Applied Logic. Lecture 3 part 1 - Fuzzy logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw

Applied Logic Lecture 3 part 1 - Fuzzy logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018 1 / 36

Lecture plan 1 Introduction Definitions and properties Operations on fuzzy sets Linguistic rules 2 Fuzzy sets and fuzzy reasoning Basic fuzzy notions revisited Fuzzy relations 3 Fuzzy logical operators Fuzzy implication Marcin Szczuka (MIMUW) Applied Logic 2018 2 / 36

The origins of fuzziness In contrast to precise, limited and constrained language that we use to describe notions, entities, and concepts while building logical models (so far), the real-life concepts and entities are described in much less rigid way. Let s consider the following example. In real life sentence: John is a tall guy may mean many things, depending of our perspective, the place we live (meaning of tall is different in e.g. Japan) and so on. But, if we want to feed John s data into computer, we have to determine his height precisely say 190 cm. But what if do not know John s height exactly? Marcin Szczuka (MIMUW) Applied Logic 2018 3 / 36

The origins of fuzziness In real life we are doing perfectly all right with the sentences like: It takes about 40 minutes to reach an airport if the traffic is not too heavy. But what if we want a computer to understand such a sentence? How do we represent about and too heavy in a machine? Marcin Szczuka (MIMUW) Applied Logic 2018 4 / 36

Fuzzy concepts and fuzzy sets In 1965 Lotfi Zadeh proposed a different way of looking at notions such as: set, containment, subset. His target was to make it possible to deal with concepts (sets) and dependencies that by nature are imprecise and vague - so called fuzzy concepts (sets). Again, the example of such concept is the natural language sentence: John is a tall guy. If we know, that John is 175 cm tall, we may start to wonder about the validity of the above sentence. In classical set theory we are forced to make definite decision whether 175 cm qualifies John as tall or not. In the fuzzy set theory we may be more subtle and express to what degree 175 cm of height makes John a tall guy. Marcin Szczuka (MIMUW) Applied Logic 2018 5 / 36

Fuzzy sets In classical set theory and with classical binary logic, that we usually employ when doing things with use of computer, the (contents of a) set A in some universe X can be expressed in the form of its characteristic (containment) function: { 1 if x A χ A (x) = 0 if x / A Such classical, rigidly defined set we will further call crisp or definite. The key step in defining fuzzy set theory is the replacement of characteristic function χ A by function µ A : X [0, 1]. µ A is called membership function or fuzzy membership. If x X µ A (x) {0, 1} then A is a classical set i.e., crisp (definite) set. If there exists x X such that 0 < µ A (x) < 1 the set A is fuzzy. Marcin Szczuka (MIMUW) Applied Logic 2018 6 / 36

Fuzzy sets - examples A classical example of a fuzzy set near zero provided by Zadeh for the concept of real number near 0. This set may be defined, for example, by the following membership function: Which looks like: µ near zero = 1 1 + x 2 Marcin Szczuka (MIMUW) Applied Logic 2018 7 / 36

Fuzzy sets - examples The previously considered notion of tall guy could be given for height x in centimetres by membership: 0 if x 125 µ tall = 1 if x 185 x 185 2 + 1 if 125 < x < 185 Which looks like: 1 µ wysoki 125 185 Wzrost w cm Marcin Szczuka (MIMUW) Applied Logic 2018 8 / 36

Fuzzy sets - examples Another example of three fuzzy sets for the notion of cold, warm and hot, where x is a temperature. Marcin Szczuka (MIMUW) Applied Logic 2018 9 / 36

Fuzzy sets definitions and properties Fuzzy set A given by membership function µ A : X [0, 1] is normal iff x X µ A (x) = 1. For the fuzzy set theory to be really a set theory we must establish some fundamental notions such as: containment, subset, empty set and so on. Fuzzy inclusion Let A, B be fuzzy sets in some universe X. Fuzzy set A is included in fuzzy set B (A B) iff x X µ A (x) µ B (x). Some properties For each fuzzy set we may establish: Height of A: height(a) = h(a) = max x X µ A (x). Support of A: Supp(A) = {x X : µ A (x) > 0}. Core of A: Core(A) = {x X : µ A (x) = 1}. Marcin Szczuka (MIMUW) Applied Logic 2018 11 / 36

