Chapter 3 Solving Problems by Searching

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Chapter 3 Solving Problems by Searching
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Outline

• Problem-solving agents
• Problem formulation
• Example problems
• Uninformed search algorithms
• Avoiding repeated states
• Searching with partial information

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Example Romania
An agent is on holiday in Romania currently in
Arad. Flight leaves tomorrow from Bucharest
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Problem-solving agents
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Problem formulation - Definition

• A problem is defined by four components
• initial state where the agent starts
• e.g., x In(Arad)
• successor function SUCCESSOR-FN(x) set of
  action-state pairs
• e.g., ltGo(Sibiu), In(Sibiu)gt,
  ltGo(Timisoara), In(Timisoara)gt,
• state space states reachable from the initial
  state graph node state, arc action.
• path a sequence of states connected by a
  sequence of actions
• goal test determines if the current state is a
  goal state
• e.g., In(Bucharest)
• path cost additive numeric cost to the path the
  performance measure
• e.g., sum of distance or time, number of actions
  executed.
• step cost c(x,a,y) taking action a to go from
  state x to state y (nonnegative!)
• solution a path from the initial state to a goal
  state
• optimal solution the solution having the lowest
  path cost

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Problem formulation - Abstraction
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Example Toy problems vacuum world
states? integer dirt and agent locations
successor function? Left, Right, Suck, NoOp
goal test? no dirt
path cost? 1 per action (0 for NoOp)
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Example Toy problems 8-puzzle
states? integer locations of tiles
successor function? move blank left, right, up,
down
goal test? given goal state
path cost? 1 per move
Optimal solution of sliding-block puzzles is
NP-hard!
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Example Toy problems 8-queens
Incremental formulation
Complete state formulation
n-queens Family
states? Any arrangement of 0 to 8 queens
successor function? Add a queen to any empty
square
goal test? 8 queens are on the board, none
attacked
path cost? no interest
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Example Real world problems

• Route-finding problem
• Special locations and transitions along links
  between them
• Commercial travel advice systems
• Touring problem
• Visit each city at least once
• State includes the current location and the set
  of cities visited
• Traveling salesperson problem (TSP)
• A touring problem each city must be visited
  exactly once
• Find the shortest tour NP-hard!
• Other problems

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Solving the problem using a search tree

• Search tree
• Search graph the same state can be reached from
  multiple paths
• Root is a search node corresponding to the
  initial state In(Arad)

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Solving the problem using a search tree (contd)

• Expanding applying the successor function to the
  current state
• Generating a new set of states In(Sibiu),
  In(Timisoara), In(Zerind)

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Solving the problem using a search tree (contd)

• Fringe the set of nodes generated but not yet
  expanded
• Leaf node each element of the fringe

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General tree search

• Strategy determine the choice of which state to
  expand
• Search algorithms

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States vs. Nodes

• A state is a representation (configuration) of
  the world
• A node is a data structure constituting part of
  the search tree

• State the state to which the node corresponds
• Parent-node the node that generated this node
• Action the action (applied to parent) that
  generated this node
• Path-cost g(n) the cost of the path from the
  initial state to this node
• Depth the number of steps along the path from
  the initial state

• Two different nodes can contain the same world
  state (different paths)!

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General tree search Implementation

• Assume the fringe is implemented as a queue
• Operations on a queue
• Make-Queue(element, )
• Creates a queue with the given element(s)
• Empty?(queue)
• Returns true if no more elements in the queue
• First(element)
• Returns the first element of the queue
• Remove-First(queue)
• Returns the first element of the queue and remove
  it from the queue
• Insert(element, queue)
• Inserts an element into the queue and returns the
  resulting queue
• First-in-first-out (FIFO) or last-in-first-out
  (LIFO) (stack)
• Insert-All(elements, queue)
• Inserts a set of elements into the queue and
  returns the resulting queue

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General tree search Implementation (contd)

• The EXPAND function creates new nodes, filling in
  the various fields using the SUCCESSOR-FN of the
  problem to create corresponding states

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Performance evaluation of the strategy

