Artificial Intelligence in Game Design

1. Artificial Intelligence in Game Design Lecture 6: Fuzzy Logic and Fuzzy State Machines

2. Problems with Discrete States • Characters may be in multiple states simultaneously • Particularly emotional states • Hit  angry • Low HP  frightened • May indicate contradictory responses • Frightened  run • Angry  attack • Need “intermediate” states and actions • “Somewhat angry” • “Somewhat frightened”

3. Problems with Discrete States • Abrupt transitions between states • Exact behavior from 100 down to 10 HP • Sudden behavior change at 9 HP • More realistic if behavior gradually changes as HP approaches 10 Angry Snarling Moving forward Frightened Wincing Backing away HP < 10

4. Problems with Discrete States • Serious problem if action is continuous value • Example: Racing gamesActions: • Speed of car • Amount to steer • … • Basic idea: • Near curve  slow speed • Far from curve  fast speed

5. Problems with Discrete States • Bad solution (based on discrete states) • Distance to curve > 10  120 mph • Distance to curve <= 10  60 mph Sudden transition!

6. Problems with Discrete States • Best solution: • As get closer to curve, gradually slow down • As get further from curve, gradually speed up

7. Fuzzy Logic • Membership functionµ S (x) • “Degree” to which entity X belongs in state S • Discrete states: µ S (x) = 1  X is in state S or 0  X is not in state S • Fuzzy states: µ S (x) may be any number between 0 and 1 • X may be “somewhat” in state S

8. Fuzzy Logic and Language • Often used to quantify “natural language” hedges • Example: • “Not at all”  µ S (x)= 0 • “A little”  µ S (x)= 0.2 • “Kind of”  µ S (x)= 0.4 • “Pretty”  µ S (x)= 0.6 • “Very”  µ S (x)= 0.8 • “Absolutely”  µ S (x)= 1.0

9. Fuzzy Logic Example • Example: Oswald the Orc • Very angryµ angry(Oswald)= 0.8 • Kind of frightened µ frightened(Oswald)= 0.4 • Not at all confident µ confident(Oswald)= 0.0 Note that fuzzy memberships don’t have to add up to 1 (unlike probabilities)

10. Fuzzification • Converting a continuous valued attribute to a fuzzy state membership • Racing example: • Attribute = car’s distance from curve • Fuzzy states: • Near curve • Far from curve • Goal: determine membership of car in these sets • First step to determine desired speed of car Start of curve

11. Fuzzification • Often represent as simple graphs • X axis: continuous attribute • Y axis: membership in set(s) • Example: current distance = 60 Near Far 0.6 0.4 distance to curve 0 20 40 60 80 100 120 µnear (car) = 0.6 µ far (car) = 0.4

12. Fuzzy Rules • Fuzzy antecedent Fuzzy conclusion • Fuzzy antecedent created from fuzzification • Requires fuzzy definitions of logical connectives AND, OR, and NOT. What is fuzzy membership in this? Given fuzzy membership in this

13. Fuzzy Connectives Usual measures: • AND: µ S and T(x)= min (µ S(x),µ T(x)) • OR: µ S or T(x)= max (µ S(x),µ T(x))Note: this also applies if combining separate rules • NOT: µ not T(x)= 1 - µ T(x)

14. Fuzzy Rule Example Rules: • IF near curve exit AND going slow THEN accelerate • IF far from curve exit AND going medium THEN accelerate Question: • If near, far, slow, medium all fuzzy measuresWhat is resulting membership in accelerate? Speed: 100 mph Distance from exit: 80

15. Fuzzy Rule Example Near Far 0.6 0.4 distance to curve 0 20 40 60 80 100 120 µnear (car) = 0.4 µ far (car) = 0.6 1.0 Slow Fast Medium 0.3 speed 0 30 60 90 120 150 180 µslow (car) = 0.3 µ medium (car) = 1.0

16. Fuzzy Rule Example • Rule 1:IF near curve exit AND going slow THEN accelerateµ accelerate(car)=µ near and slow(car)= min (µ near(car),µ slow(car)) = min(0.4, 0.3) = 0.3 • Rule 2:IF far curve exit AND going medium THEN accelerateµ accelerate(car)=µ far and medium(car)= min (µ far(car),µ medium(car)) = min(0.6, 1.0) = 0.6

17. Fuzzy Rule Example • Combining the rules:µ accelerate(car)=µ accelerate1 or accelerate2(car)= max (µ accelerate1(car),µ accelerate2(car)) = max(0.3, 0.6) = 0.6Membership of the car in accelerate = 0.6

18. Defuzzification • Converting fuzzy state membership to appropriate actions • Higher membership in state  greater degree of action implied by state • Possible problems • Conflicts if multiple actions indicated • Some actions not fuzzy (attack vs. no attack) • Many open problems!

