Fuzzy Logic and Computing with Words. Ning Xiong. School of Innovation, Design, and Engineering Mälardalen University. Motivations

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1 /3/22 Fuzzy Logic and Computing with Words Ning Xiong School of Innovation, Design, and Engineering Mälardalen University Motivations Human centric intelligent systems is a hot trend in current research, entailing more computer-human interaction. Humans are more comfortable to express opinions in vague terms in natural language; they also prefer to receive information about system behavior in abstract and comprehensible formulations. Complex, ill-defined processes difficult for description and analysis by exact mathematical techniques Precision is not always the goal to strive for. Tolerance of imprecision can lead to tractability, robustness, and short computation/reaction time

2 /3/22 Significance of Fuzzy Theory We need Fuzzy logic and fuzzy theory as powerful means of manimulation of vague and uncertaion information, and to create systems that are much closer in spirit to human thinking and reasoning Enhancing Classic AI Knowledge Base Sentences Judgment New sentence The classic AI only concerns binary statement/judgments AI augmented with fuzzy logic can reason with sentences with uncertainty or partial truth, which indeed reflect pervasive phenomena in real world. 2

3 /3/22 Outline Basic concepts of fuzzy logic and fuzzy set theory Liguistic fuzzy rules Computing with words via fuzzy logic Procedure of fuzzy reasoning and decision making Two illustrative examples of fuzzy reasoning What is Fuzzy Logic Fuzzy logic is not logic that is fuzzy, but a precise logic that is used to cope with fuzziness and approximate reasoning Fuzzy Logic is the theory of fuzzy sets, sets that Accomodate vagueness and uncertainty. 3

4 /3/22 Father of Fuzzy Logic Lotfi Zadeh, Professor in the Graduate School, Computer Science Division Department of Elec. Eng. and Computer Sciences, University of California Berkeley. In 965 Zadeh published his famous paper Fuzzy sets. In 973 Zadeh proposed his theory of Fuzzy Logic. In 996 Zadeh published the new pioneering paper Fuzzy logic=computing with words Fuzzy vs. Crisp Sets Fuzzy set indicates how much (to which extent) an element belongs to it. Crisp set indicates whether an element belongs to it 4

5 /3/22 Classical (Crisp) Set young = { x x 3 } characteristic function: m young (x) m young (x) = { : x 3 : x> 3 A= young 3 x [years] Fuzzy Set Definition : A fuzzy set F in a universe of discourse U is characterized by membership function m F, which takes values in the interval [,], i.e., m F : U [,] Example: If U contains finite number of elements, fuzzy set F can be denoted by: F={m F (u )/u, m F (u 2 )/u 2,, m F (u n )/u n } 5

6 /3/22 Fuzzy versus Classical Logic Classical Logic Element x belongs to set A or it does not: m A (x) {,} Fuzzy Logic Element x belongs to set A with a certain degree of membership: m A (x) [,] m A (x) A= young m A (x) A= young x [years] x [years] Types of Membership Functions Trapezoid: <a,b,c,d> Gaussian: N(m,s) m(x) m(x) s a b c d x m x m(x) Triangular: <a,b,b,d> Singleton: (a,) m(x) a b d x a x 6

7 /3/22 Operators on Fuzzy Sets Union (of crisp sets): whether an element belongs to either set Union (of fuzzy sets): How much an element belongs to either set m A B (x)=max{m A (x),m B (x)} m A B (x)=min{,m A (x)+m B (x)} m A (x) m B (x) m A (x) m B (x) x x Operators on Fuzzy Sets Intersection (of crisp sets): whether an element belongs to both sets Intersection (of fuzzy sets): How much an element belongs to both sets m A B (x)=min{m A (x),m B (x)} m A B (x)=m A (x) m B (x) m A (x) m B (x) m A (x) m B (x) x x 7

8 /3/22 Operators on Fuzzy Sets Complement (of a crisp set): whether an element is excluded from the set Complement (of a fuzzy set): how much an element is excluded from the set Negation: m A (x)= - m A (x) Classical law does not always hold: m A A (x) m A A (x) Example : m A (x) =.6 m A (x) = - m A (x) =.4 m A A (x) = max(.6,.4) =.6 m A A (x) = min(.6,.4) =.4 Fuzzy Relations A classical relation R is a crisp subset in the product space X Y, defined by m R (x,y) = if (x,y) R if (x,y) R Good_friend ={(Johan, Mary), (Alice,Mathias)} A fuzzy relation R is a fuzzy subset in the product space X Y, defined by m R (x,y) [,] m R (x,y) describes to which degree x and y are related by R. 8

