Problem solving and search

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Problem solving and search

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Reminders

Assignment 0 due 5pm today Assignment 1 posted, due 2/9

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Outline

♦ Problem-solving agents ♦ Problem types

♦ Problem formulation ♦ Example problems

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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)}

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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:

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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

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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

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Example: vacuum world

Single-state, start in #5. Solution??

1 2

3 4

5 6

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Example: vacuum world

[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

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Example: vacuum world

[Right, Suck]

[Right, Suck, Lef t, Suck] Contingency, start in #5

Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only.

Solution??

1 2

3 4

5 6

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Example: vacuum world

[Right, Suck]

[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]

1 2

3 4

5 6

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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

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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

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Example: vacuum world state space graph

R L S S S S R L R L R L S S S S L L L L R R R R states?? actions?? goal test?? path cost??

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Example: vacuum world state space graph

R L S S S S R L R L R L S S S S L L L L R R R R

states??: integer dirt and robot locations (ignore dirt amounts etc.)

actions?? goal test?? path cost??

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Example: vacuum world state space graph

actions??: Lef t, Right, Suck, N oOp

goal test?? path cost??

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Example: vacuum world state space graph

goal test??: no dirt

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Example: vacuum world state space graph

goal test??: no dirt

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2

5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5

goal test??: = goal state (given)

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Example: The 8-puzzle

2

5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5

goal test??: = goal state (given)

path cost??: 1 per move

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Example: robotic assembly

R R R P R R

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!

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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

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Tree search example

Rimnicu Vilcea Lugoj

Zerind Sibiu

Arad Fagaras Oradea

Timisoara

Arad

Arad Oradea

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Tree search example

Rimnicu Vilcea Lugoj

Arad Fagaras Oradea Arad Arad Oradea

Zerind Arad

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Tree search example

Lugoj Arad Arad Oradea Rimnicu Vilcea Zerind Arad Sibiu

Arad Fagaras Oradea

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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!

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

State Node depth = 6

g = 6

state

parent, action

The Expand _{function creates new nodes, filling in the various fields and}

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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}

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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/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

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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

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C

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Properties of breadth-first search

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Properties of breadth-first search

Complete?? Yes (if b is finite)

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Properties of breadth-first search

Time?? 1 + b + b2

+ b3

+ . . . + bd + b(bd − 1) = O(bd+1

), i.e., exp. in d

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Properties of breadth-first search

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)

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Properties of breadth-first search

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.

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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(bdC∗/e)

where C∗ is the cost of the optimal solution

Space?? # of nodes with g ≤ cost of optimal solution, O(bdC∗/e)

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Depth-first search

Expand deepest unexpanded node

Implementation:

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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

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Properties of depth-first search

Time?? O(bm): terrible if m is much larger than d

but if solutions are dense, may be much faster than breadth-first

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Properties of depth-first search

Space?? O(bm), i.e., linear space!

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Properties of depth-first search

Space?? O(bm), i.e., linear space!

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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/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

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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

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Iterative deepening search

l = 0

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Iterative deepening search

l = 1 Limit = 1 A B C A B C A B C A B C

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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 I J K L M N O A B C D E F G H I J K L M N O

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Properties of iterative deepening search

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Complete?? Yes

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

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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

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Repeated states

Failure to detect repeated states can turn a linear problem into an exponential one! A B C D A B B C C C C

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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)

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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