Analogical Reasoning Ron Ferguson

Youve already performed analogical problem solving in class today

Problem-solving with rules Analogy and similarity Case-based reasoning (CBR) Analogy in education Things youve already discussed

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)

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)

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?

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 analogy Spatial Feature-based Structural (including CBR)

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)

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 From Markman (1997), Knowledge Representation.

MDS results on color similarity 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 analogy Spatial Feature-based Structural (including CBR)

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)

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

Tverskys 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) – Roschs 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)

Tverskys 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 Tverskys model is more sophisticated than this, though, because it uses not just the features in common, but those that are different as well!

Tverskys Contrast Model s(a,b) = f(A^B) – f(A-B) – f(B-A).

Tverskys 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? Minimality Symmetry (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 arent always independent Need to capture relational structure

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

Need to capture relational structure Attempt#1: square circle above 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 analogy Spatial Feature-based Structural (including CBR)

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)

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 TARGET Description SME operates in polynomial time by exploiting predicate labels and by using a greedy merge algorithm Inputs = propositional descriptions, with incremental updates BASE Description

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: Dont look a gift horse in the mouth.

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

MAC/FAC: Similarity-based retrieval Memory Pool of Cases Probe case Result = memory item + SME mapping SME 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