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CS 385 Fall 2006Chapter 4
Heuristic Search
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Heuristics
eurisko ("I discover" in Greek)
"the study of the methods and rules of discovery and invention." Polya 1945
rules of thumb
"Intelligence for a system with limited resources consists in making wise choices of what to do next." Newell and Simon
Heuristics can
1. choose a most likely solution when exact is impossible (diagnosis)
2. guide a search along the most promising path when the state space is too large for complete search
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Examples
Tic tac toe: Pick the arc with the most winning paths
Chess: Use a board strength metric (pieces in danger, domination of center)
Soccer:Consider distance to goal, position of opposing team, surprise,...
Homework assignments?
Your life?
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Figure 4.3: Heuristically reduced state space for tic-tac-toe.
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Algorithms
1. Hill climbing (gradient search)Select the best child for further expansion.
Don't retain siblings or parent
2. Dynamic programming (Math 305)
3. Best firstStates on the open list sorted by a heuristic evaluation function.
When children are generated, all are added to open, in order
What do you use when you areLost and trying to find your way to Auburn?
Looking for your keys?
Integrating a function?
Other examples?
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Step EvaluateOpen Closed
1 [A5] [ ]
2 A5 [B4, C4, D6] [A5]
3 B4 [C4, E5, F5, D6] [B4, A5]
4 C4 [H3, G4, E5, F5, D6] [C4, B4, A5]
5 H3 [O2, P3, G4, E5, F5, D6] [H3, C4, B4, A5]
6 O2 [P3, G4, E5, F5, D6] [O2, H3, C4, B4, A5]
7 P3 solution found
best_first_search for Figure 4.4
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Heuristic search with open and closed states highlighted.
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Heuristics applied to the 8-puzzle
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Evaluation function f(n)
f(n) = g(n) + h(n)g(n): length of path from start state to n
h(n): heuristic estimate of the distance from n to goal
What does g do?If n is nearer the root, it is more likely to be on the shorted path to
the goal
This favors equally good states closer to the start
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Figure 4.9: The heuristic f applied to states in the 8-puzzle.
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Is there a single evaluation function?
No. Each step may have different reasoning.
E.g. chess, identical states may have different h(n) depending on history.
Real world: the pattern matcher picks the right heuristic to apply at each step.
Financial advisor: add certainty factors ( -1 to 1) to the rulessavings_account(adequate) ^ income(adequate)→ investment(stocks) confidence
0.8
savings_account(adequate) ^ income(adequate)→ investment(combination) confidence 0.5
savings_account(adequate) ^ income(adequate)→ investment(savings) confidence 0.1
What's funny here?
How would you use this?
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Minimax for Games
You: want to MAX your gains
Opponent want to MIN your gains
Traditional Operations Research:Strategies 1-n for each player
Payoff matrix (i,j)th position is payoff to 1 if 1 picks strategy i and 2 picks strategy j
Game: Each player shows 0 or 1 fingers.
Even sum: player 1 wins $1, odd: player 2 wins $1
Payoff matrix: 1\2 0 1
0
1 1 -1
-1 1
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Minimax for Games
What about this one?Payoff matrix: 1\2 0 1
0
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1's reasoning: The worst I can do with strategy 1 is -10
The worst I can do with strategy 2 is 0
The best of the worst is 0
Pick strategy 2
max (min(-10, 0)) → payoff 0
1 -10
2 0
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Nim
Start with 7 tokens in a pile
Each player divides a pile into an unequal number of tokens
The first player who cannot move, loses
Strategy?
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Figure 4.19: Minimax for nim(0 = win for MIN, 1= win for MAX)Bold lines indicate forced win for MAX
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Figure 4.21: Minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values.
Note, we seem to be using min/max inconsistently
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Alpha-Beta Pruning
Minimax investigates all paths to ply depth
Sometimes a path is obviously not worth following
Alpha-beta pruning removes those known to be worse than a possible outcome.
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Figure 4.26: Alpha-beta pruning applied to state space of Figure 4.15. States without numbers are not evaluated.
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Fig 4.30.
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Figure 4.25