PROBLEM SOLVING

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PROBLEM SOLVING

• SEARCH

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Characteristics of Problem Solving

• Much of the work in AI problem solving is based
  on work cognitive psychology, or human problem
  solving.

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Three consistent features of problem solving
episodes

• Goal directedness
• behavior is organized toward achieving a goal.
• Subgoal decomposition
• decomposing the original goal into subtasks, or
  subgoals. These subtasks or subgoals are often
  easier to accomplish, yet by achieving them we
  move towards our goal.
• Operator selection
• An operator is an action that will achieve a goal
  (or subgoal). By sequencing together several
  operators, we can achieve the goal.

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Early Work in Problem Solving

• The most prominent early work on problem solving
  was conducted by a group of mainly German
  psychologists, "Gestalt" psychologists, in the
  early 1900's.

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Wolfgang Kohler

• One of the founders of the Gestalt school of
  Psychology.
• While trapped on the island of Tenerife in the
  Atlantic Ocean during World War I, he studied the
  subjects available to him - a colony of captive
  chimpanzees.
• Sultan was his most prized subject.

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Example Experiment

• One problem was for Sultan to get bananas from
  outside his cage.
• Not difficult when given a stick long enough to
  reach the food.
• When given two short sticks, neither long enough
  to reach the food, he was not successful.
• First Sultan tried to reach bananas with the two
  sticks,
• When he realized that approach wouldn't work, he
  gave up and sulked in his cage for awhile.
• Then, he got up and went over to the sticks, put
  one inside the other to create one long pole, and
  got the bananas - problem solved.

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Kohlers Conclusion

• Believed insight
• putting the two poles together
• is necessary to solve problems.
• Solutions were often preceded by a period of
  intense thinking incubation (sulking) by the ape.
  

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Functional Fixedness

• A primary premise of Gestalt psychology is that
  human problem solvers get stuck when trying to
  solve problems because they cannot change their
  problem solving set.
• Human problem solvers can change their problem
  solving set when they are given direction
  (hints), have a flash of insight (ah-ha).

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Does Insight Occur?

• Associationist argue that insight (ah-ha) does
  not occur.
• Covert behavior.
• Thorndike experimented with cats in a Puzzle Box.
  Cats exhibited simple trial and error behavior
  to get out of the box.
• Humans substitute mental trial and error for the
  physical trial and error process.

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Problem Representation

• Join all the dots by drawing four straight lines
  without removing your pencil from the paper (all
  lines are connected).
• . . .
• . . .
• . . .

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The Importance of Problem Representation

• A key factor in problem solving.
• It is often necessary to reorganize the problem,
  changing the way it is represented to be able to
  solve it.
• Thinking processes are often unnecessarily
  restricted by a poor representation.

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Search

• In AI, problem solving is often seen as a process
  of searching the problem space for a solution.

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State

• The state of the problem space must be
  represented so that we know
• Initial State - starting point
• Goal State - destination, what the situation must
  look like to be done.
• State Transition - operators, ways to move around
  the problem space.
• Similar to Human Problem Solving

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Three consistent features of problem solving
episodes

• Goal directedness
• behavior is organized toward achieving a goal.
• Subgoal decomposition
• decomposing the original goal into subtasks, or
  subgoals. These subtasks or subgoals are often
  easier to accomplish, yet by achieving them we
  move towards our goal.
• Operator selection
• An operator is an action that will achieve a goal
  (or subgoal). By sequencing together several
  operators, we can achieve the goal.

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Other Search Definitions

• Search space - is the set of state that may be
  reached by applying legal operators.
• Problem - find a sequence of states that lead
  from the initial state to the goal state.

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Finding the Solution

• A matter of searching the problem space.
• States may
• Have already occurred at a higher level. Stop!
  In a loop!
• Identical states may appear on two branches at
  the same level. Develop only one branch.
• Be illegal. Stop!
• Goal State. Stop! Problem Solved!
• None of the above. Keep looking.

