Administrativia. Assignment 1 due tuesday 9/24/2002 BEFORE midnight. Midterm exam 10/10/2002. CS 561, Sessions 8-9 1
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1 Administrativia Assignment 1 due tuesday 9/24/2002 BEFORE midnight Midterm exam 10/10/2002 CS 561, Sessions 8-9 1
2 Last time: search strategies Uninformed: Use only information available in the problem formulation Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening Informed: Use heuristics to guide the search Best first: Greedy search queue first nodes that maximize heuristic desirability based on estimated path cost from current node to goal; A* search queue first nodes that maximize sum of path cost so far and estimated path cost to goal. Iterative improvement keep no memory of path; work on a single current state and iteratively improve its value. Hill climbing select as new current state the successor state which maximizes value. Simulated annealing refinement on hill climbing by which bad moves are permitted, but with decreasing size and frequency. Will find global extremum. CS 561, Sessions 8-9 2
3 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions 8-9 3
4 Depth-first search Node queue: initialization # state depth path cost parent # 1 A CS 561, Sessions 8-9 4
5 Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 2 B C D A CS 561, Sessions 8-9 5
6 Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 5 E F G H B C D A CS 561, Sessions 8-9 6
7 Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 5 E F G H B C D A CS 561, Sessions 8-9 7
8 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions 8-9 8
9 Breadth-first search Node queue: initialization # state depth path cost parent # 1 A CS 561, Sessions 8-9 9
10 Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 A B C D CS 561, Sessions
11 Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 A B C D E F G H CS 561, Sessions
12 Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 A B C D E F G H CS 561, Sessions
13 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions
14 Uniform-cost search Node queue: initialization # state depth path cost parent # 1 A CS 561, Sessions
15 Uniform-cost search Node queue: add successors to queue so that entire queue is sorted by path cost so far; empty queue from top # state depth path cost parent # 1 A B D C CS 561, Sessions
16 Uniform-cost search Node queue: add successors to queue so that entire queue is sorted by path cost so far; empty queue from top # state depth path cost parent # 1 A B D E F G H C CS 561, Sessions
17 Uniform-cost search Node queue: add successors to queue so that entire queue is sorted by path cost so far; empty queue from top # state depth path cost parent # 1 A B D E F G H C CS 561, Sessions
18 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions
19 Greedy search Node queue: initialization # state depth path cost total parent # cost to goal cost 1 A CS 561, Sessions
20 Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth path cost total parent # cost to goal cost 1 A B D C Sort key CS 561, Sessions
21 Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth path cost total parent # cost to goal cost 1 A B G E H F D C CS 561, Sessions
22 Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth path cost total parent # cost to goal cost 1 A B G E H F D C CS 561, Sessions
23 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions
24 A* search Node queue: initialization # state depth path cost total parent # cost to goal cost 1 A CS 561, Sessions
25 A* search Node queue: Add successors to queue, sorted by total cost. # state depth path cost total parent # cost to goal cost 1 A B D C Sort key CS 561, Sessions
26 A* search Node queue: Add successors to queue front, sorted by total cost. # state depth path cost total parent # cost to goal cost 1 A B G E H D F C CS 561, Sessions
27 A* search Node queue: Add successors to queue front, sorted by total cost. # state depth path cost total parent # cost to goal cost 1 A B G E H D F C CS 561, Sessions
28 Exercise: Search Algorithms The following figure shows a portion of a partially expanded search tree. Each arc between nodes is labeled with the cost of the corresponding operator, and the leaves are labeled with the value of the heuristic function, h. Which node (use the node s letter) will be expanded next by each of the following search algorithms? (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search (e) A* search B h=14 C D 6 h= h=15 E F G H A h=10 h=12 h=8 h=10 h=20 CS 561, Sessions
