代写代考 Problem Solving and Search

Problem Solving and Search

Chapter 3, Sections 1–4

AIMA Slides © and ; Chapter 3, Sections 1–4 1

♢ Problem-solving agents

♢ Problem types

♢ Problem formulation

♢ Example problems

♢ Basic search algorithms

AIMA Slides © and ; Chapter 3, Sections 1–4 2

Problem-solving agents

Restricted form of general agent:

function Simple-Problem-Solving-Agent( p) returns an action
inputs: p, a percept
static: s, an action sequence, initially empty

state, some description of the current world state
g, a goal, initially null
problem, a problem formulation

state←Update-State(state, p)
if s is empty then

g←Formulate-Goal(state)
problem←Formulate-Problem(state, g)
s←Search( problem)

action←Recommendation(s, state)
s←Remainder(s, state)
return action

Note: this is offline problem solving.
Online problem solving involves acting without complete knowledge of the
problem and solution.

AIMA Slides © and ; Chapter 3, Sections 1–4 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
operators: drive between cities

Find solution:
sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

AIMA Slides © and ; Chapter 3, Sections 1–4 4

Example: Romania

AIMA Slides © and ; Chapter 3, Sections 1–4 5

Single-state problem formulation

A problem is defined by four items:

initial state e.g., “at Arad”

actions (or successor function S(x))
e.g., Arad → → Sibiu etc.

goal test, can be
explicit, e.g., x = “at Bucharest”
implicit, e.g., Checkmate in chess

path cost (additive)
e.g., sum of distances, number of actions executed, etc.

A solution is a sequence of actions
leading from the initial state to a goal state

Note: we sometimes refer to actions as “operators”

AIMA Slides © and ; Chapter 3, Sections 1–4 6

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!

AIMA Slides © and ; Chapter 3, Sections 1–4 7

Example: The 8-puzzle

Start State Goal State

goal test??
path cost??

[Note: optimal solution of n-Puzzle family is NP-hard]

AIMA Slides © and ; Chapter 3, Sections 1–4 8

Example: The 8-puzzle

Start State Goal State

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]

AIMA Slides © and ; Chapter 3, Sections 1–4 9

Example: robotic assembly

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

AIMA Slides © and ; Chapter 3, Sections 1–4 10

Search algorithms

Basic idea:
offline, simulated exploration of state space
by generating successors of already-explored states

(a.k.a. expanding states)

function General-Search( problem, strategy) returns a solution, or failure
initialize the search tree using the initial state of problem

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

AIMA Slides © and ; Chapter 3, Sections 1–4 11

General search example

AIMA Slides © and ; Chapter 3, Sections 1–4 12

General search example

AIMA Slides © and ; Chapter 3, Sections 1–4 13

General search example

AIMA Slides © and ; Chapter 3, Sections 1–4 14

General search example

AIMA Slides © and ; Chapter 3, Sections 1–4 15

Implementation of search algorithms

function General-Search( problem,Queuing-Fn) returns a solution, or failure

nodes←Make-Queue(Make-Node(Initial-State[problem]))

if nodes is empty then return failure
node←Remove-Front(nodes)
if Goal-Test[problem] applied to State(node) succeeds then return node
nodes←Queuing-Fn(nodes,Expand(node,Operators[problem]))

AIMA Slides © and ; Chapter 3, Sections 1–4 16

Implementation contd: 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!

The Expand function creates new nodes, filling in various fields and using
Operators (or Actions) of problem to create the corresponding states.

AIMA Slides © and ; Chapter 3, Sections 1–4 17

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 ∞)

AIMA Slides © and ; Chapter 3, Sections 1–4 18

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

AIMA Slides © and ; Chapter 3, Sections 1–4 19

Breadth-first search

Expand shallowest unexpanded node

Implementation:
QueueingFn = put successors at end of queue

AIMA Slides © and ; Chapter 3, Sections 1–4 20

Breadth-first search

Expand shallowest unexpanded node

Implementation:
QueueingFn = put successors at end of queue

AIMA Slides © and ; Chapter 3, Sections 1–4 21

Breadth-first search

Expand shallowest unexpanded node

Implementation:
QueueingFn = put successors at end of queue

AIMA Slides © and ; Chapter 3, Sections 1–4 22

Breadth-first search

Expand shallowest unexpanded node

Implementation:
QueueingFn = put successors at end of queue

AIMA Slides © and ; Chapter 3, Sections 1–4 23

Properties of breadth-first search

Complete??

AIMA Slides © and ; Chapter 3, Sections 1–4 24

Properties of breadth-first search

Complete?? Yes (if b is finite)

Time?? 1 + b + b2 + b3 + . . . + bd = O(bd), i.e., exponential in d

Space?? O(bd) (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 1MB/sec
so 24hrs = 86GB.

AIMA Slides © and ; Chapter 3, Sections 1–4 25

Romania with step costs in km

AIMA Slides © and ; Chapter 3, Sections 1–4 26

Uniform-cost search

Expand least-cost unexpanded node

Implementation:
QueueingFn = insert in order of increasing path cost

AIMA Slides © and ; Chapter 3, Sections 1–4 27

Uniform-cost search

Expand least-cost unexpanded node

Implementation:
QueueingFn = insert in order of increasing path cost

AIMA Slides © and ; Chapter 3, Sections 1–4 28

Uniform-cost search

Expand least-cost unexpanded node

Implementation:
QueueingFn = insert in order of increasing path cost

AIMA Slides © and ; Chapter 3, Sections 1–4 29

Uniform-cost search

Expand least-cost unexpanded node

Implementation:
QueueingFn = insert in order of increasing path cost

AIMA Slides © and ; Chapter 3, Sections 1–4 30

Properties of uniform-cost search

Complete?? Yes, if step cost ≥ ϵ

Time?? # of nodes with g ≤ cost of optimal solution

Space?? # of nodes with g ≤ cost of optimal solution

Optimal?? Yes

Depth-first search

Expand deepest unexpanded node

Implementation:
QueueingFn = insert successors at front of queue

Depth-first search

Expand deepest unexpanded node

Implementation:
QueueingFn = insert successors at front of queue

Depth-first search

Expand deepest unexpanded node

Implementation:
QueueingFn = insert successors at front of queue

Depth-first search

Expand deepest unexpanded node

Implementation:
QueueingFn = insert successors at front of queue

I.e., depth-first search can perform infinite cyclic excursions
Need a finite, non-cyclic search space (or repeated-state checking)

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?? 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

Implementation:
Nodes at depth l have no successors

Iterative deepening search

function Iterative-Deepening-Search( problem) returns a solution sequence
inputs: problem, a problem

for depth← 0 to ∞ do
result←Depth-Limited-Search( problem, depth)
if result ̸= cutoff then return result

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

Bidirectional Search

Search simultaneously forwards from the start point, and backwards from
the goal, and stop when the two searches meet in the middle.

Problems: generate predecessors; many goal states; efficient check for node
already visited by other half of the search; and, what kind of search.

Properties of Bidirectional Search

Complete?? Yes

Time?? O(b

Space?? O(b

Optimal?? Yes (if done with correct strategy – e.g. breadth first).

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

Examples of skills expected:

♢ Formulate single-state search problem

♢ Apply a search strategy to solve problem

♢ Analyse complexity of a search strategy

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