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Symbolizing Quantity Praveen Paritosh
paritosh_at_northwestern.edu
Department of Computer Science, Northwestern
University, Evanston, IL 60201
Representation Representations dont arise in
vacuum. There are at least three sources of
constraints on a cognitively-plausible
representation Reasoning, Ecological, and
Psychological constraints.

• Goals
• Representation What do people know about
  quantities?
• Learning How do people learn about quantities
  from experience?

Experiment Dimensional Partitions
Size labeling subjects were asked to label each
country as SMALL/ MEDIUM/LARGE. Agreement
81.2 (plt0.01) Country naming subjects were
asked to name each of the 54 countries on the
map. Mean correctly named 6/54 sd 6.5

• Examples
• 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.

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

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

• Motivation
• 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.
• 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 doesnt know that Brazil is a large
  country.

CARVE A computational model
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.

• Structural Partitions
• 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

• 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.

• Distributional information
• Causal relationships between quantities

• 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

• 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

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
Temperature of water (degree Celsius)
Related Work
Freezing Point
Boiling Point

• Landmarks
• Distributional information.

Income of people ()

• 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!

Poverty Line
Middle Class
Upper Class
Lower Class
Size of dictionaries (Number of Pages, Weight)
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.
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Cognitive Science, 2004, Chicago