Fuzzy Reasoning Outline Introduction Bivalent & Multivalent Logics Fundamental fuzzy concepts Fuzzification Defuzzification Fuzzy Expert System Neuro-fuzzy System
Introduction Fuzzy concept first introduced by Lotfi Zadeh in the 1965 Form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. Compared to traditional binary sets, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1 Resembles human reasoning in its use of imprecise information to generate decisions, unlike classical logic which requires a deep understanding of a system, exact equations, and precise numeric values
Bivalent Logics Classical logic, often described as Aristotelian logic True or false Bayesian Reasoning and probabilistic models Each fact is either True or false Often unclear whether a given fact is true or false Probability A particular expression will turn out to be true
Multivalent Logics Three-valued logic True, false, and undetermined 1 represents true, 0 represents false, and real numbers between 0 and 1 represent degree of truth
Bivalent Logic vs. Multivalent Logic A fact has a probability value of 0.5, means it is as likely to be true as it is to be false, or it will be either true or false There is Uncertainty, (at the moment we don t know whether the proposition will be true or false, but it will definitely either be true or false not both, not neither, and not something in between) A proposition has a logical value of 0.5, means it is about the degree to which that statement is true We are Certain of the truth value of the proposition, it is just vague (it is neither true nor false, or it is both true and false)
Linguistic Variables Often used to facilitate the expression of rules and facts A linguistic variable such as height may have a value from a range of fuzzy values including tall short and medium. It may be defined over the Universe of discourse from 2 feet up to 8 feet. The values tall, short, and medium define subsets of this universe of discourse.
Fuzzy Sets vs. Traditional Sets Taditional set, Crisp set Defined by the values that are contained within it. A value is either within the set, or it is not. e.g a set of natural number Fuzzy set Each value is a member of the set to some degree, or is not a member of the set to some degree. Example: the tall people. Bill is 7 feet tall, so he is definitely included in the set of tall people, John is 4 feet tall, so most say that he is not included in the set, and Jane is 5 feet tall, some would say she is tall, but others would say she is not
Fuzzy Set Fuzzy set membership function Fuzzy set A is defined by membership function MA. Choose entirely arbitrarily, reflect a subjective view on the part of the author. A list of pairs for representing fuzzy set in computer like A = {(x1,ma(x1)),..., (xn,ma(xn))}
Fuzzy Set operator Traditional set theory Not A the complement of A, Intersection, and Union Commutative, Associative, Distributive, and DeMorgan's law Fuzzy set Complement of A, M A(x) = 1 - MA(x) Intersection, MA B (x) = MIN (MA (x),mb (x)) Union, MA B (x) = MAX (MA (x),mb (x)) Containment, B A iff x (MB (x) MA (x))
Hedges Fuzzy set qualifier such as very, quite, extremely, or somewhat Produce a new set when of them is applied to a fuzzy set Raise the set's membership function to an appropriate power. e.g a membership value of very tall people is (MA(x))², or a membership value of quite tall people is (MA(x))¹ ³
Fuzzy Logic Form of logic that applies to fuzzy variables Each fuzzy variable can take a value from 0 (not at all true) to 1 (entirely true). e.g 0.5 might indicate somewhat true, or about as true as it is false Use Min, Max for calculating the conjunction ( )and disjunction ( ) of two fuzzy variables If A and B are fuzzy logic values, A B MAX (A,B) A B MIN (A,B) A = 1- A
Classical Logic vs. Fuzzy Logic Classical logic A A = TRUE A A = FALSE Fuzzy logic A A can be to some extend false A A can be to some extend true
Fuzzy Logic Fuzzy truth table for a finite set of input. Set {0, 0.5, 1} A B A B 0 0 0 0 0.5 0.5 0 1 1 0.5 0 0.5 0.5 0.5 0.5 0.5 1 1 1 0 1 1 0.5 1 1 1 1
Fuzzy Logic A A 0 1 0.5 0.5 1 0
Fuzzy logic implication, or A B A->B 0 0 1 0 0.5 1 0 1 1 * 0.5 0 0.5 * 0.5 0.5 0.5 0.5 1 1 1 0 0 1 0.5 0.5 1 1 1
One of alternative for fuzzy implication is Godel implication A B (A B) B A B A->B 0 0 1 0 0.5 1 0 1 1 0.5 0 0 0.5 0.5 1 0.5 1 1 1 0 0 1 0.5 0.5 1 1 1
Rusell's paradox : Fuzzy Logic as Applied to Traditional Logic Paradox A barber, who himself has a beard, shaves all men who do not shave themselves. He does not shave man who shave themselves. Paradox: conclusion contradicts one or more of the premises
All Cretan are liar, said the Cretan. The Paradox can be resolved by Fuzzy logical values, instead of the two logical values true and false, the Cretan's statement is true and false, to some extend, at the same time.
