Analogical ThinkingARVIND KRISHNAA JAGANNATHAN

5

A Wise Man Once Said

Shore, Bradd. Culture in mind: Cognition, culture, and the problem of meaning. Oxford University Press, USA, 1996.

Do you see something funny here?

“That’s a really small slice you have there.”

“Oh boy! That dinosaur exhibit was huge!”

“Yeah I guess you are right.”

“The referee made a terribly wrong decision.”

Why does green signal mean let’s go ahead with something? And why does someone give you a red flag to warn about a potential danger?

“Lakoff, George, and Mark Johnson. Metaphors we live by. Vol. 111. London: Chicago, 1980.”

What is an Analogy?• It is not a mere numerical count of the number of features/attributes

which map from a source concept to a target concept.

• It is not a general account of relatedness.

“An analogy is an assertion that a relational structure that normally applies in one

domain can be applied in another domain”

Preliminary Assumptions

• Domains and situations are a system of objects, object attributes and relations between objects

• Knowledge = Propositional Network of nodes+predicates

• Attributes vs. Relations; First-Order vs. Second-Order Predicates

• Representations mirror cognitive constructs

Start

MOTION(Vehicle, Stop)

MOTION(Vehicle, Start)

COLOR(Signal,

Red)

COLOR(Signal,

Green)

CAUSES[COLOR(Signal, Red), MOTION(Vehicle, Stop)]

Street Domain

Causal Relations

The next time your advisor Red Flags you…

Stop

PROGRESS(Research, Halted)

PROGRESS(Research, Smooth)

RED_FLAG(Professor,

Student)

GREEN_FLAG(Professor, Student)

RESULTS[PROGRESS(Research, Halted), RED_FLAG(Professor, Student)]

Academic Domain

Diagnostic Relations

Structure Mapping: Interpretation Rules for Analogy

Class Hierarchy Analogy

Vehicle

Car Truck

Is-a

Is this a meta-analogy?!

A T(Target) is (like) a B(Base)

Target

Base

Is like-a

Let’s take an example

“…What light from yonder window breaks?/It is the east, and Juliet is the sun!...”

I bet you would know this:

“Twinkle, twinkle little star,How I wonder what you areUp above the world so high,Like a diamond in the sky.”

Rule 1: Discard Attributes of ObjectsStar• BRIGHTNESS: An absolute measure in

lumens. BRIGHTNESS(Star) = x lm

• DISTANCE: Distance from Earth (or some other referential celestial object). DISTANCE(Star, Earth) = y x 10z km

• LUMINOSITY. This is the amount of energy generated in the star and released as electromagnetic radiation.

• RADIUS

• CHEMICAL COMPOSITION CARBON_CONTENT(Star) = k NITROGEN_CONTENT(Star) = l

• TEMPERATURE

Diamond• BRIGHTNESS

• RADIUS (?)

• CHEMICAL COMPOSITION: Cannot obviously have a 1:1 match with a star

• TEMPERATURE

• Moh’s Scale of HARDNESS

• COST

• VALUE AS GIFT

Rule 2: Preserve Relations Between ObjectsStar

• Covered by several layers of thick, highly carbon-dense gaseous layers. SURROUNDS(Star, Carbon Layers)

• Appears to twinkle when viewed from a distance. APPEARANCE(Twinkles, Distance)

• Twinkling is caused by multiple refractions of light in differently dense layers of the atmosphere, eventually leading to total internal reflection.

1. CAUSES[SURROUNDS(Star, Carbon Layers), MULTIPLE_REFRACTIONS(Light)]

2. CAUSES[MULTIPLE_REFRACTIONS(Light), APPEARANCE(Twinkles, Distance)]

Diamond

• Covered by several layers of thick, highly carbon-dense layers. SURROUNDS(Diamond, Carbon Layers)

• Appears to twinkle when viewed from a distance. APPEARANCE(Twinkles, Distance)

• Twinkling is caused by multiple refractions of light in differently dense (solid) layers of the diamond, eventually leading to total internal reflection.

1. CAUSES[SURROUNDS(Diamond, Carbon Layers), MULTIPLE_REFRACTIONS(Light)]

2. CAUSES[MULTIPLE_REFRACTIONS(Light), APPEARANCE(Twinkles, Distance)]

Rule 3: The Systematicity Principle (aka more interesting = more appropriate) [Isomorphism Constraint]

Noteworthy Points

• Rules are purely based on the structural representations of knowledge.

• Content plays a limited role.

• Need to express representations consistently across domains.

• Establishing “seemingly correct” relationships does NOT ensure an instantiation of the concept mapping in the target.

• Technique can be used to generate hypotheses in a semi-automated manner. No scope for verification of the hypothesis.

• Experiments/observations/methods exist for generating such candidate relations for analogy mapping (e.g.., mass spectrometry in the case of an atom to estimate weight of the nucleus and electrons)

Domain Comparisons – A Continuum of Categories

• Literal Similarity: A large subset of attributes as well as relations match between the source and the target. “The sun is a star like the Alpha Centauri”.

• Analogy: There is a low attribute match, but it is possible to establish a high relation match. “The structure of the atom is similar to the solar system”.

