University of Texas at Austin CS 378 – Game Technology Don Fussell

CS 378: Computer Game Technology

AI – Fuzzy Logic and Neural NetsSpring 2012

Today

AIFuzzy LogicNeural Nets

Fuzzy Logic

Philosophical approachDecisions based on “degree of truth” Is not a method for reasoning under uncertainty – that’s probability

Crisp Facts – distinct boundariesFuzzy Facts – imprecise boundariesProbability - incomplete factsExample – Scout reporting an enemy

“Two tanks at grid NV 54“ (Crisp)“A few tanks at grid NV 54” (Fuzzy)“There might be 2 tanks at grid NV 54 (Probabilistic)

Apply to Computer Games

Can have different characteristics of playersStrength: strong, medium, weakAggressiveness: meek, medium, nastyIf meek and attacked, run away fastIf medium and attacked, run away slowlyIf nasty and strong and attacked, attack back

Control of a vehicleShould slow down when close to car in frontShould speed up when far behind car in front

Provides smoother transitions – not a sharp boundary

Fuzzy Sets

Provides a way to write symbolic rules with terms like “medium” but evaluate them in a quantified way Classical set theory: An object is either in or not in the set

Can’t talk about non-sharp distinctionsFuzzy sets have a smooth boundary

Not completely in or out – somebody 6” is 80% in the tall set tallFuzzy set theory

An object is in a set by matter of degree 1.0 => in the set 0.0 => not in the set 0.0 < object < 1.0 => partially in the set

Example Fuzzy Variable

Each function tells us how much we consider a character in the set if it has a particular aggressiveness valueOr, how much truth to attribute to the statement: “The character is nasty (or meek, or neither)?”

Aggressiveness

Membership (Degree of Truth)

1.0

0.0 1 0.5

MediumNastyMeek

-1 -0.5

Fuzzy Set Operations: Complement

The degree to which you believe something is not in the set is 1.0 minus the degree to which you believe it is in the set

Membership

Units

1.0

0.0

FS

¬FS

Fuzzy Set Ops: Intersection (AND)

If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A and B?

Eg: How much faith in “that person is about 6’ high and tall”

Does it make sense to attribute more truth than you have in one of A or B?

Membership

Height

1.0

0.0

About 6’ Tall

Fuzzy Set Ops: Intersection (AND)

Assumption: Membership in one set does not affect membership in anotherTake the min of your beliefs in each individual statementAlso works if statements are about different variables

Dangerous and injured - belief is the min of the degree to which you believe they are dangerous, and the degree to which you think they are injured

Membership

Height

1.0

0.0

About 6’ Tall

About 6’ high and tall

Fuzzy Set Ops: Union (OR)

If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A or B?

Eg: How much faith in “that person is about 6’ high or tall”

Does it make sense to attribute less truth than you have in one of A or B?

Membership

Height

1.0

0.0

About 6’ Tall

Fuzzy Set Ops: Union (OR)

Take the max of your beliefs in each individual statementAlso works if statements are about different variables

Membership

Height

1.0

0.0

About 6’ Tall

About 6’ high or tall

Fuzzy Rules

“If our distance to the car in front is small, and the distance is decreasing slowly, then decelerate quite hard”

Fuzzy variables in blueFuzzy sets in redConditions are on membership in fuzzy setsActions place an output variable (decelerate) in a fuzzy set (the quite hard deceleration set)

We have a certain belief in the truth of the condition, and hence a certain strength of desire for the outcomeMultiple rules may match to some degree, so we require a means to arbitrate and choose a particular goal - defuzzification

Fuzzy Rules Example(from Game Programming Gems)

Rules for controlling a car:Variables are distance to car in front and how fast it is changing, delta, and acceleration to applySets are:

Very small, small, perfect, big, very big - for distanceShrinking fast, shrinking, stable, growing, growing fast for deltaBrake hard, slow down, none, speed up, floor it for acceleration

Rules for every combination of distance and delta sets, defining an acceleration set

Assume we have a particular numerical value for distance and delta, and we need to set a numerical value for acceleration

Extension: Allow fuzzy values for input variables (degree to which we believe the value is correct)

Set Definitions

distance

v. small small perfect big v. big

delta

< = > >>

acceleration

slow present fast fastest

<<

brake

Instance

Distance could be considered small or perfectDelta could be stable or growingWhat acceleration?

distance

v. small small perfect big v. big

delta

< = > >>

acceleration

slow present fast fastest

<<

brake

????

