Analogical Reasoning

1
Analogical Reasoning

• CogSci 3790
• March 2003

2
Youve already performed analogical problem
solving in class today
3
Things youve already discussed

• Problem-solving with rules
• Analogy and similarity
• Case-based reasoning (CBR)
• Analogy in education

4
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

5
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

6
Rule-Based Problem Solving

• My step sister is visiting this weekend, and
  shes bringing her exchange student from Hungary.
• How do I get from here to the World of Coca-Cola?

7
Some characteristics of rule-based problem solving

• Well-defined search space
• Easy to develop a chain of operators that,
  collectively, solve the problem
• Easy to decompose the solution to explain it
• Soundness
• If the operators are sound, then the solution is
  sound
• Possible to show why some solutions are better
  than others (time, distance of alternatives)
• How would I model this?

8
Modeling rule-based problem solving

• Model using rules, of course!
• What dimensions of the task can we model?
• Solution
• Protocol of intermediate problem-solving steps
• Effect of broken rules
• Developmental effects

9
Analogical Problem Solving

• What are good places in Atlanta to take a
  Hungarian teenager?

10
What we may base our solutions on

• Other visits
• Parents, family, friends
• Other teenagers
• Visitors from foreign lands or from places really
  different from Atlanta vs. visitors from other
  U.S. cities
• Are these explanations sound? Can we show that
  some are better than others?

11
What characteristics of comparison can we use in
our models?

• Correspondences?
• Closeness or aptness of analogies?
• Inferences?

12
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

13
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

14
Spatial representations of analogies

• Suppose that each concept is a point in some
  large, multidimensional concept space
• Goose
• Duck
• Sheep
• More similar concepts are closer, more different
  are farther away

15
Creating a concept space

• Input A proximity matrix
• Output A multidimensional space with a location
  for each item
• Example How similar (1-99) are
• Green and red?
• Green and yellow?
• Blue and violet?
• And so on

16
Proximity matrix for color similarity
From Markman (1997), Knowledge Representation.
17
MDS results on color similarity
From Markman (1997), Knowledge Representation.
18
Results of MDS algorithm in numeral similarity
data
From Markman (1997), Knowledge Representation.
19
Rips, Fitts Shoben (1973)
20
Summary Spatial models of analogy

• Everything a point in a conceptual space
• Similarity and difference represented by distance
• Given sets of pairwise similarity estimates, we
  can (sometimes) automatically derive a conceptual
  space
• Higher-order spaces hard to derive and hard to
  visualize

21
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

22
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

23
Feature-based models

• Tverskys critique of spatial models
• Tverskys feature-based model of similarity

24
Tverskys Axioms

• Implications of spatial similarity models
• Minimality
• Symmetry
• Triangle Inequality
• Buteach is not true of humans.

25
Minimality

• d(x,x) d(y,y) 0.
• Everything is most similar (or proximate) to
  itself
• Each thing is as similar to itself as another
  item is similar to itself.
• Dog, Dog
• Freedom, Freedom
• George Washington, George Washington
• 1.23 , 1.23

26
Problems with minimality

• Some things are more similar to themselves than
  others
• Example Cross-mapping experiment by Gentner
  Ratterman
• When choosing between multiple potential similar
  parts, complex identity matches have a stronger
  pull than weak identity matches.

27
Symmetry

• d(x,y) d(y,x).
• A is as similar to B as B is to A.
• d(Cuba, China) d(China, Cuba)
• d(butcher, surgeon) d(surgeon, butcher)
• Experiments
• Similarity of countries (Tversky)
• Similarity of good and bad forms (Tversky)
• Roschs A is essentially B study.

28
Triangle Inequality

• d(x,y)lt d(x,z)d(y,z)
• d(atlanta,chicago) lt d(atlanta,indianapolis)
  d(indianapolis, chicago)
• d(goat,sheep) lt d(goat, pig) d(pig,
  sheep).

29
Problems with Triangle Inequality

• Difficult to falsify, but
• d(watch,bracelet)d(watch,clock) ltlt
  d(bracelet, clock)
• d(box,barrel)d(box,toy-block) ltlt d(barrel,
  toy-block)

30
Tverskys Conclusion

• Because of these three problems, spatial models
  are inadequate
• Proposed feature-based model instead

31
Example Pens and Chalk

• PEN
• Oblong
• Writing-instrument
• Marking-item
• Pointed
• Uses-ink
• Inexpensive
• Contains-cartridge
• Made-of-plastic

• CHALK
• Oblong
• Writing-instrument
• Marking-item
• Bipolar
• Made-of-chalk
• Inexpensive

32
Pens and Chalk

• PEN
• Oblong
• Writing-instrument
• Marking-item
• Pointed
• Uses-ink
• Inexpensive
• Contains-cartridge
• Made-of-plastic

• CHALK
• Oblong
• Writing-instrument
• Marking-item
• Bipolar
• Made-of-chalk
• Inexpensive