19. Simple Defuzzification • Create fuzzy set for action • Map membership in set back to value • Assumes no conflicts with other actions based on same value • Assumes monotonic function for fuzzy set • Example: µ accelerate(car) = 0.6  acceleration = 16 feet/second2 Acceleration 0.6 feet/second2 0 5 10 15 20 25 30

20. Conflicts in Defuzzification • Example rule:IF near curve entrance AND going fast THEN brake • Distance: 90  µ near(car) = 0.2 • Speed: 80  µ fast(car) = 0.15 • µ brake(car) = min (0.2, 0.15) = 0.15

21. Conflicts in Defuzzification • Conflicting fuzzy actions: • µ accelerate(car) = 0.6 • µ brake(car) = 0.15 • Intuitive idea: • Should accelerate since its membership is larger • Membership in brake should decrease acceleration to some degree

22. Conflicts in Defuzzification Methods: • Just use largest rule (µ accelerate(car) = 0.6) • Fast to compute • Leads to uneven behavior • Weighted center of mass of sets • Commonly used in AI • Slow and costly to compute Braking Acceleration 0.6 0.15 feet/second2 -15 -10 -5 -0 5 10 15

23. Conflicts in Defuzzification • Choose one value for each set • Usually its maximum value • Weight by memberships in each set • Determine average value Example: • Maximum acceleration = 30 feet/second2 • Maximum braking = -40 feet/second2 • Weighted by membership: • Acceleration = 0.6 * 30 feet/second2 = 18 feet/second2 • Braking = 0.15 * -40 feet/second2 = -6 feet/second2 • Resulting acceleration: 12 feet/second2

24. Defuzzifying Discrete Actions • Example: Attack/No attack (must do one or the other) • Thresholding: • If fuzzy membership > threshold take action • Example: If µ angry(orc) > 0.7  attackIf µ angry(orc) <= 0.7  no attack • Conflicts:Take action with highest membership • Angry  attack, µ angry(Oswald) = 0.7 • Frightened  run, µ frightened(Oswald) = 0.3 • Note: Fuzzy logic works best for continuous valued actions Oswald attacks!

25. Fuzzy State Machines • Fuzzy membership in states • May be in multiple states • Transitions between states may be fuzzy • Resulting actions may be fuzzy Example: • May be confident and fearful to some degree • Transitions based on fuzzy measures “high” and “low” HP • Action (% chance of attack) is continuous valued HP low Confident Attack 100% Fearful Attack 20% HP high

26. Fuzzy State Machines T • Transitions of formtreated like fuzzy and • Must be in state S1 • Stimulus T must be present • µ S2(orc) = min(µ S1(orc), µ T(orc)) • Note that unless µ S2(orc) = 1, it is still in S1 to some degree: µ S1(orc) = 1 - µ S2(orc) • Note that state S2 maybe indicated as a next state by many other states (including itself) • Will use max rule to determine final membership in S2 S1 S2

27. Fuzzy State Machine Example • Current state memberships • µ conf(Oswald) = 0.8 • µ fear(Oswald) = 0.4 HP low Confident Attack 100% Fearful Attack 40% HP high

28. Fuzzy State Machine Example • Current HP = 13 (just took heavy hit by player) • µ low(Oswald) = 0.7 • µ high(Oswald) = 0.2 Low High HP 0 5 10 15 20 25 30

29. Fuzzy State Machine Example • Confident  Fearful transition • µ fear(Oswald) = min (µ conf(Oswald), µ low(Oswald)) = min (0.8, 0.7) = 0.7 • µ conf(Oswald) = 1 - µ fear(Oswald) = 0.3 • Fearful  Confident transition • µ conf(Oswald) = min(µ fear(Oswald), µ high(Oswald)) = min (0.4, 0.2) = 0.2 • µ fear(Oswald) = 1 - µ conf(Oswald) = 0.8

30. Fuzzy State Machine Example • Combining effects of transitions • µ conf(Oswald) = max (0.3, 0.2) = 0.3 • µ fear(Oswald) = max (0.7, 0.8) = 0.8 • Determining current action (defuzzification) • Attack % = ( µ conf(Oswald) * 100% +µ fear(Oswald) * 20% ) / 2 = (0.4 * 100% + 0.7 * 20% ) / 2 = 46%