i exists such that the compatibility between Z i and A* is low, the compatibility between A and A* should be low, too. Thus, it seems appropriate to modelize c(A, A*) as a function of c(Ai, A* ) through a conjunctive operator as a t-norm, and by taking the minimum t-norm, the compatibility between A and A* is defined by c(A, A*) = min{c(Ai, A*)} (43) i where c(Ai, A*) was defined in (8). The negation of the vector A can be interpreted as a disjunction of the negations of each A i. So, the compatibility between ~ A and A* will be defined through a disjunctive operator, as a t-conorm, and by taking the maximum t-conorm, c(~A,A*) is defined by c( -7 A, A*) = max {c( -1 Ai, A* )} (44) i From (43) and (44) we can also define the measure on the antecedent domain as ux(a/A*) = c(A, A*) Ux( --7 A/A*) = c( -~ A, A*) (45) It can be easily proved that ux(A/A*)+ Ux(-~A/A* ) > 1 since c(A i, A* ) + c(-~ A i, A* ) > 1 Vi, and so, we can interpret u x as an upper probability measure. By using the above measure, the conditional measure defined by (6) and (7) for certain rules or defined by (34) and (38) for uncertain rules, and the propagation model, the result of the inference model for conjunctions in premises is direct: for certain rules and uB *(s) = tzB(s) • ,X (46) ~B*(s) = /xB(s ) ~ A • (1 - c~) (47) for uncertain rules, with A = max i Ai, t~ i = c(~hi, a~) and a is the uncertainty of the rule. A Fuzzy Inference Model 159 Note that (46) (and similarly (47)) can also be written as /xB*(s ) = m.ax{/xB,(s)} = max{/ZB(S ) * Ai} (48) t i where each /xB(s) = p.B(s) • A i would be the output produced by a simple rule "If X i is A i then Y is B." Disjunctions in premises Now, let us consider the following kind of fuzzy inference IfX lisA lorX 2isA 2or... orX nisA nthenYisB X 1 is A'~ and X 2 is A~ and ... and X n is A* Y is B* (49) where again txB(s) = txB(s) • A i. where X i are variables on reference sets U/, Y is a variable on reference set V, A i are fuzzy sets on U/, and B is a fuzzy set on V. A* are the inputs and B* is the output to the inference model. The solution of this problem can be solved in a similar way to that of conjunctions in premises, by using the following compatibilities c(A 1 VA, v... VAn,A T AA~ A"" AA*) = max{c(A i, A*)} (50) i c(~A 1A -~A 2 A ". A -~An,A T AA~ A'" AA*) = min{c(~ Ai,A*)} (51) i that is to say, by defining a measure u x on U A = {A, -~ A], where -1 A = (-1 A1, -~ A2,... , -n An) , as ux(A/A* ) = max{c(Ai, A*)} i Ux(-~ A/A*) = min {c(-~ A i, A*) (52) i By using the conditional measures (6) and (7) or (34) and (38) again, the propagation model, and using the measure (52) too, we obtain the output ,(s) = B(s) • A (53) where A = min i A i and }k i = C("n Zi, AT) Vi. Note that (53) may also be written as /x B *(s) = m!n{/xB,(s) } = min{/xB(s ) * Ai} (54) t i 160 Luis M. de Campos and Antonio Gonz~lez In fact, the rule (49) could be considered as equivalent to the set of rules If X i is A i then Y is B, i = 1,... n (55) Therefore, the output generated by the complex rule (49) is equivalent to the conjunction of the outputs from the simpler rules in (55). 8. EXAMPLE Finally, we are going to show a simple but representative example of the inference method presented. Suppose that we have three rules and three facts in our knowledge base. The fuzzy sets that appear in the rules all belong to the set {Very Low, Low, Medium, High, Very High}, whose elements are the fuzzy numbers represented in Figure 3 and defined by Very Low VL = (0, 0, 25) Low L = (25, 25, 25) Medium M = (50, 25, 25) High H = (75, 25, 25) Very High VH = (100, 25, 0) where the parameters of the triangular fuzzy number A = (m, a, b) have the same meaning as in example 2. The facts and rules in the knowledge base are the following: X 1 is A X z is 30 X 4 is B is 0.8 ifX 1 isVLandX 2 is Lthen X 3 is M ifX 3 isMthen X 5 is His0.9 if X 4 is VH then X 5 is VH where A = (10, 10, 20) and B = (95, 1, 1). VL L M H VH 0 25 50 75 100 Figure 3. Linguistic labels for antecedent and consequent in the rules. A Fuzzy Inference Model 161 Let us see the inference process: i) Let the values of X a and X 2 be inputs to the first rule. Since c(A, ~ VL) = 0.4, c(30, -~ L) = 0.2, the output is: X 3 is 01, with tXo,(X) = /XM(X) • 0.4. If we take this fuzzy quantity as input to the second rule, as c(O 1, -1 M) = 0.4, then the new output is X 5 is 02, with 1~o2(x) =/zH(x) • (0.4 • 0.1) =/xH(x) ~ 0.46. ii) Let the value of X 4 be the input to the third rule. Since C(B, ~ VH) = 0.2, the output is: X 5 is 03, with /xo3(x) = tZvH(X) (0.2 • 0.2) = tZvn(X) • 0.36. iii) Since X 5 has two possible values at the end of the process, we can combine them by using the min operator, that is, X 5 is 04, with tZo,(X) = min{ tzn(x) • 0.46, tZvn(X) • 0.36}. The membership functions of the outputs of X 3 and X 5 are shown in Figure 4. 