• A strategy (algorithm) picks the order of node
  expansion
• Strategies are evaluated in four ways
• Completeness Does it always find a solution if
  there is one?
• Optimality Does it find a solution having lowest
  cost path?
• Time complexity How long does it take to find a
  solution
• Number of nodes generated/expanded
• Space complexity How much memory is needed
  during the search
• Maximum number of nodes in memory
• Time and space complexity are measured in terms
  of
• b the branching factor maximum number of
  successors of any node
• d the depth of the shallowest goal node
• m the maximum length of any path in the state
  space
• Total cost search cost path cost

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Uninformed search strategies

• Uninformed search uses only the information
  available in the problem definition
• Breadth-first search
• Uniform-cost search
• Depth-first search
• Depth-limited search
• Iterative deepening search
• Bidirectional search

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Breadth-first search (BFS)

• Expand the shallowest unexpanded node
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end

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Properties of BFS

• Complete?
• Optimal?
• Time?
• Space?

Yes (if b is finite)
Yes (if the path cost is nondecreasing function
of the depth of the node)e.g., cost 1 per step
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Uniform-cost search

• Expand the node with lowest path cost
• fringe is a queue, ordered by path cost
• Equivalent to BFS if all step costs are equal

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Properties of uniform-cost search

• Complete?
• Optimal?
• Time?
• Space?

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Depth-first search (DFS)

• Expand the deepest unexpanded node
• Implementation
• fringe is a LIFO queue, i.e., put successors at
  front

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Depth-first search (DFS) (contd)
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Depth-first search (DFS) (contd)
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Properties of DFS

• Complete?
• Optimal?
• Time?
• Space?

No
O(bm), i.e., linear space!
O(m) when backtracking only one successor is
generated expanded node remembers which
successor to generate next.
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Depth-limited search

• Nodes at depth l have no successors
• Equivalent to DFS with depth limit l

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Properties of Depth-limited search

• Complete?
• Optimal?
• Time?
• Space?

No if l lt d
No
O(bl)
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Iterative-deepening search (IDS)

• Iterative deepening depth-first search
• Combines the benefits of breadth-first and
  depth-first search

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Iterative-deepening search (contd)
Limit 0
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Iterative-deepening search (contd)
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Iterative-deepening search (contd)
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Iterative-deepening search (contd)
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Properties of IDS

• Complete?
• Optimal?
• Time?
• Space?

Yes
Yes
O(bd)
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Bidirectional search

• Run two simultaneous searches
• one forward from the initial state and the other
  backward from the goal
• stop when the two searches meet in the middle

• Properties
• complete and optimal if both searches are
  breadth-first
• both time complexity and space complexity are

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Comparing uninformed search strategies
1. complete if b is finite 2. complete if step
costs for positive 3. optimal if step costs are
all identical 4. if both directions use
breadth-first search
b branching factor d depth of the
shallowest solution m maximum depth of the
search tree l depth limit
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Avoiding repeat states

• Detecting states that have already been expanded
  before
• if there is more than one path to each state
  graph
• where the actions are reversible route-finding
  problem and sliding-blocks puzzles

State space d1 states
Search tree branches

• Prune the repeated states
• cut infinite (exponential) search tree down to
  finite size (polynomial)

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General graph search

• Algorithms that forget their history are doomed
  to repeat it!
• open list (fringe of unexpanded nodes) and closed
  list (set of all expanded nodes)
• The current node is discarded is it matches a
  node on the closed list.

• Optimal solution?
• newly discovered path is always discarded miss
  an optimal solution if it is shorter
• optimality breadth-first search and
  uniform-cost search with constant step costs
• depth-first search and iterative deepening search
  no longer have linear space

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Searching with partial information
Problem types
The agent must reason about sets of states that
it might get to Each such set of states is a
belief state. e.g. vacuum agent knows the
effects of its actions, but has no sensors.
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Reachable belief state space for sensorless
vacuum world
belief state space belief states 12
reachable
physical state space S 8 states
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Searching with partial information
Problem types
The agent must reason about sets of states that
it might get to Each such set of states is a
belief state. e.g. vacuum agent knows the
effects of its actions, but has no sensors.
Adversarial if the uncertainty is caused by the
actions of another agent e.g., game