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Artificial Intelligence _Lec. 1_ Introduction to A

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					Artificial Intelligence

(Lecture 7,8,9)




         Fakhar Mangla/IIU Islamabad
 CHAPTER FOUR
HEURISTIC SEARCH




     Fakhar Mangla/IIU Islamabad
              Heuristics

§ Heuristics are formalized as:

  § Rules for choosing those branches in a state
    space that are most likely to lead toward
    the solution of the problem.




                  Fakhar Mangla/IIU Islamabad
                   Heuristics
§ Two basic situation for the use of Heuristics:
   § An inherently ambiguous problem e.g,
     medical diagnosis system
   § The cost of finding a solution by exhaustive
     methods like depth first search, is
     prohibitive
§ A heuristic is fallible: it is only an informed guess of next
  step to be taken in solving a problem

§ A heuristic can lead a search algorithm to a sub-optimal
  solution or fail to find any solution.


                        Fakhar Mangla/IIU Islamabad
Tic-Tac-Toe problem solving
       using Heuristic
§ Consider the example of state space
  § 9 nodes at the first level 8 nodes of
    each node at level–2 =9x8x7 . . 2
    (9!=326880) nodes to search
§ Need to reduce the number of nodes
  or the size of the state space
§ Analyze carefully the rules of playing
  Tic-Tac-Toe or view as expert to
  setup heuristic
                  Fakhar Mangla/IIU Islamabad
State Space Reduced by Symmetry




            Fakhar Mangla/IIU Islamabad
Tic-Tac-Toe problem solving
       using Heuristic
§ A simple heuristic can eliminate search
  entirely:
  § We may move to the board in which “X” has
    the most winning lines.
§ In case of states with equal numbers of
  potential wins, take the first such state
  found.
§ In case “X” takes the center of the grid.
  That will eliminates other alternates along
  with all of its descendants to 2/3 of the
  space.
                  Fakhar Mangla/IIU Islamabad
  Tic-Tac-Toe problem solving
         using Heuristic
   Heuristics for State Space for TIC-TAC-TOE



         X                                              X

                                       X



             3                          4                   2

Dashed lines shows the number of potential wins at each level-1


                          Fakhar Mangla/IIU Islamabad
IMPLEMENTING HEURISTIC EVALUATION FUNCTION

          8 PUZZLE PROBLEM

A)   The simple heuristic count the tiles
  out place in each state when it is
  compared with the goal.

B) Sum all the distances by which the
  tiles are out of place, one for each
  square a tile must be moved to reach its
  position in the goal state.

                Fakhar Mangla/IIU Islamabad
IMPLEMENTING HEURISTIC EVALUATION FUNCTION



2 8 3                                           1 2 3
              a          b
1 6 4                                           8   4
  7 5         5          6
                                                7 6 5
2 8 3
1   4         3          4

7 6 5
2 8 3         5          6
1 6 4
7 5               Fakhar Mangla/IIU Islamabad
A Possible Result of a Heuristic Search Procedure




                   Fakhar Mangla/IIU Islamabad
IMPLEMENTING HEURISTIC EVALUATION FUNCTION
           8 PUZZLE PROBLEM
§   If two states have same heuristic
    value it is preferable to pick the
    state that is near the start state
§   Thus our evaluation function takes
    the form
§   f(n)=g(n) + h(n)
§   h(n) is the heuristic value of the
    state n
§   g(n) is the distance of the state n
    from the start state
                Fakhar Mangla/IIU Islamabad
       Heuristic Evaluation Function


         g(n) = 0




          g(n) = 1




Values of f(n):      6                     4           6
                         Fakhar Mangla/IIU Islamabad
Heuristic Search using the
   evaluation function




        Fakhar Mangla/IIU Islamabad
      AN ALGORITHM FOR HEURISTIC SEARCH


l Heuristic search can be implemented
  through hill climbing procedure
l Generate the children of current
  node and keep only the best child
  (according to some criteria)
l Hill climbing procedure can terminate
  at some local maxima is the current
  node is better than all its children
l Hill climbing procedure cannot
  recover from a dead end situation
                 Fakhar Mangla/IIU Islamabad
       AN ALGORITHM FOR HEURISTIC SEARCH


l Hill climbing procedure can stuck in a
  loop or can go to an infinite path
l If the criteria calculation function is
  good enough then these problems
  can be avoided
l By combing hill climbing with best
  first we can have a very good
  algorithm