0 0.46 0.36 ,,' . .•' ,, 0.4 25 50 75 (a) 100 VH / .' '., • " ." ", 25 50 75 (b) Figure 4. Outputs for a) X 3, b)-X 5. 100 162 Luis M. de Campos and Antonio Gonzfilez 9. CONCLUDING REMARKS The inference approach we have proposed: • is very easy to implement and computationally efficient, • is able to do inference with different kinds of fuzzy propositions. Moreover, the proposed model has the following characteristics: • it can be interpreted, in a particular case, as a fuzzy modus ponens, • it verifies the most usual properties required by a fuzzy inference model, and • it introduces a very flexible framework within which fuzzy inference can be done, with different freedom degrees: matching process, uncertainty representation, propagation model, integration model, etc. The restriction on the uncertainty intervals [a, 1] for the rules, that forces us to use only possibilistic uncertainty, will be removed in forthcoming work. This will allow us to manage a more general kind of uncertainty (for example, probabilistic uncertainty). The implementation of this method, together with its application to real problems, will be also the object of further work. APPENDIX This appendix contains basic concepts about the Choquet integral and the expression of the propagation model (4) for this particular measure. This result is used in several sections to derive the propagated uncertainty measure on the consequent. If the lower-upper probability measures (l x, Ux) are Choquet capacities of order two (see [14]), then the resultant measure ur(B) on Dy can be obtained in the following way: uy(B) E B = ux(f ), where fn is a function on D x defined by fn(x i) -- u(B/x i) VX i ~-- Dx, and Eux(.) is the Choquet integral [12], [13]) with respect to the measure Ux(.). This result is interesting because it is easy to calculate the Choquet integral, and therefore the calculation of ur(.) is direct. The measure, defined in (12), which has been propagated in section 4 is a Choquet capacity of order two. Therefore the propagation can use the above result. Let us briefly show how to calculate the Choquet integral on finite domains: Given a fuzzy measure g on a set D x and a function f: A Fuzzy Inference Model 163 D x ~ R+, the Choquet integral of f with respect to the measure g is f0 Eg(f) = g(F~) da, where F, = {x ~ Dx/f(x) >__ a}. If the set D x is finite (D x -- {x I ..... x,}) and the values of the function f are ordered in the following way: f(xl) <_ f(x 2) < ... <f(x,), then the Choquet integral can be written as n Eg(f) = Y'~f(xi)(g(Ai) - g(Ai+l)), i=1 where A i = {Xi, Xi+ 1 ..... Xn} , An+ 1 = Q~. • References 1. Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statistics 38, 325-339, 1967. 2. Shafer, G., A Mathematical Theory of Euidence, Princeton University Press, Princeton, 1976. 3. Dubois, D., and Prade, H., Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988. 4. Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1, 3-28, 1978. 5. Zadeh, L. A., A theory of approximate reasoning, in Machine Intelligence 9 (J. E. Hayes, D. Michie, and L. I. Mikulich, Eds.), Halstead Press, New York, 149-194, 1979. 6. Zadeh, L. A., Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. Systems Man Cybernet. 3, 159-176, 1973. 7. Dubois, D., and Prade, H., Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions, Fuzzy Sets Syst. 40, 143-202, 1991. 8. Magrez, P., and Smets, P., Fuzzy modus ponens: A new model suitable for applications in knowledge-based systems, Int. J. Intell. Syst. 4, 181-200, 1989. 9. Campos, L. M., de, and Moral, S., Propagating uncertain information forward, Int. J. Intell. Syst. 7, 15-24, 1992. 10. Zadeh, L. A., The role of fuzzy logic in the management of uncertainty in expert systems, Fuzzy Sets Syst. 11, 199-227, 1983. 11. Campos, L. 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