heuristic search -...
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Heuristic Search
Chapter 3
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Outline
• Generate-and-test
• Hill climbing
• Best-first search
• Problem reduction
• Constraint satisfaction
• Means-ends analysis
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Generate-and-Test
Algorithm1. Generate a possible solution.
2. Test to see if this is actually a solution.
3. Quit if a solution has been found. Otherwise, return to step 1.
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Generate-and-Test
• Acceptable for simple problems.
• Inefficient for problems with large space.
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Generate-and-Test
• Exhaustive generate-and-test.
• Heuristic generate-and-test: not consider paths that seem unlikely to lead to a solution.
• Plan generate-test: − Create a list of candidates. − Apply generate-and-test to that list.
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Generate-and-Test
Example: coloured blocks“Arrange four 6-sided cubes in a row, with each side of each cube painted one of four colours, such that on all four sides of the row one block face of each colour is showing.”
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Generate-and-Test
Example: coloured blocks
Heuristic: if there are more red faces than other colours then, when placing a block with several red faces, use few of them as possible as outside faces.
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Hill Climbing
• Searching for a goal state = Climbing to the top of a hill
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Hill Climbing
• Generate-and-test + direction to move.
• Heuristic function to estimate how close a given state is to a goal state.
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Simple Hill Climbing
Algorithm1. Evaluate the initial state.
2. Loop until a solution is found or there are no new operators left to be applied:
− Select and apply a new operator − Evaluate the new state:
goal → quitbetter than current state → new current state
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Simple Hill Climbing
• Evaluation function as a way to inject task-specific knowledge into the control process.
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Simple Hill Climbing
Example: coloured blocks
Heuristic function: the sum of the number of different colours on each of the four sides (solution = 16).
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Steepest-Ascent Hill Climbing (Gradient Search)
• Considers all the moves from the current state.
• Selects the best one as the next state.
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Steepest-Ascent Hill Climbing (Gradient Search)
Algorithm1. Evaluate the initial state.
2. Loop until a solution is found or a complete iteration produces no change to current state:
− SUCC = a state such that any possible successor of the current state will be better than SUCC (the worst state).
− For each operator that applies to the current state, evaluate the new state:
goal → quitbetter than SUCC → set SUCC to this state
− SUCC is better than the current state → set the current state to SUCC.
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Hill Climbing: Disadvantages
Local maximumA state that is better than all of its neighbours, but not better than some other states far away.
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Hill Climbing: Disadvantages
PlateauA flat area of the search space in which all neighbouring states have the same value.
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Hill Climbing: Disadvantages
RidgeThe orientation of the high region, compared to the set of available moves, makes it impossible to climb up. However, two moves executed serially may increase the height.
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Hill Climbing: Disadvantages
Ways Out
• Backtrack to some earlier node and try going in a different direction.
• Make a big jump to try to get in a new section.
• Moving in several directions at once.
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Hill Climbing: Disadvantages
• Hill climbing is a local method: Decides what to do next by looking only at the “immediate” consequences of its choices.
• Global information might be encoded in heuristic functions.
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Hill Climbing: Disadvantages
B
C
D
A
B
C
Start Goal
Blocks World
A D
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Hill Climbing: Disadvantages
B
C
D
A
B
C
Start Goal
Blocks World
A D
Local heuristic: +1 for each block that is resting on the thing it is supposed to be resting on. −1 for each block that is resting on a wrong thing.
0 4
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Hill Climbing: Disadvantages
B
C
D
B
C
D
A
A0 2
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Hill Climbing: Disadvantages
B
C
D
A
B
C D
A B
C
DA
00
0
B
C
D
A
2
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Hill Climbing: Disadvantages
B
C
D
A
B
C
Start Goal
Blocks World
A D
Global heuristic: For each block that has the correct support structure: +1 to
every block in the support structure. For each block that has a wrong support structure: −1 to
every block in the support structure.
−6 6
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Hill Climbing: Disadvantages
B
C
D
A
B
C D
A B
C
DA
−6−2
−1
B
C
D
A
−3
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Hill Climbing: Conclusion
• Can be very inefficient in a large, rough problem space.
• Global heuristic may have to pay for computational complexity.
• Often useful when combined with other methods, getting it started right in the right general neighbourhood.
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Simulated Annealing
• A variation of hill climbing in which, at the beginning of the process, some downhill moves may be made.
• To do enough exploration of the whole space early on, so that the final solution is relatively insensitive to the starting state.
• Lowering the chances of getting caught at a local maximum, or plateau, or a ridge.
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Simulated Annealing
Physical Annealing
• Physical substances are melted and then gradually cooled until some solid state is reached.
