NON-MONOTONIC FUZZY REASONING J.L. CASTRO, E. TRILLAS, J.M. ZURITA. Technical Report #DECSAI-951

Transcription

1 NON-MONOTONIC FUZZY REASONING J.L. CASTRO, E. TRILLAS, J.M. ZURITA Technical Report #DECSAI-951 NOVEMBER, 1995

2 Non-Monotonic Fuzzy Reasoning J.L Castro, E. Trillas, J.M. Zurita Depto. Ciencias de la Computacion e Inteligencia Articial E.T.S.I. Informatica. Universidad de Granada Avenida de Andalucia, 38, 18071, Granada, (Spain) castro, zurita@robinson.ugr.es Abstract Fuzzy reasoning can provide techniques for representing and management the imprecision inherent in commonsense reasoning. But, like human reasoning, it conduces to inconsistences (inherent in the imprecise or incomplete knowledge) that might be solved in the frame of fuzzy logic, simulating the human behavior. In this paper we analyze this kind of conicts and propose a non-monotonic fuzzy logic in order to solve it. Moreover, we show that many (non-monotonic) human reasoning patterns can be modeled by means of this "non-monotonic fuzzy reasoning". 1 Introduction Fuzzy Reasoning Systems have been shown to be an important tool for problems where, due to the complexity or the imprecision, classical tools are unsuccessful. The knowledge is represented as a set of fuzzy propositions expressing in a compact way the domain knowledge. Fuzzy consequences are then obtained using fuzzy inference rules. Nevertheless, what is going on when contradictory fuzzy consequences are obtained? In a rst step we can think that knowledge is incorrect. However, it is a very simple point of view, since reasonable fuzzy propositions can allow this kind of conict due to their inherent imprecision. For example, two reasonable rules for driving a car could be: R1: If obstacle (near) then brake(strong) R2: If in-curve(sharp) then brake(nothing) or brake(very little) but, what's the matter if you nd an obstacle in a sharp curve? From rules R1 and R2 follow two inconsistent actions: brake(strong) and brake(nothing) or brake(very little). The reasonable in this case will be a "averaging" between both actions with more importance for the action of the rule R1: brake(slightly strong). Suppose now, the following also reasonable rule: R3 If obstacle(very near) then brake(very strong)

3 In the case of nding an obstacle very near in a sharp curve, the inconsistent actions derivated are brake(very strong) and brake(nothing) or brake(very little), but the reasonable in this case would be consider predominant the rule R3 and so, the action might be brake(very strong). With these examples we want to illustrate that: a) Inconsistent fuzzy consequences can be derived from reasonable fuzzy knowledge. b) The solution to those conicts will depend on the knowledge which conduce to it. Every rule is reasonable and produces an adequate answer, but when new information is added, the conclusion changes. In this sense, we say that fuzzy reasoning might incorporate some kind of non-monotonicity. In this paper, we will present a solution to this kind of conicts by means of the use of an averaging function. An averaging function should be used for every possible conict, depending on the dierent derivation path which conduce to it. Moreover, we will show that many non-monotonic reasoning patterns can be modeled by means of this approach. Another notable approach to non-monotonic fuzzy reasoning was put forward by Zadeh as "syllogistic reasoning with dispositions"[11]. Zadeh interpreted commonsense knowledge as an "elastic constraint" where the variability is represented by a fuzzy set of sigmacount associated with imprecise quantiers. The main dierence between both approaches is that Zadeh's approach does not give an eective way for choosing a particular proposition among possible candidates (as it is required in many applications), while our approach is designed in order to give such a procedure. The paper is organized as follows. In the next section we present the concept of consequence and inconsistency in fuzzy logic that we are going to use. Next we introduce the approach to non-monotonic fuzzy reasoning based in the use of averaging functions for solving conicts. Finally, we show how dierent non-monotonic reasoning patterns can be modeled by this approach. 2 Consequence and Inconsistency in Fuzzy Logic 2.1 Fuzzy Propositions, atomic and normal form >From now on, we will x a domain of problem D, that is, a nite set of variables are X 1 ; :::; X n, taking values in its respective domain of discourse U 1 ; :::; U n. Denition. We will call an atomic fuzzy proposition on D to a proposition with the form (X i1 ; :::; X im) is R being R a fuzzy relation on (U i1 ; :::; U im). The most simple case of atomic propositions are propositions of the type X is A, where X = X i, and A is a fuzzy subset of U i for some i 2 f1::ng. The set of all atomic fuzzy propositions on D will be denoted by AP rop D or more briey by AP rop. Fuzzy propositions on D are now dened as propositions compound by atomic fuzzy propositions and the logical binary connectives "and", "or", "not" and "If... then". That is: Denition. A fuzzy proposition on D is dened recursively by: i) Every atomic fuzzy proposition is a fuzzy proposition.

