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Berkeley COMPSCI 188 - Adversarial Search

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CS 188: Artificial Intelligence Fall 2006AnnouncementsMotion as SearchDecomposition MethodsApproximate DecompositionHierarchical DecompositionSkeletonization MethodsVisibility GraphsSlide 10Probabilistic RoadmapsRoadmap ExamplePotential Field MethodsPotential FieldsAdversarial SearchGame Playing State-of-the-ArtGame PlayingDeterministic Single-Player?Deterministic Two-PlayerTic-tac-toe Game TreeMinimax ExampleMinimax SearchMinimax PropertiesResource LimitsEvaluation FunctionsEvaluation for PacmanIterative Deepening- Pruning Example- Pruning- Pruning Pseudocode- Pruning PropertiesNon-Zero-Sum GamesStochastic Single-PlayerStochastic Two-PlayerSlide 36What’s Next?CS 188: Artificial IntelligenceFall 2006Lecture 7: Adversarial Search9/19/2006Dan Klein – UC BerkeleyMany slides over the course adapted from either Stuart Russell or Andrew MooreAnnouncementsProject 1.2 is up (Single-Agent Pacman)Critical update: make sure you have the most recent version!Reminder: you are allowed to work with a partner!Change to John’s section: M 3-4pm now in 4 EvansMotion as SearchMotion planning as path-finding problemProblem: configuration space is continuous Problem: under-constrained motionProblem: configuration space can be complexWhy are there two paths from 1 to 2?Decomposition MethodsBreak c-space into discrete regionsSolve as a discrete problemApproximate DecompositionBreak c-space into a gridSearch (A*, etc)What can go wrong?If no path found, can subdivide and repeatProblems?Still scales poorlyIncomplete*Wiggly pathsGSHierarchical DecompositionBut:Not optimalNot completeStill hopeless above a small number of dimensions Actually used in practical systemsSkeletonization MethodsDecomposition methods turn configuration space into a gridSkeletonization methods turn it into a set of points, with preset linear paths between themVisibility GraphsShortest paths:No obstacles: straight lineOtherwise: will go from vertex to vertexFairly obvious, but somewhat awkward to proveVisibility methods:All free vertex-to-vertex lines (visibility graph)Search using, e.g. A*Can be done in O(n3) easily, O(n2log(n)) less easilyProblems?Bang, screech!Not robust to control errorsWrong kind of optimality?qstartqgoalqstartVoronoi DecompositionAlgorithm:Compute the Voronoi diagram of the configuration spaceCompute shortest path (line) from start to closest point on Voronoi diagramCompute shortest path (line) from goal to closest point on Voronoi diagram.Compute shortest path from start to goal along Voronoi diagramProblems:Hard over 2D, hard with complex obstaclesCan do weird things:Probabilistic RoadmapsIdea: just pick random points as nodes in a visibility graphThis gives probabilistic roadmapsVery successful in practiceLets you add points where you need themIf insufficient points, incomplete, or weird pathsRoadmap ExamplePotential Field MethodsSo far: implicit preference for short pathsRational agent should balance distance with risk!Idea: introduce cost for being close to an obstacleCan do this with discrete methods (how?)Usually most natural with continuous methodsPotential FieldsCost for:Being far from goalBeing near an obstacleGo downhillWhat could go wrong?Adversarial Search[DEMO 1]Game Playing State-of-the-ArtCheckers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used an endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 443,748,401,247 positions.Chess: Deep Blue defeated human world champion Gary Kasparov in a six-game match in 1997. Deep Blue examined 200 million positions per second, used very sophisticated evaluation and undisclosed methods for extending some lines of search up to 40 ply.Othello: human champions refuse to compete against computers, which are too good.Go: human champions refuse to compete against computers, which are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.Pacman: unknownGame PlayingAxes:Deterministic or stochastic?One, two or more players?Perfect information (can you see the state)?Want algorithms for calculating a strategy (policy) which recommends a move in each stateDeterministic Single-Player?Deterministic, single player, perfect information:Know the rulesKnow what actions doKnow when you winE.g. Freecell, 8-Puzzle, Rubik’s cube… it’s just search!Slight reinterpretation:Each node stores the best outcome it can reachThis is the maximal outcome of its childrenNote that we don’t store path sums as beforeAfter search, can pick move that leads to best nodewin loseloseDeterministic Two-PlayerE.g. tic-tac-toe, chess, checkersMinimax searchA state-space search treePlayers alternateEach layer, or ply, consists of a round of movesChoose move to position with highest minimax value = best achievable utility against best playZero-sum gamesOne player maximizes resultThe other minimizes result8 2 5 6maxminTic-tac-toe Game TreeMinimax ExampleMinimax SearchMinimax PropertiesOptimal against a perfect player. Otherwise?Time complexity?O(bm)Space complexity?O(bm)For chess, b  35, m  100Exact solution is completely infeasibleBut, do we need to explore the whole tree?10 10 9 100maxmin[DEMO2: minVsExp]Resource LimitsCannot search to leavesLimited searchInstead, search a limited depth of the treeReplace terminal utilities with an eval function for non-terminal positionsGuarantee of optimal play is goneMore plies makes a BIG difference[DEMO 3: limitedDepth]Example:Suppose we have 100 seconds, can explore 10K nodes / secSo can check 1M nodes per move- reaches about depth 8 – decent chess program? ? ? ?-1 -2 4 94min minmax-2 4Evaluation FunctionsFunction which scores non-terminalsIdeal function: returns the utility of the positionIn practice: typically weighted linear sum of features:e.g. f1(s) = (num white queens – num black queens), etc.Evaluation for Pacman[DEMO4: evalFunction]Iterative DeepeningIterative deepening uses DFS as a subroutine:1. Do a DFS which only searches for paths of length 1 or less. (DFS gives up on any path of length 2)2. If “1” failed, do a DFS which


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Berkeley COMPSCI 188 - Adversarial Search

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