Fuzzy-evolutionary systems. Hybrid inference systems design. Piotr Czekalski Gliwice, 25th April 2012

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1 Utworzenie nowej specjalności Studiów Doktoranckich w Dyscyplinie Informatyka na Wydziale AEiI Politechniki Śląskiej: Eksploracja Danych (Data Mining) akronim: EkDan Fuzzy-evolutionary systems. Hybrid inference systems design. Piotr Czekalski Gliwice, 25th April 2012

2 Lecture outline: 1. Preface, AI general info. 2. Classical sets theory. 3. Fuzzy sets theory. 4. Fuzzy inference systems Fuzzy relations and linguistics variables If-then rules Fuzzy rules database and inference Mamdani-Assilian FIS model 4.4. Takagi-Sugeno-Kang FIS model 4.5. Czogała-Łęski FIS with parameterized consequents.

3 Lecture outline: 5. Evolutionary techniques 5.1. A mathematical model of mother nature Genetic algorithms review Introduction Members coding/modeling Fitness function and border conditions Genetic operators Reproduction and stop conditions Evolutionary computing Introduction Evolutionary strategies Fitness function and border conditions Members coding/modeling Evolutionary operators What is a difference between GA and EA?

4 Lecture outline: 6. Fuzzy-evolutionary techniques Fuzzy database learning models review Michigan approach Pittsburgh approach Iterative learning Data mining for fuzzy rules system learning Three stage Mamdani-Assilian FIS fuzzy rules extraction Two-stage Takagi-Sugeno-Kang FIS fuzzy rules extraction Fuzzy Inference System with parameterize d consequent fuzzy rules extraction. 7. Experiment design Learning data set and validation data set.

5 Preface Daily everyone solve problems like: Driving a car, avoiding accidents. Parking car. Doing profitable business (i.e. trading Forex or shares on NASDAQ/GPW) Reading hand-written text, with even partially damaged or intentionally misswritten text (i.e. P!CK UP THAT PHONE). Correcting plane on approach. Scheduling university time table ;-). Experts carry out (examples): Diagnose non-certain diseases. Optimize playing classic games (i.e. chess)

6 Preface (cont.) Those mentioned above constitute an intelligent approach to the problem. The AI systems are supposed to replace the human expert and/or operator. They may be considered as intelligent if human observing the behavior of such a system considers it as intelligent or cannot even distinguish machine behavior vs man. Systems may be equipped with other features that human cannot do or do much slower (i.e. high complexity computing problems).

7 Preface (cont.) Result? The number of freedoms of movement (i.e. input conditions) is huge. The computation algorithm is very complicated. Algorithms describing usually became NP or NPC (nondeterministic polynomial / complete) as stated by computational complexity theory. Real world problems are too complicated, to be described by regular, crisp approach.

8 Classical sets (Czogała, Łęski 2000, Łęski 2008). Classical sets provide crisp approach. Set is a collection of objects. Objects are distinguishable by some property (or properties). Individual objects (elements) constitute members of a set. Given the space/universe of discourse X any set A is a subset of X. If P(X) is a family of all subsets of X, there exists: empty set full set Being given finite set of elements 2 is true.

9 Classical sets (cont.). Given a set A any object x can belong to A, or can be excluded of A (not belong to A),. There are no immediate states for membership/nonmembership. That guides to the crisp, two-valued Boolean (Boole s) algebra defined by axiomatic postulates (i.e. Huntington) with the following three main operators (two binary, one unary): Conjunction (product) of A and B Disjunction (sum) of A or B Negation (complementary)

10 Classical sets (cont.). The following properties (postulates and theorems) hold true: Identity:,,, Idempotency:, Commutativity:, Associativity:, Law of excluded middle and law of contradiction Distributivity:

11 Classical sets (cont.). Absorption:, Involution: De Morgan:, Most of the above hold true for multiple objects/elements because of commutativity. The crisp membership function can be considered as characteristic function on some object s parameter: χ : 0,1 1, χ 0,

12 Classical sets (cont.). Let Ch(X) stand for all characteristic functions in X χ χ: 0,1 Then Ch(X) constitute Bool algebra with those operations: Conjunction-like (product) of A and B, χ, χ χ χ min χ, χ Disjunction (sum) of A or B χ χ mχ, χ Negation (complementary) χ 1 χ Structures,,, ) and,,, are isomorphic as Boolean algebras.