4 ii) If p and q are fuzzy propositions then (p and q), (p or q), (If p then q) and (not p) are also a fuzzy proposition. iii) All fuzzy propositions are obtained by applying i) and ii) a nite number of times. The set of all fuzzy propositions on D will be denoted by F P rop D or more briey by F P rop. Obviously, F P rop D is the free algebra on Aprop D with the operations "and", "or", "not" and "If... then". Denition. An interpretation I on D is a four-upla (T,S,I,n) where i) T is a t-norm, ii) S is a t-conorm iii) I is an implication function iv) n is a negation function. The meaning of this interpretation is represented by the translation (see []) of any fuzzy proposition to an atomic fuzzy proposition. Denition. The atomic form via I of a fuzzy proposition is dened recursively by: i) Every atomic fuzzy proposition has associated itself. ii) If p and q are fuzzy propositions with atomic forms ~ X is R p and ~ Y is R q respectively, the atomic form of a) (p and q) is ~X [ ~Y is R p^q where R p^q(~z) = T (R p (~z X ); R q (~z Y )); b) (p or q) is ~X [ ~Y is R p_q where R p_q(~z) = S(R p (~z X ); R q (~z Y )); c) (if p then q) is ~ X [ ~ Y is R p!q where R p!q(~z) = I(R p (~z X ); R q (~z Y )); d) (6 p) is ~ X is R :p where R :p (~x) = n(r p (~x)). where if ~u and ~v are two vectors, ~u [ ~v denotes the upla obtained joint the vector ~u and the vector ~v and elimating the duplicated elements, and ~u X denotes the projection of u over X. Example The atomic form associated to will be If (X 1 ; X 3 ; X 5 ) is R 1 and (X 2 ; X 3 ) is R 2 then X 4 is A being (X 1 ; X 2 ; X 3 ; X 4 ; X 5 ) is R p R p (x 1 ; x 2 ; x 3 ; x 4 ; x 5 ) = I(T (R 1 (x 1 ; x 2 ; x 5 ); R 2 (x 2 ; x 3 )); A(x 4 )). In the following, we will use the atomic form when we will refer to a fuzzy proposition, and it will be expressed by ~ X is R. On the other hand, any fuzzy proposition have a normal form of the type (X 1 ; :::; X n ) is R being R a fuzzy relation on (U 1 ; :::; U n ): Denition. The normal form associated via I to a fuzzy proposition p is dened by:

5 i) If p is an atomic fuzzy proposition X ~ is R, the formal norm of p is (X 1 ; :::; X n ) is R p where R p (x 1 ; :::; x n ) = A((x 1 ; :::; x n ) X ). R p is also called the cylindrical extension of R in (X 1 ; :::; X n ). ii) If p and q are fuzzy propositions with normal form ~X is R p and ~X is R q respectively, the normal form of a) (p and q) is ~ X is R p_q where R p_q(x 1 ; :::; x n ) = T (R p (x 1 ; :::; x n ); R q (x 1 ; :::; x n )); b) (p or q) is ~X is R p^q where R p^q(x 1 ; :::; x n ) = S(R p (x 1 ; :::; x n ); R q (x 1 ; :::; x n )); c) (if p then q) is ~X is R p!q where R p!q(x 1 ; :::; x n ) = I(R p (x 1 ; :::; x n ); R q (x 1 ; :::; x n )); d) (6 p) is ~X is R :p where R :p (x 1 ; :::; x n ) = n(r p (x 1 ; :::; x n )). Example On the domain fx 1 ; :::; X 6 g, the normal form of the proposition will be If (X 1 ; X 3 ; X 5 ) is R 1 and (X 2 ; X 3 ) is R 2 then X 4 is A being (X 1 ; X 2 ; X 3 ; X 4 ; X 5 ; X 6 ) is R p R p (x 1 ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 ) = I(T (R 1 (x 1 ; x 2 ; x 5 ); R 2 (x 2 ; x 3 )); A(x 4 )). Proposition. The normal form of a fuzzy proposition is the normal form of its atomic form. Denition. The set of all propositions in a normal form will be denoted NP rop D. Two propositions will be considered equivalents when they have the same normal form or equivalently the same atomic form. Proposition. The relation p q if and only if R p = R q is an equivalence relation on Fprop. Proposition. Every fuzzy proposition is equivalent to its atomic form and it is equivalent to its normal form. 2.2 Inference rules in fuzzy logic The basic inference rules in fuzzy logic are (see [2,10]): 1.- Entailment principle. ~X is A A B ~X is B 2.- Conjunctive rule ~X is A ~X is B

