Problem Solving

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Problem Solving
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Problem Solving

• Game Playing
• Planning

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Game Playing

• Attractive AI problem, because it is abstract
• One of the oldest domains of AI
• In most cases, the world state is fully
  accessible
• Computer representation of the situation can be
  clear and exact

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Game Playing

• Challenging uncertainty introduced by the
• opponents and the
• complexity of the problem (full search is
  impossible)
• Hard in chess, branching factor is about 35, and
  50 moves by each player 35100 nodes to search
  1040 possible legal board states.

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Games vs. Search Problems

• Unpredictable opponent ? solution is a
  contingency plan
• Time limits ? unlikely to find goal, must
  approximate.
• Plan of attack
• Algorithm for perfect play (Von Neumann, 1944)
• Finite horizon, approximate evaluation (Zuse,
  1945 Shannon, 1950 Samuel, 1952 57)
• Pruning to reduce costs (McCarthy, 1956)

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Types of Games
deterministic chance
Perfect info Chess, checkers, go Backgammon, monopoly
Imperfect info ? Bridge, poker
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Two-Person Perfect Information Game

• Two agents, players, move in turn until one of
  them wins, or the result is a draw.
• Each player has a complete and perfect model of
  the environment

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Two-Person Perfect Information Game

• initial state initial position and who goes
  first
• operators legal moves
• terminal test game over?
• utility function outcome (win1, lose-1,
  draw0, etc.)
• two players (MIN and MAX) taking turns to
  maximize their chances of winning (each turn
  generates one ply)
• one players victory is anothers defeat
• need a strategy to win no matter what the
  opponent does

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MINIMAX Strategy for Two-Person Perfect Info
Games

• The mini-max procedure is a depth first, depth
  limited search procedure. At the current state
  next possible moves are generated by plausible
  move generator.
• Then static evaluation function is applied to
  chose the best move.We assume that the static
  evaluation function returns large values for the
  player and small numbers for the opponent.

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MINIMAX PROCEDURE

• Assume that 10 means a win for the player -10
  means a win for the opponent and 0 means a tie.
  If the two players are at the same level of
  knowledge then moves will be taken as maximizing
  when it is the players turn and minimizing when
  it is the opponents turn.

We should chose move B to maximize our advantage
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MINIMAX PROCEDURE

• But, we usually carry the search for more than
  one step.

If we take move B the opponent will make move F
which is more to his advantage.
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MINIMAX PROCEDURE
We propagate the static evaluation function
values upwards and chose to make move C.
MAX
MIN
(-4)
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MINIMAX PROCEDURE

• Complete search of most game graphs is
  computationally infeasible.
• After search terminates, an estimate of the best
  first move is made.
• This can be made by applying a static evaluation
  function to the leaf nodes of the search tree.
  The evaluation function measures the worth of a
  leaf node.
• Positions favorable to MAX evaluate to a large
  positive value, where positions favorable to MIN
  evaluate to a negative value.
• Values near zero correspond to game positions not
  particularly favorable to either MAX or MIN

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MINIMAX PROCEDURE

• The backed up values are based on looking ahead
  in the game tree and therefore depend on features
  occurring near the end of the game.

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MINIMAX PROCEDURE
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Tic-Tac-Toe

• Assume MAX marks Xs, MIN marks Os
• MAX plays first
• With a depth bound of 2, we generate all the
  nodes till level 2, then apply static evaluation
  function to the positions at these nodes.
• If p is not a winning position for either of the
  players
• e(p) number of complete rows, columns or
  diagonals that are still open
• e(p) ? if p is a win for MAX
• e(p) - ? if p is a win for MIN

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Tic-Tac-Toe

• Thus if p is

We have e(p) 6 4 2
We make use of symmetries in generating successor
positions thus the following game states are all
considered identical.
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MINIMAX PROCEDURE
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MINIMAX PROCEDURE
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MINIMAX PROCEDURE
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The Alpha-Beta PROCEDURE