Matching

Relevant rules are:If distance is small and delta is growing, maintain speedIf distance is small and delta is stable, slow downIf distance is perfect and delta is growing, speed upIf distance is perfect and delta is stable, maintain speed

For first rule, distance is small has 0.75 truth, and delta is growing has 0.3 truth

So the truth of the and is 0.3Other rule strengths are 0.6, 0.1 and 0.1

Fuzzy Inference

Convert our belief into actionFor each rule, clip action fuzzy set by belief in rule

acceleration

present

acceleration

presentacceleration

slow

acceleration

fast

Defuzzification Example

Three actions (sets) we have reason to believe we should take, and each action covers a range of values (accelerations)Two options in going from current state to a single value:

Mean of Max: Take the rule we believe most strongly, and take the (weighted) average of its possible valuesCenter of Mass: Take all the rules we partially believe, and take their weighted average

In this example, we slow down either way, but we slow down more with Mean of Max

Mean of max is cheaper, but center of mass exploits more information

Evaluation of Fuzzy Logic

Does not necessarily lead to non-determinismAdvantages

Allows use of continuous valued actions while still writing “crisp” rules – can accelerate to different degreesAllows use of “fuzzy” concepts such as mediumBiggest impact is for control problems

Help avoid discontinuities in behaviorIn example problem strict rules would give discontinuous acceleration

DisadvantagesSometimes results are unexpected and hard to debugAdditional computational overheadThere are other ways to get continuous acceleration

References

Nguyen, H. T. and Walker, E. A. A First Course in Fuzzy Logic, CRC Press, 1999.Rao, V. B. and Rao, H. Y. C++ Neural Networks and Fuzzy Logic, IGD Books Worldwide, 1995.McCuskey, M. Fuzzy Logic for Video Games, in Game Programming Gems, Ed. Deloura, Charles River Media, 2000, Section 3, pp. 319-329.

Neural Networks

Inspired by natural decision making structures (real nervous systems and brains)If you connect lots of simple decision making pieces together, they can make more complex decisions

Compose simple functions to produce complex functionsNeural networks:

Take multiple numeric input variablesProduce multiple numeric output valuesNormally threshold outputs to turn them into discrete valuesMap discrete values onto classes, and you have a classifier!But, the only time I’ve used them is as approximation functions

Simulated Neuron - Perceptron

Inputs (aj) from other perceptrons with weights (Wi,j)Learning occurs by adjusting the weights

Perceptron calculates weighted sum of inputs (ini)Threshold function calculates output (ai)

Step function (if ini > t then ai = 1 else ai = 0)Sigmoid g(a) = 1/(1+e-x)

Output becomes input for next layer of perceptron aj Wi,j

aiΣ Wi,j aj = ini

ai = g(ini)

Network Structure

Single perceptron can represent AND or OR, but not XORCombinations of perceptron are more powerful

Perceptron are usually organized in layersInput layer: takes external inputHidden layer(s)Output layer: external output

Feed-forward vs. recurrentFeed-forward: outputs only connect to later layers

Learning is easierRecurrent: outputs can connect to earlier layers or same layer

Internal state

Neural network for Quake

Four input perceptron One input for each condition

Four perceptron hidden layerFully connected

Five output perceptron One output for each actionChoose action with highest outputOr, probabilistic action selection

Choose at random weighted by output

EnemySound

DeadLow Health

Attack

Retreat

Wander

ChaseSpawn

Learning Neural Networks

Learning from examplesExamples consist of input, t, and correct output, o

Learn if network’s output doesn’t match correct outputAdjust weights to reduce differenceOnly change weights a small amount (η)

Basic perceptron learningLet f be the actual output for input tWi,j = Wi,j + η(f-o)aj

If output is too high (f-o) is negative so Wi,j will be reducedIf output is too low (f-o) is positive so Wi,j will be increasedIf aj is negative the opposite happens

Neural Net Example

Single perceptron to represent ORTwo inputsOne output (1 if either inputs is 1)Step function (if weighted sum > 0.5 output a 1)

Initial state (below) gives error on (1,0) inputTraining occurs

10.1

Σ Wj aj = 0.1

g(0.1) = 0

0

0

0.6

Neural Net Example

Wj = Wj + η(f-o)aj

W1 = 0.1 + 0.1(1-0)1 = 0.2W2 = 0.6 + 0.1(1-0)0 = 0.6After this step, try (0,1)1 example

No error, so no training

00.2

Σ Wj aj = 0.6

g(0.6) = 0

1

1

0.6

Neural Net Example

Try (1,0)1 exampleStill an error, so training occurs

W1 = 0.2 + 0.1(1-0)1 = 0.3W2 = 0.6 + 0.1(1-0)0 = 0.6And so on…

10.2

Σ Wj aj = 0.2

g(0.2) = 0

0

0

0.6

Neural Networks Evaluation

AdvantagesHandle errors wellGraceful degradationCan learn novel solutions

Disadvantages“Neural networks are the second best way to do anything”Can’t understand how or why the learned network worksExamples must match real problemsNeed as many examples as possibleLearning takes lots of processing

Incremental so learning during play might be possible

References

Mitchell: Machine Learning, McGraw Hill, 1997.Russell and Norvig: Artificial Intelligence: A Modern Approach, Prentice Hall, 1995.Hertz, Krogh & Palmer: Introduction to the theory of neural computation, Addison-Wesley, 1991.Cowan & Sharp: Neural nets and artificial intelligence, Daedalus 117:85-121, 1988.