1. Problem Solving

2. Views of Problem solving
Well-defined problems
Much studied in AI
Requires search
Domain general heuristics for solving problems
What about ill-defined problems?
No real mechanisms for dealing with these
The problem may be solved suddenly by ‘seeing’ the problem differently
Often requires developing a suitable representation
Well defined problems: all aspects of problem are clearly specified:
Initial state
Set of allowable moves
Goal state (it is clear when it is reached)

3. Gestalt View of Problem Solving
Reaction against Thorndike’s (1898) trial-and-error learning
Focused on
Productive as opposed to reproductive thinking
Insight vs. trial and error
Structural understanding vs. rote memory
Ill-defined vs. well-defined problems
Structural understanding: the ability to comprehend how the problem elements fit together to form a structure in order to solve a problem.
Reproductive thinking: re-use of previous experiences

4. Kohler (1945): monkey and banana problem.
Kohler suggested that the chimpanzees suddenly restructured the problem which allowed them to rapidly solve the problem
Kohler observed that chimpanzees appeared to have an insight into the
problem before solving it

5. Kohler’s two-stick problem
Difficult to show insight in animals – what is their previous exposure to the problem? Is this the first time that they are solving this problem?
However, Birch (1945) showed that apes raised in captivity did not show insightful problem solving

6. Insight
Gestaltists claim that insight is special and involves moments of sudden comprehension resulting from:
a) extended unconscious leaps in thinking
b) greatly accelerated mental processing
c) short-circuiting of normal reasoning processes
Gestaltists were vague on these processes.
Others have argued that insight is nothing special -- Insights are merely significant products of ordinary thinking processes
(e.g. Perkins, 81; Weisberg, 1986,1995; Langley, Simon, Bradshaw & Zytkow, 1986)

7. Evidence for concept of insight: Metcalfe and Weibe (1987) experiment
Noninsight problem (algebra):
factor 16y^2 – 40yz + 25z^2
2) Insight problem (nonroutine):
A prisoner was attempting escape from a tower. He found in his cell a rope which was half long enough to permit him to reach the ground safely. He divided the rope in half and tied the two parts together and escaped. How could he have done this?
Solution:
He unraveled the rope lengthwise and tied the remaining strands together

8. Results (1)
First result: subjects “feelings of knowing” (beforehand) only predicted eventual success of solving the problem for noninsight problems.
At 15 seconds intervals, ss. rated how close they felt to solving the problem:
1=cold (nowhere close to solution) ….
7=hot (problem is virtually solved)

9. Results (2)
Number of times a particular warmth rating was given

10. Different Interpretations of Insight Problems
Problems where solvers cannot recognize when they are getting close OR
Problems that involve sudden moments of comprehension
This study does not prove that insight is special
Difficulty of relying on introspective measures to assess problem solving behavior

11. Brain Imaging of Insight Processes
Difficult to link behavioral data to internal processes -- e.g., do warmness ratings really show that insight occurs rapidly?
With brain imaging techniques, we can get converging evidence for a special insight process. Does insight occur suddenly, in the brain, as behavioral data suggests?
Jung-Beeman et al. (2004). Studied neural correlates of “aha” moment using fMRI and EEG

12. Jung-Beeman et al. (2004)
Compound word problems
Example: pine, crab, sauce
Question: what word can form a familiar compound word or phrase with the each of these words?
Solution: apple
(pineapple, crabapple, applesauce)

13. Example Problems

14. Solutions
school
turn
box
paper
clock
shower
coffee
farm

15. Experimental Setup Measure brain response here
Ss. press button when they solved a problem
Ss. press button when they felt insight during problem solving
Measure brain response here

16. Areas showing greater fMRI signal for insight than non-insight solutions
Do the fMRI results show that insight occurs rapidly?
increased activity in the right hemisphere anterior Superior Temporal Gyrus

17. Representational Change Theory
Ohlsson (1992)
A problem occurs when the problem is represented in a way that does not permit retrieval of the necessary operators or possible actions
The impasse is broken when the problem representation is changed
Changes in problem representation:
Elaboration or adding of new problem information
Remove constraints
Re-interpret old information

18. Rigidity in problem solving: Functional Fixedness
Cannot reach one string while holding the other
Maier’s (1931) two-string problem

19. Only 39% of subjects were able to see solution within 10 minutes

20. Why people get stuck solving problems
Functional Fixedness:
Subjects who utilize an object for a particular function will have more trouble in a problem-solving situation that requires a new and dissimilar function for the object.

