Chapter 18 Fuzzy Reasoning

1. Chapter 18 Fuzzy Reasoning

2. Chapter 18 Contents (1) • Bivalent and Multivalent Logics • Linguistic Variables • Fuzzy Sets • Membership Functions • Fuzzy Set Operators • Hedges

3. Chapter 18 Contents (2) • Fuzzy Logic • Fuzzy Rules • Fuzzy Inference • Fuzzy Expert Systems • Neuro-Fuzzy Systems

4. Bivalent and Multivalent Logics • Bivalent (Aristotelian) logic uses two logical values – true and false. • Multivalent logics use many logical values – often in a range of real numbers from 0 to 1. • Important to note the difference between multivalent logic and probability – P(A) = 0.5 means that A may be true or may be false – a logical value of 0.5 means both true and false at the same time.

5. Linguistic Variables • Variables used in fuzzy systems to express qualities such as height, which can take values such as “tall”, “short” or “very tall”. • These values define subsets of the universe of discourse.

6. Fuzzy Sets • A crisp set is a set for which each value either is or is not contained in the set. • For a fuzzy set, every value has a membership value, and so is a member to some extent. • The membership value defines the extent to which a variable is a member of a fuzzy set. • The membership value is from 0 (not at all a member of the set) to 1.

7. Membership function for the Fuzzy Set of Tall People • 1 8 ft

8. Membership Functions • The following function defines the extent to which a value x is a member of fuzzy set B: • This function would be stored in the computer as: B = {(0, 1), (2, 0)} • This function could represent, for example, the extent to which a person can be considered a baby, based on their age.

9. Fuzzy Set Membership Functions • If we use Mb(x) and Mc(x) to represent the membership functions for baby and child respectively, we can write : • 1 – x/2 for x <= 2 • Mb(x) = 0 for x >2 • (x-1)/6 for x <= 7 • Mc(x) = 1 for x > 7 and x <= 8 • (14 – x)/6 for x >8

10. How to represent a fuzzy set in computers? • We use a list of pairs and each pair represent a value and the fuzzy membership value for that value. • For example: • A = { (x1, MA(x1),…, (xn, MA(xn)} • The fuzzy set of babies then can be: • B = {(0,1), (2, 0)}

11. Crisp Set Operators • Not A – the complement of A, which contains the elements which are not contained in A. • A  B – the intersection of A and B, which contains those elements which are contained in both A and B. • A  B – the union of A and B which contains all the elements of A and all the elements of B. • Fuzzy sets use the same operators, but the operators have different meanings.

12. Fuzzy Set Operators • Fuzzy set operators can be defined by their membership functions • M¬A(x) = 1 - MA(x) • MA  B (x) = MIN (MA (x), MB (x)) • MA  B (x) = MAX (MA (x), MB (x)) • We can also define containment (subset operator): • B  A iff x (MB (x)  MA (x))

13. Hedges • A hedge is a qualifier such as “very”, “quite”, “somewhat” or “extremely”. • When a hedge is applied to a fuzzy set it creates a new fuzzy set. • Mathematic functions are usually used to define the effect of a hedge. • For example, “Very” might be defined as: • MVA (x) = (MA (x))2

14. Fuzzy Logic • It is a form of logic that applies to fuzzy variables. • It is non-monotonic, meaning: • If a new fuzzy fact is added to a database, this fact may contradict conclusions that previously derived from the database.

15. Fuzzy Logic • Each fuzzy variable can take a value from 0 (not at all true) to 1 (entirely true) • The values can be any real value between 0 and 1.

16. Fuzzy Logic • A non-monotonic logical system that applies to fuzzy variables. • We use connectives defined as: • A V B  MAX (A, B) • A Λ B  MIN (A, B) • ¬A  1 – A • We can also define truth tables:

17. Fuzzy Inference • Inference is harder to manage. • Since: A  B  ¬A V B • Hence, we might define fuzzy inference as: A  B  MAX ((1 – A), B) • This gives the unintuitive truth table shown on the right. • This gives us 0.5  0 = 0.5, where we would expect 0.5  0 = 0

18. Fuzzy Inference • An alternative is Gödel implication, which is defined as: A  B  (A ≤ B) V B • This gives a more intuitive truth table.

19. i.e. for some d ∈ M, R(a, d) → • A(d) = 0. It follows R(a, d) > 0 and A(d) = 0 (Godel implication!). Then • R(a, d)∧¬A(d) = R(a, d)∧1 = R(a, d) > 0, thus (R(a, d)∧¬A(d) = 0 • and (¬∃R.¬C)(a) = 0, thus our concept has value 0, a contradiction.

20. Fuzzy Rules • Fuzzy rule has the form of: • If A = x then B =y or more general, • If A op x then B = y • Here, op is some mathematical operator, such as =, >, < , …,

22. Fuzzy Inference (3) • Mamdani inference derives a single crisp value by applying fuzzy rules to a set of crisp input values. Step 1: Fuzzify the inputs. Step 2: Apply the inputs to the antecedents of the fuzzy rules to obtain a set of fuzzy outputs. Step 3: Convert the fuzzy outputs to a single crisp value using defuzzification.

23. Fuzzy Rules • A fuzzy rule takes the following form: IF A op x then B = y • op is an operator such as >, < or =. • For example: IF temperature > 50 then fan speed = fast IF height = tall then trouser length = long IF study time = short then grades = poor

24. Fuzzy Expert Systems • A fuzzy expert system is built by creating a set of fuzzy rules, and applying fuzzy inference. • In many ways this is more appropriate than standard expert systems since expert knowledge is not usually black and white but has elements of grey. • The first stage in building a fuzzy expert system is choosing suitable linguistic variables. • Rules are then generated based on the expert’s knowledge, using the linguistic variables.

25. Neuro-Fuzzy Systems • A fuzzy neural network is usually a feed-forward network with five layers: • Input layer – receives crisp inputs • Fuzzy input membership functions • Fuzzy rules • Fuzzy output membership functions • Output layer – outputs crisp values

26. Applications of Fuzzy Logic • Examples where fuzzy logic is used • Automobile subsystems, such as ABS and cruise control • Air conditioners • The MASSIVE engine used in the Lord of the Rings films, which helped show huge scale armies create random, yet orderly movements • Cameras • Digital image processing • Dishwashers • Elevators • Washing machines and other home appliances. • Video game artificial intelligence

27. Some Misconceptions on Fuzzy Logic • Fuzzy logic has suffered many misconceptions, partly due to its name. "Fuzzy" often has negative connotations, either suggesting something cute or something imprecise; the latter sometimes causes people to equate "fuzzy logic" with "imprecise logic".

28. However, fuzzy logic is not any less precise than any other form of logic: it is an organized and mathematical method of handling inherently imprecise concepts. • The concept of "coldness" cannot be expressed in an equation, because although temperature is a quantity, "coldness" is not. • However, people have an idea of what "cold" is, and agree that something cannot be "cold" at N degrees but "not cold" at N+1 degrees — a concept classical logic cannot easily handle due to the principle of bivalence.

29. Another common misconception is that fuzzy logic is a new way of expressing probability. • However, Bart Kosko has shown that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty. • He also proved a theorem demonstrating that Bayes' theorem can be derived from the concept of fuzzy subsethood. • This should not by any means suggest that all those who study probability accept or even understand fuzzy logic, however: to many, fuzzy logic is still a curiosity.

30. Fuzzy logic is also sometimes said to be used only in AI, control systems, and/or expert systems (note that these fields can have significant overlap). • These are by far the most common applications, but by no means the only possible: fuzzy logic can be applied in any situation requiring the handling of uncertainty.