9 /3/22 Fuzzy Relations Example: X = { rainy, cloudy, sunny } Y = { swimming, bicycling, camping, reading } X/Y swimming bicycling camping reading rainy..2.. cloudy sunny Linguistic Variables A linguistic variable has a set of linguistic terms/values. Each linguistic value corresponds to a fuzzy set and explained with its membership function linguistic variable : temperature linguistics terms (fuzzy sets) : { cold, warm, hot } m(x) m cold m warm m hot 2 6 x [C] 9

10 /3/22 Fuzzy Rules A fuzzy rule is a linguistic expression of causal dependencies between linguistic variables in form of if-then statements General form: If <antecedent> then <consequence> Example: if temperature is cold and oil price is cheap then heating is high linguistic variables linguistic values/terms (fuzzy sets) Fuzzy Rule Base Heating Oil price: cheap normal expensive Temperature : cold warm hot high high medium high medium low medium low low A fuzzy rule base is a collection of fuzzy if-then rules

11 /3/22 Fuzzy vs. Crisp Rules Oscillation, sharply changing behavior due to crisp rules If a robot decides its speed based on the below rules, it can only go two speeds, mph or 3mph, and no speed in between. if sensor value is between and 27, then go mph if sensor value is between 28 and 255, then go 3mph Fuzzy rules can produce continuous, gracefully changing behavior, due to the overlapping fuzzy membership functions. Fuzzy Logic and Computing with Words Computing with words is inspired by human capability to perform a variety of mental and physical tasks without measurements and computations Computing with words is a methodology in which words are the objects for computing and reasoning Words are considered as linguistic terms corresponding to fuzzy sets defined by membership functions Fuzzy logic plays a vital role in computing with words By approximation: Fuzzy Logic = Computing with Words

12 /3/22 Natural Language Processing with Fuzzy Technique Natural language is inherently imprecise young m-precisiation young With fuzzy technique we can translate natural language into a precisiated language so that it can be computed by machines. The fuzzy logic opens the door to computation with information described in natural language. Perception Based Modeling To describe the subject of interest with a set of linguistic rules, in which antecedents and consequences comprise perceptions or words. Perception-based function explanation: summarize the function with linguistic if-then rules Perception-based system modeling: describe system behavior with perceptions and linguistic rules X S Y 2

13 /3/22 Perception Based Function Explanation X If X is small then Y is small If X is medium then Y is medium If X is large then Y is small Fuzzy linguistic rules are transparent Extract fuzzy linguistic rules from neural network Modeling of a Dynamic System (Inverted Pendulum) 3

14 /3/22 Mathematical Model of the System Perception-Based (Linguistic) Model 4

15 /3/22 How to Build Perception Model Interview with experts: summarize their ideas into linguistic rules. But this approach suffers from knowledge acquisition bottleneck. Data based fuzzy modeling: Learn a linguistic fuzzy model from the available data Fuzzy Decision Making Procedure Inputs Fuzzification Rule matching Firing strengths of rules Fuzzy inference Suggestions by rules (fuzzy) Crisp output value Defuzzification Overall output fuzzy set Fuzzy aggregation Fuzzification: compute the membership degrees for each input variable respect to its linguistic terms. Rule Matching: calculating firing strengths (degrees of satisfaction) of the individual rules Fuzzy Inference: determine the suggestions of rules according to firing strengths and rule conclusions Fuzzy Aggregation: combine suggestions from individual rules into an overall output fuzzy set. Sometimes this step is also called accumulation. Defuzzification: determine a crisp value from the overall output membership function as the final result or solution. 5

16 /3/22 Fuzzification Determine the degree of membership for each term of an input variable : temperature : t=8 C m(x) m cold m warm m hot 2 6 m cold (t)=.2, m warm(t)=.6, m hot (t)= t [C] Rule Matching Calculate the firing strength of every rule by combining the individual membership degrees for all terms involved in the condition part of the rule through fuzzy AND: min-operator.5 m cold (t)=.5 5C.3 t m cheap (p)=.3 $3/barrel p R i if temperature is cold... and oil is cheap... t i = min{m cold (t), m cheap (p)} = min{.5,.3} =.3 6