6 ~X is A \ B where A \ B(x) = T (A(x); B(x)): 3.- Cartesian Product ~X is A ~Y is B ( ~X; ~Y ) is A B where A B(x; y) = T (A(x); B(y)): 4.- Proyection rule ( ~ X; ~ Y ) is R ~X is R ~X where R ~X (x) = sup y R(x; y): 5.- Compositional rule ~X is A ( ~X; ~Y ) is R ~Y is AoR where AoR(y) = sup x T (A(x); R(x; y)): 6.- Generalized Modus Ponens: If ~ X is A then ~ Y is B ~X is A 0 ~Y is B 0 where B 0 (y) = sup x T (A(x); I(A(~x); B(y))). 2.3 Consequence in Fuzzy Logic Denition. Given a set of fuzzy propositions 2 F P rop, a set of fuzzy propositions A 2 F P rop, and an fuzzy proposition p 2 F prop, a proof of p from the hypothesis A under the knowledge is a sucesion of fuzzy propositions p 1 ; :::; p m such that: i) p m = p, ii) For each i 2 f1:::mg, p i 2 [ A or there exist t < i and s < i such that p i is obtained from a rule of inference with p s and p t or there exists t < i such that p i is the atomic form of p t.

7 Denition. Given a set of fuzzy propositions 2 F P rop, a set of fuzzy propositions A 2 F P rop, and an fuzzy proposition p 2 F prop, we will say that p is a consequence of A under the knowledge, write A ` p (A ` p when = ;), if there exists a proof of p from the hypothesis A under the knowledge. Denition. Given a set of fuzzy propositions 2 F P rop, a set of fuzzy propositions A 2 F P rop, the deduction about X ~ from the hypothesis A under the knowledge, is dened as the fuzzy proposition ~X is A, being A = T fa i : A ` ( ~X is A i g. Proposition. ` is a consequence relation on FProp, i.e., satises the following conditions of reexivity G1, cumulative transitivity G2, and monotony G3: G1) A ` p whenever p 2 A G2) A ` p whenever A ` p i for all i 2 I and A [ fp i g i2i ` p G3) A [ B ` p whenever A ` p. Proof. G1 and G3 are obvious. If A[fp i g i2i ` p then there exists a proof q 1 ; :::; q m = p of p from the hypothesis A [ fp i g i2i under the knowledge. Then, changing any q i such that q i 2 fp i g i2i by one respective proof of q i from the hypothesis A under the knowledge, we obtain a proof of p from the hypothesis A under the knowledge. As it is well known, this concept can be expressed equivalently using a consequence operator, i.e., an operation on sets of fuzzy propositions to sets of fuzzy propositions, C k (A) = fp 2 F prop; A ` pg, satisfying the three conditions of inclusion (C1), idempotence (C2), and monotony (C3): C1) A C k (A), C2) If A B, then C k (A) C k (B). C3) C k (C k (A)) = C k (A). It is obvious that C k (A) = C ; (A [ ). Thus, the Fuzzy Logic represented by C = C ; is a (monotonic) logic in the "classical" sense [3]. 2.4 Inconsistency in Fuzzy Logic Denition. We will say that a set of fuzzy propositions A 2 F P rop is inconsistent under the knowledge 2 F P rop if there exist a p 2 F prop of the type ~ X is ; such that A ` p. The most usual case of inconsistence is obtained when there exist p; q 2 F prop such that p is " ~X is A 1 ", q is ~X is A 2, A ` p, A ` q, and A 1 \ A 2 = ;. For example, under the knowledge = fr1; R2g of introduction, the set of fuzzy propositions A = f obstacle is near, curve is shape g is inconsistent since brake(nothing or very little) and brake(strong) are derived, being (nothing or very little) \ strong = ;. Other denitions of inconsistency in fuzzy logic can be found in the literature [4,5]. Any of them can be chosen, and the rest of the paper would be the same. 3 Non-Monotonic Fuzzy Reasoning As it was said in the introduction, it its possible to nd a set of reasonable fuzzy propositions such that a possible additional information A will be inconsistent under. In order to avoid this kind of conicts we propose a deviant fuzzy reasoning, that we will call "Nonmonotonic Fuzzy Reasoning". The idea is very simple, when a conict c is derived