33
Pens and Chalk

• PEN
• Oblong
• Writing-instrument
• Marking-item
• Pointed
• Uses-ink
• Inexpensive
• Contains-cartridge
• Made-of-plastic

• CHALK
• Oblong
• Writing-instrument
• Marking-item
• Bipolar
• Made-of-chalk
• Inexpensive

Tverskys model is more sophisticated than this,
though, because it uses not just the features in
common, but those that are different as well!
34
Tverskys Contrast Model
s(a,b) ?f(AB) ?f(A-B) ?f(B-A).
35
Tverskys model Pens and Chalk
AB oblong, writing-instrument, marking-item,
inexpensive 4. A-B pointed, uses-ink,
contains-cartridge, made-of-plastic 2. B-A
bipolar, made-of-chalk 4.
Formula s(a,b) ?f(AB) ?f(A-B) ?f(B-A).
Assume ? 1.0, ?0.1, ?0.3. f() is a simple sum.
S(pen,chalk) 4 0.1(4) - .3(2) 3.0
S(chalk,pen) 4 0.1(2) - .3(4) 2.6
36
Does Tversky meet his own criticisms?

• Minimality
• Symmetry (or asymmetry)
• Triangle inequality

37
Other advantages of feature sets

• Independence of features
• Can be manipulated via set operations
• AND, OR, NOT, ?, ?.
• Divvies up conceptual space
• Keywords in library searches
• Canonicalization
• Can be computed in parallel (very important!)

38
Problems with feature-based models

• Features arent always independent
• Need to capture relational structure

39
Features arent always independent

• Assumption of independence isnt always true
• Some features cause others
• OBLONG, WRITING-INSTRUMENT
• Some features are categorically related
• Some features are part of a closed set of
  alternatives
• MADE-OF-PLASTIC, MADE-OF-CHALK

40
Need to capture relational structure

• Attempt1
• square
• circle
• above
• Attempt2
• above(square-a,circle-b)
• Attempt 3
• above(a,b)
• square(a)
• circle(b)

B
A
41
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

42
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)

43
How can we account for relational structure?

• Use a form of graph matching
• Match frames (Case-based reasoning)
• Match conceptual graphs (Structure Mapping)

44
SME Structure-Mapping Engine
Inputs propositional descriptions, with
incremental updates
Output Mappings (correspondences
candidate inferences)
BASE Description
TARGET Description
SME operates in polynomial time by exploiting
predicate labels and by using a greedy merge
algorithm
45
How do we test structural models?

• Correspondences
• Inferences
• Aptness
• Cross-mapping tasks!

46
Cross-mapping tasks

• Pit feature-based (a.k.a. attribute-based)
  similarity against relational similarity
• Two scenes (Gentner Markman)
• Man bringing a woman groceries
• Woman feeding a squirrel
• Do we map the woman to the woman, or the woman to
  the squirrel?
• Or, a robot repair-shop vs. a robot-repair shop.
• Key insight use of relational structure changes
  over time!

47
Cross-Mapping Experiment (Gentner, Ratterman
Forbus 1993)
Sticker-finding task for 3, 4, 5 yr olds.
48
Younger children were aided by rich structure in
the literal similarity task.
Children were consistently worse on the
cross-mapping task for rich stimuli.
49
Outline for Today

• How is solving problems by analogy different from
  solving problems via rules?
• Several broad models of analogy
• Spatial
• Feature-based
• Structural (including CBR)
• DISCUSSION

50
Models of analogy

• Not clear that humans use just one type of
  analogy
• Spatial color comparisons?
• For some processes, we may even use multiple
  comparison processes
• Good example retrieval

51
The Problem of Retrieval

• The analogies we retrieve are not always the same
  as those we find apt
• Dont look a gift horse in the mouth.

52
Dealing with the problem of retrieval

• Why dont we always retrieve the most apt
  analogy?
• Possibility We economizing on retrieval
• Comparing two cases involves only a little data
• Retrieving from a memory of millions of items
  involves a lot of data
• So maybe retrieval is different than comparison

53
MAC/FAC Similarity-based retrieval
Memory Pool of Cases
Probe case
CVmatch
Result memory item SME mapping
SME
CVmatch
SME
CVmatch
SME
CVmatch
Slower, structural matcher.
Cheap, fast, non-structural feature-based matcher
54
MAC/FAC is consistent with psychological evidence

• Primacy of the mundane
• Literal similarity gt Surface match gt True
  analogical match
• Occasional distant remindings
• Expert encoding facilitates accurate retrieval
• Expects more deeply encode causal structure
• May have a specialized set of relations to draw
  upon

55
Conclusion

• Reasoning by analogy is very different than
  rule-based reasoning
• We can still model it.
• Different models make different predictions
• Spatial, feature-based, structural
• We may use different analogical reasoning
  processes for different cognitive tasks

56
THE END