13 Classical sets (cont.). Intuitive approach (P(X), ) and methematical model (Ch(X), ) are easy to visualise using Venn Diagrams [wikipedia]:

14 Classical sets (cont.). Example set of tall persons holding true when height is more than or equal to 195cm can be defined: 195, or modelled using membership function as: χ 1 195, 0

15 Classical sets (cont.). It is obvious that crisp approach drives to some to tight boundaries when describing real world problems, as classical set are of dichotomous nature: i.e. tall presons set as described above drives to the reasoning that I m considered as tall (195cm) but someone who is 194.9cm is not! To solve this problem using classical sets it would be required to specify many sets, i.e. Low-tall: cm Medium-tall: cm High-tall: 200cm+ This drives to non-differentiable functions describing phenomena and increases algorithms complexity.

16 Fuzzy sets. The general approach of classical sets need redesign: Instead of using classical s set sharp transition that is crisp ( or and there is nothing more than inclusion or exclusion) let us use gradual membership of an element to the set. This idea constitute Fuzzy Sets (Zadeh, 1965). An element x contains a grade of membership to the set A in X given by functionµ A (x) (other notation A(x)) such as: : 0,1 or : 0,1 Thus set A is specified by the µ A (x) function directly or indirectly by a set of ordered pairs (x,µ A (x)):, µ A (x)) x

17 Fuzzy sets (cont.). The membership function can be referenced as (Łęski, 2008): Level of similarity how far value x is from the fuzzy prototype. This approach is common is fuzzy data grouping, fuzzy inference (how far is the value of the premiese of the rule involving acivation level of the rule) Preference coefficient i.e. how do I like some elements of the universe,while some other I dislike a common approach in fuzzy optimisation techniques. Level of uncertainty related to the Theory of Possibility (Zadeh, 1978) where membership function provides probability level of a fact that x has some fuzzy value.

18 Fuzzy sets (cont.). It is impossible (Łęski, 2008) to measure membership function directly, thus we use indirect methods, like: Measuring distance when membership function is considered as level of similarity, assuming the x a reference component, the can be determined as diminishing function of distance d(x,x ) (i.e. fuzzy data clustering). Measuring frequency when membership function is considered as preference coefficient, the is proportional to the probability x that component x was identified as a member of fuzzy set A (in the series of experiments).

19 Fuzzy sets (cont.). It is impossible (Łęski, 2008) to measure membership function directly, thus we use indirect methods, like (cont.): Measuring cost - is reciprocal to the cost of making x belong to set A. Cost=0 means cost=1 means.

20 Fuzzy sets (cont.). The common notation for a fuzzy set is also (Zadeh, 1973): µ A (x)/ when universe of discourse is countable µ A (x)/ when universe of discourse is uncountable When µ A (x) is restricted to Boolean 0 and 1, the fuzzy set becomes classical set.

21 Fuzzy sets (cont.). Similarly to the classical sets P(X), let F(X) denote a class of all fuzzy subsets over the universe of discourse X. As in the classical sets, fuzzy sets have their operators like: Union Intersection Complement,,, constitute so caled de Morgan / soft algebra: However some well known Boolean theorems for classical sets are not satisfied, including law of excluded middle and law of contradiction (Czogała, Łęski 2000).

22 Fuzzy sets (cont.). Fuzzy set contains some characteristic features: α-cut (α-level cut) a crisp set of elements that membership function is greater than or equal α: Strong α-cut (α-level cut) a crisp set of elements that membership function is greater than α: Support a crisp set of elements that membership function is positive: 0 Core a crisp set of elements that membership function is equal 1: 1 Thus Support is strong 0-cut and Core is 1-cut.

23 Fuzzy sets (cont.). Fuzzy set contains some characteristic features (cont.): Crossover: 0.5 Width: 0.5 Fuzzy set is convex iff the following is true:, λ 0,1 λ 1 λ min, Fuzzy sets A and B are equal iff: Fuzzy set A is subset of fuzzy set B ( ) iff:

24 Fuzzy sets (cont.). Fuzzy set contains some characteristic features (cont.): When fuzzy set A holds non-empty core then fuzzy set is Normal. Symetrical fuzzy set A: Left and right side open fuzzy set A: lim 1 lim lim 0 0 lim 1 Closed fuzzy set A: lim lim 0

