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Many-valued Similarity - Theory and Applications of Fuzzy Reasoning

Many-valued Similarity - Theory and Applications of Fuzzy Reasoning. Esko Turunen Tampere University of Technology Finland. Motivation. Lecture I. Aristotelian logic:. All human beings are mortal Sokrates is a human being. Sokrates is mortal.

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Many-valued Similarity - Theory and Applications of Fuzzy Reasoning

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  1. Many-valued Similarity- Theory and Applications of Fuzzy Reasoning Esko Turunen Tampere University of Technology Finland

  2. Motivation Lecture I Aristotelian logic: All human beings are mortal Sokrates is a human being Sokrates is mortal It took thousands of years before Aristotelian informal logic was expressed in a formal way, today known as First-order Boolean Logic In 1960’s Zadeh introdused Fuzzy Logic: More generally (fuzzy rule systems): IF x is in A1 and y is in B1 THEN z is in C1 * * * IF x is in An and y is in Bn THEN z is in Cn Red apples are ripe This apple is more or less red This apple is almost ripe What is the mathematical formalism of Zadeh’s Fuzzy Logic? We claim it is Pavelka - Lukasiewicz many-valued logic, in particular, many-valuedsimilarity.

  3. In science, we always want to minimize the set of axioms and maximize the set of the consequences of these axioms. Thus, consider the following Definition 1. Let L be a non-void set, 1 an element of L and →, * a binary and unary operation, respectively, defined on L such that, for all x, y, z in L, we have: Then the system L = L, →,*,1 is called Wajsberg algebra. Now, define on a Wajsberg algebra L a binary relation ≤ by (5) x ≤ y iff x→y = 1. Then, by (2) we have which implies (6) x→x = 1, i.e. x ≤ x. Hence, by (1) we have Let x→y = 1 and y→z = 1, that is, let x ≤ y and y ≤ z. By (2), thus (7) if x ≤ y, y ≤ z then x ≤ z. Let x→y = 1 and y→x = 1, that is, let x ≤ y and y ≤ x. By (3), 1→y = 1→x, thus, x = y. We conclude (8) if x ≤ y, y ≤ x then x = y. Equations (6) – (8) show that (5) defines an order on L. Next we show that the element 1 is the greatestelement with respect to this order, in other words (9) for all x in L, x →1 = 1, i.e. x ≤ 1.

  4. To this end, we first reason, by (3), (1) and (6), (x→1)→1 = (1→x)→x = x→x = 1, that is (10) (x→1)→1 = 1 On the other hand, by (1), (10), (1) and (2), in other words, Thus, so by (8) First exercise. Show that the arrow operation is antitone in the first variable, that is, if x ≤ y then y→z ≤ x →z. [Hint: use equation (2)] Proposition 1. In a Wajsberg algebra L, for any x, y, z in L, Proof. Since y ≤ 1 and → is antitone in the first variable we reason that 1→x ≤ y→x, therefore x ≤ y→x, thus (11) holds. To establish (12) we first verify Indeed, if x ≤ y→z then, as the arrow operation is antitone in the first variable, we have By (3), and, by (11), so that altogether we have By definition (5), Thus, (14) holds.

  5. Applying (14) to (z→x)→[(x→y)→(z→y)] = 1, which holds by (2), we conclude (x→y)→[(z→x)→(z→y)] = 1, i.e. (12) holds. Finally, by (11) and (3), and by (12), Therefore which, by (14), implies x→(y→z) ≤ y→(x→z). By a similar argument y→(x→z) ≤ x→(y→z). We conclude that (13) holds. Second exercise. Show that the arrow operation is isotone in the second variable, that is, if x ≤ y then z→x ≤ x→y. [Hint: use equation (12)] Proposition 2. In a Wajsberg algebra L, for any x in L, Proof. By (11), x* ≤ (1*)*→x*, by (4), (1*)*→x* ≤ x→1*, hence On the other hand, by (4) and (1), and, as the arrow operation is antitone in the first variable, which, by (3), implies that Next we reason, and by (13), by (4) and (1), and, by (11), thus Hence, and, by (18), The (in-)equalities (17) and (19) now imply equation (15).

