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Artificial intelligence
● Heuristic search● First-order logic● Constraint satisfaction problems
Problems without simple algorithmic solutions
Benoît Lemaire
Heuristic search
● Intuitive presentation● Search spaces● Two-player games
What is a heuristic
● A criteria to select between several alternatives the most promising to reach a goal
● Usually it works, but it might not!– If you see clouds, take your umbrella
(you might carry it all day long for nothing, but it usually works)
? ? ?
The 8-queen problem
● Which position is the most promising to solve the problem?
a? b? c?
The 8-queen problem
● Heuristic 1: maximize the number of free cells
The 8-queen problem
● f(a)=8
The 8-queen problem
● f(a)=8● f(b)=9
The 8-queen problem
● f(a)=8● f(b)=9● f(c)=10 : C seems a better solution!
a? b? c?
The 8-queen problem
● Heuristic 1: maximize the number of free cells
● Is there a better heuristic?
The 8-queen problem● Heuristic 1 takes into account the total
number of free cells, but not how they are distributed
>
f(S1)=8 f(S2)=8
a? b? c?
The 8-queen problem
● Heuristic 1: maximize the number of free cells
● Heuristic 2: maximize the number of free cells of the line with the fewest free cells
The 8-queen problem
● g(a)=1
The 8-queen problem
● g(a)=1● g(b)=1
The 8-queen problem
● g(a)=1● g(b)=1● g(c)=2 : C seems a better solution!
The taquin game● In the 8Q problem, the final state
was unknown● Here the final state is known
Initial state Final state
2 3
1
7
8 4
56
1 3
8
7
2
4
56
● Suppose an exhaustive search is intractable● The distance to the goal has to be estimated
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
● Heuristic 1: number of pieces at a wrong position
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
● Heuristic 1: number of pieces at a wrong position● “a” seems to be the best move
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
f(a)=2 f(b)=3 f(c)=4
● Heuristic 1: number of pieces at a wrong position● Is there a better heuristic?
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
● Heuristic 1: number of pieces at a wrong position● Heuristic 2: sum of Manhattan distances between
each piece and its correct position
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
Manhattan distance
Euclidian distance:
● Heuristic 2: sum of Manhattan distances between each piece and its correct position
● “a” seems the best move
Final state
5
1 3
8
7
2
4
56
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
a b c
f(a)=2 f(b)=4 f(c)=4
Search spaces
● We have to find a solution in a space which can be viewed as a tree of states
567
5
2 3
67
1 4
8
5
2 3
67
1 48
5
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
2 3
67
1 48
5
2 3
67
1 4
8
5
2 3
67
1 48
2 3
67
1 48 ......
Search spaces
● We have to find a solution in a space which can be viewed as a tree of states
root node
Nodes (vertices)
A
children nodes of A
parent node of A
goals
branch (edge)
● This search space can be huge– Checkers: 1040 states
– Chess: 10120 states
● It should be explored in an efficient way● Different techniques are available
Heuristic search methods
● Systematic search– Depth-first search
– Breadth-first search
● Heuristic search– Hill climbing
– Simulated annealing
● Best-first search
Depth-first search
● Backtracking if all subnodes are explored● Implemented with a Last In First Out strategy
(stack)● May not end in case of infinite depth
from Zhe Liu (1998)
Breadth-first search
● Implemented with a First In First Out strategy (queue)
● Find the shortest path to a solution
Hill climbing
● Heuristics are necessary to prune the search space
● Hill climbing trusts the heuristic : no backtracking!
● Very efficient: no memorization of the path
Heuristic search: hill climbing
● Similar to climbing a hill in the fog● Works well is there no local maximum
Hill climbing
Local maximum
Hill climbing● Works well with a good heuristic (no
local maximum)● Example
– Initial state: 000101
– Final state: 111111
– Legal moves: 2 adjacent numbers can be inverted (01, 10)
– 000101110101110110...
