CS 378: Computer Game Technology

1. CS 378: Computer Game Technology AI – Fuzzy Logic and Neural Nets Spring 2012

2. Today • AI • Fuzzy Logic • Neural Nets

3. Fuzzy Logic • Philosophical approach • Decisions based on “degree of truth” • Is not a method for reasoning under uncertainty – that’s probability • Crisp Facts – distinct boundaries • Fuzzy Facts – imprecise boundaries • Probability - incomplete facts • Example – 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)

4. 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

5. 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 – 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

6. 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 -1 -0.5 0.0 0.5 1 Aggressiveness

7. 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

8. 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

9. 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 Membership About 6’ Tall 1.0 About 6’ high and tall 0.0 Height

10. 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

11. Fuzzy Set Ops: Union (OR) • Take the max of your beliefs in each individual statement • Also works if statements are about different variables Membership About 6’ Tall 1.0 About 6’ high or tall 0.0 Height

12. Fuzzy Rules • “If our distance to the car in front is small, and the distance is decreasingslowly, then deceleratequite 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

13. 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)

14. Set Definitions v. small small perfect big v. big brake slow present fast fastest distance acceleration << < = > >> delta

15. Instance • Distance could be considered small or perfect • Delta could be stable or growing • What acceleration? v. small small perfect big v. big brake slow present fast fastest distance acceleration ???? << < = > >> delta

16. Matching • 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

17. Fuzzy Inference • Convert our belief into action • For each rule, clip action fuzzy set by belief in rule present slow acceleration acceleration fast present acceleration acceleration

18. 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

19. Evaluation of Fuzzy Logic • Does not necessarily lead to non-determinism • Advantages • Allows use of continuous valued actions while still writing “crisp” rules – 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 • Disadvantages • Sometimes results are unexpected and hard to debug • Additional computational overhead • There are other ways to get continuous acceleration

20. 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.

21. 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

22. 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 Σ Wi,j aj = ini ai ai = g(ini)

23. Network Structure • Single perceptron can represent AND or OR, but not XOR • Combinations of perceptron are more powerful • Perceptron are usually organized in 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

24. 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 Dead Sound Low Health Attack Wander Spawn Retreat Chase

25. 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 (η) • Basic perceptron learning • Let f be the actual output for input t • Wi,j = Wi,j + η(f-o)aj • If output is too high (f-o) is negative so Wi,jwill be reduced • If output is too low (f-o) is positive so Wi,jwill be increased • If aj is negative the opposite happens

26. 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 Wj aj = 0.1 0 g(0.1) = 0 0.6 0

27. Neural Net Example • Wj = Wj + η(f-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 Wj aj = 0.6 1 g(0.6) = 0 0.6 1

28. 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… 1 0.2 Σ Wj aj = 0.2 0 g(0.2) = 0 0.6 0

29. 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

30. 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.