Informed Search

1
Informed Search

• Use problem specific knowledge to
• Select a next node to expand
• To select next operator to use
• In the real world, we select actions based on the
  state of the world
• When Im driving a car, I vary the speed of the
  car based on traffic signs, other traffic etc.
• When I speak a word, I speak it based on the real
  world context
• So far, operators and nodes have not been
  selected using knowledge of world state
• E.g., depth-first search (DFS) always applies
  operators in same order and selects next node to
  evaluate in a non-problem specific way
• Problem solving techniques
• Heuristics, Greedy search, A Search
• Iterative improvement Hill climbing, Simulated
  annealing, genetic algorithms

2
Heuristic Evaluation Functions

• Evaluation function, h(n)
• n is a search node
• h(n) returns an numeric value measure of
  nearness to goal
• Used to decide which node to expand next
• Say we have two unexpanded nodes, n1, n2
• If h(n1) lt h(n2), then expand n1 first
• In the problem solving context
• Dont have much problem knowledge
• E.g., dont know how to solve problem
  analytically
• h(n) will typically be inexact
• An estimate, or a heuristic function
• i.e., h heuristic
• Greedy search
• Use heuristic function to select next expanded
  node to expand
• h(goal) 0

3
Example

• How can we perform a greedy search with the
  missionaries and cannibals problem?

4
Need a heuristic function

• Consider the state with the mc problem
• n ((ml cl bl) (mr cr br))
• How can we compute h(n)?
• Only constraint so far is that h(goal) 0
• One heuristic function
• Define h(((0 0 0) (3 3 1))) 0
• Otherwise, h(n) ml cl
• Note
• This heuristic function is specific for the mc
  problem
• It will not work on other problems
• Incorporates some knowledge of the problem into
  the search

5

• Informed Search Process

Input start state, goal state
Input operators
Input heuristic function
Look for goal
Search Computation
Output
sequence of operators, goal state
6
Another ExampleWater Jug Problem

• Have a three-gallon jug and a four-gallon jug,
  and an unlimited supply of water
• Problem get exactly two gallons of water in one
  jug
• Neither jug has any gallon "markers" (e.g., they
  are not like a measuring cup, with ¼, ½ and other
  gradations).
• Possible operations are to fill a jug with water
  completely, pour the contents of one jug into
  another (stopping just short of overflow), and
  pour the contents of one jug onto the ground
• Define the a state representation, and show a
  heuristic, h(n), for this problem

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A search

• Combines
• Uniform cost search
• g(n) Exact path cost from start state to node n
• Greedy search
• h(n) Heuristic path cost from node n to a goal
  state
• Heuristic function for A
• f(n) g(n) h(n)
• With a new restriction on h(n)
• choose h(n) so that it never overestimates
• This is termed an admissible heuristic
• E.g., straight-line distance for shortest
  geometric path is admissible
• A Next node to expand is node with lowest f(n)

8
Example Geometric Paths
straight-line distances

• Show an A search tree, proceeding from start (s)
  to goal (g), given the arcs as operators, the arc
  labels as operator costs, and this table giving
  straight-line distances for h(n).

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Costs for Example

• What is the
• Search Cost
• Path Cost
• For the last example?

10
A for shortest path

• Why not an O(n2) shortest path algorithm?
• E.g., Dijkstras algorithm for finding the
  shortest path on a weighted directed graph
• Shortest-path in a graph is an example only
• To illustrate operation of A

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Admissible HeuristicsAnother Example

• 8-puzzle

Goal state
Start state

• Give an admissible heuristic for the 8-puzzle,
  i.e., one that does not overestimate

12
A search Proof of OptimalityMonotonic A
heuristics

• Monotonic nondecreasing
• We can force admissible h(n) heuristics to be
  monotonic
• Method
• Start with parent node, n
• generate a child node, n
• f(n) max( f(n), g(n) h(n) )
• Following this approach, f-costs along paths do
  not decrease
• New f() is still optimistic (admissible) because
  h() is optimistic

13
A Proof of optimality

• What is optimal?
• Optimal means that when we find a goal, G, with
  the search method, G will have the shortest cost
  path possible
• I.e., for any other goal states, say, G
• It must be the case that g(G) gt g(G)
• Starting assumptions
• h(G) 0
• h is admissible
• h is monotonic

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A Proof of optimalityProof by contradiction

• Assume the opposite of what we are trying to
  prove
• Suppose that A search can reach a suboptimal
  goal state, G
• That is
• In our A search we have reached a goal state, G
• There is another reachable goal state G
• where g(G) lt g(G) 1
• G is a lower-cost goal state than G
• Reachable means we can get there from the start
  node
• And, we have not detected G prior to reaching G

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A Proof of optimality

• In A search, detect goal state by
• removing node at head of list of unexpanded nodes
• list ordered using f(n) h(n) g(n)
• checking if node is a goal state, prior to
  expansion
• if node is goal state have found goal
• Because node G detected as the goal
• G was at head of list of unexpanded nodes
• Therefore, f(G) lt f(n) for all n in list
• And, also g(G) lt f(n) for all nodes in
  list 2
• since f(G) g(G) for goal nodes
• You have to reach G from a node, n, in list
• Because G is reachable from the start state
• Therefore, f(G) gt f(n), for all n in the list
• Since we have monotonicity
• and g(G) gt f(n), for all n, since f(G) g(G)
  for goal nodes
• Reversing this f(n) lt g(G) 3

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A Proof of optimalityConclusion

• Putting together 1, 2, and 3
• g(G) lt g(G) lt f(n) lt g(G)
• But implies that g(G) lt g(G)
• Therefore, contradiction
• Therefore cannot be such a G
• Therefore A finds optimal G

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Complexity of A

• Number of nodes
• Exponential in the length of the solution
• For most problems
• Exponential complexity unless
• path cost error is logarithmically bounded
• error h(n) C(n)
• C(n) actual cost of path from node n to goal
• bound error lt O(log C(n))