A Sketch of a theory of Quantity

1
A Sketch of a theory of Quantity

• Praveen Paritosh
• Qualitative Reasoning Group,
• Department of Computer Science,
• Northwestern University, Evanston IL 60201 USA

2
Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

3
Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

4
Overarching Goal

• How do people understand quantities?
• Ubiquity of quantities Price, Height,
  Temperature, Intelligence, etc.
• For instance, we know that
• Zero degree Celsius is an important temperature
  for water
• Life below poverty line is hard
• Brazil is a large country
• Basketball players are tall
• Just how much pepper might make the meal too hot
  

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Why important?

• Why important?
• Qualitative Reasoning Qualitative
  representations of quantity.
• Knowledge representation a disconnect between
  symbolic and numeric representations, e.g., CYC.
• Psychology Dimensions, Similarity.
• Cognitive Science Computational models of
  retrieval, similarity and generalization.
• Linguistics Dimensional adjectives.

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Specific Questions

• Q1. How is our knowledge of quantities
  represented?
• Cognitively plausible representational machinery.
  
• Q2. What is the content of these representations?
  
• How do we carve up the continuous?
• Mechanisms to automatically make the
  distinctions.
• Mapping the distinctions (e.g., symbols in
  quantity space) to numeric values.

• To answer, combine evidence and constraints from
• Psychology
• Natural Language, Linguistics
• Ecological constraints
• Reasoning/Task constraints

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Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

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Qualitative Representations of Quantity

• Many frameworks
• Status Algebra
• Normal/Abnormal
• Sign Algebra
• -, 0,
• Quantity Spaces
• Tfreeze, Tboil
• (Fuzzy) Intervals
• Order of Magnitude
• Finite Algebras, etc.

• Quantizations of the continuous (Reals)
• Weaker than quantitative
• Symbolic
• Behaviorally meaningful
• Little work on
• Cognitive plausibility
• What distinctions to make?

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Models of Similarity

• Models of similarity do not take numbers into
  account appropriately
• Feature vector, or metric space
• CBR Leake, 1996
• Minkowski distance metric
• Structured
• SME Falkenhainer, Forbus and Gentner, 1989
• Ignore

• Numeric aspects of representation do not
  contribute to similarity.
• How to compute similarity along a dimension?
• How to combine similarity along different
  dimensions?

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Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

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Three pillars of support

• Reasoning/task constraints
• What kind of reasoning tasks we do with that
  quantity?
• Ecological constraints
• How does the quantity vary in the real world?
• Psychological constraints
• How do we perceive/understand the quantity?

Representations dont arise in vacuum.
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Reasoning Constraints
What do we do with quantities?

• Comparison
• Is A taller than B?
• Semantic congruity effect Flora and Banks, 1977
• Classification
• Is A tall?
• Is the water boiling?
• Estimation
• How tall is A?
• Anchoring and adjustment Tversky and Kahneman,
  1974.

• Labels like large set-up implicit ordinal
  relations, ease comparison.
• Must keep track of interesting points on the
  space of values, to classify and estimate.

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Ecological Constraints
What constrains the value a quantity can take?

• Quantities vary
• But in causally connected ways
• Structural Bundles
• e.g., as the engine mass increases, BHP, Bore,
  Displacement increases RPM decreases.

• Our representations must have
• Distributional information
• Role and relationship of a quantity in underlying
  structure/mechanism

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Psychological Constraints 1
What do we know about how we process/understand
quantities?

• Evidence regarding landmarks.

• Across landmark Harnad, 1987

a
a
b

b
Sim(a, b) gt Sim(a, b)

• Asymmetry in comparing to/from landmark Rosch,
  1975 Holyoak and Mah, 1984

Sim(a, ) gt Sim(, a)
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Psychological Constraints 2

• Evidence regarding distributional assimilation
• Malmi and Samson, 1983
• Social Psychology on stereotypes.
• Evidence regarding acquisition of dimensional
  adjectives
• Ryalls and Smith, 2000
• also analyses in Linguistics thereof Bierwish,
  1967 Kennedy, 2003).

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Summary What do we know about quantities?

• Their range, likelihood of values, variability
• Distributional information.
• A carving-up of the space of values
• Dimensionally dimensional adjectives like large,
  small, etc.
• Structurally special points like freezing point,
  poverty line, etc.

