Administrativia Assignment 1 due thursday 9232004 BEFORE midnight

Administrativia • Assignment 1 due thursday 9/23/2004 BEFORE midnight • Midterm exam 10/07/2003 in class CS 460, Sessions 8 -9 1

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 460, Sessions 8 -9 2

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 3

Depth-first search Node queue: initialization # state depth path cost parent # 1 A 0 0 -- CS 460, Sessions 8 -9 4

Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 2 3 4 1 B C D A 1 1 1 0 3 19 5 0 1 1 1 -- CS 460, Sessions 8 -9 5

Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 5 6 7 8 2 3 4 1 E F G H B C D A 2 2 1 1 1 0 7 8 8 9 3 19 5 0 2 2 1 1 1 -- CS 460, Sessions 8 -9 6

Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 5 6 7 8 2 3 4 1 E F G H B C D A 2 2 1 1 1 0 7 8 8 9 3 19 5 0 2 2 1 1 1 -- CS 460, Sessions 8 -9 7

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 8

Breadth-first search Node queue: initialization # state depth path cost parent # 1 A 0 0 -- CS 460, Sessions 8 -9 9

Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 2 3 4 A B C D 0 1 1 1 0 3 19 5 -1 1 1 CS 460, Sessions 8 -9 10

Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 2 3 4 5 6 7 8 A B C D E F G H 0 1 1 1 2 2 0 3 19 5 7 8 8 9 -1 1 1 2 2 CS 460, Sessions 8 -9 11

Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 2 3 4 5 6 7 8 A B C D E F G H 0 1 1 1 2 2 0 3 19 5 7 8 8 9 -1 1 1 2 2 CS 460, Sessions 8 -9 12

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 13

Uniform-cost search Node queue: initialization # state depth path cost parent # 1 A 0 0 -- CS 460, Sessions 8 -9 14

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 2 3 4 A B D C 0 1 1 1 0 3 5 19 CS 460, Sessions 8 -9 -1 1 1 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 2 3 5 6 7 8 4 A B D E F G H C 0 1 1 2 2 1 0 3 5 7 8 8 9 19 CS 460, Sessions 8 -9 -1 1 2 2 1 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 2 3 5 6 7 8 4 A B D E F G H C 0 1 1 2 2 1 0 3 5 7 8 8 9 19 CS 460, Sessions 8 -9 -1 1 2 2 1 17

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 18

Greedy search Node queue: initialization # state depth 1 A 0 path cost 0 cost total to goal cost 20 20 CS 460, Sessions 8 -9 parent # -- 19

Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth 1 2 3 4 A B D C 0 1 1 1 path cost 0 3 5 19 cost to goal 20 14 15 18 total cost 20 17 20 37 parent # -1 1 1 Sort key CS 460, Sessions 8 -9 20

Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth 1 2 5 7 6 8 3 4 A B G E H F D C 0 1 2 2 1 1 path cost 0 3 8 7 9 8 5 19 cost to goal 20 14 8 10 10 12 15 18 total cost 20 17 16 17 19 20 20 37 CS 460, Sessions 8 -9 parent # -1 2 2 1 1 21

Greedy search Node queue: Add successors to queue, sorted by cost to goal. # state depth 1 2 5 7 6 8 3 4 A B G E H F D C 0 1 2 2 1 1 path cost 0 3 8 7 9 8 5 19 cost to goal 20 14 8 10 10 12 15 18 total cost 20 17 16 17 19 20 20 37 CS 460, Sessions 8 -9 parent # -1 2 2 1 1 22

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 23

A* search Node queue: initialization # state depth 1 A 0 path cost 0 cost total to goal cost 20 20 CS 460, Sessions 8 -9 parent # -- 24

A* search Node queue: Add successors to queue, sorted by total cost. # state depth 1 2 3 4 A B D C 0 1 1 1 path cost 0 3 5 19 cost to goal 20 14 15 18 total cost 20 17 20 37 parent # -1 1 1 Sort key CS 460, Sessions 8 -9 25

A* search Node queue: Add successors to queue front, sorted by total cost. # state depth 1 2 5 6 7 3 8 4 A B G E H D F C 0 1 2 2 2 1 path cost 0 3 8 7 9 5 8 19 cost to goal 20 14 8 10 10 15 12 18 total cost 20 17 16 17 19 20 20 37 CS 460, Sessions 8 -9 parent # -1 2 2 2 1 26

A* search Node queue: Add successors to queue front, sorted by total cost. # state depth 1 2 5 6 7 3 8 4 A B G E H D F C 0 1 2 2 2 1 path cost 0 3 8 7 9 5 8 19 cost to goal 20 14 8 10 10 15 12 18 total cost 20 17 16 17 19 20 20 37 CS 460, Sessions 8 -9 parent # -1 2 2 2 1 27