21. Duncker’s problem: support a candle on a door
People fixate on the box’s function as a container as opposed to a platform
A box of tacks, some matches, and a candle

22. Hat Rack Problem

24. Why people get stuck solving problems (2)
Mental set
Einstellung
Entrenchment
Problem-solving set
A frame of mind involving a particular way of representing problem or solving a problem

25. Luchins Water Jar Problems
How would you use 3 jars A,B, and C with the indicated capacities to measure out the desired amount of water?
Solution A-3C

26. Luchins Water Jar Problems
How would you use 3 jars with the indicated capacities to measure out the desired amount of water?
LEFT SIDE
OF CLASS
Note that problem 5 and 6 are the same for both groups. Easy solution for 5: C-A. Easy solution for 6: A+C. Left group (experimental group) is more inclined to stick with solution of B-C-2A that worked before (and also works for problems 5 and 6). For the right group, there are multiple solutions for each problem and different solutions are required going from problem 1 to 2, etc., encouraging the exploration of novel strategies.
RIGHT SIDE
OF CLASS

27. Luchins Results Group Einstellung Solution (percent)
Direct Solution (percent)
No Solution (percent)
Control
100
Experimental 74 26

28. But… Are functional fixedness, mental sets, etc. really a problem?
Generalization from experience leads to efficiency
How often do mental sets and functional fixedness save time and computation?

29. Problem solving as search
INITIAL STATE
GOAL STATE
INITIAL STATE
GOAL STATE
Newell and Simon general problem solver ?
Play the game:

30. Solving most games involves search
Examples:
Cannibals and missionaries:
Theseus and the Minotaur:
More special mazes

31. Problem Solving is a search problem
Search Space
Problem Solving is a search problem
Initial
state
Solution
Goal
state

32. Working memory capacity limits how many states can be processed
Initial
state

33. What if the search space is too large?
It is not possible to enumerate the entire search space for many well-defined problems.
Chess: After a few moves, there are too many possible moves and counter moves to consider all of them.
We must use constraints
Often called heuristics
A heuristic is a general guideline
It is likely to lead to a good solution
Not guaranteed to work

34. Hill Climbing
Find some measure of the distance between your present state and the end state.
Take a step in the direction that most reduces that distance

35. Might lead to suboptimal solutions: local maximum
Hill Climbing
Might lead to suboptimal solutions: local maximum 3 4 5
FOOD 5 4 3
fence 1 2 3 2 1 1 2 1 1

36. Means-end analysis
Try to reduce the largest difference between the initial state and the goal state first.
That creates new sub-problems
Each of these new sub-problems needs to be solved.

37. Setting subgoals in means-end analysis
Painting your house (GOAL 1)
Apply paint (SUBGOAL 2)
Need paint and brush (SUBGOAL 3)
Go to hardware store (SUBGOAL 4)
Went to hardware store (SUBGOAL 4)
Got paint and brush (SUBGOAL 3)
Paint the house (GOAL 1)

38. Other Example Problems

39. The Handcuffs Puzzle
The Set-Up For this puzzle you need two people, some rope and some empty space to do the puzzle in. Each person will need a piece of rope with a loop tied in both ends, so it can be worn as handcuffs. The rope should be reasonably long, so that the person wearing it can easily step over it if they want.   
Each person puts on a complete set of handcuffs. Before putting them on, they loop their handcuffs around each other so they are tied together. Each person should wear a complete set of handcuffs. They then have to get themselves apart while following these rules: 
The handcuffs cannot be removed. 
Do not break, cut, saw through, bite
through or in any other way damage
the rope. Damaging each other is
probably a bad idea too.
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40. Interesting Maze
Start at phil’s house. At first, you can only make right turns through the maze. Each time you cross the red zigzag sign (under Carl’s auto repair), the direction in which you turn changes. So, after the first time you cross that sign, you can then only make left turns; after the second time, you switch back to right turns only, etc. How can Carl’s auto repair be reached?

41. True Story
A professor comes to a University to give a talk. They set up a slide projector for him, but neglect to test it. During the talk it becomes apparent that the projector is set too low. Graduate students and professors gather around the projector trying to fix the problem. They call out for a book to raise the front end of the projector. The book turns out to be too thick. They call out for a thinner book... While everyone is searching for another book, one person comes over and quietly solves the problem...
Punchline. All these academics in one room, and nobody knows how to open a book??
(from Ashcroft, 2002)

42. Another problem
A man climbs a mountain on Sat, leaving at daybreak and arriving at the top at sundown. He spends the night, gets up the next day at daybreak and heads down the mountain, following exactly the same path he climbed the day before.
Question: will there be any time during the second day when he will be at exactly the same point on the mountain he was at that time the first day? A B
SAT
SUN

43. 5 x 5 Checkerboard Problem
Start at the dot
Draw a line through all squares using only vertical and horizontal steps
Stay inside board, do not lift pencil
Visit each square just once
Can this be done?

44. Six stick problem Wrong solution: With these six sticks: Answer:
Make four of these:
Answer:

45. 9 dot problem