1. CS-378: Game Technology Lecture #17: AI Prof. Okan Arikan University of Texas, Austin Thanks to James O’Brien, Steve Chenney, Zoran Popovic, Jessica Hodgins V2005-08-1.1

2. 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)

3. 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. 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 partially in the set

5. 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. 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. 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. 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 Height 1.0 0.0 About 6’Tall About 6’ high and tall Fuzzy Set Ops: Intersection (AND)

9. 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

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

11. 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

12. 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)

13. Set Definitions for Example distance v. smallsmallperfectbigv. big delta <=>>> acceleration slowpresentfastfastest << brake

14. 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 ????

15. 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

16. 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

17. 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

18. 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

19. 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.

20. 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

21. Simulated Neuron - Perceptron

23. 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

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 Enem y Sound Dead Low Health Attack Retreat Wander Chase Spawn

25. 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

26. 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.

27. Rule Based Systems And decision trees coming next lecture … Now, some lower level issues …

28. Path Finding Very common problem in games: In FPS: How does the AI get from room to room? In RTS: User clicks on units, tells them to go somewhere. How do they get there? How do they avoid each other? Chase games, sports games, … Very expensive part of games Lots of techniques that offer quality, robustness, speed trade-offs

29. Path Finding Problem Problem Statement (Academic): Given a start point, A, and a goal point, B, find a path from A to B that is clear Generally want to minimize a cost: distance, travel time, … Travel time depends on terrain, for instance May be complicated by dynamic changes: paths being blocked or removed May be complicated by unknowns – don’t have complete information Problem Statement (Games): Find a reasonable path that gets the object from A to B Reasonable may not be optimal – not shortest, for instance It may be OK to pass through things sometimes It may be OK to make mistakes and have to backtrack

30. Search or Optimization? Path planning (also called route-finding) can be phrased as a search problem: Find a path to the goal B that minimizes Cost(path) There are a wealth of ways to solve search problems, and we will look at some of them Path planning is also an optimization problem: Minimize Cost(path) subject to the constraint that path joins A and B State space is paths joining A and B, kind of messy There are a wealth of ways to solve optimization problems The difference is mainly one of terminology: different communities (AI vs. Optimization) But, search is normally on a discrete state space

31. Brief Overview of Techniques Discrete algorithms: BFS, Greedy search, A*, … Potential fields: Put a “force field” around obstacles, and follow the “potential valleys” Pre-computed plans with dynamic re-planning Plan as search, but pre-compute answer and modify as required Special algorithms for special cases: E.g. Given a fixed start point, fast ways to find paths around polygonal obstacles

32. Graph-Based Algorithms Ideally, path planning is point to point (any point in the world to any other, through any unoccupied point) But, the search space is complex (space of arbitrary curves) The solution is to discretize the search space Restrict the start and goal points to a finite set Restrict the paths to be on lines (or other simple curves) that join points Form a graph: Nodes are points, edges join nodes that can be reached along a single curve segment Search for paths on the graph

33. Waypoints (and Questions) The discrete set of points you choose are called waypoints Where do you put the waypoints? There are many possibilities How do you find out if there is a simple path between them? Depends on what paths you are willing to accept - almost always assume straight lines The answers to these questions depend very much on the type of game you are developing The environment: open fields, enclosed rooms, etc… The style of game: covert hunting, open warfare, friendly romp, …

34. Where Would You Put Waypoints?

35. Waypoints By Hand Place waypoints by hand as part of level design Best control, most time consuming Many heuristics for good places: In doorways, because characters have to go through doors and straight lines joining rooms always go through doors Along walls, for characters seeking cover At other discontinuities in the environments (edges of rivers, for example) At corners, because shortest paths go through corners The choice of waypoints can make the AI seem smarter

36. Waypoints By Grid Place a grid over the world, and put a waypoint at every gridpoint that is open Automated method, and maybe even implicit in the environment Do an edge/world intersection test to decide which waypoints should be joined Normally only allow moves to immediate (and maybe corner) neighbors What sorts of environments is this likely to be OK for? What are its advantages? What are its problems?