29 Last time: Simulated annealing algorithm Idea: Escape local extrema by allowing bad moves, but gradually decrease their size and frequency. - Note: goal here is to maximize E. CS 561, Sessions
30 Last time: Simulated annealing algorithm Idea: Escape local extrema by allowing bad moves, but gradually decrease their size and frequency. < - - Algorithm when goal is to minimize E. CS 561, Sessions
31 This time: Outline Game playing The minimax algorithm Resource limitations alpha-beta pruning Elements of chance CS 561, Sessions
32 What kind of games? Abstraction: To describe a game we must capture every relevant aspect of the game. Such as: Chess Tic-tac-toe Accessible environments: Such games are characterized by perfect information Search: game-playing then consists of a search through possible game positions Unpredictable opponent: introduces uncertainty thus gameplaying must deal with contingency problems CS 561, Sessions
33 Searching for the next move Complexity: many games have a huge search space Chess: b = 35, m=100 nodes = if each node takes about 1 ns to explore then each move will take about millennia to calculate. Resource (e.g., time, memory) limit: optimal solution not feasible/possible, thus must approximate 1. Pruning: makes the search more efficient by discarding portions of the search tree that cannot improve quality result. 2. Evaluation functions: heuristics to evaluate utility of a state without exhaustive search. CS 561, Sessions
34 Two-player games A game formulated as a search problem: Initial state:? Operators:? Terminal state:? Utility function:? CS 561, Sessions
35 Two-player games A game formulated as a search problem: Initial state: board position and turn Operators: definition of legal moves Terminal state: conditions for when game is over Utility function: a numeric value that describes the outcome of the game. E.g., -1, 0, 1 for loss, draw, win. (AKA payoff function) CS 561, Sessions
36 Game vs. search problem CS 561, Sessions
37 Example: Tic-Tac-Toe CS 561, Sessions
38 Type of games CS 561, Sessions
39 The minimax algorithm Perfect play for deterministic environments with perfect information Basic idea: choose move with highest minimax value = best achievable payoff against best play Algorithm: 1. Generate game tree completely 2. Determine utility of each terminal state 3. Propagate the utility values upward in the three by applying MIN and MAX operators on the nodes in the current level 4. At the root node use minimax decision to select the move with the max (of the min) utility value Steps 2 and 3 in the algorithm assume that the opponent will play perfectly. CS 561, Sessions
40 minimax = maximum of the minimum 1 st ply 2 nd ply CS 561, Sessions
41 Minimax: Recursive implementation Complete:? Optimal:? Time complexity:? Space complexity:? CS 561, Sessions
42 Minimax: Recursive implementation Complete: Yes, for finite state-space Optimal: Yes Time complexity: O(b m ) Space complexity: O(bm) (= DFS Does not keep all nodes in memory.) CS 561, Sessions
43 1. Move evaluation without complete search Complete search is too complex and impractical Evaluation function: evaluates value of state using heuristics and cuts off search New MINIMAX: CUTOFF-TEST: cutoff test to replace the termination condition (e.g., deadline, depth-limit, etc.) EVAL: evaluation function to replace utility function (e.g., number of chess pieces taken) CS 561, Sessions
44 Evaluation functions Weighted linear evaluation function: to combine n heuristics f = w 1 f 1 + w 2 f w n f n E.g, w s could be the values of pieces (1 for prawn, 3 for bishop etc.) f s could be the number of type of pieces on the board CS 561, Sessions
45 Note: exact values do not matter CS 561, Sessions
46 Minimax with cutoff: viable algorithm? Assume we have 100 seconds, evaluate 10 4 nodes/s; can evaluate 10 6 nodes/move CS 561, Sessions
47 2. α-β pruning: search cutoff Pruning: eliminating a branch of the search tree from consideration without exhaustive examination of each node α-β pruning: the basic idea is to prune portions of the search tree that cannot improve the utility value of the max or min node, by just considering the values of nodes seen so far. Does it work? Yes, in roughly cuts the branching factor from b to b resulting in double as far look-ahead than pure minimax CS 561, Sessions
