fuzzy logic and neural networks full report
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28-03-2010, 11:27 PM

Fuzzy Logic
Philosophical approach
Decisions based on degree of truth
Is not a method for reasoning under uncertainty that probability
Crisp Facts distinct boundaries
Fuzzy Facts imprecise boundaries
Probability - incomplete facts
Example €œ 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 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 €œ 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 distinctions
Fuzzy sets have a smooth boundary
Not completely in or out €œ somebody 6 is 80% in the tall set tall
Fuzzy 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
Fuzzy Set Operations: Complement
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
Fuzzy Set Ops: Intersection (AND)
Assumption: Membership in one set does not affect membership in another
Take the min of your beliefs in each individual statement
Also 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
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
Fuzzy Set Ops: Union (OR)
Take the max of your beliefs in each individual statement
Also works if statements are about different variables
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

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 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)
Set Definitions for Example
Instance for Example
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

Fuzzy Inference for Example
Convert our belief into action
For each rule, clip action fuzzy set by belief in rule
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 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
Evaluation of Fuzzy Logic
Does not necessarily lead to non-determinism
Allows use of continuous valued actions while still writing crisp rules €œ can accelerate to different degrees
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
Sometimes results are unexpected and hard to debug
Additional computational overhead
There are other ways to get continuous acceleration
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 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 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
Network Structure
Single perceptron can represent AND or OR, but 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
Neural network for Quake
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
Learning Neural Networks
Learning from examples
Examples consist of input, t, and correct output, o
Learn if network „¢s output doesn „¢t 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
Neural Net Example
Single perceptron to represent OR
Two inputs
One 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) input
Training occurs
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

Neural Net Example
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

Neural Networks Evaluation
Handle errors well
Graceful degradation
Can learn novel solutions
Neural networks are the second best way to do anything
Can„t 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
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.
By Monday, Nov 3, Stage 3 demo
Thurs Nov 6, Midterm
Everything up to and including lecture 15

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