37. Grid Example Note that grid points pay no attention to the geometry Method can be improved: Perturb grid to move closer to obstacles Adjust grid resolution Use different methods for inside and outside building Join with waypoints in doorways

38. Waypoints From Polygons Choose waypoints based on the floor polygons in your world Or, explicitly design polygons to be used for generating waypoints How do we go from polygons to waypoints? Hint: there are two obvious options

39. Waypoints From Polygons ! Could also add points on walls

40. Waypoints From Corners Place waypoints at every convex corner of the obstacles Actually, place the point away from the corner according to how wide the moving objects are Or, compute corners of offset polygons Connects all the corners that can see each other Paths through these waypoints will be the shortest However, some unnatural paths may result Particularly along corridors - characters will stick to walls

41. Waypoints From Corners NOTE: Not every edge is drawn Produces very dense graphs

42. Getting On and Off Typically, you do not wish to restrict the character to the waypoints or the graph edges When the character starts, find the closest waypoint and move to that first Or, find the waypoint most in the direction you think you need to go Or, try all of the potential starting waypoints and see which gives the shortest path When the character reaches the closest waypoint to its goal, jump off and go straight to the goal point Best option: Add a new, temporary waypoint at the precise start and goal point, and join it to nearby waypoints

43. Getting On and Off

44. Best-First-Search Start at the start node and search outwards Maintain two sets of nodes: Open nodes are those we have reached but don’t know best path Closed nodes that we know the best path to Keep the open nodes sorted by cost Repeat: Expand the “best” open node If it’s the goal, we’re done Move the “best” open node to the closed set Add any nodes reachable from the “best” node to the open set Unless already there or closed Update the cost for any nodes reachable from the “best” node New cost is min(old-cost, cost-through-best)

45. Best-First-Search Properties Precise properties depend on how “best” is defined But in general: Will always find the goal if it can be reached Maintains a frontier of nodes on the open list, surrounding nodes on the closed list Expands the best node on the frontier, hence expanding the frontier Eventually, frontier will expand to contain the goal node To store the best path: Keep a pointer in each node n to the previous node along the best path to n Update these as nodes are added to the open set and as nodes are expanded (whenever the cost changes) To find path to goal, trace pointers back from goal nodes

46. Expanding Frontier

47. Definitions g(n): The current known best cost for getting to a node from the start point Can be computed based on the cost of traversing each edge along the current shortest path to n h(n): The current estimate for how much more it will cost to get from a node to the goal A heuristic: The exact value is unknown but this is your best guess Some algorithms place conditions on this estimate f(n): The current best estimate for the best path through a node: f(n)=g(n)+h(n)

48. Using g(n) Only Define “best” according to f(n)=g(n), the shortest known path from the start to the node Equivalent to breadth first search Is it optimal? When the goal node is expanded, is it along the shortest path? Is it efficient? How many nodes does it explore? Many, few, …? Behavior is the same as defining a constant heuristic function: h(n)=const Why?

49. Breadth First Search

51. Using h(n) Only (Greedy Search) Define “best” according to f(n)=h(n), the best guess from the node to the goal state Behavior depends on choice of heuristic Straight line distance is a good one Have to set the cost for a node with no exit to be infinite If we expand such a node, our guess of the cost was wrong Do it when you try to expand such a node Is it optimal? When the goal node is expanded, is it along the shortest path? Is it efficient? How many nodes does it explore? Many, few, …?

52. Greedy Search (Straight-Line-Distance Heuristic)

53. A* Search Set f(n)=g(n)+h(n) Now we are expanding nodes according to best estimated total path cost Is it optimal? It depends on h(n) Is it efficient? It is the most efficient of any optimal algorithm that uses the same h(n) A* is the ubiquitous algorithm for path planning in games Much effort goes into making it fast, and making it produce pretty looking paths More articles on it than you can ever hope to read

54. A* Search (Straight-Line-Distance Heuristic)

55. Note that A* expands fewer nodes than breadth-first, but more than greedy It’s the price you pay for optimality See Game Programming Gems for implementation details. Keys are: Data structure for a node Priority queue for sorting open nodes Underlying graph structure for finding neighbors

56. Heuristics For A* to be optimal, the heuristic must underestimate the true cost Such a heuristic is admissible Also, not mentioned in Gems, the f(n) function must monotonically increase along any path out of the start node True for almost any admissible heuristic, related to triangle inequality If not true, can fix by making cost through a node max(f(parent) + edge, f(n)) Combining heuristics: If you have more than one heuristic, all of which underestimate, but which give different estimates, can combine with: h(n)=max(h1(n),h2(n),h3(n),…)

57. Inventing Heuristics Bigger estimates are always better than smaller ones They are closer to the “true” value So straight line distance is better than a small constant Important case: Motion on a grid If diagonal steps are not allowed, use Manhattan distance General strategy: Relax the constraints on the problem For example: Normal path planning says avoid obstacles Relax by assuming you can go through obstacles Result is straight line distance is a bigger estimate than

58. Non-Optimal A* Can use heuristics that are not admissible - A* will still give an answer But it won’t be optimal: May not explore a node on the optimal path because its estimated cost is too high Optimal A* will eventually explore any such node before it reaches the goal Non-admissible heuristics may be much faster Trade-off computational efficiency for path-efficiency One way to make non-admissible: Multiply underestimate by a constant factor See Gems for an example of this