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CS 380: ARTIFICIAL INTELLIGENCE PROBLEM SOLVING: UNINFORMED SEARCH
10/2/2013 Santiago Ontañón [email protected] https://www.cs.drexel.edu/~santi/teaching/2013/CS380/intro.html
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• Problem Solving: • Problem Formulation • Finding a solution
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Outline • Problems and Problem Solving • Example Problems • Solving Problems
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Uninformed Search • Recall:
• Initial state: initial configuration of the problem • Actions: set of actions the agent can perform to solve the problem • State space: set of possible states that are reachable form the
initial space by any sequence of actions
• Uninformed Search: • Family of strategies to find solutions • Based on systematically exploring the state space until finding a
sequence of actions that achieves the goal
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Illustration 8 1
3 2 6
7 4 5
Initial state
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Illustration 8 1
3 2 6
7 4 5
8 1
3 2 6
7 4 5
8 2 1
3 6
7 4 5
8 1
3 2 6
7 4 5
Initial state
Set of states reachable by executing only one action
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Illustration 8 1
3 2 6
7 4 5
8 1
3 2 6
7 4 5
8 2 1
3 6
7 4 5
8 1
3 2 6
7 4 5
Initial state
Set of states reachable by executing only two actions
3 8 1
2 6
7 4 5
8 1
3 2 6
7 4 5
8 1
3 2 6
7 4 5
8 1 6
3 2
7 4 5
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Illustration 8 1
3 2 6
7 4 5
8 1
3 2 6
7 4 5
8 2 1
3 6
7 4 5
8 1
3 2 6
7 4 5
Initial state
Set of states reachable by executing only two actions
3 8 1
2 6
7 4 5
8 1
3 2 6
7 4 5
8 1
3 2 6
7 4 5
8 1 6
3 2
7 4 5
This is a SEARCH TREE
If the problem has a solution, it will be one of the branches of the tree.
Uninformed search
strategies are different ways to explore the
search tree.
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The rest of these slides are borrowed from the slides in Russell and Norvig’s website
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Implementation: states vs. nodes
A state is a (representation of) a physical configurationA node is a data structure constituting part of a search tree
includes parent, children, depth, path cost g(x)States do not have parents, children, depth, or path cost!
1
23
45
6
7
81
23
45
6
7
8
State Node depth = 6
g = 6
state
parent, action
The Expand function creates new nodes, filling in the various fields andusing the SuccessorFn of the problem to create the corresponding states.
Chapter 3 29
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Search strategies
A strategy is defined by picking the order of node expansion
Strategies are evaluated along the following dimensions:completeness—does it always find a solution if one exists?time complexity—number of nodes generated/expandedspace complexity—maximum number of nodes in memoryoptimality—does it always find a least-cost solution?
Time and space complexity are measured in terms ofb—maximum branching factor of the search treed—depth of the least-cost solutionm—maximum depth of the state space (may be !)
Chapter 3 31
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Uninformed search strategies
Uninformed strategies use only the information availablein the problem definition
Breadth-first search
Uniform-cost search
Depth-first search
Depth-limited search
Iterative deepening search
Chapter 3 32
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Breadth-first search
Expand shallowest unexpanded node
Implementation:fringe is a FIFO queue, i.e., new successors go at end
A
B C
D E F G
Chapter 3 33
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Breadth-first search
Expand shallowest unexpanded node
Implementation:fringe is a FIFO queue, i.e., new successors go at end
A
B C
D E F G
Chapter 3 34
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Breadth-first search
Expand shallowest unexpanded node
Implementation:fringe is a FIFO queue, i.e., new successors go at end
A
B C
D E F G
Chapter 3 35
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Breadth-first search
Expand shallowest unexpanded node
Implementation:fringe is a FIFO queue, i.e., new successors go at end
A
B C
D E F G
Chapter 3 36
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Properties of breadth-first search
Complete??
Chapter 3 37
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Properties of breadth-first search
Complete?? Yes (if b is finite)
Time??
Chapter 3 38
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Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd ! 1) = O(bd+1), i.e., exp. in d
Space??
Chapter 3 39
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Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd ! 1) = O(bd+1), i.e., exp. in d
Space?? O(bd+1) (keeps every node in memory)
Optimal??
Chapter 3 40
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Properties of breadth-first search
Complete?? Yes (if b is finite)
Time?? 1 + b + b2 + b3 + . . . + bd + b(bd ! 1) = O(bd+1), i.e., exp. in d
Space?? O(bd+1) (keeps every node in memory)
Optimal?? Yes (if cost = 1 per step); not optimal in general
Space is the big problem; can easily generate nodes at 100MB/secso 24hrs = 8640GB.
