Problem Solving and Search

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Problem Solving and Search

• Introduction to Artificial Intelligence
• CS440/ECE448
• Lecture 2

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This Lecture

• Problem representation
• Problem solving through search
• Reading
• Chapter 2
• Announcements
• My office hours Weds. From 2 to 3pm.

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The 8-puzzle
Start
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The corresponding search tree
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Toy Problems and Real Problems

• 8-puzzle
• Vacuum World
• Cryptarithmetic
• 8-queens
• The water jug problem
• Missionaries and Cannibals
• Towers of Hanoi

• Traveling salesman
• Robot navigation
• Process or assembly planning
• VLSI Layout

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Problem Solving

• World State values of all attributes of
  interest in the world.
• State Space the set of all possible world
  states.
• Operators change one state into another cost
  of applying operator.
• Goal An (often partial) world state or states
  in an agent, often implemented as a function of
  state and current percept.
• Initial State The values of attributes that are
  in effect at the beginning of a problem before
  any operators have been applied.
• Note The states and the operators define a
  directed (possibly weighted) graph.

• Solution (path) a sequence of operators
  leading from the initial state to a goal state.
• Path cost e.g. sum of distances, number of
  operators executed

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In the real world

• The real world is absurdly complex.
• Real state space must be abstracted for problem
  solving.
• An abstract state is equivalent to a set of real
  states.
• Abstract operator is equivalent to a complex
  combination of real actions.
• Robot operator Move down hall In practice,
  this might involve a complex set of sensor and
  motor activities.
• An abstract solution is equivalent to a set of
  real paths that are solutions in the real world.

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Example The 8-puzzle

• States
• Operators
• Goal Test
• Path Cost
• Constraints

3 3 array of integer values Move tile number i
left, right, up, down goal state (given) 1 per
move Can only move in a direction if that space
is empty
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Example The 8-puzzle

• States
• Operators
• Goal Test
• Path Cost
• Constraints

Integer location of tiles (ignore intermediate
positions) Move blank left, right, up, down
goal state (given) 1 per move Can only move blank
in a direction if it stays in puzzle
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Example The 8-puzzle
1 2 3 4 5 6 7 8 9
Initial State 4, 1, 3, 6, 9, 5, 7, 2 Goal
State 1, 2, 3, 6, 9, 8, 7, 4
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Missionaries and cannibals

• Three missionaries and three cannibals are on the
  left bank of a river.
• There is one canoe which can hold one or two
  people.
• Find a way to get everyone to the right bank,
  without ever leaving a group of missionaries in
  one place outnumbered by cannibals in that place.

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Missionaries and cannibals

• States three numbers (i,j,k) representing the
  number of missionaries, cannibals, and canoes on
  the left bank of the river.
• Initial state (3, 3, 1)
• Operators take one missionary, one cannibal, two
  missionaries, two cannibals, one missionary and
  one cannibal across the river in a given
  direction (I.e. ten operators).
• Goal Test reached state (0, 0, 0)
• Path Cost Number of crossings.

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Missionaries and Cannibals
(3,3,1)
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Missionaries and Cannibals
A missionary and cannibal cross
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Missionaries and Cannibals
(2,2,0)
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Missionaries and Cannibals
One missionary returns
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Missionaries and Cannibals
(3,2,1)
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Missionaries and Cannibals
Two cannibals cross
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Missionaries and Cannibals
(3,0,0)
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Missionaries and Cannibals
A cannibal returns
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Missionaries and Cannibals
(3,1,1)
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Missionaries and Cannibals
Two missionaries cross
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Missionaries and Cannibals
(1,1,0)
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Missionaries and Cannibals
A missionary and cannibal return
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Missionaries and Cannibals
(2,2,1)
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Missionaries and Cannibals
Two Missionaries cross
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Missionaries and Cannibals
(0,2,0)
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Missionaries and Cannibals
A cannibal returns
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Missionaries and Cannibals
(0,3,1)
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Missionaries and Cannibals
Two cannibals cross
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Missionaries and Cannibals
(0,1,0)
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Missionaries and Cannibals
A cannibal returns
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Missionaries and Cannibals
(0,2,1)
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Missionaries and Cannibals
The last two cannibals cross
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Missionaries and Cannibals
(0,0,0)
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Water Jugs

• You are given
• a spigot,
• a 3 Gallon jug,
• a 4 Gallon jug.
• The goal Get 2 gallons of water in the 4 gallon
  jug.
• Actions Filling jugs from spigot, dumping water
  in jugs onto ground, dumping 4 gallon into 3
  gallon jug until 3 gallon jug is full. Dumping 3
  gallon jug into 4 gallon jug until empty or until
  4 gallon is full, etc, etc.

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Water Jugs

• States How full are the two jugs?
• State Representation
• 4G ?
• 3G ?
• Constraints
• 0 ? 4G ? 4
• 0 ? 3G ? 3

• Initial State
• 4G 0
• 3G0
• Goal State
• 4G2

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Operators

• F3 Fill the 3 Gallon jug from the tap.
• F4 Fill the 4 Gallon jug from the tap.
• E4 Empty the 4-Gallon jug on the ground.
• P43 Pour water from 4G jug into the 3G jug until
  3G jug is full.
• P34 Pour water from 3G jug into the 4G jug until
  4G jug is full or 3G is empty.

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Partial State GraphAnd Solution Path

• F3 Fill the 3 Gallon jug from the tap.
• F4 Fill the 4 Gallon jug from the tap.
• E4 Empty the 4-Gallon jug on the ground.
• P43 Pour water from 4G jug into the 3G jug until
  3G jug is full.
• P34 Pour water from 3G jug into the 4G jug until
  4G jug is full or 3G is empty.

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Search Methods
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A Toy Example A Romanian Holiday

• State space Cities in Romania
• Initial state Town of Arad
• Goal Airport in Bucharest
• Operators Drive between cities
• Solution Sequence of cities
• Path cost number of cities, distance, time, fuel

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The state space
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Search Algorithms

• Basic Idea Off-line exploration of state space
  by generating successors of already-explored
  states (also known as expanding states).

Function GENERAL-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
node contains goal state then return
solution else expand node and add resulting
nodes to search tree. end
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General Search Example
Arad
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The solution
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Tree search example
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Tree search example
Expanded node
Fringe
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Tree search example
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Implementation of Search Algorithms
Nodes state, parent-node,operator, depth, path
cost
Function GENERAL-SEARCH (problem,
queing-fn) returns a solution or failure
queue ? MAKE-QUEUE (MAKE-NODE(INITIAL-STATEproble
m)) loop do if queue is empty, then return
failure node ? Remove-Front(queue) if
GOAL-TEST problem applied to STATE(node)
succeeds then return node else
queue?QUEING-FN(queue,EXPAND(node,operatorsproble
m)) end
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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(n).
• States do not have parents, children, depth, or
  path cost!

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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?
• optimality does it always find a least-cost
  solution?
• time complexity number of nodes
  generated/expanded
• space complexity maximum number of nodes in
  memory
• 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
  infinite)

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Uninformed Search Strategies

• Uninformed (blind) strategies use only the
  information available in the problem definition.
• Informed search techniques which might have
  additional information (e.g. a compass).
• Breadth-first search
• Uniform-cost search
• Depth-first search
• Depth-limited search
• Iterative deepening search