17 /3/22 Fuzzy Inference Inference step: For each rule apply the firing strength to modify its consequent fuzzy set, resulting in a new output fuzzy set as the suggestion of the rule. m high (h) m resp. (h)... t =.3 h... then heating is high min-inference: m resp. = min{t, m high } m high (h) m resp. (h)... t =.3 h prod-inference: m resp. = t m high Fuzzy Aggregation Aggregation: combine suggestions of individual rules to yield an overall output fuzzy set. The overall output fuzzy set is the union of suggestions (output fuzzy sets) from individual rules F F F F m F 2 ( F F2 n y) max m ( y), m ( y),, m ( y) F n F F2 F3 h 7

18 /3/22 Defuzzification Determine crisp value from output membership function. Commonly used is the Center of Gravity method: m consequent (h) COG 73 h COG b a m ( x) xdx b a A m ( x) dx A COG b x a b x a m ( x) x A m ( x) A An Example of Fuzzy Reasoning R: if temp is cold then valve is open R2: if temp is warm then valve is half R3: if temp is hot then valve is close m open m half m close v m cold m warm m hot min(.7, (m open )) Overall fuzzy set measured temperature t=.7 t3 =. t2=.2 temp.2 min(.2, (m half )) v v Center of gravity as crisp output for valve setting 8

19 /3/22 Other example: Financial Advisor Financial Recommendation Fuzzy Rules: a) If saved amount is inadequate, then percentage in saving is High b) If saved amount is adequate and income is adequate, then the percentage in saving is Low c) If saved amount is adequate and income is inadequate, then the percentage in saving Medium. Low Medium High Percentage To Classify Saved Amount Fuzzy rules to classify saved amount:.if a saved amount is Small, it is inadequate 2.If a saved amount is Large, it is adequate The meanings of Small and Large for saved amount are defined for the individual as follows:. Small Large Dollar Given the saved amount x=22 (for the individual), we have t =m Small (x)=.3, t 2 =m Large (x)=.7, m Inadequate (x)=.3, m Adequate (x)=.7 9

20 /3/22 To Classify Income Fuzzy rules to classify income: 3. If income is Small and Steady, it is inadequate 4. If income is Large and Steady, it is adequate 5. If income is Unsteady, it is inadequate The meanings of Small and Large for income are defined for the individual as follows:. Small Large Dollar Given steady income y=25 (for the individual), we have t 3 =m Small (y)=.8, t 4 =m Large (y)=.2, t 5 = m Inadequate (y)=.8, m Adequate (y)=.2 Reasoning about Investment Rewriting Financial Recommendation Fuzzy Rules: a) If x is inadequate, then percentage v is High b) If x is adequate and y is adequate, then percentage v is Low c) If x is adequate and y is inadequate, then percentage v is Medium The firing strength of the above fuzzy rules are as follows: t a =m Inadequate (x)=.3, t b =min(m Adequate (x), m Adequate (y))=.2 t c =min(m Adequate (x),m Inadequaate (y))=.7 The output fuzzy sets from the rules are: m Fa (v)=t a m High (v)=.3 m High (v) m Fb (v)=t b m Low (v)=.2 m Low (v) m Fc (v)=t c m Medium (v)=.7 m Medium (v) 2

21 /3/22 Deciding Final Percentage The overall output fuzzy set is derived as: F=F a F b F c Finally the percentage in saving is decided by calculating the center of gravity: v* b a m ( v) vdv b a F m ( v) dv F Summary Fuzzy sets are characterized by membership functions between and Fuzzy rules describes causal relations between variables in terms of fuzzy sets Fuzzy decision making based on fuzzy rules includes: fuzzification, rule matching, fuzzy inference, aggregation and defuzzification. Fuzzy logic enables computing with words (word has to be m-precisiated before computation) Fuzzy systems are receiving huge applications, including fuzzy control, fuzzy modeling, function explanation, pattern recognition, decision analysis, 2

22 /3/22 Reading Guidance Please read the lecture slides carefully. You can also study the relevant knowledge from some other books. Examples are:.chapter 4 Fuzzy expert systems in Artificial Intelligence A guide to intelligent systems, by Michael Negnevitsky, Addison Wesley, 22 2.Chapters 2-22, in Computational Intelligence: An Introduction, by Andries P. Engelbrecht, John Wiley &Sons, 27. Some additional reading material is available under the folder \content\fuzzy Supplementary Reading\ in blackboard. 22