Hill climbing
● Heuristic 1: maximize the number of 1
000101 (2)
110101 (4) 011101 (4) 001001 (2) 000011 (2) 000110 (2)
Hill climbing
● Heuristic 1: maximize the number of 1
000101 (2)
110101 (4) 011101 (4) 001001 (2) 000011 (2) 000110 (2)
101101 (4) 010001 (2) 011011 (4) 011110 (4)000101 (2)
Local maximum!
Hill climbing
● Heuristic 2: minimize the maximum distance between 0
000101 (4)
110101 (2) 011101 (4) 001001 (4) 000011 (3) 000110 (5)
Hill climbing
● Heuristic 2: minimize the maximum distance between 0
000101 (4)
110101 (2) 011101 (4) 001001 (4) 000011 (3) 000110 (5)
000101 (4) 111001 (1) 110011 (1) 110110 (3)101101 (3)
Hill climbing
● Heuristic 2: minimize the maximum distance between 0
000101 (4)
110101 (2) 011101 (4) 001001 (4) 000011 (3) 000110 (5)
000101 (4) 111001 (1) 110011 (1) 110110 (3)101101 (3)
111111 (0)
Simulated annealing
● Analogy with annealing in metallurgy● Use a temperature parameter to control
the choice of the next state– Temperature is slightly decreasing
– High temperature → next state can be far from the current state
– Low temperature → next state is close to the current state
Best-first search
● Select the best (according to the heuristic) move from all nodes already processed
● It is like climbing a hill with several people communicating with mobile phones. The one with the best chance to reach the top is doing the next move.
What is the cost of a solution passing by node n?
● Suppose k(ni,n
j) = minimal cost to go from n
i to n
j
● Let g*(n) = k(no,n)
● Let h*(n) = k(n,Goal)● f*(n) = g*(n) + h*(n)
nO
n
GoalIf we could compute f*(n), we would know which path to follow... Goal
Goal
...but we can only rely on estimates
● g(n): estimate of g*(n)– Number of branches from the root to n
● h(n): estimate of h*(n)– Heuristic function
● f(n) = g(n) + h(n)
5
2 3
67
1 4
8
5
2 3
67
1 4
8
5
2 36
7
1 4
8
5
2 36
7
1 4
8
1+5=6
Heuristic 1:h(n)=number of pieces
at wrong position
1+3=4 1+5=6
5
1 2
67
483
Final state
5
2 3
67
1 4
8
5
2 3
67
1 4
8
5
2 36
7
1 4
8
5
2 36
7
1 4
8
1+5=6
Heuristic 1:h(n)=number of pieces
at wrong position
1+3=4 1+5=6
5
2 3
67
1 4
8
5
2 3
67
1 4
8
5
2 3
67
1 48
2+3=5 2+3=5 2+4=6
5
1 2
67
483
Final state
5
2 3
67
1 4
8
5
2 36
7
1 4
8
5
2 36
7
1 4
8
1+5=6 1+3=4 1+5=6
5
2 3
67
1 4
8
5
2 3
67
1 4
8
5
2 3
67
1 48
2+3=5 2+3=5 2+4=6
5
1 2
67
483
Final state
5
2 3
67 1 4
8
52
3
67
1 4
8
3+3=6 3+4=7
5
2 3
67
1 4
8
5
2 3
67
1 4
8
5
2 3
67
1 48
2+3=5 2+3=5 2+4=6
backtracking!
A* algorithm
● What if ∀n h(n) = 0 ?
A* algorithm
● What if ∀n h(n) = 0 ?– The algorithm is a breadth-first search (BFS)
● What if ∀n g(n) = 0 ?
A* algorithm
● What if ∀n h(n) = 0 ?– The algorithm is a breadth-first search (BFS)
● What if ∀n g(n) = 0 ?– The algorithm may be fooled by a wrong heuristic
that leads to a path that never gets to a goal
● If h(n) ≤ h*(n), the algorithm is admissible (it finds the shortest path to the goal). It is an A* algorithm.