                  Fakhar Mangla/IIU Islamabad
         Best-First Search
§ Depth first Algorithm uses LIFO style list
  to implement search
§ Breadth First Search uses FIFO style list to
  search
§ Best-first is an heuristic search that uses a
  priority queue to order the states on the
  OPEN



                  Fakhar Mangla/IIU Islamabad
    Best-First Search
                                  A-5



                                                        D-6
              B-4            C-4



    E-5       F-5       G-4         H-3           I           J



K     L                       N                               Q   R
                M                    O-2          P-3




S         T
                    Fakhar Mangla/IIU Islamabad
        Best-First Search
Evaluat Opened                                Closed
e
        [A5]                                  []
A5      [B4,C4,D6]                            [A5]
B4      [C4,E5,F5,D6]                         [B4,A5]
C4      [H3,G4,E5,F5,D6]                      [C4,B4,A5]
H3      [O2,P3,G4,E5,F5,D6]                   [H3,C4,B4,A5]
O2      [P3,G4,E5,F5,D6]                      [O2,H3,C4,B4,A5]
P3      The solution

                Fakhar Mangla/IIU Islamabad
        Characteristics of Heuristics
  We can evaluate the behavior of heuristics
  on the basses of following Characteristics:
§ Admissible : (Shortest Path)
§ Informed: (have much more accurate information for next step to be taken)
§ Monotonic: ( min-Cost to find Shortest path)




                              Fakhar Mangla/IIU Islamabad
               Admissibility
§ A search algorithm is admissible if , for any
  graph it always terminates in the optimal
  solution path.
§ The ‘h’ function must never overestimate
  the cost to reach the goal. h is called an
  admissible heuristic
§ Admissible functions are optimistic, as they
  always think the cost to the goal is less
  than it actually is


                  Fakhar Mangla/IIU Islamabad
              Informedness
§ For two A heuristics h1, h2, if
                  h1(n) < h2(n)
  for all states n in the search space, h2 is
  said to be more informed than h1
§ Means if h2 have much more accurate and
  effective information for the next step to be
  taken




                  Fakhar Mangla/IIU Islamabad
               Monotonicity
§ A heuristic function h is monotonic if
   § For all states ni and nj where nj is
     descendent of ni
   § h(ni) - h(nj)  cost (ni, nj)
   § where cost (ni, nj) is the actual cost of
     going from state ni to nj.
   § The heuristic evaluation of the goal state
     is zero or h(goal)=0.
§ Consistently find the minimal path to each
  state they encounter in the search.
                   Fakhar Mangla/IIU Islamabad
     Heuristic Search Algorithms
§ Heuristic Search Algorithms
  § Best-First Search
     § Maintains a priority queue to shift to heuristically
       promising paths

  § A* Search
     § Uses algorithm A (best-first search + evaluation
        function f(n)) with a constraint that h(n) never
        overestimates h*(n) -- the actual cost to reach the
        goal from any state ‘n’.
  § Hill-Climbing

     § Often called greedy search algorithm, does not
       maintain any information to backtrack to other paths
       in the state space if once adopts a fruitless
       “opportunist” path.
                        Fakhar Mangla/IIU Islamabad
      ADMISSIBILITY MEASURES
l A search algorithm is admissible if it
  is guaranteed to find a minimal path
  to solution whenever such a path
  exists
l Breadth first is admissible because
  it looks at every state level n before
  considering level n+1



                Fakhar Mangla/IIU Islamabad
      ADMISSIBILITY MEASURES
l In the formula f(n)=h(n)+g(n), g(n)
  measures the depth of the node n
l h(n) measures the heuristic
  estimate of the distance from n to
  goal state
l Thus f(n) estimates the cost of the
  path from start through n to the
  goal state
l Note that h(n) is not actual cost to
  path from n to goal state
               Fakhar Mangla/IIU Islamabad
      ADMISSIBILITY MEASURES
l Let f*(n)=h*(n)+g*(n)
l Where g*(n) is the cost of the
  shortest path from start to n
l And h*(n) is the actual cost of the
  shortest path from n to goal
l Therefore f*(n) is the actual cost of
  the shortest path from start to goal
  state that passes through n


                Fakhar Mangla/IIU Islamabad
      ADMISSIBILITY MEASURES
l g(n)>=g*(n)
l g(n) is good estimate of g*(n)
l g(n)=g*(n) if the algorithm has
  found the shortest path from start
  to n