• ? Cuts
• When the current maximum value is greater than
  the successors min value, dont look further on
  that min subtree

Right subtree can be at most 2, so MAX will
always choose the left path regardless of what
appears next.
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The Alpha-Beta PROCEDURE

• ? Cuts
• When the current min value is les than the
  successors max value, dont look further on
  that max subtree

Right subtree can be at least 5, so MIN will
always choose the left path regardless of what
appears next.
MIN lt 3
MAX gt5
3
5
3
1
discard
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The Alpha-Beta PROCEDURE
3
C
MAX
MIN
3
0
A
D
E
2
3
0
B
2
MAX
2
3
5
0
2
1
1) A has ? 3 (A will be no larger than 3 )
D is ? pruned, since 0lt3 E is ? pruned, since 2lt3
B is ? pruned
2) C has ? 3 (C will be no smaller than 3
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Search Efficiency of ? - ? Procedure

• Pruning does not affect the final result
• In order to perform ? - ? cutoffs, at least some
  part of the search tree must be generated to the
  max. depth, because alpha and beta values must be
  based on the static values of the tip nodes.
• Good move ordering improves effectiveness of
  pruning

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Ordering is important for good Pruning
For MIN, sorting successors utility in an
increasing order is better (shown above
left). For MAX, sorting in decreasing order is
better.
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? - ?Pruning Properties

• A game of m moves can be represented with a tree
  of depth m. If the branching factor is b then we
  have bm tip nodes.
• With perfect ordering (lowest successor values
  first for MIN nodes and the highest successor
  values first for MAX nodes), the number of cut
  offs is maximized and the number of tip nodes
  generated is minimized.
• time complexity bm/2.
• bm/2 (b1/2)m ,thus b 35 in chess reduces to 6.

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? - ?Pruning Properties

• That is, the number of tip nodes at depth d that
  would be generated by optimal alpha-beta search
  is about the same as the number of tip nodes that
  would have been generated at depth d/2 without
  alpha-beta

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Planning Means Ends Analysis

• In some problems like guiding a robot around a
  house description of a single state is very
  complex.
• A given action on the part of the robot will
  change only very small part of the total state.
• Instead of writing rules that describe
  transformations between states, we would like to
  write rules that describe only the affected parts
  of the state description. The rest can be assumed
  to stay constant.
• These methods focus on ways of decomposing the
  original problem into appropriate subparts and on
  ways of recording and handling interactions among
  the subparts as they are detected during the
  problem solving process.

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Planning Means Ends Analysis

• The means ends analysis centers around the
  detection of the difference between the current
  state and the goal state. Once such a difference
  is isolated, an operator that can reduce the
  difference (means) must be found. In other words
  it relies on a set of rules that transform one
  problem state into another (ends).
• Example Suppose a robot is given a task of
  moving a desk with two things on it from one room
  to another.

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Means-Ends Analysis
Operator (means) Preconditons Results (ends)
PUSH(obj, loc) at(robot, obj), large(obj), clear(obj), armempty at(obj, loc), at(robot, loc)
CARRY(obj, loc) at(robot, obj), small(obj) at(obj, loc), at(robot, loc)
WALK(loc) none at(robot, loc)
PICKUP(obj) at(robot, obj) armempty holding(obj)
PUTDOWN(obj) holding(obj) holding(obj)
PLACE(obj1, obj2) at(robot, obj2), holding(obj1) on(obj1, obj2)
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Means-Ends Analysis

• The main difference between the start state and
  the goal state is the location of the desk
• To reduce this difference PUSH or CARRY operators
  are available.
• CARRY reaches a dead-end obj is not small
• PUSH

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Means-Ends Analysis
This difference can be reduced by using WALK to
get the robot back to the objects followed by
PICKUP and CARRY
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References

• Nilsson, N.J. Artificial Intelligence A new
  Synthesis, Morgan Kaufmann, 1998
• Luger, Stubblefield. Artificial Intelligence
• Rich, Artificial Intelligence