Today

1
Today

• AI
• Fuzzy Logic
• Neural Nets

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Fuzzy Logic

• Philosophical approach
• Ontological commitment based on degree of truth
• Is not a method for reasoning under uncertainty
• Crisp Facts distinct boundaries
• Fuzzy Facts imprecise boundaries
• Probability - incomplete facts
• Example Scout reporting an enemy
• Two to three tanks at grid NV 123456 (Crisp)
• A few tanks at grid NV 123456 (Fuzzy)
• There might be 2 tanks at grid NV 54
  (Probabilistic)

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Apply to Computer Games

• Can have different characteristics of players
• Strength strong, medium, weak
• Aggressiveness meek, medium, nasty
• If meek and attacked, run away fast
• If medium and attacked, run away slowly
• If nasty and strong and attacked, attack back
• Control of a vehicle
• Should slow down when close to car in front
• Should speed up when far behind car in front
• Provides smoother transitions not a sharp
  boundary

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Fuzzy Sets

• Classical set theory
• An object is either in or not in the set
• Sets with smooth boundary
• Not completely in or out somebody 6 is 80
  tall
• Fuzzy set theory
• An object is in a set by matter of degree
• 1.0 gt in the set
• 0.0 gt not in the set
• 0.0 lt object lt 1.0 gt partially in the set
• Provides a way to write symbolic rules with terms
  like medium but evaluate them in a quantified
  way

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Example Fuzzy Variable

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

Membership (Degree of Truth)
Meek
Nasty
1.0
Medium
0.0
1
0.5
-1
-0.5
Aggressiveness
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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
FS
1.0
FS
0.0
Units
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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
About 6
Tall
1.0
0.0
Height
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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
About 6
Tall
1.0
0.0
Height
9
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 blue
• Fuzzy sets in red
• Conditions are on membership in fuzzy sets
• Actions 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 outcome
• Multiple rules may match to some degree, so we
  require a means to arbitrate and choose a
  particular goal - defuzzification

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Fuzzy Rules Example (from Gems)

• Rules for controlling a car
• Variables are distance to car in front and how
  fast it is changing, delta, and acceleration to
  apply
• Sets are
• Very small, small, perfect, big, very big - for
  distance
• Shrinking fast, shrinking, stable, growing,
  growing fast for delta
• Brake 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)

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Set Definitions for Example
v. small
small
perfect
big
v. big
slow
present
fast
fastest
brake
distance
acceleration
lt

gt
gtgt
ltlt
delta
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Instance for Example
v. small
small
perfect
big
v. big
slow
present
fast
fastest
brake
distance
acceleration
????
lt

gt
gtgt
ltlt

• Distance could be considered small or perfect
• Delta could be stable or growing
• What acceleration?

delta
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Matching for Example

• Relevant rules are
• If distance is small and delta is growing,
  maintain speed
• If distance is small and delta is stable, slow
  down
• If distance is perfect and delta is growing,
  speed up
• If 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 0.3
• Other rule strengths are 0.6, 0.1 and 0.1

14
Fuzzy Inference for Example

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

present
slow
acceleration
acceleration
present
fast
acceleration
acceleration
15
Defuzzification Example

• We have three things sets we have reason to
  believe we are in, and each set covers a range of
  values
• 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 values
• Center 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

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Evaluation of Fuzzy Logic

• Does not necessarily lead to non-determinism
• Advantages
• Allows use of numbers while still writing crisp
  rules
• Allows use of fuzzy concepts such as medium
• Biggest impact is for control problems
• Help avoid discontinuities in behavior
• In example problem strict rules would give
  discontinuous acceleration
• Disadvantages
• Sometimes results are unexpected and hard to
  debug
• Additional computational overhead
• There are other ways to get continuous
  acceleration

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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.

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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
  functions
• Neural networks
• Take multiple numeric input variables
• Produce multiple numeric output values
• Normally threshold outputs to turn them into
  discrete values
• Map discrete values onto classes, and you have a
  classifier!
• But, the only time Ive used them is as
  approximation functions

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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 gt t then ai 1 else ai
  0)
• Sigmoid g(a) 1/1e-x
• Output becomes input for next layer of perceptron

aj
Wi,j
S Wi,j aj ini
ai
ai g(ini)
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Network Structure

• Single perceptron can represent AND, OR not XOR
• Combinations of perceptron are more powerful
• Perceptron are usually organized on layers
• Input layer takes external input
• Hidden layer(s)
• Output layer external output
• Feed-forward vs. recurrent
• Feed-forward outputs only connect to later
  layers
• Learning is easier
• Recurrent outputs can connect to earlier layers
  or same layer
• Internal state

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Neural network for Quake
Enemy
Dead

• Four input perceptron
• One input for each condition
• Four perceptron hidden layer
• Fully connected
• Five output perceptron
• One output for each action
• Choose action with highest output
• Or, probabilistic action selection
• Choose at random weighted by output

Sound
Low Health
Attack
Wander
Spawn
Retreat
Chase
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Learning Neural Networks

• Learning from examples
• Examples consist of input and correct output
• Learn if networks output doesnt match correct
  output
• Adjust weights to reduce difference
• Only change weights a small amount (?)
• Basic perceptron learning
• Wi,j Wi,j ?(t-o)aj
• If output is too high (t-o) is negative so Wi,j
  will be reduced
• If output is too low (t-o) is positive so Wi,j
  will be increased
• If aj is negative the opposite happens

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Neural Net Example

• Single perceptron to represent OR
• Two inputs
• One output (1 if either inputs is 1)
• Step function (if weighted sum gt 0.5 output a 1)
• Initial state (below) gives error on (1,0) input
• Training occurs

1
0.1
S Wj aj 0.1
0
g(0.1) 0
0.6
0
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Neural Net Example

• Wj Wj ?(t-o)aj
• W1 0.1 0.1(1-0)1 0.2
• W2 0.6 0.1(1-0)0 0.6
• After this step, try (0,1)?1 example
• No error, so no training

0
0.2
S Wj aj 0.6
1
g(0.6) 0
0.6
1
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Neural Net Example
1
0.2
S Wj aj 0.2
0
g(0.2) 0

• Try (1,0)?1 example
• Still an error, so training occurs
• W1 0.2 0.1(1-0)1 0.3
• W2 0.6 0.1(1-0)0 0.6
• And so on
• What is a network that works for OR?
• What about AND?
• Why not XOR?

0.6
0
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Neural Networks Evaluation

• Advantages
• Handle errors well
• Graceful degradation
• Can learn novel solutions
• Disadvantages
• Neural networks are the second best way to do
  anything
• Cant understand how or why the learned network
  works
• Examples must match real problems
• Need as many examples as possible
• Learning takes lots of processing
• Incremental so learning during play might be
  possible

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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 11785-121, 1988.