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: • 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),

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.

1 1 31 1073741824 61 11529215046068500002 2 32 2147483648 62 23058430092136900003 4 33 4294967296 63 46116860184273900004 8 34 8589934592 64 92233720368547800005 16 35 17179869184 65 184467440737096000006 32 36 34359738368 66 368934881474191000007 64 37 68719476736 67 737869762948382000008 128 38 137438953472 68 1475739525896760000009 256 39 274877906944 69 29514790517935300000010 512 40 549755813888 70 59029581035870600000011 1024 41 1099511627776 71 118059162071741000000012 2048 42 2199023255552 72 236118324143482000000013 4096 43 4398046511104 73 472236648286965000000014 8192 44 8796093022208 74 944473296573929000000015 16384 45 17592186044416 75 1888946593147860000000016 32768 46 35184372088832 76 3777893186295720000000017 65536 47 70368744177664 77 7555786372591430000000018 131072 48 140737488355328 78 15111572745182900000000019 262144 49 281474976710656 79 30223145490365700000000020 524288 50 562949953421312 80 60446290980731500000000021 1048576 51 1125899906842620 81 120892581961463000000000022 2097152 52 2251799813685250 82 241785163922926000000000023 4194304 53 4503599627370500 83 483570327845852000000000024 8388608 54 9007199254740990 84 967140655691703000000000025 16777216 55 18014398509482000 85 1934281311383410000000000026 33554432 56 36028797018964000 86 3868562622766810000000000027 67108864 57 72057594037927900 87 7737125245533630000000000028 134217728 58 144115188075856000 88 15474250491067300000000000029 268435456 59 288230376151712000 89 30948500982134500000000000030 536870912 60 576460752303423000 90 618970019642690000000000000

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

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.

Time of daySunrise 3:30 Sunset

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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

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?

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.

18 43 10 5

Use these three bottles to pour the perfect amount into the glass

9 42 6 21

18 48 4 22

(1)

(2)

(3)

28 76 3 25(4)

fill bottle B, pour into bottle A, then pour into bottle C twice…5 oz

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:

Did subjects realize the connection?

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 weight-sensitive mines. The general’s solution was to divide his soldiers into many small platoons and approach the fortress from different directions.

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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Strong Hint No Hint

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 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

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)

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 •