Fuzzy Logic

1. Fuzzy Logic

2. Fuzzy Logic • LotfiZadeh (professor at UC Berkeley) wrote his original paper on fuzzy set theory. In various occasions, this is what he said… • “Fuzzy logic is a means of presenting problems to computers in a way akin to the way humans solve them” • “The essence of fuzzy logic is that everything is a matter of degree” • What do these statements really mean?

3. Fuzzy Logic • Very often, we humans analyze situations and solve problems in a rather imprecise manner • Do not have all the facts • Facts might be uncertain • Maybe we only generalize facts without having the precise data or measurements… • Real-life example: Playing a game of basketball

4. Everything is a matter of degree? • Is your basketball opponent tall, or average or short? (use of linguistic terms to measure degree) • Is 7 feet tall? Is 6 feet 10 inches tall? Are they both considered tall? (overlapping degrees) • Problem with traditional Boolean logic • You are forced to define a point above which we will consider the guy to be tall or just average, e.g. > 7 ft • Fuzzy Logic  allows gray areas or degrees of being considered “tall”

5. The degree of truth • So…you can think of fuzzy logic as classifying something as being TRUE, but to varying degrees • Real-life control applications (air-conditioning, household appliances): • Traditional Boolean logic will result in abrupt switching of response functions • Fuzzy logic alleviates this problem  Responses will vary smoothly given the degree of truth or strength of the input conditions

6. Fuzzy logic for games • A previous game AI example… • An AI character makes his decision to chase (using FSM or DT) based on traditional Boolean logic, e.g. distance of player < 20 units, and player health < 50% • In fuzzy logic, we can represent these input conditions using a few “membership” degrees of measure • Distance: (“Far”, “Average”, “Near”) • Health: (“Good”, “Normal”, “Poor”) • The output actions can also be represented with different membership degrees (“Chase Fast”, “Chase Slow”)

7. How to use Fuzzy Logic in Games? • 3 possible ways how fuzzy logic can be used in games • Control • Modulating steering forces, travelling/moving towards target • Threat Assessment • Assessing player’s strengths/weaknesses for deploying units and making moves • Classification • Identifying the combat prowess of characters in the game based on a variety of factors in order to choose opponent • There are many other possibilities…

8. Fuzzy Logic Basics • Fuzzy control or fuzzy inference process – 3 basic steps

9. Step 1: Fuzzification • Fuzzification: Process of mapping/converting crisp data (real numbers) to fuzzy data • Find degree of membership of the crisp input in predefined fuzzy sets • E.g. given a character’s health, determine the degree to which it is “Good”, “Fair” or “Poor”. • Mapping is achieved using membership functions

10. Membership Functions • Membership Functions • Map input variables to a degree of membership, in a fuzzy set, between 0 and 1. Degree 1  absolutely true, degree 0  absolutely false, any degree in between  true or false to a certain extent • “Boolean logic membership function”

11. Membership Functions • Fuzzy Membership Functions • Enables us to transition gradually from false to true • Grade membership function

12. Membership Functions • Triangular m/f • Reverse grade m/f • Equations are just the inverse of the grade m/f

13. Membership Functions • Trapezoid m/f • Other nonlinear m/f • Gaussian or Sigmoid ‘S’-shaped curves

14. Membership Functions • Typically, we are interested in the degree of which an input variable falls within a number of qualitative sets

15. Membership Functions • Setting up collections of fuzzy sets for an input variable is a matter of judgment and trial-and-error  not uncommon to “tune” the sets • While tuning, one can try different membership functions, increase or decrease number of sets • Some fuzzy practitioners recommend 7 fuzzy sets to fully define a practical working range (?!?!?)

16. Membership Functions • One rule of thumb for ensuring smooth transitions (in later steps) is to enforce overlapping between neighboring sets

17. Hedge Functions • Hedge functions are sometimes used to modify the degree of membership • Provide additional linguistic constructs that you can use in conjunction with other logical operations. • Two common hedges: • VERY(Truth(A)) = Truth(A)2 • NOT_VERY(Truth(A)) = Truth(A)0.5 (Truth(A) is the degree of membership of A in some fuzzy set)

18. Step 2: Fuzzy Rules • Next, construct a set of rules, combining the input in some logical manner, to yield some output • If-then style rules (if A then B) – A being the antecedent/premise and B being the consequent/conclusion • Fuzzy input variables are combined logically to form premise • Conclusion will be the degree of membership of some output fuzzy set

