Problem Solving as Search

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Problem Solving as Search
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Problem Types

• Deterministic, fully observable ? single-state
  problem
• Non-observable ? conformant problem
• Nondeterministic and/or partially observable ?
  contingency problem
• Unknown state space ? exploration problem

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A Starter Activity

• Cannibals and Missionaries
• Three missionaries and three cannibals
• Want to cross a river using one canoe.
• Canoe can hold up to two people.
• Can never be more cannibals than missionaries on
  either side of the river.
• Get all safely across the river without any
  missionaries being eaten.
• Find a solution

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Single-State Problem Formulation

• A problem is defined by four items
• initial state
• successor function (which actually defines all
  reachable states)
• goal test
• 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

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Problem Representation Cannibals and
Missionaries

• Initial State
• We can show number of cannibals, missionaries and
  canoes on each side of the river.
• Start state is therefore
• 3,3,1,0,0,0

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Problem Representation Cannibals and
Missionaries

• Initial State
• However, since the system is closed, we only need
  to represent one side of the river, as we can
  deduce the other side.
• We will represent the finishing side of the
  river, and omit the starting side.
• So start state is
• 0,0,0

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Problem Representation Cannibals and
Missionaries

• Successor Function
• There are five actions we can represent
• Move one cannibal across the river.
• Move two cannibals across the river.
• Move one missionary across the river.
• Move two missionaries across the river.
• Move one missionary and one cannibal.

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Problem Representation Cannibals and
Missionaries

• Successor Function
• To be a little more mathematical/computer like,
  we want to represent this in a true successor
  function format
• S(state) ? state
• Move one cannibal across the river.
• S(x,y,0) ? x1,y,1
• S(x,y,1) ? x-1,y,0
• Note, this is a slight simplification.
• Obviously, 0 ? x, y, x1, and x-1 ? 3

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Problem Representation Cannibals and
Missionaries

• Successor Function
• S(x,y,0) ? x1,y,1
• S(x,y,1) ? x-1,y,0
• S(x,y,0) ? x2,y,1
• S(x,y,1) ? x-2,y,0
• S(x,y,0) ? x,y1,1
• S(x,y,1) ? x,y-1,0
• S(x,y,0) ? x,y2,1
• S(x,y,1) ? x,y-2,0
• S(x,y,0) ? x1,y1,1
• S(x,y,1) ? x-1,y-1,0

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Problem Representation Cannibals and
Missionaries

• Successor Function
• S(x,y,0) ? x1,y,1 , x2,y,1 , x,y1,1 ,
  x,y2,1 , x1,y1,1
• S(x,y,1) ? x-1,y,0 , x-2,y,0 , x,y-1,0 ,
  x,y-2,0 , x-1,y-1,0
• Where valid states a,b,c are those such that
• 0 ? a ? 3
• 0 ? b ? 3
• 0 ? c ? 1

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Problem Representation Cannibals and
Missionaries

• Successor Function
• S(x,y,0) ? x1,y,1 , x2,y,1 , x,y1,1 ,
  x,y2,1 , x1,y1,1
• S(x,y,1) ? x-1,y,0 , x-2,y,0 , x,y-1,0 ,
  x,y-2,0 , x-1,y-1,0
• Where valid states a,b,c are those such that
• 0 ? a ? 3
• 0 ? b ? 3
• 0 ? c ? 1
• Actually one more set of constraints we will
  get to later.

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Problem Representation Cannibals and
Missionaries

• Goal State
• 3,3,1

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Problem Representation Cannibals and
Missionaries

• Path Cost
• One unit per trip across the river.

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Problem Representation Cannibals and
Missionaries

• Initial State
• 0,0,0
• Successor Function
• S(x,y,0) ? x1,y,1 , x2,y,1 , x,y1,1 ,
  x,y2,1 , x1,y1,1
• S(x,y,1) ? x-1,y,0 , x-2,y,0 , x,y-1,0 ,
  x,y-2,0 , x-1,y-1,0
• Goal State
• 3,3,1
• Path Cost
• One unit per trip across the river.

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Tree Search Algorithms

• Basic ideaoffline, simulated exploration of
  state spaceby generating successors of
  already-explored states (a.k.a. expanding states)

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Tree Search Example
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Tree Search Example
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Tree Search Example
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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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Implementation States vs. Nodes

• The EXPAND function creates new nodes, filling in
  the various fields and using the SUCCESSORFN of
  the problem to create the corresponding states.

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

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Search Strategies

• Time and space complexity are measured in terms
  of b maximum branching factor of the search
  treed depth of the least-cost solutionm
  maximum depth of the state space (may be infinite)

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Uninformed Search Strategies

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

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Uninformed Search Strategies

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

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Breadth-First Search

• Expanding shallowest unexpanded
  nodeImplementation fringe is a FIFO queue,
  i.e., new successors go at end

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Breadth-First Search

• Expanding shallowest unexpanded
  nodeImplementation fringe is a FIFO queue,
  i.e., new successors go at end

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Breadth-First Search

• Expanding shallowest unexpanded
  nodeImplementation fringe is a FIFO queue,
  i.e., new successors go at end

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Breadth-First Search

• Expanding shallowest unexpanded
  nodeImplementation fringe is a FIFO queue,
  i.e., new successors go at end

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Properties of Breadth-First Search

• Complete??

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Properties of Breadth-First Search

• Complete?? Yes (if b is finite)
• Time??

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Properties of Breadth-First Search

• Complete?? Yes (if b is finite)
• Time?? 1 b b2 b3 bd b(bd 1) O(
  bd1 ), ie, exp. in d
• Space??

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Properties of Breadth-First Search

• Complete?? Yes (if b is finite)
• Time?? 1 b b2 b3 bd b(bd 1) O(
  bd1 ), ie, exp. in d
• Space?? O( bd1 ) (keep every node in memory)
• Optimal??

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Properties of Breadth-First Search

• Complete?? Yes (if b is finite)
• Time?? 1 b b2 b3 bd b(bd 1) O(
  bd1 ), ie, exp. in d
• Space?? O( bd1 ) (keep 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 10MB/sec, so 24hours 860GB.

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Depth-First Search

• Expand deepest unexpanded nodeImplementation f
  ringe LIFO queue, I.e., put successors
  at front

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Properties of Depth-first Search

• Complete??

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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??

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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
  dbut if solutions are dense, may be much faster
  than breadth-first
• Space??

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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
  dbut if solutions are dense, may be much faster
  than breadth-first
• Space?? O(bm), I.e., linear space!
• Optimal??

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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
  dbut if solutions are dense, may be much faster
  than breadth-first
• Space?? O(bm), I.e., linear space!
• Optimal?? No.