Transcript Symbolizing Quantity Praveen Paritosh [[email protected]] Department of Computer Science, Northwestern University, Evanston, IL 60201 Goals Representation 1.

Symbolizing Quantity
Praveen Paritosh [[email protected]]
Department of Computer Science, Northwestern University, Evanston, IL 60201
Goals
Representation
1. Representation: What do people know about quantities?
2. Learning: How do people learn about quantities from experience?
Experiment: Dimensional Partitions
Representations don’t arise in vacuum. There are at least three sources of constraints on a cognitively-plausible representation – Reasoning,
Ecological, and Psychological constraints.
Size labeling: subjects were asked to
label each country as SMALL/
MEDIUM/LARGE.
Agreement = 81.2% (p<0.01)
Examples
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.
• Labels like large setup
implicit ordinal relations,
ease comparison.
• Must keep tract of
interesting points to classify
and estimate
Dimensional Partitions
Symbols like Large and Small, which arise from
distributional information about how the quantity varies.
CARVE: A computational model
Ci
height
short
average
tall
Difficulties
Varied 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!
Add these facts
to original cases
Quantity 1
Cj
Ecological
Quantities vary
In range and distribution of values
But in causally connected ways
Structural bundles: e.g., as the
engine mass increases, BHP, Bore,
Displacement increases; RPM
decreases.
L1
Structural Partitions
• Distributional information
• Causal relationships
between quantities
Symbols 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]
Symbolizations of Quantity
Named 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
(isa Algeria
(HighValueContextualizedFn
Area AfricanCountries)
.
.
.
Dimensional
partitioning for
each quantity
Temperature of water (degree Celsius)
Psychological
Landmark 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 assimilation
Malmi and Samson, 1983
Social psychology on stereotypes
Acquisition of dimensional adjectives
Ryalls and Smith, 2000
S1
Theories/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.
Comparison
Is John taller than Chris?
Semantic Congruity Effect [Flora
and Banks, 1977]
Classification
Is John tall?
Is the water boiling?
Estimation
How tall is John?
Anchoring and adjustment [Tversky
and Kahneman, 1974]
Country naming: subjects were asked
to name each of the 54 countries on the
map.
Mean correctly named = 6/54
sd = 6.5
Freezing Point
• Landmarks
• Distributional information.
Boiling Point
Income of people ($)
S2
Motivation
Reasoning
Structural
clustering
using SEQL
S3
Quantities: 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.
L2
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
Related Work
Psychological Theories
Computational Models
• Quantitative Estimation: Peterson
• Similarity and Generalization:
MAC/FAC [Forbus, Gentner and
Law, 1995], SME [Falkenhainer,
Forbus and Gentner, 1989], and
SEQL [Kuehne, Forbus, Gentner and
Quinn, 2000]
and Beach (1967), Tversky and
Kahneman (1974), Brown and Siegler
(1993), Linder (1999).
Poverty Line
• Similarity: Spatial [Shepard, 1962],
Lower
Class
Middle
Class
Upper
Class
Ci*
Set-theoretic [Tversky, 1977], and
Structural [Gentner, 1983].
Size of dictionaries (Number of Pages, Weight)
Pocket
Editions
Desktop
Editions
Library
Editions
Acknowledgements
This 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