Problem Solving

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Problem Solving
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Outline

• Well vs. ill-defined problems
• Heuristics for problem solving
• Hill climbing
• Means-Ends analysis
• Working Backwards
• representation of problems
• Fixedness
• Analogical Reasoning
• In ordinary and scientific reasoning
• role of expertise

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Well defined vs. ill defined Problems

• Well defined
• Examples
• geometry proofs,
• logical puzzles
• a clearly specified goal (clear criterion on
  whether the goal has been achieved )
• Necessary information is spelled out in the
  statement of the problem

• Ill defined
• Examples
• finding a perfect mate,
• writing a great novel
• not obvious when a goal has been reached,
• Not obvious which is the relevant information
• One strategy to solve ill-defined problems is to
  add constraints (e.g. operationally define the
  goal),

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General Problem-Solving

• Problem-solving as search Each problem has
• an initial state
• a goal state
• a set of operators (actions that change the
  current state into a new state)
• a path constraint
• a problem space set of all possible paths

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A sample well-defined problem The Tower of Hanoi
Goal move the tower from the left peg to the
rightmost peg,
Restrictions - never placing a larger disk on
top of a smaller one - only move one disk at a
time.
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Problem space the set of all states that can be
achieved during the course of solving a problem.

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Heuristics for problem solving
Hill climbing strategy For any particular state,
carry out the operation that moves you closest to
the final goal state. (often not a good strategy)
Means-end analysis 1. Break down the current
difference between initial state and goal into
subgoals with sub-differences. 2. Choose the
most important difference, then 3. find an
operator that will reduce this.
Working backwards 1. Start at the goal state
and 2. work backwards via means-end analysis,
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Working backwards Heuristic Example
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One (painful) way to solve the water lilies
problem

• Initial number of water lilies 1
• double the initial value 90 times
• Record each of these values
• Find the value that is 1/2 of the 90th day value.

Working backwards - value doubling every day
is equivalent to say that the value is halved
each preceding day - the field was full Day
90th - the field was half full on day 89th
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Representations of the Problem
Some problems are more easily understood and
solved if they are represented in concrete terms
(e.g. a mental image), others are more easily
solved in abstract terms. Finding the right
representation of a problem can be crucial for
finding the solution.
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descent
ascent
top
Position
bottom
Sunrise 330 Sunset
Time of day
A visual representation of the monk problem makes
it obvious that the monk MUST have occupied the
same spot at the same time during the two trips...
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Possible or Impossible?
Starting in the square marked by the circle, draw
a line through all the squares without picking up
your pencil, without passing through a square
more than once, without diagonal lines and
without leaving the checkerboard.
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Functional Fixedness A Problem of Representation

• People fixate on one potential function of an
  object (box container)
• Fail to consider other functions (box holder)
• If box is displayed empty, the second function is
  highlighted, better performance.

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Use these three bottles to pour the perfect
amount into the glass
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5
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43
(1)
fill bottle B, pour into bottle A, then pour into
bottle C twice5 oz
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21
9
42
(2)
(3)
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22
18
48
3
25
(4)
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Rigidity in use of the same strategy
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Analogical reasoning

• Analogy is a common and powerful form of
  reasoning.
• In ordinary reasoning (love is a journey, war
  on drugs)
• In scientific reasoning (attentional spotlight,
  storehouse memory)
• In problem solving
• Analogy is a mapping of knowledge from one domain
  to another.
• Base domain --gt target domain (journey -gt
  love)
• What is being mapped?
• Elements of each map (e.g, nucleus of the atom
  -gt sun electrons -gt planets)
• Attributes of the elements
• Relations among elements rotation (planet, sun)
  rotation (electron, nucleus)
• The structural relations are much more important
  than the surface attributes
• knowledge from the base domain is then applied to
  understand the target domain and to generate
  inferences about it

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Analogical reasoning is a 4-step process

• 1. Access the base.
• 2. Align base and target (Match Attributes
  Relations)
• 3. Evaluate the match.
• 4. Make inferences about the target

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Analogical Reasoning in problem solving

• Literal.
• Collapsing stars spin faster as their size
  shrinks. This occurs because of a principle
  called conservation of angular momentum.

