Problem solving and search
Chapter 3
Reminders
Assignment 0 due 5pm today Assignment 1 posted, due 2/9
Section 105 will move to 9-10am starting next week
Outline
♦ Problem-solving agents
♦ Problem types
♦ Problem formulation
♦ Example problems
♦ Basic search algorithms
Problem-solving agents
Restricted form of general agent:
function Simple-Problem-Solving-Agent(percept) returns an action static: seq, an action sequence, initially empty
state, some description of the current world state goal, a goal, initially null
problem, a problem formulation state←Update-State(state, percept) if seq is empty then
goal←Formulate-Goal(state)
problem←Formulate-Problem(state, goal) seq←Search(problem)
action←Recommendation(seq, state) seq←Remainder(seq, state)
return action
Example: Romania
On holiday in Romania; currently in Arad.
Flight leaves tomorrow from Bucharest Formulate goal:
be in Bucharest Formulate problem:
states: various cities
actions: drive between cities Find solution:
sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest
Example: Romania
Urziceni
Hirsova Neamt
Oradea
Zerind Arad
Timisoara
Lugoj Mehadia Dobreta
Sibiu Fagaras
Pitesti
Vaslui Iasi
Rimnicu Vilcea
Bucharest 71
75
118
111
70 75
120 151 140
99 80
97
101 211
138
146 85
98 142
92 87
86
Problem types
Deterministic, fully observable =⇒ single-state problem
Agent knows exactly which state it will be in; solution is a sequence Non-observable =⇒ conformant problem
Agent may have no idea where it is; solution (if any) is a sequence Nondeterministic and/or partially observable =⇒ contingency problem
percepts provide new information about current state solution is a contingent plan or a policy
often interleave search, execution
Unknown state space =⇒ exploration problem (“online”)
Example: vacuum world
Single-state, start in #5. Solution??
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3 4
5 6
7 8
Example: vacuum world
Single-state, start in #5. Solution??
[Right, Suck]
Conformant, start in {1, 2, 3,4,5, 6, 7,8}
e.g., Right goes to {2, 4, 6,8}. Solution??
1 2
3 4
5 6
7 8
Example: vacuum world
Single-state, start in #5. Solution??
[Right, Suck]
Conformant, start in {1, 2, 3,4,5, 6, 7,8}
e.g., Right goes to {2, 4, 6,8}. Solution??
[Right, Suck, Lef t, Suck]
Contingency, start in #5
Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only.
Solution??
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3 4
5 6
7 8
Example: vacuum world
Single-state, start in #5. Solution??
[Right, Suck]
Conformant, start in {1, 2, 3,4,5, 6, 7,8}
e.g., Right goes to {2, 4, 6,8}. Solution??
[Right, Suck, Lef t, Suck]
Contingency, start in #5
Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only.
Solution??
[Right,if dirt then Suck]
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3 4
5 6
7 8
Single-state problem formulation
A problem is defined by four items:
initial state e.g., “at Arad”
successor function S(x) = set of action–state pairs
e.g., S(Arad) = {hArad → Zerind, Zerindi, . . .}
goal test, can be
explicit, e.g., x = “at Bucharest”
implicit, e.g., N oDirt(x) path cost (additive)
e.g., sum of distances, number of actions executed, etc.
c(x, a, y) is the step cost, assumed to be ≥ 0 A solution is a sequence of actions
Selecting a state space
Real world is absurdly complex
⇒ state space must be abstracted for problem solving (Abstract) state = set of real states
(Abstract) action = complex combination of real actions e.g., “Arad → Zerind” represents a complex set
of possible routes, detours, rest stops, etc.
For guaranteed realizability, any real state “in Arad”
must get to some real state “in Zerind”
(Abstract) solution =
set of real paths that are solutions in the real world
Each abstract action should be “easier” than the original problem!
Example: vacuum world state space graph
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S S
S L
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states??
actions??
goal test??
path cost??
Example: vacuum world state space graph
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S S
S S
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L S
S S
S L
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states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??
goal test??
path cost??
Example: vacuum world state space graph
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S S
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L S
S S
S L
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states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Lef t, Right, Suck, N oOp
goal test??
path cost??
Example: vacuum world state space graph
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L
S S
S S
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L
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L S
S S
S L
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states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Lef t, Right, Suck, N oOp
goal test??: no dirt path cost??
Example: vacuum world state space graph
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S S
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S L
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states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Lef t, Right, Suck, N oOp
goal test??: no dirt
path cost??: 1 per action (0 for N oOp)
Example: The 8-puzzle
2
Start State Goal State
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4 6
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5 1
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8 5
states??
actions??
goal test??
path cost??
Example: The 8-puzzle
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Start State Goal State
51 3
4 6
7 8
5 1
2
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4 6 7
8 5
states??: integer locations of tiles (ignore intermediate positions) actions??
goal test??
path cost??
Example: The 8-puzzle
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Start State Goal State
51 3
4 6
7 8
5 1
2
3
4 6 7
8 5
states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??
path cost??
Example: The 8-puzzle
2
Start State Goal State
51 3
4 6
7 8
5 1
2
3
4 6 7
8 5
states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given)
path cost??
