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德明科技大學資訊科技系 1 Problem & Solving Unit 5. 德明科技大學資訊科技系 2...
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Transcript of 德明科技大學資訊科技系 1 Problem & Solving Unit 5. 德明科技大學資訊科技系 2...
![Page 1: 德明科技大學資訊科技系 1 Problem & Solving Unit 5. 德明科技大學資訊科技系 2 You Will Know Problem and solving Production system Search Game Theory.](https://reader033.fdocuments.net/reader033/viewer/2022061503/56649ed35503460f94be341c/html5/thumbnails/1.jpg)
德明科技大學資訊科技系 1
Problem & Solving
Unit 5
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德明科技大學資訊科技系 2
You Will Know
• Problem and solving
• Production system
• Search
• Game Theory
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德明科技大學資訊科技系 3
Problem and Solving
• In general– Problem : what goals do you want to reach?– Solving: how do you reach the goals?
• We want to build problem-solving agents for our AI problems
• Problem formulation– the process of deciding what actions and states
to consider, given a goal
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德明科技大學資訊科技系 4
Problem Formulation
• Four components in a problem– Initial state the agent starts in it.– Actions: possible actions available to the agent– Goal test: determine whether a given state is the
goal– Path cost: assign a numeric cost to each path
• Optimal solution– The lowest path cost among all solutions
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德明科技大學資訊科技系 5
Example: Vacuum CleanerGoal:
clean the housein state 7 or 8
Initial:
Actions:
Solution path:
in state 5
{left, right, suck}
{right, suck}
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德明科技大學資訊科技系 6
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德明科技大學資訊科技系 7
Example: Vacuum Cleaner• Problem formulation
– States: possible world states
– Initial state: any state can be designated as the initial state
– Successor function (or actions): (left, right, suck)
– Goal test: This check whether all the square are clean
– Path cost: Each step costs 1, so the path cost is the number of steps in the path
822 2
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德明科技大學資訊科技系 8
Example: 8-puzzleGoal:
goal state
Initial:
Actions:
Path cost:The start state
black move {left, right, up, down}
unit cost
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德明科技大學資訊科技系 9
Example: 8-puzzle• Problem formulation
– States: the location of each of the eight tiles and the black in one of the nine square
– Initial state: any state can be designated as the initial state
– Successor function (or actions): (left, right, up, down)
– Goal test: This check whether the state matches the goal
– Path cost: Each step costs 1, so the path cost is the number of steps in the path
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德明科技大學資訊科技系 10
Example: DiagnosisGoal:
disease state
Initial:
Actions:
Path cost:symptom
test, diagnose
test cost
patient symptoms(cough, fever)
disease state(pneumonia)
diagnose
test (x-ray)
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德明科技大學資訊科技系 11
N-Queen Problem Goal:
No two queens attacking
State:
Actions:
Path cost:Placement of n queens
Place a queen on square (i, j)
zero cost
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德明科技大學資訊科技系 12
N-Queen Problem
• Problem formulation (please fill it)– States:– Initial state:– Successor function (or actions):– Goal test:– Path cost:
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德明科技大學資訊科技系 13
Production System
• One type of problem-solving systems
• Three components in production system– Global database– Set of rules– Control system
• Data driven and data adaptive
• Take an example: chess
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德明科技大學資訊科技系 14
Global Database
• Information of the problem– Environment description– Facts– Parameters of rule inferences
• In practice, global database means designing data structure for environment
• What’s the global database for chess?
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德明科技大學資訊科技系 15
Set of Rules
• Rule– Cover the ability and possibility of the agent– Rule inference can produce new facts
• The general form of a ruleConditions => Actions
If Condition then Action A
else Action B
Facts => Results
What’s the rule for chess?
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德明科技大學資訊科技系 16
Control System
• Inference engine for rules– Select proper rules for observing facts– Enable rules– Disable rules– Goal: looking for an optimal path for solutions
What’s the control system for chess?
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德明科技大學資訊科技系 17
Search
• Why do we need search– Many many possible solutions for a problem– An optimal solution: minimum cost
• It is often impossible to test all possible solutions in difficult problems– Question: Who is the tallest in the world?– We need smarter strategies for search– How smart? What’s smart?
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德明科技大學資訊科技系 18
Search
• Complexity– The cost of the search approach– In most cases, we use “complexity” to measure
the cost of an algorithm– O(n), O(n2), O(n log n), …– Two kinds of complexity
• Time complexity
• Space complexity
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德明科技大學資訊科技系 19
Complexity Example
• Given a data sequence with size 1000000 items, does it contain the number “1205”?– sequential search:
• It may be success at the first time if you are so lucky
• O(n): 1000000 times at the worst case
– binary search:• It may be success at the first time if you are so lucky
• O(n log n): 1000 times at the worst case
• But the data sequence need to be sorted
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德明科技大學資訊科技系 20
Complexity
• Consider the example at the previous page– If the data size is 100 items?– If the data size is 10000 items?– If the data size is 100000000 items?