Fuzzy sets definitions and properties Empty fuzzy set Fuzzy set is said to be an empty set iff x X µ (x) = 0 In case of classical set theory the size of the set is measured in terms of the number of its elements. In case of fuzzy set elements may be only partly in the set, so we use membership function to determine the P ower of set the equivalent of cardinality for classic sets. Power of fuzzy set For a fuzzy set A its cardinality is given as { n P ower(a) = A = i=1 µ A(x) if X = {x 1,..., x n } X µ A(x)dx otherwise To complete the picture, we have to define the way of constructing fuzzy sets by summing, superposing, and complementing the fuzzy sets. For that, we need fuzzy set operators. Marcin Szczuka (MIMUW) Applied Logic 2018 12 / 36

Fuzzy sets operations In classical set theory we perform operations on sets, e.g. union, intersection, complement, difference. In case of fuzzy sets there are actually several ways of defining set operators. In majority (over 90%) of practical applications we use minimum of membership values (as membership for intersection) and maximum of membership values (as membership for union). Fuzzy operators For fuzzy sets A and B with corresponding memberships µ A and µ B, we have: Sum of fuzzy sets A B: µ A B = max(µ A, µ B ). Intersection of fuzzy sets A B: µ A B = min(µ A, µ B ). Complement of fuzzy set \A: µ \A = 1 µ A. Marcin Szczuka (MIMUW) Applied Logic 2018 14 / 36

Fuzzy sets operations µ B (x) µ B (x) µ A (x) µ A (x) x x Alternative ways of defining operations on fuzzy sets include: µ A B = max(0, µ A + µ B 1), µ A B = min(1, µ A + µ B ) so called Łukasiewicz operators. µ A B = max(0, µ A + µ B µ A µ B ), µ A B = µ A µ B - so called product operators. Marcin Szczuka (MIMUW) Applied Logic 2018 15 / 36

Important notes on fuzzy sets It is very important to realise that: 1 Fuzzy set theory is NOT an alternative to classical set theory. It is an extension of classical set theory that cannot exist independently. We have to use the apparatus of classical set theory in order to define fuzzy sets. Hence, fuzzy set theory is not independent of classical one. 2 Fuzzy set theory, despite an immediate intuitive association, is NOT a replacement for probabilistic reasoning. This association is caused by the fact that both methodologies are founded on the interval [0, 1]. However, it may be the case that no reasonable probability distribution corresponds to the layout of a family of fuzzy sets in a given space. Marcin Szczuka (MIMUW) Applied Logic 2018 16 / 36

Linguistic rules Linguistic rules are the statements of the form: IF A 1 AND A 2 AND... AND A k THEN D where conditions A 1,..., A k and decision D correspond to fuzzy sets. For example: IF waether is good AND traffic is light AND we have enough fuel THEN we will reach the airport in about 30 minutes such rules we may obtain form a human expert or discover (mine) from data. In order to use them in the context of fuzzy sets, we will employ fuzzy set operators. Marcin Szczuka (MIMUW) Applied Logic 2018 18 / 36

Fuzzy operators As mentioned before, there are in fact (infinitely) many ways for defining operations on fuzzy sets. That comes form the fact, that there is more than one: 1 Function that satisfies conditions for a T-norm, i.e., equivalent to intersection. 2 Function that satisfies conditions for a T- co-norm (S-norm), i.e., equivalent to union. 3 Function that satisfies conditions complement (negation). We will see several examples of such functions. Marcin Szczuka (MIMUW) Applied Logic 2018 21 / 36

Fuzzy intersection T-norm The whole class of functions that are called T-norms can be used as fuzzy intersection. Definition T-norm For any a, b, c, d [0, 1] T-norm is a function T : [0, 1] 2 [0, 1] such that: Commutativity: T (a, b) = T (b, a); Associativity: T (a, T (b, c)) = T (T (a, b), c); Monotonicity: T (a, b) T (c, d) whenever a c, b d; Invariance for 1: T (a, 1) = a It is quite easy to see that the intersection of fuzzy sets defined by T (a, b) = min(a, b) is a proper T-norm. In fact, the function min(.,.) is a maximal element in the class of T-norms. Marcin Szczuka (MIMUW) Applied Logic 2018 22 / 36