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 (a) Depth-first search (b) Breadth-first search (c) Uniform-cost search (d) Greedy search 3 B h=14 5 4 (e) A* search E F h=10 h=12 h=20 19 C 5 G h=8 CS 460, Sessions 8 -9 6 5 D h=18 h=15 H h=10 28

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 460, Sessions 8 -9 29

Last time: Simulated annealing algorithm • Idea: Escape local extrema by allowing “bad moves, ” but gradually decrease their size and frequency. < CS 460, Sessions 8 -9 Algorithm when goal is to minimize E. 30

This time: Outline • Game playing • • The minimax algorithm Resource limitations alpha-beta pruning Elements of chance CS 460, Sessions 8 -9 31

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 game-playing must deal with contingency problems CS 460, Sessions 8 -9 32

Searching for the next move • Complexity: many games have a huge search space • Chess: b = 35, m=100 nodes = 35 100 if each node takes about 1 ns to explore then each move will take about 10 50 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 460, Sessions 8 -9 33

Two-player games • A game formulated as a search problem: • • Initial state: ? Operators: ? Terminal state: ? Utility function: ? CS 460, Sessions 8 -9 34

Two-player games • A game formulated as a search problem: • • Initial state: Operators: Terminal state: Utility function: board position and turn definition of legal moves conditions for when game is over a numeric value that describes the outcome of the game. E. g. , -1, 0, 1 for loss, draw, win. (AKA payoff function) CS 460, Sessions 8 -9 35

Game vs. search problem CS 460, Sessions 8 -9 36

Example: Tic-Tac-Toe CS 460, Sessions 8 -9 37

Type of games CS 460, Sessions 8 -9 38

Type of games CS 460, Sessions 8 -9 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 460, Sessions 8 -9 40

Generate Game Tree CS 460, Sessions 8 -9 41

Generate Game Tree x x CS 460, Sessions 8 -9 42

Generate Game Tree x x o o x x o CS 460, Sessions 8 -9 x o 43

Generate Game Tree x 1 ply 1 move x o o x x o CS 460, Sessions 8 -9 x o 44

A subtree x ox o win lose x o x x ox o x o x ox x ox o ox oo x o x o ox x xo x o x ox x oo x o x o ox x x o CS 460, Sessions 8 -9 draw x ox o x o x x ox o 45

What is a good move? x ox o win lose x o x x ox o x o x ox x ox o ox oo x o x o ox x xo x o x ox x oo x o x o ox x x o CS 460, Sessions 8 -9 draw x ox o x o x x ox o 46

Minimax 3 12 8 2 4 6 14 5 2 • Minimize opponent’s chance • Maximize your chance CS 460, Sessions 8 -9 47

Minimax 3 2 2 MIN 3 12 8 2 4 6 14 5 2 • Minimize opponent’s chance • Maximize your chance CS 460, Sessions 8 -9 48

Minimax 3 MAX 3 2 2 MIN 3 12 8 2 4 6 14 5 2 • Minimize opponent’s chance • Maximize your chance CS 460, Sessions 8 -9 49

Minimax 3 MAX 3 2 2 MIN 3 12 8 2 4 6 14 5 2 • Minimize opponent’s chance • Maximize your chance CS 460, Sessions 8 -9 50

minimax = maximum of the minimum 1 st ply 2 nd ply CS 460, Sessions 8 -9 51

Minimax: Recursive implementation Complete: ? Optimal: ? Time complexity: ? Space complexity: ? CS 460, Sessions 8 -9 52

Minimax: Recursive implementation Complete: Yes, for finite state-space Time complexity: O(bm) Optimal: Yes Space complexity: O(bm) (= DFS Does not keep all nodes in memory. ) CS 460, Sessions 8 -9 53

Do We Have To Do All That Work? MAX MIN 3 12 8 CS 460, Sessions 8 -9 54

Do We Have To Do All That Work? 3 MAX 3 MIN 3 12 8 CS 460, Sessions 8 -9 55

Do We Have To Do All That Work? 3 MAX 3 2 MIN 3 12 8 2 Since 2 is smaller than 3, then there is no need for further search CS 460, Sessions 8 -9 56

Do We Have To Do All That Work? 3 MAX 3 X 2 MIN 3 12 8 14 5 2 More on this next time: α-β pruning CS 460, Sessions 8 -9 57

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 460, Sessions 8 -9 58

Evaluation functions • Weighted linear evaluation function: to combine n heuristics f = w 1 f 1 + w 2 f 2 + … + wnfn 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 460, Sessions 8 -9 59

Note: exact values do not matter CS 460, Sessions 8 -9 60

Minimax with cutoff: viable algorithm? Assume we have 100 seconds, evaluate 104 nodes/s; can evaluate 106 nodes/move CS 460, Sessions 8 -9 61