25 Fuzzy sets (cont.). Using α-cut and decomposition theorem (Zimmermann,1985) fuzzy set A can be represented as a sum over it s α-cuts:, Thus membership function can be presented (using decomposition) as (Łęski, 2008):,, where for every 0 otherwise

26 Fuzzy sets (cont.). Usually, a membership function for continuous universe can be represented as one of the curves: Triangular (a,b,c): 0 y a b c x

27 Fuzzy sets (cont.). (cont.): Trapezoidal(a,b,c,d): y 1 0 a b c d x Gaussian (m,ρ), ρ >0: y 1 0 m x

28 Fuzzy sets (cont.). (cont.): General normal ( bell ) curve (m,ρ,γ), ρ>0 & γ>0 : 1 1 Sigmoidal (sigma) curve (c,β): 1 1 Fuzzy singleton (x,x 0 ), Core=Support:, 1, 0, The list above do not exhaust all possibilities but represents common ones.

29 Fuzzy sets (cont.). Sample fuzzy sets: Discrete, unordered: Universe X={ all colour names } A - MyPreferedColours:): (blue,0.9),(yellow,0.8),(black,0.6),(white,0.4 or using Zadeh notation: A=0.9/blue+0.8/yellow+0.6/Black+0.4/White Discrete, ordered: Universe: reasonable number of displays attached to one PC: X={0,1,2,3,4,6} A MyPreferedNumberOfLCDs attached ;-): 0,0.0, 1,0.25, 2,0.8, 3,0.81, 4,1, 6,0.8

30 Fuzzy sets (cont.). Sample fuzzy sets (cont.): Continuous: Universe: A human body blood pressure (Łęski, 2008): 1 1 /

31 Fuzzy set operations. Operations on fuzy sets are functions: T, S: 0,1 0,1 0,1 : 0,1 0,1 Operations on fuzzy sets A, B of X universe are: Intersection (t-norm):, Union (s-norm, t-conorm):, Negation:

32 Fuzzy set operations (cont.). The following axioms are accepted for fuzzy operators of t-norm and s-norm: Boundary conditions:, 1,, 0 0,, 1 1,, 0 0,1 Commutativity:,,,,, 0,1 0,1 Monotonicity: 0,1 0,1 0,1 0,1,,,,,,,,

33 Fuzzy set operations (cont.). (cont.): Associativity:,,,, 0,1 0,1 0,1,,,, If all four conditions are satisfied, t- and s-norms are considered to be triangular. The following conditions should be held to let the nonincreasing function : 0,1 0,1 become negation operator on fuzzy set: 1. n is continuous and strictly decreasing 2. 0,1 n is strict negation if 1 is satisfied. n is strong negation if 1 and 2 are satisfied.

34 Fuzzy set operations (cont.). (cont.): Strict negation n is strictly decreasing and continuous function over [0,1], its inverse function n -1 is also strict negation, however different than n. For every strict negation there exists: 0,1 If negation is strong, the graphical presentation of n and n -1 is symetrical vs. y=x line, thus n -1 (x)=n(x). The most common strong negation (standard negation) is: 1.

35 Fuzzy set operations (cont.). (cont.): If t-norm T, s-norm S and strong negation n constitute de Morgan triple (T,S,n) then de Morgan laws are true:,, 0,1,, 0,1 T and S are called n-duals The t- and s-norms presented on next page with standard negation are de Morgan triples.

36 Fuzzy set operations (cont.). Popular t- and s- norms on 0,1, 0,1: Name t-norm s-norm Zadeh, min, max, Probabilistic (algebraic) Bounded (Łukasiewicz) Π, W, max 1,0 Π, W, min, 1 Fodor Drastic Einstein, min, 1 0 1, min, max, 1 0, 2, m, 1 1 1, m, m, 0 1, 1

37 Fuzzy sets (cont.). Fuzzy set operations (cont.). t-norm / s-norm is continuous if T / S is continuous over 0,1 0,1 t-norm / s-norm is strict if T/S is strictly monotonic t-norm is Archimedian if 0,1, s-norm is Archimedian if 0,1, Not all of the norms presented previously satisfy all conditions.

