Artificial Intelligence

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Artificial Intelligence
Search 3

• Ian Gent
• ipg_at_cs.st-and.ac.uk

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Artificial Intelligence
Search 3

• Part I Best First Search
• Part II Heuristics for 8s Puzzle
• Part III A

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Search Reminder

• Search states, Search trees
• Dont store whole search trees, just the frontier

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Best First Search

• All the algorithms up to now have been hard wired
• I.e. they search the tree in a fixed order
• use heuristics only to choose among a small
  number of choices
• e.g. which letter to set in SAT / whether to be A
  or a
• Would it be a good idea to explore the frontier
  heuristically?
• I.e. use the most promising part of the frontier?
• This is Best First Search

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Best First Search

• Best First Search is still an instance of general
  algorithm
• Need heuristic score for each search state
• MERGE merge new states in sorted order of score
• I.e. list always contains most promising state
  first
• can be efficiently done if use (e.g.) heap for
  list
• no, heaps not done for free in Lisp, Prolog.
• Search can be like depth-first, breadth-first, or
  in-between
• list can become exponentially long

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Search in the Eights Puzzle

• The Eights puzzle is different to (e.g.) SAT
• can have infinitely long branches if we dont
  check for loops
• bad news for depth-first,
• still ok for iterative deepening
• Usually no need to choose variable (e.g. letter
  in SAT)
• there is only one piece to move (the blank)
• we have a choice of places to move it to
• we might want to minimise length of path
• in SAT just want satisfying assignment

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Search in the Eights Puzzle

• Are the hard wired methods effective?
• Breadth-first very poor except for very easy
  problems
• Depth-first useless without loop checking
• not much good with it, either
• Depth-bounded -- how do we choose depth bound?
• Iterative deepening ok
• and we can use increment 2 (why?)
• still need good heuristics for move choice
• Will Best-First be ok?

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Search in the Eights Puzzle

• How can we use Best-First for the Eights puzzle?
• We need good heuristic for rating states
• Ideally want to find guaranteed shortest solution
• Therefore need to take account of moves so far
• And some way of guaranteeing no better solution
  elsewhere

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Manhattan distance heuristic

• There is an easy lower bound on moves required
• Just calculate how far each piece is from its
  goal
• add up this for each piece
• sum is minimum number of moves possible
• This is Manhattan distance
• because pieces move according to Manhattan
  geometry
• Manhattan is not exact
• Why not?
• Can use it as heuristic as estimate of distance
  to solution
• makes sense to explore apparently nearest first

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The Eights Puzzle

• Inaccuracy of Manhattan
• Manhattan distance ?
• optimal solution 18

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Using Heuristics

• Take the Manhattan distance as an example
• In Best first, order all states in list by
  Manhattan
• In Depth first, order only new states by
  Manhattan
• still hope to explore most promising first
• In Breadth first, similarly
• Heuristics important to all search algorithms
• Almost all problems solved by search solved by
  good heuristics
• Excepting small problems like 8s puzzle

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Manhattan Distance in 8s

• Manhattan distance in 8s puzzle is NOT a good
  heuristic
• It can be misled
• Suppose we have a small Manhattan distance for
  move A
• but any solution for move A must reverse move A
  eventually (e.g. to allow a vital move B)
• We have in reality made the solution 2 moves
  longer
• moving piece A and then putting it back again
• Heuristic thinks we are closer to a solution
• Infinite loops can occur in Best First Manhattan

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Total distance Heuristic

• Can use Manhattan as basis of excellent heuristic
• The result will in fact be the A algorithm
• sorry about the name
• pronounced A star
• Total distance heuristic takes account of moves
  so far
• Manhattan distance moves to reach this position
• This must be a lower bound on moves from start
  state to goal state via the current state

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A-ghastly name-

• Actually the name is just A
• The Total distance heuristic has a guarantee
• 1. heuristic score is guaranteed lower bound on
  true path cost via the current state
• 2. heuristic score of solution is the true cost
  of solution
• A Best First heuristic with this guarantee
• A guarantees that first solution found is
  optimal
• Helpful because we can stop searching immediately
• otherwise must continue to find possible better
  solutions
• e.g. in Depth First for 8s puzzle.

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The A Guarantee

• A guarantees to find optimal solution
• Proof suppose not, and we derive a contradiction
• Then there is a solution with higher cost found
  first
• must be earlier in list than precursor of optimal
  solution
• heuristic cost true cost (by guarantee 2)
• true cost of worse solution gt true cost of
  optimal
• true cost of optimal ? heuristic cost of
  precursor (guar. 1)
• ? true cost of worse solution gt heuristic cost of
  precursor
• ? precursor of optimal earlier in list than worse
  solution
• Contradiction, w5 (which was what was wanted)

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Branch and Bound

• BnB is not always bed and breakfast
• Branch and bound is similarly inspired to A
• Unlike A may not guarantee optimal solution
  first
• As in A, look for a bound which is guaranteed
  lower than the true cost
• Search the branching tree in any way you like
• e.g. depth first (no guarantee), best first
• Cut off search if cost bound gt best solution
  found
• If heuristic is cost bound, search best first
  
• then BnB A

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BnB example TSP

• Consider the Travelling Salesperson Problem
• Branch and Bound might use depth-first search
• Cost so far is sum of costs of chosen edges
• Bound might be cost of following minimum spanning
  tree of remaining nodes
• MST tree connected to all nodes of min cost
  among all such trees
• all routes have to visit all remaining nodes
• cant possibly beat cost of MST
• Bounds often much more sophisticated
• e.g. using mathematical programming optimisations

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Summary and Next Lecture

• Summary
• Best first tries to explore most promising node
  first
• In 8s puzzle, Manhattan distance is one heuristic
• Total distance is much better and has guarantees
• Best First Guarantees A
• Branch and Bound also uses guaranteed bounds
• Next Lecture Heuristics in decision problems
• so far looked at heuristics for optimisation
• what about when just want any old solution, e.g.
  SAT
• Look at heuristics in these situations