  6. The (in-)equality (16) follows by (11), (4) and (1), indeed, Remark. Condition (16) implies that 1* is the least element in the corresponding Wajsberg algebra L and will therefore denoted by 0. We write x** instead of (x*)*. Third exercise. Show that, for all elements x, y in a Wajsberg algebra L, hold [Hint: Apply (13) and (4) for (20), moreover (4), (20) and (4) for (21) and finally, (20), (4) and (20) for (22).] Till now we have seen that Wajsberg algebra axioms generate an order relation on L. Our aim is to show that, after a suitable stipulation, L becomes a lattice, that is, all pairs x, y of elements of L have the greatest lower bound in L and the least upper bound in L with respect to the order relation given by (5). For l.u.b{x,y} we set First we realize, by (11), that and then, by (11) and (3), Let now z be such an element of L that x, y ≤ z. Then x→z =1 thus, by (1), (x→z)→z = z. Since the arrow operation is antitone on the first variable, we first reason z→x ≤ y→x, and then (y→x) →x ≤ (z→x) →x = z. We conclude that (y→x) →x coincide with l.u.b{x,y}, i.e. that (23) is a correct definition.

  7. For g.l.b{x,y} we set First we realize that which is the case. On the other hand, if z is such an element of L that Then therefore and so We conclude that (24) is a correct definition. Fourth exercise. Show that, for all elements x, y in a Wajsberg algebra L, hold de Morgan laws [Hint: Apply (24) and (20).] Define on a Wajsberg algebra L a binary operation for each x, y, z in L via Then we have Proposition 3. In a Wajsberg algebra L, for any x,y,z in L, Lecture II commutativity associativity isotonity Proof. By (26), (20), (21) and (26), respectively, we have Thus, (27) holds. therefore We have established (29). For (28) we reason in the following way

  8. Fifth exercise. Show that, for all elements x, y, z in a Wajsberg algebra L, hold Galois connection Remark. Equations (27) – (31) mean that lattice L generated by Wajsberg algebra axioms is a residuated lattice. Thus, all properties valid in a residuated lattice hold in Wajsberg algebras, too. For example, the meet and join operations are associative, commutative and absorption holds. For all x, y, z in L, By (24), (23), (21) and (26) we reason that and, by commutativity of Λ, we have

  9. Proposition 4. In a Wajsberg algebra L, for any x, y in L, Proof (ofprelinearity). By (32) and (6), and, similarly, therefore Thus, Therefore (34b) holds. Sixth exercise. Show that, for all elements x, y, z in a Wajsberg algebra L, holds [Hint: (21), (25), (23), (3), (13), (21)] Remark. Residuated lattices such that (34a) and (34b) hold are called BL-algebras (Basic Logic algebras by P. Hajek 1997), moreover, BL-algebras such that a double negation low x**= x holds are known as MV-algebras (Multi Valued algebras by C.C. Chang 1957). Hence, Wajsberg algebras are MV-algebras. Even more is true, these two structures coincide: each MV-algebra generates a Wajsberg algebras and vice versa.

  10. In an MV-algebra, there is a binary operation In a Wajsberg algebra, a sum operation is introduced by a formula Seventh exercise. Show that, for all elements x, y, z in a Wajsberg algebra L, hold For the sake of completeness, we present the original MV-algebra axioms of C.C. Chang. It will be an extra exercise to show that they hold in Wajsberg-algebras!