38 Fuzzy sets (cont.). The following table represents the features of t- and s-norms described before: Name t-norm s-norm Zadeh continuous, not Archimedian continuous not Archimedian Probabilistic (algebraic) Bounded (Łukasiewicz) Π continuous and Archimedian W continuous and Archimedian Π continuous and Archimedian W continuous and Archimedian Fodor left continuous, not Archimedian right continuous, not Archimedian Drastic non-continuous but Archimedian non-continuous but Archimedian

39 Fuzzy sets (cont.). The large amount of t- and s- norms drives to the question about relations among them. To present relation between norms it is a good idea to define pseudometric length over [0,1] between two operators and : 0,1 0,1 0,1 (Czogała, Łeski, 2000): d,

40 Fuzzy sets (cont.). Fuzzy set operations (cont.). Measuring distances between appropriate norms presented (Łęski, 2008) it is possible to classify norms: Products: Averages: Sums: Where is another class of operators, including: Arithmetic mean: Geometric mean: Harmonic mean: Z W Π M Σ M Π W Z 0 1/6 1/4 1/3 1/2 2/3 3/4 5/6 1 products averages sums

41 Fuzzy sets (cont.). T- and s-norms presented before are class of non-parameterized functions. There exists a vast amount of parameterized operators as well (Fodor, Roubens, 1994): Frank family 0, 1:, log , 1 log when 0 then Zadeh operators when 1 then Algebraic operators when then Łukasiewicz operators

42 Fuzzy sets (cont.). (cont.). Schweizer and Sklar family (p>0):, , Yager family (q>0, s>0):, 1 min 1, 1 1, min 1, 1 Hamacher family 0):, 1, 2 1 1

43 Fuzzy sets (cont.). (cont.). Dubois and Prade family 0,1:, max,, min,, 1, max 1, 1, Sugeno family 1, 1, max 0, 1 1, min 1, 1 1

44 Fuzzy sets (cont.). (cont.). Dombi family 0: 1, , Compensatory AND (Zimmermann, Zysno, 1980) general t- and s-norm, expotential (Θ ) and linear combination (Θ ): Θ,,, Θ, 1,, where 0,1 denotes operators to be more t-norm like (0) or s-norm like (1).

45 Fuzzy sets (cont.). Fuzzy set operations (cont.) - unary. By the de Morgan triples there exist more operations over fuzzy sets, including (Łęski, 2008; Wang, Zhou): Concetration: Dilatation: Contrast intensity 1 typically=2: 2, , 1 2

46 Fuzzy sets (cont.). Fuzzy set operations (cont.) - unary. Contrast dilatation 1 typically=2: 2, , 1 2

47 Fuzzy sets (cont.). Extension principle (Czogała,Łęski, 2000) how to extend the math function domain from crisp to fuzzy set? Point to point mapping is extended to mapping between sets. Let s have fuzzy sets,, over, universes of discourse. Let s have function,, from. The extension of g that operates on,, and returns fuzzy set B on Y is given by:,,,, when set, is restricted,

48 Fuzzy sets (cont.). Extension principle (cont.) sample (single universe X): X is countable universe:,,, fuzzy set (Zadeh notation): a function :, Fuzzy set B (extension) is: where + denotes logical sum. However for non-injection functions, it is necessary to calculate s-norm of and. When set, is finite, we obtain: sup

49 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Fuzzy relation is a fuzzy subset over product spaces. Let s use binary relation, as generalization to the n-dimensional space is easy by intuition: having two sets X, Y fuzzy relation R is a subset of Cartesian product of : or : 0,1 That leads to fuzzy relation construction as a set of ordered pairs: R,,, x,,, 0,1, or using Zadeh form:, /,,,

50 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Expanding relation to n-dimensions (given sets,, ) :,,,,,,,, 0,1 or using Zadeh notation:,, /,,

51 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Fuzzy relation may be a subject of algebraic operations: Let s have three relations on sets X,Y,Z,U: : : : The sup-t-norm composition of R 1 and R 2 ( ) is:,, sup,,,, When space is finite, sup switches to max operator. In this case relation can be considered as matrix and relation composition is a matrix multiplication, where product is t- norm and sum is max.

52 Fuzzy relations (cont.) (Łęski, 2008) algebraic operations (cont.): The inf-s-norm composition of R 1 and R 2 ( ) is:,, inf,,,,

53 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Fuzzy relation (cont.). The following theorems of fuzzy relation compositions are true (Fodor, Roubens, 1994): Associativity: Monotonicity: Distributivity (vs. sum): Distributivity (vs. product) weak!:

54 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Fuzzy relation (cont.). Some of the basic relation examples over binary (X,Y) sets are: Cartesian product:, Dual Cartesian product:,

55 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Cylindrical extension of fuzzy sets is a method to extend n- dimensional fuzzy set into n+m dimensions: Cylindrical extension (binary): Let A is a fuzzy set on X: : 0,1 then : : /, or directly:,, Ce can be imagined as parallel projection of a membership function curve over next dimension (i.e. from 2D to 3D).