19. Fuzzy Axioms • Since we are writing “logical” rules with fuzzy input, we need a way to apply logical operators to fuzzy input (just like with Boolean input) • Logical OR (disjunction) • Truth(A OR B) = MAX(Truth(A), Truth(B)) • Logical AND (conjunction) • Truth(A AND B) = MIN(Truth(A), Truth(B)) • Logical NOT (negation) • Truth(NOT A) = 1 – Truth(A)

20. Fuzzy Axioms • Example, given a person is overweight to the degree of 0.7 and tall to the degree of 0.3: • Overweight AND tall = MIN(0.7, 0.3) = 0.3 • Overweight OR tall = MAX(0.7, 0.3) = 0.7 • NOT overweight = 1 – 0.7 = 0.3 • NOT tall = 1 – 0.3 = 0.7 • NOT(overweight AND tall) = 1 – MIN(0.7, 0.3) = 0.7 • There are other definitions for these logical operators…

21. Rule Evaluation • Unlike traditional Boolean logic, • Rules in fuzzy logic can evaluate into any number between 0 and 1 (not just 0 or 1) • All rules are evaluated in parallel (not in series that the first one that is true gets fired). Each rule always fires, to various degrees • The strength of each rule represents the degree of membership in the output fuzzy set

22. Rule Evaluation • Example: Evaluating whether an AI should attack player • Rules can be written like: • If (in melee range AND uninjured) AND NOT hard then attack • Set up as many rules to handle all possibilities in the game

23. Rule Evaluation • Given specific degrees for the input variables, you might get outputs (conclusions of the rules) that look something like this: Attack to degree: 0.2 Do nothing to degree: 0.4 Flee to degree: 0.7 • The most straightforward way to interpret these outputs is to take the action associated with the highest degree (in this case, the action will be flee)

24. Step 3: Defuzzification • In some cases, you might want to use the fuzzy output degree to determine a crisp value (real number), which can be useful for further calculations • Defuzzification: Process of converting the results from the fuzzy rules to get a crisp number as an output • Opposite of fuzzification (you can say that, although the purpose and methods are different!)

25. Step 3: Defuzzification • Previous example: Instead of determining some finite action (do nothing, flee, attack), we also want to use the output to determine the speed to take the action • To get a crisp number, aggregate the output strengths on the predefined output membership functions

26. Step 3: Defuzzification • With the numerical output from the earlier example (0.2 degree attack, 0.4 degree do nothing, 0.7 degree flee), we have the composite membership function below

27. Defuzzifying composite m/f • Truncate each output set to the output degree of membership for that set. Then combine all output sets by disjunction • A crisp number can be arrived from such an output fuzzy set in many ways • Geometric centroid of the area under the output fuzzy set, taking its horizontal axis coordinate as the crisp output

28. Using “predefuzzified” output • A less computationally expensive method is the use of singleton output membership function or a “predefuzzified” output function • Instead of doing lots of calculation, assign speeds to each output action (-10 for flee, 1 for do nothing, 10 for attack). • E.g. The resulting speed for flee is simply the preset value of -10 times the degree to which the output action flee is true (-10 x 0.7 = -7)

29. Using “predefuzzified” output • Aggregate of all outputs with a simple weighted average • In our example, we might have: Output = [(0.7)(-10) + (0.4)(1) + (0.3)(10)] / (0.7+0.4+0.3) = -2.5 • This output would result in the creature fleeing, but not earnestly in full extent • Naturally, we can obtain various output (crisp) values depending on the different input conditions

30. Further Examples • There are 2 good examples in the textbook, showing the full process of using fuzzy logic to model game AI characters

31. Using Fuzzy Logic in FSMs? • If we want to add some fuzzy logic into FSMs, how can that be accomplish? Is it possible? • Remember: Each state defines a behavior or action, and each state is reached by transition from another state on the basis of fulfilling some input conditions… • Conditions for transition are normally in Boolean logic, how do we accommodate fuzzy logic?

32. Fuzzy State Machines • Different AI developers regard Fuzzy State Machines differently • State machine with fuzzy states • State transitions that use fuzzy logic to trigger • Both • Find out more about how these different variations can be worked out and implemented (refer to Millington book)

33. … • Next up • Homework 2 (due in Week 10, submission via mail) • Milestone #2 (due in Week 11, 11.00am 23/8, Thurs) • Upcoming lectures • Probabilities and Uncertainty Techniques • Tactical and Strategic AI