Chapter 3 41
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Uniform-cost search
Expand least-cost unexpanded node
Implementation:fringe = queue ordered by path cost, lowest first
Equivalent to breadth-first if step costs all equal
Complete?? Yes, if step cost ! !
Time?? # of nodes with g " cost of optimal solution, O(b!C!/!")
where C# is the cost of the optimal solution
Space?? # of nodes with g " cost of optimal solution, O(b!C!/!")
Optimal?? Yes—nodes expanded in increasing order of g(n)
Chapter 3 42
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 43
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 44
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 45
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 46
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 47
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 48
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 49
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 50
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 51
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 52
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 53
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Depth-first search
Expand deepest unexpanded node
Implementation:fringe = LIFO queue, i.e., put successors at front
A
B C
D E F G
H I J K L M N O
Chapter 3 54
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Properties of depth-first search
Complete??
Chapter 3 55
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Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path
! complete in finite spaces
Time??
Chapter 3 56
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Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path
! complete in finite spaces
Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first
Space??
Chapter 3 57
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Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path
! complete in finite spaces
Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first
Space?? O(bm), i.e., linear space!
Optimal??
Chapter 3 58
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Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loopsModify to avoid repeated states along path
! complete in finite spaces
Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first
Space?? O(bm), i.e., linear space!
Optimal?? No
Chapter 3 59
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Depth-limited search
= depth-first search with depth limit l,i.e., nodes at depth l have no successors
Recursive implementation:
function Depth-Limited-Search(problem, limit) returns soln/fail/cuto!Recursive-DLS(Make-Node(Initial-State[problem]),problem, limit)
function Recursive-DLS(node,problem, limit) returns soln/fail/cuto!cuto!-occurred?! falseif Goal-Test(problem,State[node]) then return node
else if Depth[node] = limit then return cuto!
else for each successor in Expand(node,problem) do
result!Recursive-DLS(successor,problem, limit)if result = cuto! then cuto!-occurred?! trueelse if result "= failure then return result
if cuto!-occurred? then return cuto! else return failure
Chapter 3 60
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Iterative deepening search
function Iterative-Deepening-Search(problem) returns a solutioninputs: problem, a problem
for depth! 0 to " do
result!Depth-Limited-Search(problem, depth)if result #= cuto! then return result
end
Chapter 3 61
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Iterative deepening search l = 0
Limit = 0 A A
Chapter 3 62
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Iterative deepening search l = 1
Limit = 1 A
B C
A
B C
A
B C
A
B C
Chapter 3 63
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Iterative deepening search l = 2
Limit = 2 A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
A
B C
D E F G
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Iterative deepening search l = 3
Limit = 3
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H I J K L M N O
A
B C
D E F G
H J K L M N OI
A
B C
D E F G
H I J K L M N O
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Properties of iterative deepening search
Complete??
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Properties of iterative deepening search
Complete?? Yes
Time??
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Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d ! 1)b2 + . . . + bd = O(bd)
Space??
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Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d ! 1)b2 + . . . + bd = O(bd)
Space?? O(bd)
Optimal??
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Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d ! 1)b2 + . . . + bd = O(bd)
Space?? O(bd)
Optimal?? Yes, if step cost = 1Can be modified to explore uniform-cost tree
Numerical comparison for b = 10 and d = 5, solution at far right leaf:
N(IDS) = 50 + 400 + 3, 000 + 20, 000 + 100, 000 = 123, 450
N(BFS) = 10 + 100 + 1, 000 + 10, 000 + 100, 000 + 999, 990 = 1, 111, 100
IDS does better because other nodes at depth d are not expanded
BFS can be modified to apply goal test when a node is generated
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Summary of algorithms
Criterion Breadth- Uniform- Depth- Depth- IterativeFirst Cost First Limited Deepening
Complete? Yes! Yes! No Yes, if l ! d YesTime bd+1 b"C
!/!# bm bl bd
Space bd+1 b"C!/!# bm bl bd
Optimal? Yes! Yes No No Yes!
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Repeated states
Failure to detect repeated states can turn a linear problem into an exponentialone!
A
B
C
D
A
BB
CCCC
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Graph search
function Graph-Search(problem, fringe) returns a solution, or failure
closed! an empty setfringe! Insert(Make-Node(Initial-State[problem]), fringe)loop do
if fringe is empty then return failurenode!Remove-Front(fringe)if Goal-Test(problem,State[node]) then return node
if State[node] is not in closed then
add State[node] to closed
fringe! InsertAll(Expand(node,problem), fringe)end
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Summary
Problem formulation usually requires abstracting away real-world details todefine a state space that can feasibly be explored
Variety of uninformed search strategies
Iterative deepening search uses only linear spaceand not much more time than other uninformed algorithms
Graph search can be exponentially more e!cient than tree search
Chapter 3 74