8 A ` " X ~ is A 1 ", A ` " X ~ is A 2 ", A 1 \ A 2 = ;. we propose to use an "averaging" function (depending on the type of conict) m c in such a way that consequences " X ~ is A 2 "and " X ~ is A 1 " are eliminated and changed by m c (A 1 ; A 2 ). 3.1 Conicts In order to make precise this idea, we will start making precise the concept of conict. Denition A conict on D is dened as a triple (p 1 ; p 2 ; X) such that p 1 ; p 2 2 F prop, X is a variable in D, and there exist a set of atomic propositions A 2 AP rop verifying: A `fp1 g "X is A 1 ", A `fp2 g "X is A 2 ", A 1 \ A 2 = ;. Thus, we only consider conicts between two fuzzy propositions. A particular case is when p 2 is the trivial proposition "X" is U", U being the universe of discourse of X. In such a case, we say that p 1 is a conictive proposition. Examples. i) R1 and R2 are conictive fuzzy propositions, since that A = f obstacle(near), in-curve(sharp) g produce inconsistent answer: A `fr1g brake(strong), A `fr2g brake(nothing or very slightly), strong \ nothing or very slightly = ;. ii) Every rule of the type "If X is A then Y is B" can be a conictive proposition since that A = f "X is A", "Y is B 0 "g where B \ B 0 = ; is inconsistent under it: A `f"if X is A then Y is Bg "Y is B", A `fy is B'g "Y is B0 ", B \ B 0 = ;. 3.2 Averaging Functions Now, we will dene the concept of "averaging" function. Let a partial order over the set P (U) of all fuzzy subsets of U. For example, if U is a real interval, we can consider any of the known ranking methods of the real fuzzy numbers [12]. Denition A averaging function over the partial ordered set (P (U); ) will be a mapping m from P (U) P (U) into P (U) such that A \ B m(a; B) A [ B: The set of all averaging functions on (P (U); ) will be denoted F (U;). Very simple examples of averaging functions are the fuzzy extension of real averaging functions. A real averaging is a function m from RR into R such that a^b m(a; b) a_b for every a; b 2 R. Among them are well known [1]: i) weighted Arithmetic Means m(x; y) = px + (1? p)y), and ii) weighted Geometric Means m(x; y) = x p y (1?p), where p 2 [0; 1]. A fuzzy real averaging is the extension of a real averaging to P (R) by means of the extension principle. For example,

9 i) generalized weighted Arithmetic Means m 1 (A; B)(x) = sup A(a) ^ B((x? (1? p)a)=p); a and ii) generalized weighted Geometric means m 2 (A; B)(x) = sup A(a) ^ B((x=a (1?p) ) 1?p ; a where p 2 [0; 1]: Other important examples of averaging function (over any kind of universe U) are the agregations of fuzzy sets: i) fuzzy weighted Arithmetic Means and ii) fuzzy weighted Geometric means m 3 (A; B)(x) = pa(x) + (1? p)b(x); m 4 (A; B)(x) = A(x) p B(x) 1?p ; where p 2 [0; 1]. Finally, very useful examples in non-monotonic reasoning are: ( A \ B if A \ B 6= ; m 5 (A; B) = B otherwise and ( A if B = U m 6 (A; B) = B otherwise 3.3 Non-monotonic Fuzzy Inference Denition A non-monotonic fuzzy rule is a pair (c; f c ) where c = (p 1 ; p 2 ; X) is a conict and f c is an averaging function on the universe of discourse U of X. Denition. Let F P rop, and let NM be a set of non-monotonic fuzzy rules. Given a set of fuzzy propositions A 2 F P rop, and a fuzzy proposition p 2 F prop, a non-monotonic proof of p form the hypothesis A under the knowledge and NM is a sucesion of fuzzy propositions p 1 ; :::; p m such that: i) p m = p, ii) For each i 2 f1:::mg, it is veried one of the following conditions: a) p i 2 [ A, b) there exist t < i and s < i such that p i = " X ~ is A" is obtained from a fuzzy rule of inference with p s and p t, being neither p s nor p t propositions appearing in a conict of the rules of NM. c) there exist t < i and s < i such that p i = "X is A", (c; f c ) 2 NM where c = (p t ; p s ; X), A = f i (A 1 ; A 2 ), "X is A 1 is obtained form a fuzzy rule of inference with p t and the deduction