  11. We needed only four equational axioms to establish a rich structure. However, to be able to introduce fuzzy inference in an axiomatic way, we will still need two more axioms. Unfortunately, they are not equational. First consider a completeness axiom where L is an MV-algebra, called complete MV-algebra. For such algebras we have e.g. Proposition 5. In a complete MV-algebra L, for any xL, {yi | iG }Í L. Proof. Since the operation is isotone, we have, for each i in Г, therefore Conversely, for each i, by the Galois connection, equivalent to Therefore Again by the Galois connection we conclude We have demonstrated equation (49). Equation (50) can be shown in a quite similar manner. Indeed, since for each i, we have Conversely, trivially

  12. Thus, by the Galois connection, We shall conclude which is equivalent to This completes the proof of equation (50). To establish (51) we first realize, as the arrow operation is antitone in the first variable, Therefore therefore Conversely, thus hence whence which is equivalent to The proof is complete. Eighth exercise. Prove in a complete MV-algebra Ninth exercise. Prove in a complete MV-algebra

  13. An element b of an MV-algebra L is called an n-divisor of an element a of L, if If all elements have n-divisors for all natural n, then L is called divisible. An MV-algebra L is called injective if it is complete and divisible. We will see that the six axioms of an injective MV-algebra are sufficient to construct fuzzy IF-THEN inrefence systems. A canonical example of an injective MV-algebra is the Lukasiewicz algebra defined on the real unit interval [0,1]: 1 = 1, x* = 1 – x, x→y = min{1, 1 – x + y}. Di Nola and Sessa proved in 1995 that an MV-algebra L is injective if, and only if L is isomorphic to F(L), where F(L) is the MV-algebara of all continous [0,1]-valued functions on the set of all maximal ideals of L, and 1(M) = 1, (f→g)(M) = min{1, 1 – f (M) + g(M)}, f*(M) = 1 – f(M), for any maximal ideal M of L. Tenth exercise. Write the MV-operations on the Lukasiewicz structure, that is

  14. Proposition 5. In an injective MV-algebra L, any n-divisor is unique. Lecture III Proof. It is enough to show that the statement holds in any injective MV-algebra F(L). To this end, let n a natural number and g, h two n-divisors of f. Let M be a maximal ideal of L. If f(M) = a < 1, then n[g(M)] = (ng)(M) = a = (nh)(M) = n[h(M)]. Thus, g(M) = h(M). Now assume f(M) = 1. Then If (n-1)g(M) would be equal to 1, then g(M) should be equal to 0, which it is clearly not. Let a counter assumption Therefore (n-1)g(M) < 1. Similarly (n-1)h(M) = 1-h(M) < 1. g(M) < h(M) hold. Then which implies a contradiction h(M) < g(M). An assumption h(M) < g(M) leads to a similar contradiction, too.Therefore h(M) = g(M). We conclude h = g and the proof is complete. By Proposition 5, we may denote the unique n-divisor of an element a by a/n. For any maximal ideal M of an injective MV-algebra L, it holds that n(f(M)/n) = f(M), moreover, We therefore conclude (f(M)/n) = f/n(M), that is, in F(L), ’map first, then divide equals to divide first, then map’.

  15. Eleventh exercise. Prove that in the Lukasiewicz structure, Clearly, in the Lukasiewicz structure, we have Thus, in F(L), Summarizing Proposition 6. In any injective MV-algebra L,

  16. Definition 2. Let L be an injective MV-algebra and let A be a non-void set. A fuzzy • similarity S on A is such a binary fuzzy relation that, for each x, y, and z in A, • S(x,x) = 1 (everything is similar to itself), • S(x,y) = S(y,x) (fuzzy similarity is symmetric), • (iii)S(x,y)○S(y,z) ≤ S(x,z) (fuzzy similarity is weakly transitive). Recall an L-valued fuzzy subset X of A is an ordered couple (A,μX), where the member- ship function μX:A→L tells the degree to which an element a in A belongs to the fuzzy subset X. Given a fuzzy subset (A,μX), define a fuzzy relation S on A by (54) S(x,y) = μX(x)↔μX(y). This fuzzy relation is trivially symmetric, by (6) it is reflexive and, by (2), transitive. So, Any fuzzy set generates a fuzzy similarity [this is true for L being any residuated lattice] Proposition 7. Consider n injective MV-algebra L valued fuzzy similariteis Si, i = 1,...,n on a set A. Then a fuzzy binary relation S on A defined by is an L valued fuzzy similarity on A. More generally, any weighted meanSIM is an L valued fuzzy similarity, where Proof. (i) & (ii) obvious, (iii) by Proposition 6. Example 1 Countries, Example 2 Functionality