• Scalability issue– The performance, either time or space, of an
algorithm depends on the size of the data
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德明科技大學資訊科技系 21
Complexity• The search problem is
– P problem• There exists a polynomial-time algorithm to solve
the search problem• This is not very difficult
– NP-Complete• No polynomial-time solution can be found• That means that we can solve the problem if the data
size is not small• We need to find a non-optimal solution• e.g., 0-1 knapsack
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德明科技大學資訊科技系 22
Search Space
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德明科技大學資訊科技系 23
Quality of A Search Algorithm• Complete
– Is it a correct solution?
• Optimality– Is it high quality?
• Time complexity– How long to find a solution?
• Space complexity– How much memory is required to find a solution?
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德明科技大學資訊科技系 24
4-Queen Problem
X
X X
(12, 24, 31, 43)
(13, 21, 34, 42)
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德明科技大學資訊科技系 25
4-Queen Problem• Action
– Rij: put a chess at (i, j), where
– total: 16 different actions– Tree structure: 44=256 nodes
41 ,41 ji
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德明科技大學資訊科技系 26
Depth First Search (DFS)
• DFS Algorithm1. 以某一頂點 v 為起點2. 在所有與 v 相鄰且未被走過的其他頂點中,選擇一個頂點 w 為新的起
點3. 從 w 開始再度遞迴的執行深度優先搜尋法4. 當所有相鄰頂點都拜訪過,回溯到上一層繼續深度優先搜尋其他位被
拜訪過的頂點procedure DFS(v) {
visited(v) = 1; // 紀錄 v 已拜訪過
for each vertex w adjacent to v
//loop for 所有 v 的相鄰頂點 w
if visited(w)==0
then DFS(v);
}
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德明科技大學資訊科技系 27
Depth First Search (DFS)
)(
...10
d
d
bO
bbb
Complete
Optimal
Time
Space
The solution will be found if it exists
Find the shallowest goal
)(
...10
d
d
bO
bbb
b: degree, or # of childrend: depth of search
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德明科技大學資訊科技系 28
Breadth First Search (BFS)
• BFS Algorithm1. 以某一頂點 v 為起點
2. 依序拜訪 v 的所有相鄰頂點
3. 當一層的相鄰頂點拜訪完畢,才繼續拜訪下一層的頂點
BFS 是 level-by-level 的,所以使用 queue 的概念; DFS 是使用遞迴,所以是 stack
DFS 與 WFS 的結果都跟使用之資料結構有關,採用資料結構不同,即使起點相同,結果也不一定會一樣
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德明科技大學資訊科技系 29
Breadth First Search (BFS)Complete
Optimal
Time
Space
May not find the best solution if ∞ depth
May not optimal if ∞ depth
)( mbO
)(mbO
b: degree, or # of childrenm: max depth
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德明科技大學資訊科技系 30
DFS vs. BFS
DFSv1, v2, v4, v8, v5, v6, v3, v7
v1
v2
v3
v4
v5
v6
v7
v8
BFSv1, v2, v3, v4, v5, v6, v7, v8
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德明科技大學資訊科技系 31
Exercise
• Start from A– Travel by DFS– Travel by BFS
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德明科技大學資訊科技系 32
Which Strategy to Use
• Depends on your problem.
• If there are infinite paths– DFS is bad
• If goal is at know depth– DFS is good almost
• If there exist large branching factor– BFS is bad
• Consider data structure in your program.