Example: The 8-puzzle
2
Start State Goal State
51 3
4 6
7 8
5 1
2
3
4 6 7
8 5
states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given)
path cost??: 1 per move
[Note: optimal solution of n-Puzzle family is NP-hard]
Example: robotic assembly
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states??: real-valued coordinates of robot joint angles parts of the object to be assembled
actions??: continuous motions of robot joints
goal test??: complete assembly with no robot included!
path cost??: time to execute
Tree search algorithms
Basic idea:
offline, simulated exploration of state space
by generating successors of already-explored states (a.k.a. expanding states)
function Tree-Search(problem, strategy) returns a solution, or failure initialize the search tree using the initial state of problem
loop do
if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy
if the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree
end
Tree search example
Rimnicu Vilcea Lugoj
Zerind Sibiu
Arad Fagaras Oradea
Timisoara
Arad
Arad Oradea
Arad
Tree search example
Rimnicu Vilcea Lugoj
Arad Fagaras Oradea Arad Arad Oradea
Zerind Arad
Sibiu Timisoara
Tree search example
Lugoj Arad
Arad Oradea
Rimnicu Vilcea
Zerind Arad
Sibiu
Arad Fagaras Oradea
Timisoara
Implementation: states vs. nodes
A state is a (representation of) a physical configuration
A 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!
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State Node depth = 6
g = 6
state
parent, action
The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.
Implementation: general tree search
function Tree-Search(problem, fringe) returns a solution, or failure fringe←Insert(Make-Node(Initial-State[problem]),fringe) loop do
if fringe is empty then return failure node←Remove-Front(fringe)
if Goal-Test(problem,State(node)) then return node fringe←InsertAll(Expand(node,problem),fringe)
function Expand(node, problem) returns a set of nodes successors←the empty set
for each action, result in Successor-Fn(problem,State[node]) do s←a new Node
Parent-Node[s]←node; Action[s]←action; State[s]←result Path-Cost[s]←Path-Cost[node] + Step-Cost(node,action,s) Depth[s]←Depth[node] + 1
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/expanded space complexity—maximum number of nodes in memory optimality—does it always find a least-cost solution?
Time and space complexity are measured in terms of b—maximum branching factor of the search tree d—depth of the least-cost solution
m—maximum depth of the state space (may be ∞)
Uninformed search strategies
Uninformed strategies use only the information available in the problem definition
Breadth-first search Uniform-cost search Depth-first search Depth-limited search
Iterative deepening search
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
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
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
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
Properties of breadth-first search
Complete??
Properties of breadth-first search
Complete?? Yes (if b is finite) Time??
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??
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??
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/sec so 24hrs = 8640GB.
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(bdC∗/e) where C∗ is the cost of the optimal solution
Space?? # of nodes with g ≤ cost of optimal solution, O(bdC∗/e) Optimal?? Yes—nodes expanded in increasing order of g(n)
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
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
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
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
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
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
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
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
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
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
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
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
Properties of depth-first search
Complete??
Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path
⇒ complete in finite spaces Time??
Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path
⇒ complete in finite spaces
Time?? O(bm): terrible if m is much larger than d
but if solutions are dense, may be much faster than breadth-first Space??
Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path
⇒ complete in finite spaces
Time?? O(bm): terrible if m is much larger than d
but if solutions are dense, may be much faster than breadth-first Space?? O(bm), i.e., linear space!
Optimal??
Properties of depth-first search
Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path
⇒ complete in finite spaces
Time?? O(bm): terrible if m is much larger than d
but if solutions are dense, may be much faster than breadth-first Space?? O(bm), i.e., linear space!
Optimal?? No
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/cutoff Recursive-DLS(Make-Node(Initial-State[problem]),problem,limit)
function Recursive-DLS(node,problem,limit) returns soln/fail/cutoff cutoff-occurred?←false
if Goal-Test(problem,State[node]) then return node else if Depth[node] = limit then return cutoff
else for each successor in Expand(node,problem) do result←Recursive-DLS(successor,problem,limit) if result = cutoff then cutoff-occurred?←true
else if result 6= failure then return result
Iterative deepening search
function Iterative-Deepening-Search(problem) returns a solution inputs: problem, a problem
for depth← 0 to ∞ do
result←Depth-Limited-Search(problem, depth) if result 6= cutoff then return result
end
Iterative deepening search
l = 0Limit = 0 A A
Iterative deepening search
l = 1Limit = 1 A
B C
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B C
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B C
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B C
Iterative deepening search
l = 2Limit = 2 A
B C
D E F G
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D E F G
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D E F G
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B C
D E F G
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D E F G
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D E F G
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D E F G
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B C
D E F G
Iterative deepening search
l = 3Limit = 3
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H I J K L M N O
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D E F G
H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
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H I J K L M N O
Properties of iterative deepening search
Complete??
Properties of iterative deepening search
Complete?? Yes Time??
Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd) Space??
Properties of iterative deepening search
Complete?? Yes
Time?? (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd) Space?? O(bd)
Optimal??
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 = 1
Can 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
Summary of algorithms
Criterion Breadth- Uniform- Depth- Depth- Iterative
First Cost First Limited Deepening
Complete? Yes∗ Yes∗ No Yes, if l ≥ d Yes
Time bd+1 bdC∗/e bm bl bd
Space bd+1 bdC∗/e bm bl bd
Optimal? Yes∗ Yes No No Yes∗
Repeated states
Failure to detect repeated states can turn a linear problem into an exponential one!
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C C
Graph search
function Graph-Search(problem, fringe) returns a solution, or failure closed←an empty set
fringe←Insert(Make-Node(Initial-State[problem]),fringe) loop do
if fringe is empty then return failure node←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
Summary
Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored
Variety of uninformed search strategies
Iterative deepening search uses only linear space
and not much more time than other uninformed algorithms
Graph search can be exponentially more efficient than tree search