lec 25 problem solving

7/27/2019 Lec 25 Problem Solving

1/28

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


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),


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


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.


6/28

Problem space:the set of all states that can be achieved

during the course of solving a problem.


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,


8/28

Working backwards Heuristic: Example


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


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.


11/28


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


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?


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.


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


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


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


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.


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)


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.


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


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


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


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


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


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)


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.


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

lec 25 problem solving

Documents