COSC 4350 and 5350 Artificial Intelligence

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COSC 4350 and 5350Artificial Intelligence

• Problem solving using uninformed search
• Dr. Lappoon R. Tang

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Overview

• Define what is problem solving by searching
• See how we can formalize a problem as a search
  problem
• Examples of search problems
• Explain a number of search algorithms
• Depth first search
• Breadth first search
• Depth bounded search
• Iterative deepening search
• Bidirectional search

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Whats Problem Solving by Searching?
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Whats Problem Solving by Searching? (Contd)

• Try to imagine you are working out a route from
  Brownsville, TX to Strasbourg in France
• Whatd be the first problem in your mind that you
  need to solve?
• Whatd be the next problem in your mind after the
  first was solved?
• When will you stop solving the problem?

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Whats Problem Solving by Searching? (Contd)

• Lets try to make our thinking a bit more
• formal

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Whats Problem Solving by Searching? (Contd)

• Initial state where you are at the beginning
  before you even attempted to solve the problem
  (i.e. in Brownsville)
• Current state where you are at the moment (e.g.
  Chicago)
• Next state where you WILL be after you make a
  move from the current state (e.g. Paris)
• Final state (or Goal state) where you WANT to
  be at the end
• Action how to go from one state to another
  (e.g. taking a plane from Brownsville to Houston)

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Whats Problem Solving by Searching? (Contd)

• Solving problem by searching means the following
• Finding a sequence of actions that will bring you
  from the initial state to the goal state in the
  most cost effective way
• It is a fun problem that we solve many many times
  each day not knowing that we are doing it! ?

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Formalizing a problem as a state space search

• To formalize a problem as a search problem, one
  needs to specify
• a set of states
• a start state
• a set of goal states
• a set of operators (a successor function)
• i.e. the actions
• Optional path costs

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Uninformed Search

• A.k.a. brute force search
• Exhaustive searching of the search space until a
  solution is found keep looking for the next
  state until the goal state is finally reached
• Mechanical and unintelligent
• Domain knowledge is not utilized even if
  available (we dont use knowledge to guide our
  choice of which action to take)
• Termination might not be guaranteed for some of
  the search methods
• It could happen if the search space is not
  bounded (hold on, you will find out more about it)

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Example A Water Jug Problem

• You are given two jugs, a 4-gallon one and a
  3-gallon one. Neither has any measuring markers
  on it. There is a pump that can be used to fill
  the jugs with water. How can you get exactly 2
  gallons of water into the 3-gallon jug?
• States (x, y) where x is content of 4-gallon
  jug, y 3-gallon jug
• Start state (0, 0)
• Goal state (n, 2)
• Operators
• 1) Fill x from pump, 2) Fill y from pump
• 3) Pour x to y, 4) Pour y to x
• 5) Drain x (pour water to the drain), 6) drain y

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Example A Water Jug Problem (Contd)
You are given two jugs, a 4-gallon one and a
3-gallon one. Neither has any measuring markers
on it. There is a pump that can be used to fill
the jugs with water. How can you get exactly 2
gallons of water into the 4-gallon jug? (4, 0) /
Fill x from pump / (Start state of the
world) (1, 3) / Pour x to y / (1, 0) / Drain y
/ (0, 1) / Pour x to y / (4, 1) / Fill x from
pump / (2, 3) / Pour x to y / (2, 0) / Drain
y / (0, 2) / Pour x to y / (Goal state final
state of the world) Is there a better solution? ?
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State Space Representation

• State space is usually a tree or a graph

(4, 0)
(1, 3)
(4, 3)
(0, 0)

(1, 0)


(0, 1)
In this case, it is a tree

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Example Moving blocks in a block world
There are a number of blocks on a table. The
task is to rearrange them so that they line up in
a certain order.
A
A
B
C
B
C

• States on(x,y) block x is on top of block y
• Start state on(a,c) and on(c,b)
• Goal state on(a,b) and on(b,c)
• Operators
• put(x,y) place block x on top of y if top of
  both blocks are clear
• put(x,table) move block x to the table if top
  of x is clear

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Example Moving blocks in a block world
There are a number of blocks on a table. The
task is to rearrange them so that they line up in
a certain order.
A
A
B
C
B
C

• What would be the sequence of actions that take
  us from the initial
• state to the goal state?

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Example Moving blocks in a block world
There are a number of blocks on a table. The
task is to rearrange them so that they line up in
a certain order.
A
A
B
C
B
C

• put(A, table)
• put(C, table)
• put(B, C)
• put(A, B)

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Searching for a solution

• Two key decisions involved
• Use a tree or a graph (state space
  representation)?
• How to choose which node to expand next (search
  strategy)?
• Example

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Problem solving performance

• Completeness
• Is the search guaranteed to find a solution if
  one exists?
• Optimality
• Does it find the solution with the best path
  cost?
• Is it the shortest path from the start state to
  the goal state?
• Time complexity
• How long does it take to find a solution (or of
  nodes generated by search operators)?
• Space complexity
• How much space are needed to stores the nodes to
  find a solution?

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The Outline of a Basic Tree Search(informal
version)

• If the current state is already the goal state,
  then we quit and return the solution path that
  connects the initial state to the goal state
• Otherwise, we apply all the possible actions to
  the current state and expand the current state
  into all possible next states
• Select the next state according to the search
  strategy from all the states that have not been
  explored (i.e. not yet expanded)
• Make next state the current state
• Go back to step 1

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The Outline of a Basic Tree Search(formal
version)

• Given an initial state x, a goal state y, a set
  of operators O, a search strategy
• current-state x, Q current-state
• If current-state y, return the solution path
  and quit
• Expand node current-state into a set of children
  nodes C by applying the operators in O on the
  node current-state
• Q C U (Q current-state)
• Select the node next-state from Q according to
  the search strategy
• current-state next-state
• Goto step 2

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Breadth-First Search
Goal State

• Generate children nodes for the current node
• If the current node goal state, you stop
• Otherwise, current node the next sibling node
• Is this a good idea?

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Depth-First Search
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Bounded Depth-First Search

• It is simply DFS with a depth bound
• Searching is not permitted beyond the depth bound
• Pros Termination is guaranteed
• Cons If the solution is beneath the depth bound,
  the search cannot find the goal (hence this
  search algorithm is incomplete).

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Iterative Deepening
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Is Iterative Deepening a Win?
Suppose the solution is at depth d of the tree
the last node on that level (i.e. worst case
scenario)
N(BFS) b b2 bd (bd1-b) O(bd1)
N(IDS) (d)b (d-1)b2 (1)bd O(bd)
Example Let b 10 and d 5 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 is actually faster!
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Bi-directional Search
Idea Expand the start state AND the goal state
until a common middle point is reached. Note Not
always applicable depending on if one can
expand the goal state Complexity O(bd/2) both
time and space
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Comparing uninformed search strategies

• Only depth first search and depth limited search
  are not complete nor optimal
• Only depth first search, depth limited search,
  and IDS have linear space complexity
• All uninformed searches have an exponential time
  complexity hopeless as a viable problem solving
  mechanism (unless you have a quantum computer!)

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Avoiding repeating states

• If the state space is organized as a tree, there
  are no repeated states.
• Some problem state spaces can be more readily
  represented as a graph instead of a tree and
  repeated states can be produced IF the problem
  state space was organized as a tree
• Example the number grid problem
• Solution Remember search nodes expanded before
• Maintain a CLOSED list that contains all nodes
  expanded before and an OPEN list containing nodes
  yet to be expanded
• If the current node is in CLOSED, discard it
  because it is a repeated state