Fuzzy union T- co-norm Analogously, the whole class of S-norms (T- co-norms) can be used as fuzzy union. Definition S-norm For any a, b, c, d [0, 1] S-norm (T- co-norm) is a function S : [0, 1] 2 [0, 1] such that: Commutativity: S(a, b) = S(b, a); Associativity: S(a, S(b, c)) = S(S(a, b), c); Monotonicity: S(a, b) S(c, d) whenever a c, b d; Invariance for 0: S(a, 0) = a. We have already seen examples of T-norms and S-norms such as: T (a, b) = max(0, a + b 1), S(a, b) = min(1, a + b) - so called Łukasiewicz operators. T (a, b) = a + b ab, S(a, b) = ab - so called product operators. Marcin Szczuka (MIMUW) Applied Logic 2018 23 / 36

Fuzzy complement The complement of a fuzzy set (fuzzy negation) we can also define in great number of ways. All we need, is that negation conforms to a set of conditions. Definition fuzzy complement (negation) For any a, b [0, 1] function N : [0, 1] [0, 1] is called a complement (negation) operation if the following holds: Preservation of constants: N(0) = 1;N(1) = 0; Reversing of the order: N(a) N(b) iff b a; Involution: N(N(a)) = a. There are several functions that may be used as complement, but in 99.9% of applications (and further in this lecture) the only function used is N(x) = 1 x (µ \A = 1 µ A ). If in the definition above we cannot assure involution, the resulting operator is called an intuitionistic negation. Marcin Szczuka (MIMUW) Applied Logic 2018 24 / 36

Duality of T-norms and S-norms Once we have a negation operator, we can define an S-norm (T - co-norm) dual to a given T-norm. Definition dual S-norm Given a T-norm T : [0, 1] 2 [0, 1] we can define its dual co-norm (S-norm), and vice versa, by: S(a, b) = N(T (N(a), N(b))) As an exercise one may check if the examples of T-norms and S-norms presented previously are dual to each other. As mentioned before, the pair of operations min(.,.) and min(.,.) play a special role. They are only idempotent operators in the class of T-norms and S-norms, respectively. They are also ones that conform to distributive laws: A (B C) = (A B) (A C) and A (B C) = (A B) (A C). Marcin Szczuka (MIMUW) Applied Logic 2018 25 / 36

Fuzzy relations In classical set theory a (binary) relation is defined as a subset of a Cartesian product. In fuzzy case the definition is analogous. We will consider only binary (two argument) relations, but is worth mentioning that the definitions presented in this section are easily extensible to the case of n-ary relations for any finite n. Definition - fuzzy relation Any fuzzy subset of X Y is a fuzzy relation defined over X Y. Fuzzy relation defined in such a way possesses all the required features. However, please note that in order to introduce fuzzy relation we have not used the notion of Cartesian product of fuzzy sets. Marcin Szczuka (MIMUW) Applied Logic 2018 27 / 36

Cartesian product of fuzzy sets We want to define the Cartesian product of two fuzzy sets and check how it relates to the previously introduced notion of fuzzy relation. Definition - Cartesian product of fuzzy sets Let A, B be fuzzy sets in universes X and Y, respectively. Cartesian product A B is a relation R (notation: R = A B) defined on X Y by: In general (n-ary) case: where R = A 1 A 2... A n µ R (x, y) = min(µ A (x), µ B (y)) µ R (x 1,..., x n ) = min i (µ Ai (x i )) Marcin Szczuka (MIMUW) Applied Logic 2018 28 / 36

