CS计算机代考程序代写 data structure assembly algorithm chapter03.dvi

chapter03.dvi

Problem solving and search

Chapter 3

Chapter 3 1

Outline

♦ Problem-solving agents

♦ Problem types

♦ Problem formulation

♦ Example problems

♦ Basic search algorithms

Chapter 3 2

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

Note: this is offline problem solving; solution executed “eyes closed.”
Online problem solving involves acting without complete knowledge.

Chapter 3 3

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

Chapter 3 4

Example: Romania

Giurgiu

Vilcea

Bucharest

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Chapter 3 5

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”)

Chapter 3 6

Example: vacuum world

Single-state, start in #5. Solution??
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Chapter 3 7

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??

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Chapter 3 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, Left, Suck]

Contingency, start in #5
Murphy’s Law: Suck can dirty a clean carpet
Local sensing: dirt, location only.
Solution??

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Chapter 3 9

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, Left, 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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Chapter 3 10

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) = {〈Arad→ Zerind, Zerind〉, . . .}

goal test, can be
explicit, e.g., x = “at Bucharest”
implicit, e.g., NoDirt(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
leading from the initial state to a goal state

Chapter 3 11

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!

Chapter 3 12

Example: vacuum world state space graph
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states??
actions??
goal test??
path cost??

Chapter 3 13

Example: vacuum world state space graph
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states??: integer dirt and robot locations (ignore dirt amounts etc.)
actions??
goal test??
path cost??

Chapter 3 14

Example: vacuum world state space graph
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states??: integer dirt and robot locations (ignore dirt amounts etc.)
actions??: Left, Right, Suck, NoOp
goal test??
path cost??

Chapter 3 15

Example: vacuum world state space graph
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S S

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states??: integer dirt and robot locations (ignore dirt amounts etc.)
actions??: Left, Right, Suck, NoOp
goal test??: no dirt
path cost??

Chapter 3 16

Example: vacuum world state space graph
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states??: integer dirt and robot locations (ignore dirt amounts etc.)
actions??: Left, Right, Suck, NoOp
goal test??: no dirt
path cost??: 1 per action (0 for NoOp)

Chapter 3 17

Example: The 8-puzzle

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Start State Goal State

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states??
actions??
goal test??
path cost??

Chapter 3 18

Example: The 8-puzzle

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Start State Goal State

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states??: integer locations of tiles (ignore intermediate positions)
actions??
goal test??
path cost??

Chapter 3 19

Example: The 8-puzzle

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Start State Goal State

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states??: integer locations of tiles (ignore intermediate positions)
actions??: move blank left, right, up, down (ignore unjamming etc.)
goal test??
path cost??

Chapter 3 20

Example: The 8-puzzle

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Start State Goal State

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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??

Chapter 3 21

Example: The 8-puzzle

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Start State Goal State

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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]

Chapter 3 22

Example: robotic assembly

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RR
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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

Chapter 3 23

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

Chapter 3 24

Tree search example

Arad

Chapter 3 25

Tree search example

LugojArad AradArad

Arad

Chapter 3 26

Tree search example

Timisoara

Chapter 3 27

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.

Chapter 3 28

Implementation: general tree search

Chapter 3 29

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(State[node],action,

result)

Depth[s]←Depth[node] + 1

add s to successors

return successors

Chapter 3 30

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 ∞)

Chapter 3 31

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

Chapter 3 32

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

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

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

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

Properties of breadth-first search

Complete??

Chapter 3 37

Properties of breadth-first search

Complete?? Yes (if b is finite)

Time??

Chapter 3 38

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

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

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.

Chapter 3 41

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 ≥ ǫ

Time?? # of nodes with g ≤ cost of optimal solution, O(b⌈C
∗/ǫ⌉)

where C∗ is the cost of the optimal solution

Space?? # of nodes with g ≤ cost of optimal solution, O(b⌈C
∗/ǫ⌉)

Optimal?? Yes—nodes expanded in increasing order of g(n)

Chapter 3 42

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

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

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

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

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

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

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

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

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

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

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

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

Properties of depth-first search

Complete??

Chapter 3 55

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??

Chapter 3 56

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??

Chapter 3 57

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??

Chapter 3 58

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

Chapter 3 59

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

if cutoff-occurred? then return cutoff else return failure

Chapter 3 60

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

Chapter 3 61

Iterative deepening search l = 0

Limit = 0 A A

Chapter 3 62

Iterative deepening search l = 1

Limit = 1 A

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Chapter 3 63

Iterative deepening search l = 2

Limit = 2 A

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Chapter 3 64

Iterative deepening search l = 3

Limit = 3

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H I J K L M N O

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Chapter 3 65

Properties of iterative deepening search

Complete??

Chapter 3 66

Properties of iterative deepening search

Complete?? Yes

Time??

Chapter 3 67

Properties of iterative deepening search

Complete?? Yes

Time?? (d + 1)b0 + db1 + (d− 1)b2 + . . . + bd = O(bd)

Space??

Chapter 3 68

Properties of iterative deepening search

Complete?? Yes

Time?? (d + 1)b0 + db1 + (d− 1)b2 + . . . + bd = O(bd)

Space?? O(bd)

Optimal??

Chapter 3 69

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

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

Chapter 3 70

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 b⌈C

∗/ǫ⌉ bm bl bd

Space bd+1 b⌈C
∗/ǫ⌉ bm bl bd

Optimal? Yes∗ Yes No No Yes∗

Chapter 3 71

Repeated states

Failure to detect repeated states can turn a linear problem into an exponential
one!

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CCCC

Chapter 3 72

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

Chapter 3 73

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

Chapter 3 74