               Fakhar Mangla/IIU Islamabad
       ADMISSIBILITY MEASURES
l A best first algorithm that uses the
  evaluation function f(n)=g(n)+h(n) is
  called algorithm A
l A search algorithm is admissible if for
  any graph it always finds the optimal
  solution path whenever such a path
  exists
l If h(n) <= h*(n) in an A algorithm
  then it is called algorithm A*
l All A* algorithms are admissible
l A* algorithms may not be locally
  admissible
                Fakhar Mangla/IIU Islamabad
           MONOTONICITY

A heuristic function h is monotone if
1. For all states ni & nj, where nj is
   descendent of ni
   h(ni)-h(nj)<=cost(ni, nj)
2. The heuristic evaluation of the
   goal state is zero i.e. h(goal)=0



              Fakhar Mangla/IIU Islamabad
           MONOTONICITY
Statement:-Any monotonic
   heuristic h is A* and admissible
Proof:-
l We will show that h is every
   where admissible
l Let s1 be the start state and sg
   be the goal state
l Let us trace the path from start
   to goal
              Fakhar Mangla/IIU Islamabad
MONOTONICITY




  Fakhar Mangla/IIU Islamabad
          INFORMEDNESS

l Let h1 and h2 be two A*
  heuristics such that
  h1(n)<=h2(n) for all states n
  then h2 is said to be more
  informed then h1
l We can say that h2 is more
  informed then h1
l h2 is better than h1

              Fakhar Mangla/IIU Islamabad
  USING
HEURISTICS
 IN GAMES
   Fakhar Mangla/IIU Islamabad
       ADVERSARIAL SEARCH
l The searches we have discussed so
  far involves only one person
l You can say that you are the only
  player in the game
l But in two person games the
  situation is different
l The ordinary search methods cannot
  be used in these situations
l The search used in two person
  games is called adversarial search
              Fakhar Mangla/IIU Islamabad
        MINIMAX PROCEDURE
l The opponents in a game are
  referred to as MIN & MAX
l We suppose that MIN uses the same
  strategy as MAX is using
l MAX is the person trying to maximize
  its score while MIN will try to
  minimize the score of MAX
l Each level in the search is labeled
  according to whose move it is at that
  point in the game (MIN or MAX)
               Fakhar Mangla/IIU Islamabad
        MINIMAX PROCEDURE
l Each state of the game is assigned a
   heuristic value according to some
   evaluation function
l The minimax procedure propagates
   these values up the graph according
   to following rules
1. If the parent is a MAX node, give it
   the maximum value among its
   children
2. If the parent is a MIN node, give it
   the minimum value among its
   children
               Fakhar Mangla/IIU Islamabad
        MINIMAX PROCEDURE
                 7
                       A                                MAX


                                                        MIN
    7                             5
         B                               C

                                                        MAX

D            E                F                    G

7            1              5                      15
             2



                     Fakhar Mangla/IIU Islamabad
      ALPHA BETA PROCEDURE
l Straight minimax procedure requires
  two pass analysis of the search
  space, the first to descend to the ply
  depth and there apply the heuristic
  and the second to propagate value
  back up the tree
l Minimax checks all the branches
  including many that can be ignored
l Alpha beta procedure is an
  improvement to minimax that
  eliminates those branches
                Fakhar Mangla/IIU Islamabad
             ALPHA BETA PRUNING

                 7
                       A                                MAX


                                                        MIN
    =7                      <=5
         B                               C

                                                        MAX

D            E                F                    G

7            1              5                      15
             2



                     Fakhar Mangla/IIU Islamabad
                 ALPHA BETA PRUNING
                       50
                             MAX -α
           50
                                 30                                 20         MIN-β
     50                                                        20
                 70        30                                                  MAX-α




50        40    70    10   60          30          80         50     20   90    70   60




                                Fakhar Mangla/IIU Islamabad
      ALPHA BETA PROCEDURE
Rules for terminating search
1. Search can be stopped below any
   MIN node having a beta value less
   than or equal to the alpha value of
   any of its MAX ancestors
2. Search can be stopped below any
   MAX node having an alpha value
   greater than or equal to the beta
   value of any of its MIN node
   ancestors
                Fakhar Mangla/IIU Islamabad

				
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