• Abstraction: Source and Target concepts are not instantiated. “The main driving force in an atom is the centrifugal force of rotation along a fixed orbit”.

few

many

Attribute Matches

few

many

Relation Matches

Literal Similarity

Analogy

Abstraction

Anomaly

“Twinkle, twinkle little star,

How I wonder what you are

Up above the world so high,

Like an iPod Touch in the sky.”

few

many

Attribute Matches

few

many

Relation Matches

Anomaly

• Hardly any (or no) attribute as well as relational matches.

• “Totally unrelatable”• A conceptual fallacy if

assumed to be true.Its ABSURD!

Empirical Support of the Structure Mapping Theory

• Interpretation of rules = Meaning(Parti)

• Rules clearly demarcate the boundaries between different categories of domain comparisons.

• Semantic relationships during the mapping process are established syntactically (i.e., according to a well-defined set of rules and a consistent notation)

Related Research• Merlin System: Mechanism for viewing a target as a similar

object to a source. Involves explicitly comparing their shared and non-shared predicates.

• Winston’s propositional representation: Perform an algorithm similar to forward chaining to derive certain general (hidden) rules from established analogies.

• Similar work by Gick & Holyoak: Constructing general schemas representing the transformation in problem-solving techniques in parallel to analogical matching.

• Theory of Analogical Shift Conjecture: Adapting the solution of a problem in a different domain, to solve a “similar” problem in one’s domain.

Analogical Shift Conjecture

Problem Statement

Abstraction“Similar” Problem

Existing Solution

AbstractionAbstracted

Solution

Adapt/Modify

Solution to Problem

Domain A Domain X

1. Understand causal relations between domains

2. A new relational model/data store required to store sematic relations between objects

Select problem from X where problem_type “LIKE”

A.problem_type

Analogical Mapping by Constrain Satisfaction

The Mapping Question

Components of Analogy1. Selecting a “feasible” source

2. Mapping

3. Analogical Inference/Knowledge Transfer

4. Learning

• Correct conceptual mapping is central to the establishment of “meaningful” analogies.

• Is there a common set of principles that govern mapping across different domains?

• This cannot be established without taking into account goals and purposes of the cognitive system.

Knowledge required depends on the type of analogy

Qualitative traits

1. Ruling Style2. Control of the

“parliament”3. Popularity among peopleFidel Castro Daniel

Ortega

Cuba Nicaragua

Sugarcane production

Sugarcane production1. Temperature

2. Rainfall3. Other relevant weather

patterns

Quantitative Results/Observations

A Constraint-Satisfaction Theory• Assuming two potentially analogous components each have predicates and constants,

then

Total possible number of mappings:

• No apriori basis for selecting a “right” mapping

• Clearly a case of an under constrained CSP!

1. Ensuring structural consistency: Allow only isomorphic mapping from a target to a source(i.e., the mapping needs to be one-to-one and onto; more formally a bijection). Thus an analogy will be,

2. Ensure semantic consistency: There should be sufficiently high feature overlap between source and target (affects source retrieval more).

3. Pragmatic Centrality: Identify the crucial similarities; discard the “insignificant ones” (don’t focus on solar winds; focus on the sun!). Similar to the task faced by unsupervised learning algorithms; which semantic tags weigh more than everything else?

ACME: A Cooperative Algorithm for Mapping1. Governed by two principles of information

processing:(a)Graceful degradation: As input degrades,

output should at least be partial (not non-existent).

(b)Least Commitment: Perform lazy updates. Do not perform a mapping/update which may have to be undone. (“If it ain’t broken don’t fix it).

2. It is co-operative in the sense that the algorithm can be executed parallel in scenarios where the final analogy can simply be represented as a composition of the analogies of individual sub-units of the source and target.

3. Supports two distinct types of queries:(a)Cross-structure queries: Apply the

inference from one model to answer a question in the other.

(b)Internal queries: Form a hypothesis of the source model from the target’s attributes.

ACME Mapping Network. Combinatorial explosion of the states is prevented by implementing correspondence constraints.

Applications of ACME

AKA Functions served by Analogical Reasoning

1. Problem solving: Also in coming up with solutions to design problems in association with a TRIZ-like approach (recall the laser surgery of cancer and the army attacking a fort exercise).

2. Argumentation: Argue that the likelihood of two “similar” events is pretty close. States/resources/objects true in one event are most probably true in the other.

3. Understand less familiar topics by drawing a parallel with more familiar ones (teaching a KG kid laws of refraction from the nursery rhyme!)

4. Explain formal analogies and proofs in mathematics. (Mathematical Induction)

5. Use of metaphors to improve the aesthetic quality of language.

Scope for Improvement

1. Richer semantic information can be built automatically into the constraint graph.

2. Allow for re-representations: Different propositions can be established on the same set of predicates at different times or based on the context of knowledge transfer.

3. Allow for m-to-n mapping. i.e., Allow relations from the source/target to map to more than one relation of the other.

4. Flexible/dynamically modifiable set of constraints.

Objectives of the Presentation

Presenter should have fun presenting the content of the papers.

Content should be useful to the listeners.

Stimulating discussions emerge from the ideas of the paper.

? Listeners should have fun during the presentation.