● In A*, h has to be always optimistic!
● Heuristic 1: number of pieces at a wrong position● Heuristic 2: sum of Manhattan distances between
each piece and its correct position
Is H1 leading to an admissible algorithm?Is H2 leading to an admissible algorithm?
Which one is better?
A* algorithm
● h(n) ≤ h*(n) (the algorithm is admissible: it finds the shortest path to the goal)
● h(n)=0 (BFS) is admissible but it is too long● h(n)=h*(n) is perfect, but impossible to find● Find h(n) as big as possible but less than h*(n)
Exercise● Go from D to A● Heuristic= minimize Manhattan distance
between current location and goal● Generate the search space
Two-player games
● Example– Two players, one group of 7 objects initially
– Each player has to separate any group into unequal parts. The one who cannot play loses the game.
Player 1 Player 2 Player 1
Player 2 looses!
A tree of possible states
● Odd levels → Player1● Even levels → Player2● Each leaf node is marked:
– +1 if Player1 wins
– -1 if Player1 looses
– 0 (draw)
7
6,1 5,2 4,3
5,1,1 4,2,1 3,2,2 3,3,1
4,1,1,1 3,2,1,1 2,2,2,1
3,1,1,1,1
2,1,1,1,1,1
3,1,1,1,1
P1
P1
P1
P2
P2
P2 +1
-1
+1
A tree of possible states
● Values are propagated up the tree
7
6,1 5,2 4,3
5,1,1 4,2,1 3,2,2 3,3,1
4,1,1,1 3,2,1,1 2,2,2,1
3,1,1,1,1
2,1,1,1,1,1
3,1,1,1,1
P1
P1
P1
P2
P2
P2 +1
-1
+1
A tree of possible states
● Values are propagated up the tree
● Player2 can always win the game!
7
6,1 5,2 4,3
5,1,1 4,2,1 3,2,2 3,3,1
4,1,1,1 3,2,1,1 2,2,2,1
3,1,1,1,1
2,1,1,1,1,1
3,1,1,1,1
P1
P1
P1
P2
P2
P2 +1
-1
+1Player1 can choose between a winning state and a loosing state. Obviously, (5,1,1) is a winning state
The MiniMax algorithm
● Problem: the tree cannot be fully developed● A heuristic function is used at a given level
(at least 12 plies for Deep Blue)● The value of each node is the maximum of
its child nodes, for Player1 (MAX)● The value of each node is the minimum of
its child nodes, for Player2 (MIN)
Example: the tic-tac-toe game
● Heuristic– f(State) = +∞ if Max wins
– f(State) = -∞ if Max looses
– f(State) =
O OX
X
Example: the tic-tac-toe game
● Heuristic– f(State) = +∞ if Max wins
– f(State) = -∞ if Max looses
– f(State) = (number of rows open for Max) - (number of rows open for Min)
O OX
X
O
XX X
X X X X XXO
O OO O X
O
O
XX X
X X X X XXO
O OO O X
O
6-5=1 5-5=0 6-5=1 5-5=0 4-5=-1 6-4=2 5-4=1
O
XX X
X X X X XXO
O OO O X
O
6-5=1 5-5=0 6-5=1 5-5=0 4-5=-1 6-4=2 5-4=1
MAX
MIN
-1 1 -2
1
Best move according tothe heuristic
-1
OX
MIN
MAX
OX
OX
OX
OX
XX
X X1 0 1 0
Exercise
pruning
...
10
14
MAX
MIN
MAX
pruning
...
10
14
MAX
MIN
MAX pruning
pruning
10
5
MAX
MIN
MAX
pruningMIN
Algorithm● The algorithm maintains two values, and ● is the minimum score that the maximizing
player is assured of● is the maximum score that the minimizing
player is assured of respectively.● Initially =-∞, =+∞
● When >, the current position cannot be the result of best play by both players and hence need not be explored further
Exercise
● What is the best move (A, B or C)?● Which branches could be cut by an pruning?