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Introduction to Fuzzy Logic: Understanding Ambiguity in Problem-Solving

This chapter introduces fuzzy logic, a powerful methodology for dealing with uncertainty and imprecision in expert knowledge representation. Distinguishing itself from classical logic, fuzzy logic utilizes human language rules to create mathematical equivalents that can express vagueness. Developed by Lofti Zadeh in 1965, fuzzy logic addresses concepts that are often ambiguous, such as 'tall' or 'fast.' Through examples and applications in engineering and consumer products, this work highlights the significance of fuzzy systems in improving performance and flexibility in decision-making processes.

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Introduction to Fuzzy Logic: Understanding Ambiguity in Problem-Solving

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  1. CCSB354ARTIFICIAL INTELLIGENCE Chapter 9.2 Introduction to Fuzzy Logic (Chapter 9, pp. 353-363, Textbook) (Chapter 7, Ref. #1) Instructor: Alicia Tang Y. C.

  2. Fuzzy Logic & Fuzzy Thinking • Fuzzy logic is used to describe fuzziness. • It is not a logic that is fuzzy • Fuzzy logic is the theory of fuzzy sets • sets that calibrate vagueness • Experts rely on common sense when they solve problems • How can we represent expert knowledge that uses vague and ambiguous terms in a computer?

  3. What is Fuzzy Logic? • It is a powerful problem-solving methodology • Builds on a set of user-supplied human language rules • Fuzzy systems convert these rules to their mathematical equivalents • Introduced by Lofti Zadeh (1965)

  4. Fuzzy Logic • It deals with uncertainty • It deals with ambiguous criteria or values • Example: “ the girl is tall” • but, how tall is tall? • What do you mean by tall? • is 5’3” tall? • A particular height is tall to one person but is not to another • It depends on one’s relative definition of tall

  5. Degree of membership of a“tall”man Height, cm Crisp Fuzzy 208 1 1.00 205 1 1.00 198 1 0.98 181 1 0.82 179 0 0.78 172 0 0.24 167 0 0.15 158 0 0.06 155 0 0.01 152 0 0.00 Numeric data Just ‘yes’ or ‘no’ In terms of probability

  6. Uncertainty terms and their interpretations Uncertainty term CF Definitely not -1.0 Almost certainly not -0.8 Probably not -0.6 Maybe not -0.4 Unknown -0.2 to +0.2 Maybe +0.4 Probably +0.6 Almost certainly +0.8 Definitely +1.0

  7. What is not Fuzzy Logic ? • Classical logic or Boolean logic has two values • Example: • true or false • yes or no • on or off • black or white • start or stop

  8. Differences between Fuzzy Logic and Crisp Logic • Crisp Logic • precise properties • Full membership • YES or NO • TRUE or FALSE • 1 or 0 • Crisp Sets • Jane is 18 years old • The man is 1.6m tall • Fuzzy Logic • Imprecise properties • Partial membership • YES ---> NO • TRUE ---> FALSE • 1 ---> 0 • Fuzzy Sets • Jane is about 18 years old • The man is about 1.6m

  9. Boolean Logic (for ‘Temperature’) Boolean logic s discrete… Hot 100.0 Temperature (C º) Cold 0.0

  10. Fuzzy Logic (for ‘Temperature’) Extremely Hot 100.0 Hot Quite Hot Temperature (C º) Quite Cold Cold Extremely Cold 0.0

  11. Why Fuzzy Logic? • Fuzzy Logic can: • represent vague language naturally • enrich not replace crisps sets • allow flexible engineering design • improve model performance • are simple to implement • they often work!

  12. Brief History of Fuzzy Logic • 1965 - Fuzzy Sets ( Lofti Zadeh, seminar) • 1966 - Fuzzy Logic ( P. Marinos, Bell Labs) • 1972 - Fuzzy Measure ( M. Sugeno, TIT) • 1974 - Fuzzy Logic Control (E.H. Mamdani) • 1980 - Control of Cement Kiln (F.L. Smidt, Denmatk) • 1987 - Sendai Subway Train Experiment ( Hitachi) • 1988 - Stock Trading Expert System (Yamaichi) • 1989 - LIFE ( Lab for International Fuzzy Eng)

  13. Fuzzy Logic Success • Fuzzy Logic success is mainly due to its introduction into consumer products such as: • temperature controlled in showers • air conditioner • washing machines • refrigerators • television • rice cooker • camcorder • heaters • brake control of vehicles

  14. Fuzzy logic applied to asubway control system • Fuzzy Control used in the subway in Sendai, Japan • fuzzy control system is used to control the train'sacceleration, deceleration and braking • has proven to be superior to both human and conventional automated controllers • reduced the energy consumption been by 10% • passengers hardly notice when the train is actually changing its velocity

  15. Fuzzy Rule Example • A fuzzy rule can be defined as a conditional statement in the form: If x is A Then y is B where x and y are linguistic variables; A and B are linguistic values determined by fuzzy sets on the universe of discourses x and y, respectively

  16. What is the difference between classical and fuzzy rules? Consider the rules in fuzzy form, as follows: Rule 1 Rule 2 IF speed is fast IF speed is slow THEN stop_distance long THEN stop_distance short In fuzzy rules, the linguistic variable speed can have the range between 0 and 220 km/h, but the range includes fuzzy sets, such as slow, medium,fast. Linguistic variable stop_distance can take either value: long or short. The universe of discourse of the linguistic variable stop_distance can be between 0 and 300m and may include such fuzzy sets as short, medium, and long.

