1. 11/6/2001CS 638, Fall 2001 Today AI –Fuzzy Logic –Neural Nets

2. 11/6/2001CS 638, Fall 2001 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)

3. 11/6/2001CS 638, Fall 2001 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

4. 11/6/2001CS 638, Fall 2001 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 => in the set – 0.0 => not in the set – 0.0 partially in the set Provides a way to write symbolic rules with terms like “medium” but evaluate them in a quantified way

5. 11/6/2001CS 638, Fall 2001 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)?” Aggressiveness Membership (Degree of Truth) 1.0 0.0 1 0.5 Medium Nasty Meek -0.5

6. 11/6/2001CS 638, Fall 2001 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

7. 11/6/2001CS 638, Fall 2001 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

8. 11/6/2001CS 638, Fall 2001 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

9. 11/6/2001CS 638, Fall 2001 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

10. 11/6/2001CS 638, Fall 2001 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)

11. 11/6/2001CS 638, Fall 2001 Set Definitions for Example distance v. smallsmallperfectbigv. big delta <=>>> acceleration slowpresentfastfastest << brake

12. 11/6/2001CS 638, Fall 2001 Instance for Example Distance could be considered small or perfect Delta could be stable or growing What acceleration? distance v. smallsmallperfectbigv. big delta <=>>> acceleration slowpresentfastfastest << brake ????

13. 11/6/2001CS 638, Fall 2001 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. 11/6/2001CS 638, Fall 2001 Fuzzy Inference for Example Convert our belief into action –For each rule, clip action fuzzy set by belief in rule acceleration present acceleration present acceleration slow acceleration fast

15. 11/6/2001CS 638, Fall 2001 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

16. 11/6/2001CS 638, Fall 2001 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

17. 11/6/2001CS 638, Fall 2001 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.

18. 11/6/2001CS 638, Fall 2001 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

19. 11/6/2001CS 638, Fall 2001 Simulated Neuron - Perceptron Inputs (a j ) from other perceptrons with weights (W i,j ) –Learning occurs by adjusting the weights Perceptron calculates weighted sum of inputs (in i ) Threshold function calculates output (a i ) –Step function (if in i > t then a i = 1 else a i = 0) –Sigmoid g(a) = 1/1+e -x Output becomes input for next layer of perceptron ajaj W i,j aiai Σ W i,j a j = in i a i = g(in i )

20. 11/6/2001CS 638, Fall 2001 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

21. 11/6/2001CS 638, Fall 2001 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 Enemy Sound Dead Low Health Attack Retreat Wander Chase Spawn

22. 11/6/2001CS 638, Fall 2001 Learning Neural Networks Learning from examples –Examples consist of input and correct output Learn if network’s output doesn’t match correct output –Adjust weights to reduce difference –Only change weights a small amount (η) Basic perceptron learning –W i,j = W i,j + η(t-o)a j –If output is too high (t-o) is negative so W i,j will be reduced –If output is too low (t-o) is positive so W i,j will be increased –If a j is negative the opposite happens

23. 11/6/2001CS 638, Fall 2001 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 1 0.1 Σ W j a j = 0.1 g(0.1) = 0 0 0 0.6

24. 11/6/2001CS 638, Fall 2001 Neural Net Example Wj = Wj + η(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 Σ W j a j = 0.6 g(0.6) = 0 1 1 0.6

25. 11/6/2001CS 638, Fall 2001 Neural Net Example Try (1,0)  1 example –Still an error, so training occurs W 1 = 0.2 + 0.1(1-0)1 = 0.3 W 2 = 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? 1 0.2 Σ W j a j = 0.2 g(0.2) = 0 0 0 0.6

26. 11/6/2001CS 638, Fall 2001 Neural Networks Evaluation Advantages –Handle errors well –Graceful degradation –Can learn novel solutions Disadvantages –“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

27. 11/6/2001CS 638, Fall 2001 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.