56 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Cylindrical extension (cont.): Cylindrical extension (multi-dimensional): Let: A:,, /,,,,,

57 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Projection of fuzzy sets is a method to detract n+m-dimensional fuzzy set into n dimensions (is somehow opposite to the cylindrical extension): Projection (binary): Let A is a fuzzy set: : 0,1 then : : sup, / A projection of a fuzzy set A in X,Y on Y can be imagined as a 2D shadow of a 3D membership function curve over a wall parallel to the y axis.

58 Fuzzy relations (cont.) (Czogała, Łęski, 2000) Projection of fuzzy (cont.): Projection (multi-dimensional): Let,, A: then sup,,,,, /,,,, There exists m+n projections for a n+m space.

59 Fuzzy relations (cont.). Measure of fuzziness how fuzzy is a fuzzy set (Pal, Bezdek, 1994) ;-)? Fuzziness measure H properties (entrophy based approach): for classical set A C : 0 If fuzzy set A is: 0.5 H should hit unique maximum value If is sharper than : then For standard negation,

60 Fuzzy relations (cont.). Measure of fuzziness (cont.) sample function:, 0,0.5 1, 0.5,1 another one: 2 1, 0,1 and another one: 0.5 log log 1, 0,1 much more in (Łęski, 2008). There exist more approaches: Based on similarity coefficient between fuzzy set and it s negation. Distance measure.

61 Fuzzy sets (cont.). Fuzzy sets extensions (Łęski, 2008). Fuzzy sets type-2. The fuzzy sets described before are so-called fuzzy sets type-1. They suffer from crisp approach ;-) the membership function is actually fuzzyless The answer is to let the value of membership function is fuzzy instead of crisp, however that causes the next level to be crisp and so on. This leads to generalization of fuzzy sets type-m where 1 however the computation and algorithm complexity grows rapidly when m increases. It is also hard to represent them in geometric spaces understandable by human. Most work is done on fuzzy sets type-1 and type-2.

62 Fuzzy sets (cont.). Fuzzy sets extensions (Łęski, 2008). Fuzzy sets type-2. Let,, denote fuzzy sets type-2., Membership function of such a fuzzy set is:,, 0,1 The membership function of type-2 fuzzy set is type-1 fuzzy set. The exists a set of operators for type-2 fuzzy sets, i.e. t- norm and s-norm as well as complementary (negation) by the extension principle provided just before. See (Łęski, 2008) for details.

63 Fuzzy numbers. Fuzzy number A is a fuzzy set A with restricted criteria (Zadeh, 1977): Defined over R Limited support Nonempty core (normal) and 1 Convex At least segmentally continuous* * - wikipedia Fuzzy number represents a value which is somehow uncertain, i.e.: the length of the route is about 23km, temperature outside is about -23C.

64 Fuzzy numbers (cont). Operations on fuzzy sets require new set of operators, supporting fuzzy instead of crisp arguments. Having two fuzzy numbers, we obtain (using extension principle): adding two fuzzy numbers:, subtracting them:, multiplying:, and dividing:, sup, sup, sup, sup, /

65 Fuzzy numbers (cont). The following is true: 0 0 only when both positive or negative. = 1 Mixing fuzzy and real number arguments leads to fuzzying reals: Let s treat as fuzzy number - b as a singleton, : /

66 Fuzzy numbers (cont). So what is a fuzzy number in reality then? A parameter of a real object that is uncertain: i.e. color, that depends on light source A parameter of a real object that is crisp but it is hard to measure it, i.e. speed of the car observed by human, when crossing the street

67 Fuzzy sets (cont.) (Zadeh, 1973) Linguistic variable. When describing a property we (human) usually choose a word that describes best the uncertain value ( speed is low ) low is a linguistic value. We choose a word among other words that may apply to the phenomena (speed may be low, average, high ). This constitute an available set of linguistic values that the linguistic variable may be equal to. Every linguistic value is related to the fuzzy set that mathematically describes this uncertain term.

