Goals1. Representation: What do people know about quantities? 2. Learning: How do people learn about quantities from experience?

ExamplesQuantities: Price, Height, Temperature, Intelligence, etc.

• Basketball players are tall. • Life below poverty line is hard. • Canada is larger in area than US. • Kia makes cheap cars.

Experiment: Dimensional Partitions

Symbolizing QuantityPraveen Paritosh [paritosh@northwestern.edu]

AcknowledgementsThis research is supported by the Computer Science Division of the Office of Naval Research. The authors would like to thank Ken Forbus, Dedre Gentner, Chris Kennedy, Lance Rips and Sven Kuehne for insightful comments and discussion on the work presented here.

Cognitive Science, 2004, Chicago

Department of Computer Science, Northwestern University, Evanston, IL 60201

Related WorkPsychological Theories Computational Models• Quantitative Estimation: Peterson and Beach (1967), Tversky and Kahneman (1974), Brown and Siegler (1993), Linder (1999).• Similarity: Spatial [Shepard, 1962], Set-theoretic [Tversky, 1977], and Structural [Gentner, 1983].

• Similarity and Generalization: MAC/FAC [Forbus, Gentner and Law, 1995], SME [Falkenhainer, Forbus and Gentner, 1989], and SEQL [Kuehne, Forbus, Gentner and Quinn, 2000]

MotivationTheories/computational models of similarity, retrieval and generalization do not take quantities into account in a psychologically plausible manner.

Similarity• How to compute similarity/difference along a dimension?• How to combine similarity/differences across multiple dimensions?

Retrieval• A bird with wingspan of 1m should remind me of other large birds as much as a red object reminds me of other red objects.

Generalization• Generating qualitatively important distinctions and learning distributional information from experience.

Knowledge representation There is a disconnect between symbolic and numerical representations of quantity, e.g. CYC has the notion of large and knows the area of Brazil, but doesn’t know that Brazil is a large country.

Symbolizations of QuantityNamed points and intervals on the space of values –

• Freezing point/ Boiling Point• Poverty line/ Lower class/ Middle class/ Upper class• Short/ Average/ Tall• Cheap/ Expensive

DifficultiesVaried sources

• Personal experience: what spicy?• Science: phase transitions.• Society: poverty line.

Context variability: What is expensive for me, or this place might not be true for someone else or somewhere else. Vague: Sorites paradox [Varzi, 2003]

But people get along!

RepresentationRepresentations don’t arise in vacuum. There are at least three sources of constraints on a cognitively-plausible representation – Reasoning, Ecological, and Psychological constraints.

ReasoningComparison

Is John taller than Chris? Semantic Congruity Effect [Flora and Banks, 1977]

ClassificationIs John tall?Is the water boiling?

EstimationHow tall is John?Anchoring and adjustment [Tversky and Kahneman, 1974]

EcologicalQuantities vary

In range and distribution of valuesBut in causally connected ways

Structural bundles: e.g., as the engine mass increases, BHP, Bore, Displacement increases; RPM decreases.

PsychologicalLandmark effects

Similarity across landmarks higher than on the same side of landmark [Goldman, 1972].Asymmetry in comparing to/from landmarks [Rosch, 1975, Holyoak and Mah, 1984].

Distributional assimilationMalmi and Samson, 1983Social psychology on stereotypes

Acquisition of dimensional adjectivesRyalls and Smith, 2000

• Labels like large setup implicit ordinal relations, ease comparison. • Must keep tract of interesting points to classify and estimate

• Distributional information• Causal relationships between quantities

• Landmarks• Distributional information.

Dimensional PartitionsSymbols like Large and Small, which arise from distributional information about how the quantity varies.

heightshort average tall

Structural PartitionsSymbols like Boiling Point and Poverty Line, that denote changes in quality, usually changes in underlying causal story and structural aspects of objects in concern. Builds upon, and generalizes the ideas of:

• Limit points [Forbus 1984]• Phase transitions [Sethna 1992]• Attribute co-variation or Feature correlation [Malt and Smith, 1984]

Temperature of water (degree Celsius)

Income of people ($)

Size of dictionaries (Number of Pages, Weight)

Freezing Point Boiling Point

Poverty Line

Lower Class

Middle Class

Upper Class

Pocket Editions

Desktop Editions

Library Editions

Size labeling: subjects were asked to label each country as SMALL/ MEDIUM/LARGE.

Agreement = 81.2% (p<0.01)

Country naming: subjects were asked to name each of the 54 countries on the map.

Mean correctly named = 6/54sd = 6.5

CARVE: A computational model

Dimensional Partitions• K-means clustering of values on each quantitative dimension. • High/Medium/LowValueContextualizedFn (isa Algeria

(HighValueContextualizedFn Area AfricanCountries))

• 74% agreement on the Countries data.

Structural Partitions• Projection of structural clusters generated by SEQL [Skorstad et al, 1988; Kuehne et al, 2000] onto quantities. • No interesting structural partitions found because of lack of rich causal knowledge in knowledge base

Dimensional partitioning for each quantity

(isa Algeria (HighValueContextualizedFn Area AfricanCountries)...

Add these facts to original cases

Structural clustering using SEQL

S1 S2 S3Cj

Ci

Ci*

Quantity 1

L2L1