Rules Ordinary rule: IF A THEN B Fuzzy rule : IF A=x THEN B=y IF A op x THEN B=y e.g. IF temperature > 50 then fan speed = fast IF study time = short then grades = poor
Fuzzy Inference Mamdani implication : an alternative to Godel implication It allows a system to take in a set of crisp input values and apply a set of fuzzy rules to those values, in order to derive a single, crisp, output value or action recommendation.
Fuzzy Logic System
How this form of reasoning work? Example: Braking system for a car to cope when the roads are icy and the wheels lock.
Step 1 Define the Rules Rule 1: IF pressure on brake pedal is medium THEN apply the brake Rule 2: IF pressure on brake pedal is high AND car speed is fast AND wheel speed is fast THEN apply the brake Rule 3: IF pressure on brake pedal is high AND car speed is fast AND wheel speed is low THEN release the brake Rule 4: IF pressure on brake pedal is low THEN release the brake
Step 2 : Fuzzification Define fuzzy set for various linguistic variables Pressure from 0 to 100, so brake measure can be defined such as having 3 linguistic values, such as High(H),Medium(M), Low(L). H={(50,0),(100,1)} M={(30,0),(50,1),(70,0)} L={(0,1),(50,0)} Suppose pressure value is 60, so fuzzy membership for the 3 sets: MH(60)=0.2, MM(60)=0.5, ML(60)=0
Step 2 : Fuzzification Define wheel speed with having 3 linguistic values: Slow, Medium, Fast Membership function: S={(0,1),(60,0)} M={(20,0),(50,1),(80,0)} F={(40,0),(100,1)} If wheel speed is 55 then MS(55)=0.083, MM(55)=0.833, MF(55)=0.25
Step 2 : Fuzzification Define car speed with having 3 linguistic values: Slow, Medium, Fast Membership function: S={(0,1),(60,0)} M={(20,0),(50,1),(80,0)} F={(40,0),(100,1)} If car speed is 80 then MS(80)=0, MM(80)=0, MF(80)=0.667
Step3: Apply Fuzzy Values To The System's Rules Rule 1: MM(60)=0.5, it shows Apply the brake Rule 2: MH(60)=0.2, MF(80)=0.667, MF(55)=0.25, So fuzzy value of 0.2 for Apply the brake Rule 3: MH(60)=0.2, MF(80)=0.667, MS(55)=0.083, So fuzzy value of 0.083 for Release the brake Rule 4: ML(60)=0, So fuzzy value of 0.083 for Release the brake
How to combine the differing values for each of the two fuzzy variables? Sum the values So we have 0.7 for Apply the brake and 0.083 for Release the brake
Clip the membership function to the values, the member function of A has been clipped to 0.7 and the member function of R has been clipped to 0.083
Step 4: Defuzzification Process of obtaining the crisp value from a set of fuzzy variables This can be done by the center of gravity C= (MA(x)*x)/ MA(x), =((5*0.083)+(10*0.1)+(15*0.15)+...+(100*1))/(0.083+0.1+0.15+...+ 1) = 68.13 C shows the pressure applied by the brake to the wheel in the car
Fuzzy expert system Expert system contains a set of rules that are developed in collaboration with an expert The fuzzy expert system can be built by choosing a set of linguistic variables appropriate to the problem and defining membership functions for those variables. Rules are then generated based on the expert s knowledge and using the linguistic variables. The fuzzy rules can then be applied as described above using Mamdani inference.
Create The Fuzzy Expert System Obtain information from one or more experts. Define the fuzzy sets. Define the fuzzy rules
Use The Fuzzy Expert System Relate observations to the fuzzy sets. Evaluate each case for all fuzzy rules. Combine information from the rules. Defuzzify the results.
Neuro-fuzzy System Neural network that learns to classify data using fuzzy rules and fuzzy classification A fuzzy neural network is a five-layer feed-forward network Layer 1: input layer- Receives crisp inputs Layer 2: fuzzy input membership functions Layer 3: fuzzy rules Layer 4: fuzzy output membership functions Layer 5: output layer- outputs crisp values
References 1. Ben Coppin, Artificial intelligence illuminated, 2004 2. A Short Fuzzy LogicTutorial,,April 2010 3. Robert Fuller, Fuzzy Reasoning and Fuzzy Optimization,, 1998 4. Fuzzy logic,, http://en.wikipedia.org/wiki/fuzzy_logic 5. Walter Banks, Linguistic Variables: Clear Thinking with Fuzzy Logic,, Waterloo, Ontario
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