(3-Ply/turns). >=2. Alpha-Beta Pruning . (3-Ply/turns). Alpha-Beta Pruning . (3-Ply/turns). >=3. [3]. Alpha-Beta Pruning . (3-Ply/turns). <=. 3. [3]. [3]. >= 3. <=. 3. [3]. >=5. >= 3. ID: 547103
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Alpha-Beta Pruning
(3-Ply/turns)
Slide2>=2
Alpha-Beta Pruning
(3-Ply/turns)
Slide3Alpha-Beta Pruning
(3-Ply/turns)
>=3
Slide4[3]
Alpha-Beta Pruning
(3-Ply/turns)
Slide5<=
3
[3]
Slide6[3]
>= 3
<=
3
Slide7[3]
>=5
>= 3
<=
3
Slide8[3]
>=5
>= 3
<=
3
Slide9[3]
[3]
>=5
>= 3
Slide10[3]
[3]
[0]
>=5
>= 3
Slide11[3]
[3]
[0]
<=
0
>=5
>= 3
Slide12[3]
[3]
[0]
>=5
>= 3
<=
0
Slide13[3]
[3]
[0]
>=5
>= 3
<=
0
>=
2
Slide14[3]
[3]
[0]
>=5
>= 3
<=
0
[
2]
Slide15[3]
[3]
[0]
>=5
>= 3
<=
0
<=2
[
2]
Slide16[3]
[3]
[0]
>=5
>= 3
<=
0
<=2
[
2]
Slide17[3]
[3]
[0]
>=5
<=
0
<=2
[3]
[
2]
Slide18Node order
Ordering of nodes affects the number of nodes explored
Slide19How much does pruning help?
If perfect ordering: O(
b
d
) becomes
O(
b
d
/2
)
half as much time!
Typical ordering:
O(
b
3d/4
)
Chess
, b ≈ 35, d ≈100 for reasonable
games
O(35
75
) still unfeasible!
Slide20How much does pruning help?
If perfect ordering: O(
b
d
) becomes
O(
b
d
/2
)
half as much time!
Typical ordering:
O(
b
3d/4
)
Chess
, b ≈ 35, d ≈100 for reasonable
games
O(35
75
) still unfeasible!
Other ideas??
Slide21Evaluation Functions
Stop search early (at a depth limit k)
Slide22Evaluation Functions
Stop search early (at a depth limit k)
Now have to propagate utility values from non-leaf nodes
Slide23Evaluation Function
eval
(state) – estimates the utility of a state
Should:
Be efficient to compute
Order nodes at depth k in (roughly) same order as corresponding leaf nodes
Slide24>
>
Slide25Evaluation Function
eval(state) – estimates the utility of a stateShould:Be efficient to computeOrder nodes at depth k in (roughly) same order as corresponding leaf nodes
Ideas?
X
O
X
O
Slide26Evaluation Function for T3
Assuming MAX is playing as X
Difference in available wins
eval
(state)
If state is win for MAX, return
∞
Else if state is loss for MAX, return -∞
Else return:
#rows + # cols + # diagonals available to MAX – #
rows + # cols + # diagonals available to
MIN
Slide27Difference in available wins for T3
X
O
Slide28Difference in available wins for T3
X
O
=
6
–
Slide29Difference in available wins for T3
X
O
=
6
– 4
= 2
Slide30Difference in available wins for T3
X
O
X
O
X
O
=
6
– 4
= 2
=
4
– 3 =
1
Slide31MIN
MAX
All possible moves not shown
Difference in available wins for T
3
Slide32Chess evaluation functions
Ideas?
Slide33Chess EVAL
Assume each piece has the following valuespawn = 1;knight = 3;bishop = 3;rook = 5;queen = 9;EVAL(state) = sum of the value of white pieces on board – sum of the value of black pieces
= 31
-
36
=
-5
Slide34Chess EVAL
Assume each piece has the following valuespawn = 1;knight = 3;bishop = 3;rook = 5;queen = 9;EVAL(state) = sum of the value of white pieces on board – sum of the value of black pieces
= 31
- 36 = -5
Any problems with this?
Slide35Chess EVAL
Ignores actual positions!
Most valuable
Less valuable
Least valuable
Slide36Chess EVAL
Ignores actual positions!Actual heuristic functions are oftena weighted combination of features
Slide37Chess EVAL
A feature can be any numerical information about the boardas general as the number of pawnsto specific board configurationsDeep Blue: 8000 features!
number of pawns
number of attacked knights
1 if king has knighted, 0 otherwise
Slide38Chess EVAL
number of pawns
number of attacked knights
1 if king has knighted, 0 otherwise
How can we determine the weights (especially if we have 8000 of them!)?
Slide39Chess EVAL
number of pawns
number of attacked knights
1 if king has knighted, 0 otherwise
Machine
learning/Genetic algorithms!
play/examine lots of games
adjust the weights so that the
EVAL(s
) correlates with the actual utility of the states
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