Cylindrical extension and projection In some situations it is useful to consider the properties of fuzzy relation w.r.t. particular argument. To do that, we introduce two more notions cylindrical extension and projection. Definition cylindrical extension and projection Let A be a fuzzy set in universe X. Cylindrical extension of set A over Cartesian product X Y is a fuzzy relation Â = A Y defined by fuzzy membership function: µâ(x, y) = T (µ A (x), µ Y (y)) = T (µ A (x), 1) = µ A (x), where T is a T-norm. Let R be a fuzzy relation defined over X Y. Projection of R on X (analogously on other co-ordinates) is a fuzzy set A in X, denoted by A = P roj x (A) and given by membership function: µ A (x) = max (µ A(x, y)). y Marcin Szczuka (MIMUW) Applied Logic 2018 29 / 36

Classic vs. fuzzy logic In the case of classical set theory, set operators are connected 1-to-1 to logical operators (logical connectives). In fuzzy case the situation is more complicated. To begin with, we can have several different set operators. Therefore, when we consider logical operators associated with fuzzy set operations, we have to use different meaning of the truth/falsity for logical formulæ. In the classical logic the valuation is a function v : V AR {0, 1}, so the formula has either value equal to 0 (falsity) or 1 (truth). In fuzzy logic we will permit the formula to have logical value between 0 and 1. More precisely, [φ] v [0, 1]. Marcin Szczuka (MIMUW) Applied Logic 2018 31 / 36

Fuzzy logical connectives In fuzzy case logical connectives (logical operators) can no longer be defined with use of truth-tables. They are now functions [0, 1] 2 [0, 1] ([0, 1] [0, 1] in the case of negation). Using T-norms and S-norms (T- co-norms) we can introduce fuzzy conjunction and fuzzy alternative (disjunction) in quite natural way as: [φ ψ ] v = T ([φ] v, [ψ ] v ), [φ ψ ] v = S([φ] v, [ψ ] v ) Usually, we will assume the T-norm T and S-norm S in the above definitions to be dual to each other. Similarly, for negation we can use any function that meets requirements, but almost always we will use: [ φ] v = 1 [φ] v. Marcin Szczuka (MIMUW) Applied Logic 2018 32 / 36

Fuzzy logical connectives Equivalence is defined using implication and conjuction: By assuming that [φ ψ ] v = [(φ ψ) (ψ φ)] v [φ] v [ψ ] v [φ ψ ] v = 1, we obtain [φ ψ ] v = 1 lub [ψ φ] v = 1. Hence, for conjunction operator defined by means of T-norm we finally get [φ ψ ] v = min([φ ψ)] v, [ψ φ)] v ) regardless of our choice on conjunction operator (T-norm). Marcin Szczuka (MIMUW) Applied Logic 2018 33 / 36

Fuzzy implication To complete the picture we need to define the meaning of φ ψ (the value of [φ ψ ] v ) in fuzzy case. It is not much of a surprise that we can do it in several ways. Some examples are presented below. Name of implication Value of [φ ψ ] v Łukasiewicz min(1 { [φ] v + [ψ ] v, 1) 1 if [φ]v [ψ ] Gödel v { [ψ ] v otherwise 1 if [[φ]v = 0 Goguen min(1, [[ψ]] v [[φ]] v ) otherwise Kleene-Dienes max(1 [φ] v, [ψ ] v ) Zadeh max(1 [φ] v, min([ψ ] v, [φ] v )) Reichenbach 1 [φ] v + [[ψ ] v [φ] v Marcin Szczuka (MIMUW) Applied Logic 2018 35 / 36

Fuzzy connectives - peculiarities In case of fuzzy connectives we can define ones by means of others using tautologies, just as we did in classical case. We have to be careful, though. Depending on the tautology used, the results can be different. For example, using Łukasiewicz implication we can define two different alternative operators as follows: [φ 1 ψ ] v = [ φ ψ ] v = min([φ] v + [[ψ ] v, 1) (1) [φ 2 ψ ] v = [ φ (ψ φ)] v = max([φ] v, [ψ ] v ) (2) In classical logic both definitions would yield the same result since both formulas: φ ψ and φ (ψ φ) are equivalent to alternative. In fuzzy case with Łukasiewicz implication we should only use negation and alternative, i.e., only formula (1). Marcin Szczuka (MIMUW) Applied Logic 2018 36 / 36