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Search Example

• Consider Kohler's experiments with Sultan.
• Suppose a bunch of bananas is hanging from the
  center of the ceiling, out of Sultan's reach.
• A chair is under the window and Sultan is at the
  door the room.
• How can Sultan get the bananas?

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Search Example

• In the actual experiments, the chimpanzees had
  difficulty until they reorganized the problem
  space by moving the crate under the bananas.
• This is called the monkey and banana problem, it
  appears in most Prolog and many AI texts.

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How to Solve in Prolog

• First, design a state representation
• What do we need to keep track of?
• horizontal position of monkey
• vertical position of monkey
• position of box
• has/has not banana
• state (horizontal, vertical, box, banana)

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Initial and Goal States

• Initial Sultan is at the door and on the floor,
  the box is at the window and he does not have the
  bananas
• state (atdoor, onfloor, underwindow, hasnot)
• Goal Sultan is under the bananas, on the chair,
  the box is under the bananas, and he has the
  bananas.
• state(middle, onbox, middle, has).

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Operators

• Allow changes of state
• Key is Prolog is to define the possible state and
  allow Prolog to find the path (string of
  operators) that yields the goal state.
• Monkey and Banana operators
• walk on the floor
• climb the box
• push the box around
• grasp the banana

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See Prolog Handout
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Direction of Search

• Two general search approaches
• Data-driven search
• Goal-driven search

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Data-driven search

• also called "forward chaining" or "forward
  reasoning"
• start with the initial state, and work forward
  applying the various operators until the goal
  state is reached - if a solution exists.

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Goal-driven search

• reverse of data driven
• also called "backward chaining" or "backward
  reasoning"
• start at the goal and work backward to find a
  path to the initial state
• operators must be inverted

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Comparison of Data and Goal Driven Strategies

• Goal-driven (or backward chaining) search
  strategies are often more efficient because they
  search a smaller portion of the search space.
• For efficiency, prefer to search as small a part
  of the search space as necessary to find a
  solution.
• In human problem solving, we find experts
  generally work forward, while novices work
  backward.

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Search Strategies

• Two general search strategies
• breadth first
• depth first

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Breadth-first Search

• Each node is examined in order
• Each level of a search is filled out completely
  before proceeding to the next level.
• Requires a great deal of memory
• The solution path that is found is guaranteed to
  be the shortest path.
• But, rarely used in AI due to memory requirements.

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Depth-first Search

• The most recently encountered node is expanded
  first.
• Search downward all possible levels before
  backtracking to a node on a higher level.
• Exploring one path completely before looking at
  other nodes, requires less memory (to record open
  states).
• However, fully exploring branches that do not
  hold a solution is very time consuming.

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Human Problem Solving

• Studies of computer programmers found that
  experts study a computer program in a
  breadth-first fashion - study all modules on one
  level before proceeding further down the calling
  hierarchy.
• Novice programmers study programs in a depth
  first fashion, studying all lower modules in a
  calling hierarchy before studying other modules
  at the higher level.

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Sample Structure Chart
Expert programmer using breadth-first search
would search in the following pattern Top 1 2
3 1.1 1.2 2.1 2.2 3.1 3.2 1.1.1 1.1.2
3.1.1 3.1.2. Novice programmers using
depth-first search would search in the following
pattern Top 1 1.1 1.1.1 1.1.2 1.2 2 2.1
2.2 3 3.1 3.1.1 3.1.2.3.2.
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Bounded Depth-First Search

• Alternative to the basic strategies to gain some
  of the advantages of both
• Follow depth-first search, but limit the number
  of levels that are searched down before
  terminating the search on that node.
• If reach X levels without finding a solution,
  stop and backtrack.
• Weakness
• if the solution is below X levels will not find
  the solution.
• Modify to search X 1 levels if all X levels are
  search without yielding a solution.