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德明科技大學資訊科技系 33
Improvement• Either in DFS or BFS
– These are basic approaches for graph traversal– Cost is very high if the graph is huge– It is necessary to cost down
• Set constraints on search– Take some limited criteria, not optimal solution– Example: looking for a pair of shoes at shopping
• Optimal: looking for the cheapest one• Constraint: looking for the one less than NT$1000
– Depth-limited Depth-first Search
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德明科技大學資訊科技系 34
Depth-limited Depth-first Searchconstraint: depth d 4≦
Complete: No, unless at depth d≦Optimal: No
Time: O(bd)Space: O(bd)
5
6
7 8
9
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德明科技大學資訊科技系 35
Game Theory• Game
– A set of game rules– Players’ decision to win the game
• 單人遊戲的問題– 找出最佳的結果– 可用一般的搜索技術進行求解– 8-puzzle; n-queen; 接龍
• 雙人遊戲– 對方同時也想取得最佳的結果– 可用與或樹 ( 或圖 ) 來描述,因而搜索就是尋找最
佳解樹 ( 或圖 ) 的問題– Chess; Chinese chess; Go; Bridge
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德明科技大學資訊科技系 36
X
X
X
O OOO O OO O
X
X
X
XX
O O
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O O
O O O
O X
O
O
XO
OX
O
O
XO
O
XO
O
XO X X X
X X X
O
O
O OO O
O O O
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德明科技大學資訊科技系 37
Playing
• You will play a best step – Also, your opponent will do best– Past: you and your opponent knew all– Future: you and your opponent can guess
• Find a path from start to a goal– Search all possible paths: infeasible– Maximize your step (score), and minimize your op
ponent’s step (score).– MiniMax Algorithm
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德明科技大學資訊科技系 38
Score Function
• We need to define a score function to measure the “value” of a step p– f(p)=(number of possible O lines - number of p
ossible X lines)– Example: (you are “O”)
O
f = 8-4 = 4
O
X
f = 6-4 = 2
O
X
f = 5-4 = 1
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德明科技大學資訊科技系 39
X
X
X
O OOO O OO O
X
X
X
XX
O
O
O OO O
O O Of=8-5=3 f=3 f=3 f=3f=4f=2 f=2 f=2 f=2
f=2f=2f=2 f=2f=1 f=1 f=1 f=1
O O
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O O
O O Of=4 f=4 f=3 f=3 f=3f=4 f=4
O X
O
O
XO
OX
O
O
XO
O
XO
O
XO X X X
X X Xf=2 f=2 f=1 f=1 f=2 f=1
MAX
MIN
MAX
MIN
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德明科技大學資訊科技系 40
MinMax
• What’s min?
• What’s max?
• Top-down or bottom-up
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德明科技大學資訊科技系 41
X
X
X
O OOO O OO O
X
X
X
XX
O
O
O OO O
O O Of=8-5=3 f=3 f=3 f=3f=4f=2 f=2 f=2 f=2
f=2f=2f=2 f=2f=1 f=1 f=1 f=1
O O
O
X
O
X
O
X
O
X
O
X
O
X
O
X
O O
O O Of=3 f=2 f=2 f=2 f=2f=3 f=3
O X
O
O
XO
OX
O
O
XO
O
XO
O
XO X X X
X X Xf=2 f=1 f=2 f=1 f=1 f=1
MAX
MIN
MAX
MIN
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德明科技大學資訊科技系 42
極小極大搜索程序
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德明科技大學資訊科技系 43
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德明科技大學資訊科技系 44
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德明科技大學資訊科技系 45
Properties of MiniMax
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德明科技大學資訊科技系 46
Resource Limits
Solutions:
1. cutoff test, eg: depth limit2. evaluation function
or estimated desirability of position
• Example: Chess( 西洋棋 )– a reasonable game: O(35100) nodes– suppose we have 10 seconds/move– If explore 109 nodes/second1010 nodes per movenot enough
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德明科技大學資訊科技系 47
Cutoff Search• Depth-limit in Minimax search
– e.g. depth < 6
– means the player can think less than 6 steps
• Does it work in practice?
– m=4 => the computer can think less than 4 steps
• In chess– m=4, human novice
– m=8, typical pc, human master
– m=12, deep blue, kasparov
4 ,35 106 mbbm
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德明科技大學資訊科技系 48
α-β Pruning
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德明科技大學資訊科技系 49
α-β Pruning: Example
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德明科技大學資訊科技系 50
α-β Pruning: Example
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德明科技大學資訊科技系 51
α-β Pruning: Example
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德明科技大學資訊科技系 52
Properties of α-β
• Pruning doesn’t affect the final result
• Good move ordering can improve effectiveness of pruning
• With “perfect ordering”– Time complexity: O(bm/2) – Double depth for search– Easily reach depth 8 => human master
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德明科技大學資訊科技系 53
Why is it called α-β
• α: the best value to Max– If v is worse than α, Max will avoid it– i.e. prune that branch
• β: the best value to Min– or the worst value to MAX
• α for Max, and β for Min
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德明科技大學資訊科技系 54
α-β Algorithm
Basic MiniMax + tracking of α-β + pruning
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德明科技大學資訊科技系 55
Now
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德明科技大學資訊科技系 56
Non-determined Game• You cannot exactly know which step you can pl
ay– Backgammon: dice rolls determine legal moves– Coin-flipping– 暗棋
expected values