Analogical Reasoning Ron Ferguson. Youve already performed analogical problem solving in class...

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)

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



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


Results of MDS algorithm in numeral similarity data


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



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



Pens and Chalk



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 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


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

Analogical Reasoning Ron Ferguson. Youve already performed analogical problem solving in class...

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