Conceptual dimensions

1
Conceptual dimensions

• Harry Delugach
• CS 635/796 - Computational models of cognition
• Summer 2006

2
Definitions

• Cognition
• The mental faculty or process of acquiring
  knowledge by the use of reasoning, intuition or
  perception
• Knowledge that is acquired through processes such
  as reasoning, intuition or perception
• Cognitive
• Relating to the process of acquiring knowledge by
  the use of reasoning, intuition or perception
• Relating to thought processes (in general)
• Cognitive science
• Scientific study of knowledge and how it is
  acquired, combining elements of philosophy,
  psychology, linguistics, anthropology and
  artificial intelligence

3
Cognition and Representations

• Goals of Cognitive Science
• Explanatory - develop theories that can explain
  various perceptions and processes
• Constructive - build artifacts (robots, programs)
  that can accomplish cognitive tasks
• Two main approaches to representations
• Symbolic - design a computing machine that
  operates on symbols
• Associationist - relationships between elements
  define the representation
• Connectionist - neural network whose structure
  and weights define a representation

4
Conceptual Spaces

• Concept acquisition not adequately modeled by
  either symbolic or associative approaches
• Concept learning seems closely tied to similarity
• Conceptual spaces - a geometric approach
• Based on quality dimensions
• Concepts are located as points in an
  n-dimensional space

5
Similarity

• Consider shapes and colors
• Members of each class have something in common
• Shapes have particular form, color has particular
  shade
• Members all differ in that same respect
• Shapes have different shapes, colors are
  different colors
• Members exhibit a resemblance order
• Red is more like orange than blue, etc.
• Members qualities are mutually exclusive
• A thing cannot be both a triangle and a circle

6
Quality dimension

• Shape and color are quality dimensions
• It is possible to consider distances along the
  dimensions
• Shorter the distance, the more similar
• Whats the basis of dimensions?
• Phenomenal - psychological perceptions and
  memories
• Testable by experiment
• Scientific - taken from some scientific theory
• Useful in constructive goal of cognitive science

7
Properties of quality dimensions

• Ordering
• Points along the dimension are ordered
• Continuous or discrete
• E.g., weight vs. kinship relations
• Similarity is defined by distance along the
  dimensional axis

8
Colors - a conceptual space

• Familiar color circle

brightness
hue
chromaticness (saturation)
Several color models All of them three-dimensional
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Defining a space

• Between-ness
• B(a,b,c) where a,b,c are points in the space S
• point b lies between points a and c
• Axioms
• If B(a,b,c) then not B(b,a,c)
• If B(a,b,c) and B(b,c,d) then B(a,b,d)
• Density
• For any two points a and c in S, there is some
  point b such that B(a,b,c)
• Doesnt always hold! Not all spaces are dense.

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Defining a space

• Equidistance
• E(a,b,c,d)
• Point a is just as far from point b as point c
  is from point d.
• If E(a,a,p,q) then p q.
• E(a,b,b,a) for all a and b in S
• If E(a,b,c,d) and E(a,b,e,f), then E(c,d,e,f)
• If B(a,b,c), B(d,e,f), E(a,b,d,e) and E(b,c,e,f)
  then E(a,c,d,f)

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Metric space

• Equidistance is a qualitative notion
• Consider a distance function
• A real-valued function d(a,b) such that for all
  points a, b, and c in S
• d(a,b) 0
• d(a,b) 0 only if a b (minimality)
• D(a,b) d(b,a) (symmetry)
• D(a,b) d(b,c) d(a,c) (triangle inequality)
• If a space has a distance function, then it is a
  metric space

12
Euclidean vs. city block distance
d3
d3
d2
d2
d1
d1
c
c

• Euclidean
• Straight line distance

• City Block
• Must travel along axes

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Similarity

• A function of distance
• Similarity S from point i to point j
• Similarity between two objects drops quickly when
  the distance is relatively small
• Similarity drops slowly when the distance is
  relatively large

14
Identifying dimensions

• Constructive dimensions
• Geometric or topological structure chosen by
  scientist proposing theory or building the system
• Phenomenal dimensions
• Not obtainable directly
• Inferred from subjects behavior
• Various techniques
• E.g., multi-dimensional scaling

15
Integral vs. separable dimensions

• Consider conceptual space as a set of quality
  dimensions with a geometrical structure
• Integral dimensions
• Set of dimensions that all get values at the same
  time
• e.g., Musical note has pitch and volume
• Separable dimensions
• Set of dimensions that can be considered
  independently of other dimensions
• e.g., size and color

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Determining integral vs. separable

• Integral dimensions are where you cannot assign
  an object a value on one dimensions without
  assigning it a value on another
• Ask people to assign values of one dimension and
  see whether another dimension interferes

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Example tests

• Consider size, shape and color

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Evaluating test results

• Plot dimensions on axes
• If the Euclidean metric works best, consider them
  integral
• If the city-block metric works best, consider
  them separable
• Answer not always absolute
• e.g., re time and 3-D space separable?
• Classical Newtonian mechanics says yes
• Relativity theory says no

19
Domain

• A domain is a set of integral dimensions that are
  separable from all other dimensions
• Three color dimensions are a domain
• Why decompose a cognitive structure into domains?
• the assumption that an object can be assigned
  certain properties independent of other
  properties. An object can be assigned the weight
  of one kilo independent of its temperature or
  color

20
Conceptual Space

• Conceptual space - A collection of one or more
  domains
• Not all domains in a given space are necessarily
  metric
• Sometimes a domain might just be ordering (e.g.,
  process steps)
• A point is a possible object whose properties are
  determined by its location in the space

21
How do we learn dimensions?

• Some are innate (evolved over time)
• Parts of the brain associated with spatial
  reasoning
• evidence suggests that dimensions that are
  easily separable by adults are treated together
  by children
• E.g., volume vs. size