68 Fuzzy sets (cont.) Linguistic variable (cont.) A quintuple,,,, where: N is name of the variable L(G) a family of labels generated by grammar G X is universe of discourse G is grammary (syntactic rules) M is semantic a mapping between particular label and representing fuzzy set.

69 Fuzzy sets (cont.) Linguistic variable (cont.) The variable values/labels (terms),, are usually groupping universe (i.e. small, medium, high ). but human use adjectives, like very small. we need modifiers functions,, of fuzzy set presenting original label, modifying it s membership function (imagine filter, applied to the ): Samples: very (concentration): more-less (dilution): drunker s high speed feeling: Fuzzy sets related to the should satisfy: Coverage: max 0 Exclusion: min, 1

70 Fuzzy sets (cont.) Linguistic variable (cont.) The grammary G is context-free, defined as ordered quadruple:, Σ,, where, Σ denote terminal and non-terminal symbols:,,,,, (linguistic terms and modifiers) Σ,,, ; 1,, ; 1,, stands for production rules on initial symbol (BNF notation):

71 Fuzzy sets (cont.) Linguistic variable (cont.) Linguistic variables represent disjunctive information: i.e. if car is running fast it is not running slow in the same moment of time. Linguistic variable receives one, particular (fuzzy or classical) set (value, label) but cannot have more than one.

72 Fuzzy sets (cont.) Veristic variable (Zadeh, 1997, Yager, 2000) Sometimes it is necessary to provide conjunctive approach, that brings the value which is multiple : i.e. languages i speak (English, German, French, Norsk classical sets example) i.e. it is snowing in the beginning of the year and at the end of the year (perhaps somewhere in november, december, january, february and march fuzzy approach). This denotes a new approach to the is meaning: Having X veristic variable, attaching a fuzzy set A is: where denotes attachment type (open positive, open negative, closed, closed negative) More info and samples on (Łęski, 2008).

73 Fuzzy implications. Classical reasoning was introduced by Boole. He found that people are thinking the way that it is possible to write mathematical equation presenting premise and conclusion, using simple operators like and, or, not: IF <premise> THEN <conclusion> Example: IF P 1 or P 2 and P 3 or P 4 THEN C 5 where are P 1 P 4 propositions, C 5 conclusion.

74 Fuzzy implications(cont.). This approach is valid up to day, but nowadays we believe that out mind processes are processing fuzzy instead of crisp values as Boole invented. Write a set of conditions for a terminating a process of parking your car at the university ;-) (good luck) But you did it, when you came here for this lecture, with ease! Rules (among others): obstacle hitting possibility, parking space is not utilised to let others use it you may be penalized pigeon may dirty your perfect, clean car ;-) there is not enough space you won t be able to get out of the car parking place is to far from target.

75 Fuzzy implications(cont.). A fuzzy rule is a conditional statement (fuzzy if-then rule): IF X is A THEN Y is B where A and B denote linguistic values of linguistic variables X and Y, respectively. Proposition X is A is a premiese, Y is B is a conclusion (consequence). A, B are fuzzy sets given by. conjunction form of premiese is: (X 1 is A 1 ) and (X 2 is A 2 ) disjunction form of premiese is: (X 1 is A 1 ) or (X 2 is A 2 ) In classical the if a then b is a b a=>b an implication giving not a or b or by de Morgan principle: ((a and b) or not a)

76 Fuzzy implications(cont.). When in fuzzy premise and conclusion de Morgan is not held staright thus previous models of implication are not equal. In approximate reasoning we use t-norm (as intersection), s-norm (as union), so: Substituting AND with t-norm and OR with s-norm we have general form: Conjunctive (MIMO):,,, Disjunctive (MIMO):,,,

77 Fuzzy implications(cont.). The rule of decomposition states that outputs are calculated independently, thus any MIMO system can be represented as a set of p MISO reasonings: Conjunctive (MISO): 1 Disjunctive (MISO): 1 Using above and vectors we obtain canonical fuzzy rule form:

78 Fuzzy implications(cont.). Fuzzy if-then rule may be considered as a relation R on product space: that is called fuzzy implication. Canonical fuzzy rule may be presented as relation over cartesian product : A B using membership functions we obtain:,,,,,, where,,

79 Fuzzy implications(cont.). We need the method to compute fuzzy implication, anyway! There is axiomatic approach (logical interpretation) to fuzzy implication (Fodor 1991,1995,1996, Fodor and Roubens 1994), and also conjunction interpretation of fuzzy implication (Fodor,1991, Fodor and Roubens 1994, Kerre 1992, Maeda 1996, Yager 1996, Dubois and Prade 1996).