  17. The idea of partial similarity is not new. Indeed, in 1988 Niiniluoto quoted from Mill (1843) by defining: If two objects A and B agree on k attributes and disagree on m attributes, then the number can be taken to measure the degree of similarity or partial identity between A and B. Obviously, sim is a reflexive and symmetric fuzzy relation. It is weakly transitive with respect to the Lukasiewicz t-norm (and, therefore, can be considered as an injective MV-algebra valued similarity). To see this, assuming there are Nattributes, study the following Venn-diagram: B It is easy to see that k + t + r ≤ N, 0 ≤ s. Then we have A m k p t s r q C which holds true. It is worth noting that, among all BL-algebras (in particular, among continuous t-norms) injective MV-algebras are the only structures where ’the average of similarities is a similarity’. Therefore the following consideration can be done only in such a structure.

  18. An Algorithm to Construct Fuzzy IF-THEN Inference Systems Let us now return to our starting point, a fuzzy rule based system Rule 1: IF x1 is in A11andx2 is in A12 and .... and xm is in A1mTHEN y is in B1 Rule 2: IF x1 is in A21andx2 is in A22and .... and xm is in A2mTHEN y is in B2 * * * Rule n: IF x1 is in An1andx2 is in An2and .... and xm is in AnmTHEN y is in Bn Here all Aij.s and Bj are fuzzy but can be crips actions, too. As usual, it is not necessary that the rule base is complete, some rule combinations can be missing without causing any difficulties. It is also possible that different IF-part causes equal THEN-part, but it is not possible that a fixed IF-part causes two different THEN-parts. We will not need any kind of defuzzification methods, everything is based on an experts knowledge and properties of injective MV-algebra valued similarity. Step 1. Create the dynamics of the inference system, i.e. define the IF-THEN rules and give shapes to the corresponding fuzzy sets. Step 2. If necassary, give weights to various IF-parts to emphasize their importance. Step 3. List the rules with respect to the mutual importance of their IF-parts. Step 4. For each THEN-part, give a criteria on how to distinguish outputs with equal degree of membership.

  19. A general framework for the inference system is now ready. • Asssume then that we have an actual input Actual = (X1,...,Xm). A corresponding output • Y is counted in the following way. • Consider each IF-part of each rule as a crisp case, that is μAij(xj) = 1holds. • Count the degree of similarity between Actual and the IF-part of Rule i, i = 1,...,n. • Since μAij(Xj)↔μAij(xj) = μAij(Xj)↔1 = μAij(Xj), we only need to calculate averages • or weighted averages of membership degrees! • (3) Fire an output Y such that μBk(Y) = Similarity(Actual, Rule k) corresponding to the • greatest similarity degree between the input Actual and the IF-part of a Rule k. If such • a maximal rule is not unique, then use the preference list given in Step (3), and if there • are several such outputs Y, use a creteria given in Step (4). Note that counting the actual output can be viewed as an instance of Generalized Modus Ponens in the sense of (injective MV-algebra valued) Lukasiewicz-Pavelka logic; where α corresponds to the IF-part of a Rule, β corresponds to the THEN-part of the Rule, a is the value Similarity(Actual, Rule k) and b = 1. This gives a many-valued logic based theoretical justification to fuzzy inference. In the rest part of the lecture deals with real world case studies where we have applied the above metodology and algorithm.

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