Goals Representation: What do people know about quantities?

1. height short average tall Dimensional partitioning for each quantity (isa Algeria (HighValueContextualizedFn Area AfricanCountries) . . . Ci Add these facts to original cases Quantity 1 S2 Structural clustering using SEQL Cj S1 S3 Ci* L1 L2 Symbolizing Quantity Praveen Paritosh [paritosh@northwestern.edu] Department of Computer Science, Northwestern University, Evanston, IL 60201 Representation Representations don’t 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% (p<0.01) • 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 doesn’t 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. Library Editions Pocket Editions Desktop Editions Cognitive Science, 2004, Chicago