• Metaphorical (analogical). Collapsing stars spin
  faster as their size shrinks. Stars are thus
  like ice skaters, who pirouette faster as they
  pull in their arms. Both stars and skaters
  operate by a principle called conservation of
  angular momentum.

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Analogical Reasoning in problem solvingThe
radiation problem (alone)

• Very hard to come up with solution
• Would an analogous problem (of easier solution)
  help?

(Duncker, 1945)
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A problem with an analogous solution
A general and his troops approached a fortress
accessible by many heavily mined roads. If the
generals troops took only one road to the
fortress, the entire column of soldiers would be
killed, and the attack foiled. However, smaller
groups could pass safely over the
weight-sensitive mines. The generals solution
was to divide his soldiers into many small
platoons and approach the fortress from different
directions.
Did subjects realize the connection?
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Analogical Reasoning in problem solving

• Read Attack problem (Base domain)
• Next, read Radiation problem (Target domain)
• Would the base problem help?
• Half the subjects received a hint The solution
  to the attack problem might be helpful as you
  work on the radiation problem.
• The other half received no hint
• Results people could see the analogy if they
  were directed to do so, but noticing of this
  relation spontaneous was rare

Gick Holyoak (1980)
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Gick and Holyoak (1983) highlighted the
underlying concept of convergence by presenting
two analogous stories (the additional story
involved the cooperation of many small hoses to
put out a blaze) subjects tried to solve the
tumor problem. Subjects were much more likely
to spot the analogy in this situation.
Presumably, the repetition of the theme drew
subjects attention to that aspect of the
stories. Why do people sometimes fail to use
analogy? - Emphasis on superficial similarities
rather than relational similarities - Clustering
of problems based on such superficial features
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Expertise in Problem Solving
Experts tend to notice the crucial aspects of the
situation, rather than focusing on superficial
features. Task categorize simple physics
problems. Subjects novices vs. Ph.D. physics
students Results Novices grouped problems
based on surface features (having an inclined
plane, using a spring), Experts sorted according
to the physical principles relevant to the
problems. As a result, experts are better able to
notice and make use of analogies when a common
conceptual structure characterizes a set of
problems.
Chi, Feltovich and Glaser
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Analogical reasoning in science

• ATTENTION AS SPOTLIGHT Examples
• "The beam of a spotlight (1) moves from one
  location to another, (2) moves in analogue
  fashion . . . , and (3) is characterized by a
  specific size." (Umiltà, 1988)
• The spotlight . . . cannot select one or two
  (or more) objects that fall within the beam, or
  select different properties of a single object"
  (Logan, 1995, p. 106).
• MEMORY AS A STOREHOUSE

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ATTENTION AS SPOTLIGHT Inferential structure

• source domain
• An agent moves her spotlight, which sheds light
  on part of the field.
• When the spotlight sheds light on the target
  object, the object becomes visible to the agent.

• target domain .
• Homunculus controls attention system, which
  expresses attention over some brain areas.
• When the attentional system expresses attention
  on a representation the representation becomes
  conscious.(can be seen by the homunculus)

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Storehouse memory metaphor

• information is held in a short-term store with
  very limited span. From this store it may be
  passed selectively to be stored for long periods"
  (Broadbent, 1958)
• Entailments
•  Memory is a mental space, where
•  Items (discrete units of information) are
  stored.
•  There are several stages - input, -
  storage,- retrieval
• Topic of study
•  How much the subject forgets
•  Formal aspects of memory process
• Measure Quantification of memory (i.e., items)
• Type of questions asked (Controlled and
  Generalizable)
• internal architecture of the store,
• transfer of units from among departments
• information loss.

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Memory as Perception of the Past

• the act of remembering involves the
  re-perception of internal representations that
  are created from experiences in the world (Payne
  et al., p. 59)
• Entailments
• No static snapshots of the past
•  Memories can be imperfect
•  Memory is a reconstructive process
•  Memories are shaped by beliefs and desires
• Topic of study
•  What the subject remembers
•  Content of the memories
•  Errors and distortions
• Measure Accuracy of memory
• Type of questions asked (Ecologically valid)
•  Autobiographical memory Eyewitness testimony
  Memory for faces