lec 25 problem solving
Post on 14-Apr-2018
217 Views
Preview:
TRANSCRIPT
-
7/27/2019 Lec 25 Problem Solving
1/28
Problem Solving
-
7/27/2019 Lec 25 Problem Solving
2/28
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
-
7/27/2019 Lec 25 Problem Solving
3/28
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 isspelled out in the statement of
the problem
I l l def ined Examples:
finding a perfect mate,
writing a great novel
not obvious when a goal has been
reached,
Not obvious which is the relevantinformation
One strategy to solve ill-defined
problems is to add constraints (e.g.
operationally define the goal),
-
7/27/2019 Lec 25 Problem Solving
4/28
General Problem-Solving
Problem-solving as search Each problem has:
an ini tial state
a goal state: a set ofoperators(actions that change the current state
into a new state)
a path constraint
a problem space: set of all possible paths
-
7/27/2019 Lec 25 Problem Solving
5/28
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.
-
7/27/2019 Lec 25 Problem Solving
6/28
Problem space:the set of all states that can be achieved
during the course of solving a problem.
-
7/27/2019 Lec 25 Problem Solving
7/28
Heuristics for problem solving
Hil l 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,
-
7/27/2019 Lec 25 Problem Solving
8/28
Working backwards Heuristic: Example
-
7/27/2019 Lec 25 Problem Solving
9/28
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.
1 1 31 1073741824 61 1152921504606850000
2 2 32 2147483648 62 2305843009213690000
3 4 33 4294967296 63 4611686018427390000
4 8 34 8589934592 64 9223372036854780000
5 16 35 17179869184 65 18446744073709600000
6 32 36 34359738368 66 36893488147419100000
7 64 37 68719476736 67 73786976294838200000
8 128 38 137438953472 68 147573952589676000000
9 256 39 274877906944 69 295147905179353000000
10 512 40 549755813888 70 590295810358706000000
11 1024 41 1099511627776 71 1180591620717410000000
12 2048 42 2199023255552 72 2361183241434820000000
13 4096 43 4398046511104 73 4722366482869650000000
14 8192 44 8796093022208 74 9444732965739290000000
15 16384 45 17592186044416 75 18889465931478600000000
16 32768 46 35184372088832 76 37778931862957200000000
17 65536 47 70368744177664 77 75557863725914300000000
18 131072 48 140737488355328 78 151115727451829000000000
19 262144 49 281474976710656 79 302231454903657000000000
20 524288 50 562949953421312 80 604462909807315000000000
21 1048576 51 1125899906842620 81 1208925819614630000000000
22 2097152 52 2251799813685250 82 2417851639229260000000000
23 4194304 53 4503599627370500 83 483570327845852000000000024 8388608 54 9007199254740990 84 9671406556917030000000000
25 16777216 55 18014398509482000 85 19342813113834100000000000
26 33554432 56 36028797018964000 86 38685626227668100000000000
27 67108864 57 72057594037927900 87 77371252455336300000000000
28 134217728 58 144115188075856000 88 154742504910673000000000000
29 268435456 59 288230376151712000 89 309485009821345000000000000
30 536870912 60 576460752303423000 90 618970019642690000000000000
Working backwards:- value doubling every day is
equivalent to say that the value ishalved each preceding day
- the field was full Day 90th
- the field was half full on day 89th
-
7/27/2019 Lec 25 Problem Solving
10/28
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 representationof a problem can
be crucial for finding the solution.
-
7/27/2019 Lec 25 Problem Solving
11/28
-
7/27/2019 Lec 25 Problem Solving
12/28
Time of day
Sunrise 3:30 Sunsetbottom
top
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...
Position
descent ascent
-
7/27/2019 Lec 25 Problem Solving
13/28
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.
Possible or Impossible?
-
7/27/2019 Lec 25 Problem Solving
14/28
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.
-
7/27/2019 Lec 25 Problem Solving
15/28
18 43 10 5
Use these three bottles to pour the perfect amount into the glass
9 42 6 2118 48 4 22
(1)
(2)(3)
28 76 3 25(4)
fill bottle B, pour into bottle A, then pour into bottle C twice5 oz
Rigidity in use of the same strategy
-
7/27/2019 Lec 25 Problem Solving
16/28
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
-
7/27/2019 Lec 25 Problem Solving
17/28
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
-
7/27/2019 Lec 25 Problem Solving
18/28
Analogical Reasoning in problem solving
Literal.
Collapsing stars spin faster as their
size shrinks. This occurs because of a
principle called conservation ofangular 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.
-
7/27/2019 Lec 25 Problem Solving
19/28
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)
-
7/27/2019 Lec 25 Problem Solving
20/28
A problem with an analogous solution:
Did subjects realize the connection?
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 hissoldiers into many small platoons and approach the
fortress from different directions.
-
7/27/2019 Lec 25 Problem Solving
21/28
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: Thesolution to the attack problem might behelpful as you work on the radiationproblem.
The other half received no hint
Results:people could see the analogy ifthey were directed to do so, but noticingof this relationspontaneous was rare
Gick & Holyoak (1980)
92
20
0
25
50
75
100
Strong Hint No Hint
-
7/27/2019 Lec 25 Problem Solving
22/28
Gick and Holyoak (1983) highlighted the underlying
concept of convergence by presenting two analogousstories(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 analogyin 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
-
7/27/2019 Lec 25 Problem Solving
23/28
Expertise in Problem Solving
Experts tend to notice the crucial aspectsof the situation,
rather than focusing on superficial features.
Task: categorize simple physics problems.
Subjects: novices vs. Ph.D. physics studentsResults:
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
analogieswhen a common conceptual structure characterizes
a set of problems.Chi, Feltovich and Glaser
-
7/27/2019 Lec 25 Problem Solving
24/28
Analogical reasoning in science
ATTENTION AS SPOTLIGHTExamples
"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."
(Umilt, 1988)
The spotlight . . . cannot select one or two (or more)
objects that fall within the beam, or select differentproperties of a single object" (Logan, 1995, p. 106).
MEMORY AS A STOREHOUSE
-
7/27/2019 Lec 25 Problem Solving
25/28
ATTENTI ON AS SPOTLIGHT Mapping
SOURCE DOMAIN TARGET DOMAIN
(SPOTLIGHT) (ATTENTION)
Spotlight ---------------------------> Mechanism of attention
Agent --------------------------------> Executive System
(who controls the spotlight)
Agent ---------------------------------> Awareness System
(who sees the field)
Visual field -------------------------> Representational Space
Illuminated area --------------------> Attended area
-
7/27/2019 Lec 25 Problem Solving
26/28
ATTENTION AS SPOTLIGHT Inferential structure
source domain
An agent moves her spotlight, whichsheds 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 attentionsystem, which expresses attention
over some brain areas.
When the attentional system
expresses attention on a
representation the representationbecomes conscious.(can be seen
by the homunculus)
-
7/27/2019 Lec 25 Problem Solving
27/28
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 longperiods" (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.
-
7/27/2019 Lec 25 Problem Solving
28/28
Memory as Perception of the Past
the act of remembering involves the re-perception of internal representationsthat 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
top related