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

Artificial Intelligence in Game Design

Lecture 6: Fuzzy Logic and

Fuzzy State Machines

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”

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

AngrySnarling

Moving forward

FrightenedWincing

Backing away

HP < 10

Problems with Discrete States

• Serious problem if action is continuous value– Example: Racing games

Actions:• Speed of car• Amount to steer• …

– Basic idea:• Near curve slow speed• Far from curve fast speed

Problems with Discrete States

• Bad solution (based on discrete states)– Distance to curve > 10 120 mph– Distance to curve <= 10 60 mph

Sudden transition!

Problems with Discrete States

• Best solution:– As get closer to curve, gradually slow down– As get further from curve, gradually speed up

Fuzzy Logic

• Membership function µ 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

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

Fuzzy Logic Example

• Example: Oswald the Orc– Very angry

µ 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)

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

Fuzzification• Often represent as simple graphs

– X axis: continuous attribute– Y axis: membership in set(s)– Example: current distance = 60

distance to curve0 20 40 60 80 100 120

Near Far

0.4

0.6

µ near (car) = 0.6 µ far (car) = 0.4

Fuzzy Rules

• Fuzzy antecedent Fuzzy conclusion

• Fuzzy antecedent created from fuzzification• Requires fuzzy definitions of logical connectives

AND, OR, and NOT.

Given fuzzy membership in this

What is fuzzy membership in this?

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)

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 measures

What is resulting membership in accelerate?

Speed: 100 mph

Distance from exit: 80

Fuzzy Rule Example

distance to curve0 20 40 60 80 100 120

Near Far

0.4

0.6

µ near (car) = 0.4 µ far (car) = 0.6

speed0 30 60 90 120 150 180

Slow Fast

0.3

1.0

µ slow (car) = 0.3 µ medium (car) = 1.0

Medium

Fuzzy Rule Example

• Rule 1:IF near curve exit AND going slow THEN accelerate

µ 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 accelerate

µ accelerate (car) = µ far and medium (car) = min (µ far (car), µ medium (car))

= min(0.6, 1.0) = 0.6

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) = 0.6

Membership of the car in accelerate = 0.6

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!

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

feet/second2

0 5 10 15 20 25 30

Acceleration 0.6

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

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

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

-15 -10 -5 -0 5 10 15

Acceleration

0.6

feet/second2

Braking

0.15

Conflicts in Defuzzification

1. Choose one value for each set– Usually its maximum value

2. Weight by memberships in each set

3. 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

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 attack

If 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!

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

ConfidentAttack 100%

HP lowFearful

Attack 20%HP high

Fuzzy State Machines

• 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 may be indicated as a next state by many other states (including itself)– Will use max rule to determine final membership in S2

S1 S2T

Fuzzy State Machine Example

• Current state memberships

– µ conf (Oswald) = 0.8

– µ fear (Oswald) = 0.4

ConfidentAttack 100%

HP lowFearful

Attack 40%HP high

Fuzzy State Machine Example

• Current HP = 13 (just took heavy hit by player)

– µ low (Oswald) = 0.7

– µ high (Oswald) = 0.2

HP

Low High

0 5 10 15 20 25 30

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

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%