1. Artificial Intelligence and Adaptive Systems
Fuzzy Logic
Artificial Intelligence and Adaptive Systems

2. Overview
Technique for representing and manipulating uncertain information
Does not restrict the number of truth values to only two (0 or 1). Instead, they allow for a larger set W of truth degrees.
0 and 1 are extreme cases of truth
0.8
tallness
0.6
tallness
0.4
tallness
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3. Comparison Fuzzy Logic and Probabilistic Theory Similarities
Differences
Both attach numeric values between 0 and 1
Probabilistic theory measures how likely the proposition is to be correct.
Fuzzy logic measures the degree to which the proposition is correct.
Fuzzy Logic and Traditional Logic and Set Theory
Similarities
Differences
Use of three logic operations: AND, OR and NOT
Traditional set theory assigns either membership or non-membership in a class or group (0 or 1).
In fuzzy logic, the operations return a degree of membership between 0 and 1

4. How It Works Crisp inputs Fuzzifier Rules Inference Defuzzifier
Fuzzy input set
Fuzzy output set
Crisp outputs

5. Linguistic Variables and Rule Set
Example: Tipping Problem (How much to tip?)
Linguistic Variables
Service: {poor, good, excellent}
Food: {rancid, delicious}
Tip: {cheap, average, generous}
Rules
IF service is poor OR food is rancid, THEN tip is cheap
IF service is good, THEN tip is average
IF service is excellent OR food is delicious, THEN tip is generous

6. Membership Functions Image credit: McNeill, F., & Thro, E. (1994)
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7. Fuzzification Crisp Input Fuzzy Input
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Image credit: Siler, W., & Buckley, J. (2004)

8. Inference OR
AND
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9. Inference
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10. Inference
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11. Defuzzification
Is the process of producing a result in the Crisp Logic Domain
Can be thought of as turning fuzzy sets membership functions into a real value or specific decision

12. (Common) Methods for Defuzzification
Highest Membership Method (Simple, but loses information)
Centre of Area (Useful, but computationally inefficient)
Weighted Average Method (Useful, but requires symmetry)

13. Highest Membership Method
Defuzzification
Cheap
Average
Generous
Highest Membership Method
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14. Highest Membership Method
Defuzzification
Cheap
Average
Generous
15%
Highest Membership Method
50%
35%
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15. Highest Membership Method
Defuzzification
Cheap
Average
Generous
15%
Highest Membership Method
50%
35%
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16. Centre of Area Calculate the area
• Under the scaled membership function
• Within the range of the output variable (x)
Find Centre of Area
Widely used
Computationally inefficient with complex membership functions

17. Defuzzification Centre Of Area Centroid Principle Centre of Gravity
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18. Centre of Area Calculate the area
• Under the scaled membership function
• Within the range of the output variable (x)
Find Centre of Area
Widely used/Most prevalent
Computationally inefficient with complex membership functions

19. Weighted Average Method
Multiply each membership function weight by it’s corresponding membership value and sum these elements.
Divide this sum by the sum of all maximum membership values to gain our result.
Very simple, very effective.
Computationally efficient, but requires membership function symmetry (usually).

20. Weighted Average Method
Defuzzification
Weighted Average Method
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21. Weighted Average Method
Defuzzification
Weighted Average Method
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22. Defuzzification
Necessary to defuzzify results to obtain a usable output
Techniques vary greatly depending on what is required (20+ methods)
• Maxima methods -> Fuzzy reasoning systems
• Distribution and Area methods -> Fuzzy Controllers
The result of defuzzification is a usable specific decision or real value.

23. Advantages Easy to translate expert knowledge into software
Linguistic rules can be developed by expert
Translated into fuzzy rules
Smooths out behaviour of system
This helps in learning systems where predictability needs to be managed
Image credit: Mukaidono, M. (2001).

24. Advantages Suited for vagueness Easy to modify
Measurements taken may contain error, noise, etc
Membership sets models can overlap at edges
Output accommodates for imprecision in inputs
Easy to modify
Rules are non-sequential
Easy to remove or modify rules compared to if-then-else nested block
Neural networks, evolutionary algorithms, etc can be used to modify weightings on the fly

25. Disadvantages Computationally heavy Risks combinatorial explosion!
Each input requires three computations (Fuzzification, membership and IF-THEN inference, Defuzzification)
Must check each rule against each input at each computation cycle
Risks combinatorial explosion!
N inputs with S membership sets requires S^N IF-THEN rules to cover every possible case
The system can be approximated to avoid this, but may yield unwanted values
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26. Disadvantages Not suited for precision
Crisp values are lost in fuzzification process
Outputs are based on inference, not calculations
May require expert knowledge to implement
Setting ranges may require justification
Defuzzification methods depend on given problem
Image credit: Fuzzy Logic For Beginners
Image credit: Mukaidono, M. (2001).

27. Conclusion
Fuzzy Logic allows for a linguistic approach to programming.
While computationally heavy, it offers an intuitive and flexible approach to a variety of applications where precision is not a priority.
There are a multitude of fuzzification and defuzzification methods, and the “best” method must be judged on a case-by- case basis.
I just made that image in GIMP. It’s hardly worth referencing if I made the damn thing.

28. References
Scientific American. (2017). What is ‘fuzzy logic’? Are there computers that inherently fuzzy and do not apply the usual binary logic?. Retrieved from
Bilkent University. (2010). A Short Fuzzy Logic Tutorial. Retrieved from
Jantzen, J. (2013). Fuzzy Reasoning (2nd ed.). Chichester, UK: John Wiley & Sons.
MathWorks. (2017). Fuzzy Inference Process. Retrieved from
McNeill, F., & Thro, E. (1994). Fuzzy Logic : A Practical Approach. Cambridge, MA: Elsevier Science.
Mukaidono, M. (2001). Fuzzy Logic For Beginners. Singapore: World Scientific.
Pirovano, M. (2012). The use of Fuzzy Logic for Artificial Intelligence in Games. Milano, Italy: University of Milano.
Princeton University. (2007). Fuzzy Inference Systems. Retrieved from
Ross, T. J. (2004). Fuzzy Logic with Engineering Applications (2nd ed.). West Sussex, England: John Wiley & Sons.
Siler, W., & Buckley, J. (2004). Fuzzy Expert Systems and Fuzzy Reasoning. Hoboken, NJ: John Wiley & Sons.

29. Questions?