10 of one X ~ 1 from the hypothesis A under the knowledge, and "X is A 2 is obtained from a fuzzy rule of inference with p s and the deduction of one X ~ 2 from the hypothesis A under the knowledge. Denition. Let F P rop, and let NM be a set of non-monotonic fuzzy rules. Given a set of fuzzy propositions A 2 F P rop, and a fuzzy proposition p 2 F prop, we will say that p is a non-monotic consequence of A under the knowledge and NM, write A `N M p (A `N M p when = ;), if there exist a non-monotonic proof of p form the hypothesis A under the knowledge and NM. Example. Let us consider the following base composed by one fuzzy rule and one nonmonotonic fuzzy rule: r: If X is A then Y is B; nmr: ((If X is A 1 then Y is B 1, If X is A 2 then Y is B 2, Y), m). Then, from the knowledge A= f X is A' g we can derive: a) From r, Y is B' where B' is obtained by If X is A then Y is B X is A 0 Y is B 0 b) From nmr: Y is m(b 0 1,B 0 2) where B 0 1,B 0 2 are obtained by: If X is A 1 then Y is B 1 X is A 0 Y is B 0 1 If X is A 2 then Y is B 2 X is A 0 Y is B 0 2 c) From a and b: Y is B 0 \ m(b 0 1; B 0 2). Example. Let us consider the following base composed by three fuzzy rule and one nonmonotonic fuzzy rule: r 1 : If X is A 1 then Y is B 1 ; r 2 : If X is A 2 then Y is B 2 ; r 3 : If Y is B then Z is C; nmr: ((If Y is B 1 then Z is C 1, If Y is B 2 then Z is C 2, Z), m). Then, from the knowledge A= f X is A' g we can derive: d1) From r 1 : Y is B 0 1; d2) From r 2 : Y is B 0 2;

11 d3) From d1 and d2: Y is B 0 1 \ B0 2; (it is the deduction of Y from the hypothesis A under the knowledge r 1 ; r 2 ; r 3. d4) From d3 and r 3 : Z is C 1;2 d5) From d3 and nmr: Z is m(c 1 1;2; C 2 1;2) d6) From d4 and d5: Z is C 1;2 \ m(c 1 1;2; C 2 1;2). Thus, we only re a non-monotonic fuzzy rule when all the monotonic inference relative to this rule is made. The reason is that non-monotonic fuzzy rules are thought to extend the monotonic inference. Moreover, if we re this rules with any possible partial deduction, we will nd inconsistent non-monotonic consequences. In the above example: d7) From d1 and nmr: Z is m(c 1 1; C 2 1); d8) From d2 and nmr: Z is m(c 1 2; C 2 2); d9) From d7, d8 and d6: Z is C 1;2 \ m(c 1 1;2; C 2 1;2) \ m(c 1 1; C 2 1) \ m(c 1 2; C 2 2), which could be inconsistent. Theorem. `N M is a cumulative inference relation over FProp, i.e., it satises the conditions of reexivity (G1), cumulative transitivity (G2), and cumulative monotony (G3'): G1) A `N M p whenever p 2 A G2) A `N M M p whenever A `N p i for all i 2 I and A [ fp i g i2i `N M p G3') A [ fp i g i2i `N M M M p whenever A `N p and A `N p i for all i 2 I. Proof. G1) It is obvious. G2) If A [ fp i g i2i `N M p then there exists a non-monotonic proof q 1 ; :::; q m = p of p from the hypothesis A [ fp i g i2i under the knowledge and NM. Then, changing any q i such that q i 2 fp i g i2i by one respective non-monotonic proof of q i from the hypothesis A under the knowledge and NM, we obtain a proof of p from the hypothesis A under the knowledge and NM. G3') It is obvious that G3) does not hold in general (see before example). If A `N M p then there exists a non-monotonic proof q 1 ; :::; q m = p of p from the hypothesis A under the knowledge and NM. The only cause for q 1 ; :::; q m = p not being a non-monotonic proof of p from the hypothesis A [ fp i g i2i under the knowledge and NM is that the deduction about a X ~ i from the hypothesis A under the knowledge is not the deduction about X ~ i from the hypothesis A [ fp i g i2i under the knowledge, but it is a contradiction with A `N M p i for all i 2 I. It is obvious that A `N M p if and only if A [ `N M p. Thus, the non-monotonic Fuzzy Logic represented by `N M is a non-monotonic logic in the sense of Gabbay [6]. 4 Patterns of Non-monotonic Fuzzy Rules In this section we will model some patterns of non-monotonic reasoning [7,8] using our approach.