2. - pruning: search cutoff • Pruning: eliminating a branch of the search tree from consideration without exhaustive examination of each node • - 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 460, Sessions 8 -9 62

- pruning: example 6 MAX MIN 6 6 12 8 CS 460, Sessions 8 -9 63

- pruning: example 6 MAX MIN 2 6 6 12 8 2 CS 460, Sessions 8 -9 64

- pruning: example 6 MAX MIN 6 12 5 2 6 8 2 CS 460, Sessions 8 -9 5 65

- pruning: example 6 MAX Selected move MIN 6 12 5 2 6 8 2 CS 460, Sessions 8 -9 5 66

- pruning: general principle Player Opponent m If > v then MAX will chose m so prune tree under n Similar for MIN Player n Opponent CS 460, Sessions 8 -9 v 67

Properties of - CS 460, Sessions 8 -9 68

The - algorithm: CS 460, Sessions 8 -9 69

More on the - algorithm • Same basic idea as minimax, but prune (cut away) branches of the tree that we know will not contain the solution. CS 460, Sessions 8 -9 70

More on the - algorithm: start from Minimax CS 460, Sessions 8 -9 71

Remember: Minimax: Recursive implementation Complete: Yes, for finite state-space Time complexity: O(bm) Optimal: Yes Space complexity: O(bm) (= DFS Does not keep all nodes in memory. ) CS 460, Sessions 8 -9 72

More on the - 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: • : Best choice so far for MAX • : Best choice so far for MIN CS 460, Sessions 8 -9 73

More on the - algorithm: start from Minimax Note: These are both Local variables. At the Start of the algorithm, We initialize them to = - and = + CS 460, Sessions 8 -9 74

More on the - algorithm = - = + MAX MIN In Min-Value: Max-Value loops over these … Min-Value loops over these MAX 5 10 6 2 8 7 = - =5 = - 460, = 5 Sessions 8 -9 CS 75

More on the - algorithm = - = + MAX MIN In Max-Value: Max-Value loops over these =5 = + … MAX 5 10 6 2 8 7 = - =5 = - 460, = 5 Sessions 8 -9 CS 76

More on the - algorithm In Min-Value: = - = + MAX =5 = + MIN … Min-Value loops over these MAX 5 10 6 2 8 7 = - =5 =5 = - End loop and return 5 460, = 5 Sessions 8 -9 = 2 CS 77

More on the - algorithm = - = + MAX MIN In Max-Value: Max-Value loops over these =5 = + … MAX 5 10 6 2 8 7 = - =5 =5 = - End loop and return 5 460, = 5 Sessions 8 -9 = 2 CS 78

Another way to understand the algorithm • From: http: //yoda. cis. temple. edu: 8080/UGAIWWW/lectures 95/search/alpha-beta. html • For a given node N, is the value of N to MAX is the value of N to MIN CS 460, Sessions 8 -9 79

Example CS 460, Sessions 8 -9 80

- algorithm: CS 460, Sessions 8 -9 81

Solution NODE A B C D E D F D C G H G C B J K L K … TYPE Max Min Max Min Max Min ALPHA -I -I 10 -I 11 -I 10 10 9 10 10 -I -I -I 14 -I BETA +I +I 10 10 11 10 +I +I 9 9 +I 10 10 10 14 10 SCORE 10 11 10 9 9 10 14 10 NODE … J B A Q R S T S R V W V R Q A CS 460, Sessions 8 -9 TYPE ALPHA BETA Max Min Max Min Max 10 -I 10 10 5 10 10 10 4 10 10 10 +I +I 5 5 +I +I 4 4 +I 10 10 SCORE 10 10 5 5 4 4 10 10 10 82

State-of-the-art for deterministic games CS 460, Sessions 8 -9 83

Nondeterministic games CS 460, Sessions 8 -9 84

Algorithm for nondeterministic games CS 460, Sessions 8 -9 85

Remember: Minimax algorithm CS 460, Sessions 8 -9 86

Nondeterministic games: the element of chance expectimax and expectimin, expected values over all possible outcomes ? CHANCE 0. 5 ? 3 ? 8 17 CS 460, Sessions 8 -9 8 87

Nondeterministic games: the element of chance 4 = 0. 5*3 + 0. 5*5 CHANCE Expectimax 0. 5 5 3 Expectimin 5 8 17 CS 460, Sessions 8 -9 8 88

Evaluation functions: Exact values DO matter Order-preserving transformation do not necessarily behave the same! CS 460, Sessions 8 -9 89

State-of-the-art for nondeterministic games CS 460, Sessions 8 -9 90

Summary CS 460, Sessions 8 -9 91

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? Max A C B E F H G Min D I J Max K Min L 2 M 3 N 8 O 5 P 7 Q 6 CS 460, Sessions 8 -9 R 0 S 1 T 5 U 2 V 8 W 4 X Y 10 2 92