• The goal is to produce a minimal-energy state.
• Annealing schedule: if the temperature is lowered sufficiently slowly, then the goal will be attained.
• Nevertheless, there is some probability for a transition to a higher energy state: e−∆E/kT.
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Simulated Annealing
Algorithm1. Evaluate the initial state.
2. Loop until a solution is found or there are no new operators left to be applied:
− Set T according to an annealing schedule− Selects and applies a new operator
− Evaluate the new state:goal → quit∆E = Val(current state) − Val(new state)
∆E < 0 → new current state
else → new current state with probability e−∆E/kT.
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Best-First Search
• Depth-first search: not all competing branches having to be expanded.
• Breadth-first search: not getting trapped on dead-end paths.
⇒ Combining the two is to follow a single path at a time, but switch paths whenever some competing path look more promising than the current one.
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Best-First Search
A
DCB
FEHG
JI
5
66 5
2 1
A
DCB
FEHG5
66 5 4
A
DCB
FE5
6
3
4
A
DCB53 1
A
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Best-First Search
• OPEN: nodes that have been generated, but have not examined.
This is organized as a priority queue.
• CLOSED: nodes that have already been examined.
Whenever a new node is generated, check whether it has been generated before.
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Best-First Search
Algorithm1. OPEN = {initial state}.
2. Loop until a goal is found or there are no nodes left in OPEN:
− Pick the best node in OPEN− Generate its successors
− For each successor:new → evaluate it, add it to OPEN, record its parentgenerated before → change parent, update successors
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Best-First Search
• Greedy search:h(n) = estimated cost of the cheapest path from node n to a goal state.
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Best-First Search
• Uniform-cost search:g(n) = cost of the cheapest path from the initial state
to node n.
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Best-First Search
• Greedy search:h(n) = estimated cost of the cheapest path from node n to a goal state.
Neither optimal nor complete
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Best-First Search
• Greedy search:h(n) = estimated cost of the cheapest path from node n to a goal state.
Neither optimal nor complete
• Uniform-cost search:g(n) = cost of the cheapest path from the initial state
to node n.
Optimal and complete, but very inefficient
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Best-First Search
• Algorithm A* (Hart et al., 1968):
f(n) = g(n) + h(n)
h(n) = cost of the cheapest path from node n to a goal state.
g(n) = cost of the cheapest path from the initial state to node n.
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Best-First Search
• Algorithm A*:
f*(n) = g*(n) + h*(n)
h*(n) (heuristic factor) = estimate of h(n).
g*(n) (depth factor) = approximation of g(n) found by A* so far.
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Problem Reduction
Goal: Acquire TV set
AND-OR Graphs
Goal: Steal TV set Goal: Earn some money Goal: Buy TV set
Algorithm AO* (Martelli & Montanari 1973, Nilsson 1980)
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Problem Reduction: AO*
A
DCB43 5
A5
6
FE44
A
DCB43
10
9
9
9
FE44
A
DCB4
6 10
1112
HG75
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Problem Reduction: AO*
A
G
CB 10
5
11
13
ED 65 F 3
A
G
CB 15
10
14
13
ED 65 F 3
H 9Necessary backward propagation
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Constraint Satisfaction
• Many AI problems can be viewed as problems of constraint satisfaction.
Cryptarithmetic puzzle:
SEND
MORE
MONEY
+
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Constraint Satisfaction
• As compared with a straightforard search procedure, viewing a problem as one of constraint satisfaction can reduce substantially the amount of search.
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Constraint Satisfaction
• Operates in a space of constraint sets.
• Initial state contains the original constraints given in the problem.
• A goal state is any state that has been constrained “enough”.
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Constraint Satisfaction
Two-step process:
1. Constraints are discovered and propagated as far as possible.
2. If there is still not a solution, then search begins, adding new constraints.
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M = 1S = 8 or 9O = 0N = E + 1C2 = 1N + R > 8E ≠ 9
N = 3R = 8 or 92 + D = Y or 2 + D = 10 + Y
2 + D = YN + R = 10 + ER = 9S =8
2 + D = 10 + YD = 8 + YD = 8 or 9
Y = 0 Y = 1
E = 2
C1 = 0 C1 = 1
D = 8 D = 9
Initial state:• No two letters have the same value.
• The sum of the digits must be as shown.
SEND
MORE
MONEY
+
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Constraint Satisfaction
Two kinds of rules:
1. Rules that define valid constraint propagation.
2. Rules that suggest guesses when necessary.
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Homework
Exercises 1-14 (Chapter 3 – AI Rich & Knight)
Reading Algorithm A* (http://en.wikipedia.org/wiki/A%2A_algorithm)