  17. More Fuzzy Rules IF project_duration is short AND project_staffing is medium AND project_funding is inadequate THEN risk is high IF project_duration is long AND project_staffing is large AND project_funding is adequate THEN risk is low IF project_duration is short AND project_staffing is large AND project_funding is adequate THEN risk is medium IF service is excellent OR food is delicious THEN tip is generous :

  18. Example • The temperature of room is too hot/cold… • How to designed an automatic air-conditioner which will be able to set temperature: • Hotter(warm) when it is too cold • Colder(cool) when it is too hot

  19. Methodology: Boolean • Using Boolean: • Determine 2 discrete values which is mutually exclusive • E.g. hot or cold • Couldn’t cater for continuous value • Problems: • How if too many students or very few students in the room ? • How hot or how cold the room should be?

  20. Bivalent Sets to Characterize the Temperature of a room Membership Function 1 0 ºC 10 20 30 -10 0 Cold Cool Warm Hot

  21. Fuzzy Logic Methodology • Set the boundaries between two values(cold and hot) which will show the degrees of temperature • Use fuzzy set operations to solve the problem: IF temperature iscoldTHENset fan tozero IF temperature iscoolTHENset fan tolow IF temperature iswarmTHENset fan to medium IF temperature ishotTHEN setfan tohigh

  22. Fuzzy Sets to Characterize the Temperature of a room Membership Function 1 0 ºC 10 20 30 -10 0 Cold Cool Warm Hot Expresses the shift of temperature more natural and smooth

  23. Exercise: A question combiningfuzzy rules & truth values and resolution proof

  24. FUZZY RULES AND RESOLUTION PROOF • Given the following fuzzy rules and facts with their Truth Values (TV) indicated in brackets:  Q ( TV = 0.3) TVs for facts W ( TV = 0.65) Q  P  S (TV = 1.0) S  U ( TV = 1.0) TVs for fuzzy rules W  R ( TV = 0.9) W  P ( TV = 0.6) • You are required to find (or compute) the Truth Value of U by using the fuzzy refutation and resolution rules.

  25. Combining resolution proof and fuzzy refutation Steps • Convert facts and rules to clausal forms. [in our case, there are 4 rules that need conversion]. • By resolution & refutation proof , we negate the goal. [in our case, this is U. assign a TV = 1.0 for it] • For those fuzzy rules, check to see if there is any Truth Value less than 0.5 (i.e. 50%); invert the clause and compute new TV for inverted clause using formula (1 – TV(old-clause)). [we have the clause  Q which is < 0.5, in our example] • Apply resolution proof to reach at NIL (i.e. a direct contradiction). • Each time when two clauses are resolved (combined to yield a resolvent), the minimum of the TVs is taken & assigned it to the new clause.

  26. Supplementary slides

  27. Applications in Fuzzy logic decision making • The most popular area of applications • fuzzy control • industrial applications in domestic appliances • process control • automotive systems

  28. Fuzzy Decision Making in Medicine - I • Medicine • the increased volume of information available to physicians from new medical technologies • the process of classifying different sets of symptoms under a single name and determining appropriate therapeutic actions becomes increasingly difficult

  29. Fuzzy Decision Making in Medicine - II • The past history offered by the patient may be subjective, exaggerated, underestimated or incomplete • In order to understand better and teach this difficult and important process of medical diagnosis, it can be modeled with the use of fuzzy sets

  30. Fuzzy Decision Making in Medicine - III • The models attempt to deal with different complicating aspects of medical diagnosis • the relative importance of symptoms • the varied symptom patterns of different disease stages • relations between diseases themselves • the stages of hypothesis formation • preliminary diagnosis • final diagnosis within the diagnostic process itself.

  31. Fuzzy Decision Making in Medicine - IV • Its importance emanates from the nature of medical information • highly individualized • often imprecise • context-sensitive • often based on subjective judgment • To deal with this kind of information without fuzzy decision making and approximate reasoning is virtually impossible

  32. Fuzzy Decision Making in Information Systems • Information systems • information retrieval and database management has also benefited from fuzzy set methodology • expression of soft requests that provide an ordering among the items that more or less satisfy the request • allow for the presence of imprecise, uncertain, or vague information in the database

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