Problem Solving
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
Well defined vs. ill defined Problems
• Well defined:
• Ill defined
• Examples:
• Examples:
– geometry proofs,
– logical puzzles
– finding a perfect mate,
– writing a great novel
a clearly specified goal (clear
criterion on whether the goal
has been achieved )
• not obvious when a goal has been
reached,
• Necessary information is
spelled out in the statement of
the problem
• Not obvious which is the relevant
information
•
• One strategy to solve ill-defined
problems is to add constraints (e.g.
operationally define the goal),
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
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.
Problem space: the set of all states that can be achieved
during the course of solving a problem.
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,
Working backwards Heuristic: Example
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
1
1 31
1073741824
2
2 32
2147483648
3
4 33
4294967296
4
8 34
8589934592
5
16 35
17179869184
6
32 36
34359738368
7
64 37
68719476736
8
128 38
137438953472
9
256 39
274877906944
10
512 40
549755813888
11
1024 41
1099511627776
12
2048 42
2199023255552
13
4096 43
4398046511104
14
8192 44
8796093022208
15
16384 45
17592186044416
16
32768 46
35184372088832
17
65536 47
70368744177664
18
131072 48
140737488355328
19
262144 49
281474976710656
20
524288 50
562949953421312
21
1048576 51
1125899906842620
22
2097152 52
2251799813685250
23
4194304 53
4503599627370500
24
8388608 54
9007199254740990
25 16777216 55 18014398509482000
26 33554432 56 36028797018964000
27 67108864 57 72057594037927900
28 134217728 58 144115188075856000
29 268435456 59 288230376151712000
30 536870912 60 576460752303423000
61
1152921504606850000
62
2305843009213690000
63
4611686018427390000
64
9223372036854780000
65
18446744073709600000
66
36893488147419100000
67
73786976294838200000
68
147573952589676000000
69
295147905179353000000
70
590295810358706000000
71
1180591620717410000000
72
2361183241434820000000
73
4722366482869650000000
74
9444732965739290000000
75
18889465931478600000000
76
37778931862957200000000
77
75557863725914300000000
78
151115727451829000000000
79
302231454903657000000000
80
604462909807315000000000
81
1208925819614630000000000
82
2417851639229260000000000
83
4835703278458520000000000
84
9671406556917030000000000
85 19342813113834100000000000
86 38685626227668100000000000
87 77371252455336300000000000
88 154742504910673000000000000
89 309485009821345000000000000
90 618970019642690000000000000
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.
top
descent
ascent
Position
bottom
Sunrise
3:30
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...
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.
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.
Use these three bottles to pour the perfect amount into the glass
(1)
18
43
10
5
fill bottle B, pour into bottle A, then pour into bottle C twice…5 oz
(2)
(3)
9
42
6
21
18
48
4
22
(4)
28
76
3
25
Rigidity in use of the same strategy
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 --> ‘target’ domain (journey -> love)
• What is being mapped?
– Elements of each map (e.g, nucleus of the atom -> sun; electrons -> 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
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
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.”
Analogical Reasoning in problem solving:
The radiation problem (alone)
• Very hard to come up with solution
• Would an analogous problem (of easier solution) help?
(Duncker, 1945)
A problem with an analogous solution:
A general and his troops approached a fortress accessible
by many heavily mined roads. If the general’s 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 weightsensitive mines. The general’s solution was to divide his
soldiers into many small platoons and approach the
fortress from different directions.
Did subjects realize the connection?
Analogical Reasoning in problem solving
• Read Attack problem
(‘Base’ domain)
• Next, read Radiation problem
(‘Target’ domain)
• Would the base problem help?
100
92
75
50
– 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
20
25
0
Strong Hint
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)
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
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
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
ATTENTION AS SPOTLIGH T Mapping
SOURCE DOMAIN
(SPOTLIGHT)
TARGET DOMAIN
(ATTENTION)
Spotlight ---------------------------> Mechanism of attention
Agent --------------------------------> Exe cutive System
(who controls the spotlight)
Agent ---------------------------------> Awareness System
(who sees the field)
Visual field -------------------------> Representational Space
Illuminated area --------------------> Attended area
ATTENTION AS SPOTLIGHT Inferential structure
• source domain
• target 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.
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)
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.
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
•