12 Default fuzzy rules They are rules where the inference is made only if it do not produce an inconsistency: p 1 : If "X is A then Y is B" p 2 : "Y is U" ( B1 \ B m 5 (B 1 ; B 2 ) = 2 if B 1 \ B 2 6= ; otherwise B 2 An example can be "If X is student then X is Young". For any student we will infer that he is young unless we know that X is not Young. In Reiter default logic we must be careful with this kind of rules because they can produce circularity and so inconsistency [9], but in non-monotonic fuzzy logic it is not possible because the inference with non-monotonic fuzzy logic can not be used to re another non-monotonic fuzzy rule. If we want do it, then circularity and inconsistency will be possible and then, G3') does not hold. Prototypical Assumptions. When we do not know anything about the value of a variable X, it is often reasonable to assume that it takes the most common fuzzy value. p 1 : "X is A" p 2 : "X is U" ( B1 if B m 6 (B 1 ; B 2 ) = 2 = U otherwise For example, if we don't know anything about the size of a house we can assume that it is medium size. Minimum cost Assumptions. When we don't know anything about the value of a variable X, and there exist an associate cost to any possible fuzzy value, it is often assumed that it take the minimum cost fuzzy value. p 1 : "X is A" p 2 : "X is U" B 2 ( B1 if B m 6 (B 1 ; B 2 ) = 2 = U otherwise An example can be, if we don't know how of dangerous is an unknown animal, we usually assume that it is very dangerous. Complementary fuzzy rules. A reasonable fuzzy rule is usually a generic rule which can be conictive with another reasonable fuzzy rule. In this case it is reasonable to make an averaging between both inferences. p 1 : If "X is A 1 then Y is B 1 " p 2 : If "X is A 2 then Y is B 2 " B 2

13 m(b 1 ; B 2 ), where m is one of the examples of fuzzy mean m 1, m 2 or m 3. Examples of this kind of rules are the examples of the introduction. 5 Conclusion In this paper we have introduce a non-monotonic fuzzy reasoning with the following properties: - It is a non-monotonic logic in the sense of Gabbay [6]. - A non-monotonic fuzzy rule is red only when all the monotonic inference relative to this rule is made. - The inference with non-monotonic fuzzy logic can not be used to re another nonmonotonic fuzzy rule, so that circularity is not allowed. - Usual patterns of non-monotonic rules can be modeled in this approach. Taken these ideas into acount, for future work it would be interesting to develop the semantic of this logic, and then to study the completeness and the soundness between syntactic and semantic. 6 References [1] Aczel, J., Lectures on Functional Equations and their Applications, Academic Press, [2] Bellman, R.E. and Zadeh, L.A., Local and Fuzzy Logic, in J.M. Dunn and G. Epstein, eds, Modern Uses of Multiple-Valued Logic (D. Reidel, Dordrecht), , [3] Brown, D.J. and Suszko, R., Abstract Logics. Dissertations Mathematicae 102, 9-42, [4] Castro, J.L. and Trillas, E., The management of the inconsistency in Expert Systems, Fuzzy Sets and Systems 58, 51-57, [5] Castro, J.L. and Trillas, E., Constraints as incompatibility relations in BS, Int. J. of Uncertainty, Fuzziness and nowledge-based Systems 2, , [6] Gabbay, D., Theoretical Foundations for Non-Monotonic Reasoning in Expert Systems, in.r. Apt ed., Logic and Models of Concurrent Systems, Berlin, Springer-Verlag, [7] Ginsberg, M.L., Readings in Nonmonotonic Reasoning, Morgan aufmann, [8] Lukaszewicz, W. Non-monotonic Reasoning. Formalization of Commonsense Reasoning. Ellis Horwoord, Chichester, [9] Reiter, R., A Logic for Default Reasoning Articial Intelligence 13, , [10] Zadeh, L.A., A theory of Approximate Reasoning, in J.E. Hayes, M. Michie and L.I. Mickulich, eds, Machine Intelligence (John Wiley & Sons, New York), , [11] Zadeh, L.A., Syllogistic reasoning in fuzzy logic and its applications to usuality and reasoning with dispositions. IEEE Transactions on Systems, Man and Cybernetics, 15(6): , [12] Zhu, Q. and Lee, E.S., Comparison and Ranking of Fuzzy Numbers. In J. acpryzk and M. Fedrizzi (eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw, pp , 1992.