Analogical Reasoning
Ron Ferguson
You’ve already performed analogical problem solving in class today
Problem-solving with rules Analogy and similarity Case-based reasoning (CBR) Analogy in education
Things you’ve already discussed
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Rule-Based Problem Solving
My step sister is visiting this weekend, and she’s bringing her exchange student from Hungary.
How do I get from here to the World of Coca-Cola?
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?
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
Analogical Problem Solving
What are good places in Atlanta to take a Hungarian teenager?
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?
What characteristics of comparison can we use in our models?
Correspondences? “Closeness” or “aptness” of analogies? Inferences?
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
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
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…
Proximity matrix for color similarity
99 85 40 25 70
85
40
25
70
99 70 25 25
70 99 99 10
25 55 55 55
25 10 55 99
Violet
Blue
Green
Yellow
Red
Violet
Blue
Green
Yellow
Red
From Markman (1997), Knowledge Representation.
MDS results on color similarity
Yellow
Orange
Gre
en
Red
Blue
Violet
From Markman (1997), Knowledge Representation.
Results of MDS algorithm in numeral similarity data
From Markman (1997), Knowledge Representation.
Rips, Fitts & Shoben (1973)
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
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Feature-based models
Tversky’s critique of spatial models Tversky’s feature-based model of similarity
Tversky’s Axioms
Implications of spatial similarity models:– Minimality– Symmetry– Triangle Inequality
But…each is not true of humans.
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
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.
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)– Rosch’s “A is essentially B” study.
Triangle Inequality
d(x,y)<= d(x,z)+d(y,z) d(atlanta,chicago) <=
d(atlanta,indianapolis) + d(indianapolis, chicago)
d(goat,sheep) <= d(goat, pig) + d(pig, sheep).
Problems with Triangle Inequality
Difficult to falsify, but… d(watch,bracelet)+d(watch,clock) <<
d(bracelet, clock) d(box,barrel)+d(box,toy-block) << d(barrel,
toy-block)
Tversky’s Conclusion
Because of these three problems, spatial models are inadequate
Proposed feature-based model instead
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
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
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
Tversky’s model is more sophisticated than this, though, because it uses not just the features in common, but those that are different as well!
Tversky’s Contrast Model
s(a,b) = f(A^B) – f(A-B) – f(B-A).
Tversky’s model: Pens and Chalk
Formula: s(a,b) = f(A^B) – f(A-B)
– f(B-A).
A^B = {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.
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
Does Tversky meet his own criticisms?
MinimalitySymmetry (or asymmetry)Triangle inequality
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!)
Problems with feature-based models
Features aren’t always independent Need to capture relational structure
Features aren’t always independent
Assumption of independence isn’t 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
Need to capture relational structure
Attempt#1:squarecircleabove
Attempt#2:above(square-a,circle-b)
Attempt #3:above(a,b)square(a)circle(b)
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
Outline for Today
How is solving problems by analogy different from solving problems via rules?
Several broad models of analogySpatial Feature-basedStructural (including CBR)
How can we account for relational structure?
Use a form of graph matching– Match frames (Case-based reasoning)– Match conceptual graphs (Structure Mapping)
SME: Structure-Mapping Engine
Output = Mappings (correspondences + candidate inferences)
SME
TARGETDescription
SME operates in polynomial time by exploiting predicate labels and by using a greedy merge algorithm
Inputs = propositional descriptions, with incremental updates
BASEDescription
How do we test structural models?
Correspondences Inferences Aptness
Cross-mapping tasks!
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!
Cross-Mapping Experiment (Gentner, Ratterman & Forbus 1993)
Sticker-finding task for 3, 4, & 5 yr olds.
Children were consistently worse on the cross-mapping task for rich stimuli.
Younger children were aided by rich structure in the literal similarity task.
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
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
The Problem of Retrieval
The analogies we retrieve are not always the same as those we find apt:
“Don’t look a gift horse in the mouth.”
Dealing with the problem of retrieval
Why don’t 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
MAC/FAC: Similarity-based retrieval
Memory Pool of Cases
Probe case
Result = memory item+ SME mapping
SME
SME
SME
CVmatch
CVmatch
CVmatch
CVmatch
Cheap, fast, non-structural feature-based matcher
Slower, structural matcher.
MAC/FAC is consistent with psychological evidence
Primacy of the mundane– Literal similarity > Surface match > 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
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
THE END