48 α-β pruning: example MAX 6 MIN CS 561, Sessions
49 α-β pruning: example MAX 6 MIN CS 561, Sessions
50 α-β pruning: example MAX 6 MIN CS 561, Sessions
51 α-β pruning: example MAX 6 Selected move MIN CS 561, Sessions
52 α-β pruning: general principle Player Opponent m α If α > v then MAX will chose m so prune tree under n Player Similar for β for MIN Opponent n v CS 561, Sessions
53 Properties of α-β CS 561, Sessions
54 The α-β algorithm CS 561, Sessions
55 More on the α-β algorithm Same basic idea as minimax, but prune (cut away) branches of the tree that we know will not contain the solution. CS 561, Sessions
56 More on the α-β algorithm: start from Minimax CS 561, Sessions
57 Remember: Minimax: Recursive implementation Complete: Yes, for finite state-space Optimal: Yes Time complexity: O(b m ) Space complexity: O(bm) (= DFS Does not keep all nodes in memory.) CS 561, Sessions
58 More on the α-β algorithm Same basic idea as minimax, but prune (cut away) branches of the tree that we know will not contain the solution. Because minimax is depth-first, let s consider nodes along a given path in the tree. Then, as we go along this path, we keep track of: α : Best choice so far for MAX β : Best choice so far for MIN CS 561, Sessions
59 More on the α-β algorithm: start from Minimax Note: These are both Local variables. At the Start of the algorithm, We initialize them to α = - and β = + CS 561, Sessions
60 More on the α-β algorithm In Min-Value: MAX α = - β = + MIN Max-Value loops over these MAX Min-Value loops over these α = - β = 5 α = - β = 5 α = - β = 5 CS 561, Sessions
61 More on the α-β algorithm In Max-Value: MAX α = - β = + MIN Max-Value loops over these α = 5 β = + MAX α = - β = 5 α = - β = 5 α = - β = 5 CS 561, Sessions
62 More on the α-β algorithm In Min-Value: MAX MIN α = 5 β = + α = - β = + MAX Min-Value loops over these α = - β = 5 α = - β = 5 α = - β = 5 α = 5 β = 2 End loop and return 5 CS 561, Sessions
63 More on the α-β algorithm In Max-Value: MAX α = - β = + MIN Max-Value loops over these α = 5 β = + α = 5 β = + MAX α = - β = 5 α = - β = 5 α = - β = 5 α = 5 β = 2 End loop and return 5 CS 561, Sessions
64 Another way to understand the algorithm From: For a given node N, α is the value of N to MAX β is the value of N to MIN CS 561, Sessions
65 Example CS 561, Sessions
66 α-β algorithm: CS 561, Sessions
67 Solution NODE TYPE ALPHA BETA SCORE A Max -I +I B Min -I +I C Max -I +I D Min -I +I E Max D Min -I 10 F Max D Min -I C Max 10 +I G Min 10 +I H Max G Min C Max 10 +I 10 B Min -I 10 J Max -I 10 K Min -I 10 L Max K Min -I NODE TYPE ALPHA BETA SCORE J Max B Min -I A Max 10 +I Q Min 10 +I R Max 10 +I S Min 10 +I T Max S Min R Max 10 +I V Min 10 +I W Max V Min R Max 10 +I 10 Q Min A Max CS 561, Sessions
68 State-of-the-art for deterministic games CS 561, Sessions
69 Nondeterministic games CS 561, Sessions
70 Algorithm for nondeterministic games CS 561, Sessions
71 Remember: Minimax algorithm CS 561, Sessions
72 Nondeterministic games: the element of chance expectimax and expectimin, expected values over all possible outcomes CHANCE? ?? CS 561, Sessions
73 Nondeterministic games: the element of chance Expectimax CHANCE 4 = 0.5* * Expectimin CS 561, Sessions
74 Evaluation functions: Exact values DO matter Order-preserving transformation do not necessarily behave the same! CS 561, Sessions
75 State-of-the-art for nondeterministic games CS 561, Sessions
76 Summary CS 561, Sessions
77 Exercise: Game Playing Consider the following game tree in which the evaluation function values are shown below each leaf node. Assume that the root node corresponds to the maximizing player. Assume the search always visits children left-to-right. (a) Compute the backed-up values computed by the minimax algorithm. Show your answer by writing values at the appropriate nodes in the above tree. (b) Compute the backed-up values computed by the alpha-beta algorithm. What nodes will not be examined by the alpha-beta pruning algorithm? (c) What move should Max choose once the values have been backed-up all the way? B E F G H I J K C L M N O P Q R S T U V W X Y A D Max Min Max Min CS 561, Sessions
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