80 Fuzzy implications(cont.). Fuzzy implication axiomatic definition approach (Czogała, Łęski, 2000, Łęski 2008): Fuzzy implication is a function : 0,1 0,1 The following is/can be satisfied: 1.,,,, 0,1 2.,,,, 0,1 3. 0, 1 0,1 (falsity implies anything) 4., 1 1 0,1 (anything implies tautology) 5. 1,0 0 (booleanity) 6. 1, 0,1 (tautology cannot justify anything) 7.,,,,,, 0,1 (exchange princ.) 8., 1, 0,1 (implication is defining order) 9., 0 0,1 (N is strong negation, see next slide) 10.,, 0,1 11., 1 0,1 (identity)

81 Fuzzy implications(cont.). Fuzzy implication axiomatic definition approach (Czogała, Łęski, 2000, Łęski 2008) (cont.).: The following is satisfied (cont.).: 12.,,, 0,1 (N is strong negation) 13., is continuous The implication function should hold at least conditions 1 5 and should (but not necessarily need) hold true for If negation N is strong, the N-reciprocal of I(x,y) is a fuzzy implication as well:,, ;, 0,1

82 Fuzzy implications(cont.). With respect to the conditions presented before, we can classify implication with respect to their origin: S-implications I S that come from classical logic, where is (see truth table for implication), giving:, 0,1, where N(x) is strong negation, employing s-norm. R-implications I R that come from classical logic and set theory, where in set theory is: \B thus we have:, 0,1, where N(x) is strong negation, employing t-norm.

83 Fuzzy implications(cont.). With respect to the conditions (cont.): QL-implications I QL that come from classical logic, where is (see truth table for implication), giving:, 0,1, where N(x) is strong negation, employing S-norm and T- norm. QL stands for Quantum Logic.

84 Name Fuzzy implications(cont.). Common logical fuzzy implications (, ): Form Łukasiewicz Fodor Reichenbach Kleene-Dienes Zadeh Gougen Gödel Rescher min 1, 1 1, max 1,, 1 max 1, max 1, min, min, 1 1,, 1, 0,

85 Fuzzy implications(cont.). The conjunction interpretation of the fuzzy rules brings the following relation to live:, The most common t-norms employed are Mamdani (minimum operator) and Larsen (product operator). Switching to vectors we have:,,, where stands for conjunction t-norm. the premiese part,, is a firing strength (FS) of the fuzzy rule.

86 Fuzzy implications(cont.). There is possibility to join logical and conjuncion interpretation back and forth (Dubois and Prade,1996), more in (Łęski, 2008): The generalisation of interpretation of fuzzy implications drives to: conjunction interpretation is lower limit of representation of fuzzy set rules, logical interpretation is upper limit of representation of fuzzy set rules. The pseudometric measure of distance of logical implications as juxtaposed before with drastic product: Resher Zadeh Klene- Dienes Gödel Fodor Reichenbach Gougen Łukasiewicz Drastic product 1/2 5/8 2/3 3/4 5/6 1

87 Fuzzy implications(cont.). Compositional rule of inference (Zadeh, 1973). In classical math we have: when X,Y are universes of discourse and relation is given by function: if we know that: then using compositional rule of inference we get:

88 Fuzzy implications(cont.). Compositional rule (cont.): Switching to fuzzy sets and fuzzy relations (implications) we re given set A on X and having fuzzy relation R we want the set B on Y. To get B we perform: 1. Cylindrical extension of A on :,, 2. Concatenation (t-norm) of Ce(A ) and R:,,,,,

89 Fuzzy implications(cont.). Compositional rule (cont.): Switching (cont.) 3. Projection on Y space: sup, The above formula is sup-star composition / compositional rule of inference (Zadeh, 1973). Thus we obtain (using composition operator):

90 Fuzzy reasoning. The classical reasoning brings us the generalisation of the knowledge we have, i.e.: If we re given apple is red and rule if apple is red then apple is ripe we usually intuitively exted the fact that apple is very red using previous rule, to conclusion: apple is very ripe even if we do not have the strict rule: if apple is very red then apple is very ripe. The above reasoning is called Modus Ponens (modus ponendo ponens): Premise 1 (fact): Premise 2 (rule): Conclusion: A A B B