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Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

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Structural Limit Points (SLP)

• Builds upon, and generalizes the following ideas
  
• Limit points (Forbus 1984)
• Phase transitions (see Sethna 1992)
• Attribute co-variation, or, Feature correlation
  (Malt and Smith, 1984)

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Structural Limit Points
Structurally un-predictive quantity.
structural clustering
C3
C2
C1
Structurally predictive quantity, e.g., GDP
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Structural Limit Points Examples
Temperature of water (degree Celsius)
Freezing Point
Boiling Point
Income of people ()
Poverty Line
Middle Class
Upper Class
Lower Class
Size of dictionaries (Number of Pages, Weight)
Library Editions
Pocket Editions
Desktop Editions
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Structural Limit Points

• Change in underlying causal story, mechanism,
  essentially, relational structure.
• Processes (QP Theory) a special case of
  structural bundles.
• Phase transition analogy
• structure of relationships equations of phase
• Crisp/soft SLPs first-/second- order phase
  transitions.

Discontinuities in the underlying reality as
captured by the structure in the representation.
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Distributional Partitions

• Many (most?) references to quantities in NL are
  dimensional adjectives (e.g., large, small,
  expensive).
• QR, and Fuzzy Logic have indeed used linguistic
  symbols, but very little emphasis on their having
  similar meaning to that in NL.
• Goal Algorithm that answers the following
  questions as people would
• How big is a large flamingo?
• How big is a large animal?
• Possible criticism Ill-founded Are people
  consistent?

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Distributional Partitions

• A large ltxgt acquires its meaning by identifying
  the large tail-end of the distribution of values
  of size of ltxgt.
• Most common distributions uniform, normal, zipf,
  skewed.
• Open question How do we carve up the
  distributions?

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Why Distributional Partitions?

• Bootstrapping the first-cut partitions, based
  purely on variance of quantity values.
• Parameters that are not structurally central
  (height of professors, in contrast with height of
  basketball players).
• No structural variance in the class of examples
  (size of adult male penguins).

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The Sketch

• Compiled by SEQL, and used to build
  distributional partitions and structural limit
  points.

Distributions

• First-cut symbolic representations, e.g., large.
• Makes retrieval (MAC/FAC) more quantity-aware.

Distributional Partitions

• Relational representations, e.g., someones
  income is below poverty line.
• Makes structural similarity more quantity-aware.
  
• Helps make analogical inferences about
  quantities.

Structural Limit Points
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Proposed Evaluation

• Build an Analogical Estimator, which will
  estimate a quantitative value by retrieving most
  similar example for which we know the value.
• A key part of Back of the Envelope reasoner
  Paritosh and Forbus, 2003.
• Compare representations to experts
  representations.
• Test if improved retrieval and similarity with
  these representations.

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Outline

• Goals and Questions
• Background and Motivation
• Desiderata and Arguments
• Building Cognitively Plausible Representations
• Conclusions

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Conclusions

• Cognitively plausible symbolic representations of
  quantity a key part of answer to
• How do we understand quantities?
• How are quantities implicated in retrieval,
  similarity and generalization?
• The key aspects of such representations
• Distributions
• Distributional partitions
• Structural Limit Points

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Specific Questions Redux

• Q1. How is our knowledge of quantities
  represented?
• Quantity space augmented.
• Q2. What is the content of these representations?
  
• How do we carve up the continuous?
• Distributional Partitions
• Heuristics of how people make these
• Structural Limit Points
• Starting from SEQL, implement these ideas into an
  incremental learning algorithm

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Acknowledgements

• Ken Forbus
• Dedre Gentner
• Lance Rips
• Chris Kennedy
• Sven Kuehne
• Office of Naval Research

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Extra Slides
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Open Questions
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SEQQL
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Scratch slides
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• Key concern is reasoning, little work that
  establishes cognitive plausibility.
• Little work on the content of our
  representations, or, What distinctions to make?
• Necessary and Relevant, as chosen by the modeler,
  for the task (but see Sachenbacher and Struss,
  2000).

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Retrieval, Similarity and Generalization

• Analogical matching
• SME Falkenhainer, Forbus and Gentner, 1989
• Similarity-based retrieval
• MAC/FAC Forbus, Gentner and Law, 1995
• Generalization from examples
• SEQL Kuehne, Forbus, Gentner and Quinn, 2000

• Numeric aspects of representation do not
  contribute to similarity.
• How to compute similarity along a dimension?
• How to combine similarity along different
  dimensions?
• Red versus Large
• Learning the sense of the quantity, e.g., a trip
  to the zoo.