1. CSCTR – Session 6 Dana Retová
Conceptual spaces
CSCTR – Session 6
Dana Retová

2. Conceptual spaces (Gärdenfors)
Consist of a number of quality dimensions
Building blocks of representations
Weight, temperature, brightness, pitch, height, width, depth
Abstract non-sensory dimensions
Represent various qualities of objects
Independent of symbolic representations (language)
Abstract representation for modeling
Do not claim to have any immediate physical realization

3. Dimensions Innate – hardwired in nervous system Learned
Learning involves expanding conc. Space with new quality dimensions
Culturally dependent
Time
Scientific
Weight vs. mass
“Evidence suggests that dimensions that are easily separated by adults, such as the brightness and size of a square, are treated as fused together for children .... For example, children have difficulty identifying whether two objects differ on their brightness or size even though they can easily see that they differ in some way. Both differentiation and dimensionalization occur throughout one's lifetime.” (Goldstone & Bartalou 1998)

4. Dimension of ‘time’ In our culture and in science In other cultures
One-dimensional structure isomorphic to the line of real numbers
In other cultures
Circular structure

5. Dimension of ‘pitch’ 1-D structure from low tones to high
Logarithmic scale
Acoustic frequency is spatially coded in chochlea

6. Color space
Hue
Brightness
Color

7. Contrast classes Skin color
Possible colors are the subset of the full color space
Can be irregular
Subset “stretched” to form a space with the same topology
Color terms can be used even if they do not correspond to the original hues
“Metaphor”
In case of contrast classes, one set of dimensions is not really mapped onto another set but the conceptual space is mapped onto a subspace of itself retaining the same topological structure.

8. Conceptual spaces
Similarity - defined via distance between representing points
Object – point in a conceptual space
Property /Concept – region of a conceptual space

9. Metaphors in conceptual space
A metaphor expresses a similarity in topological or metrical structure between different quality dimensions
A word that represent a particular structure in one quality dimension can be used as a metaphor to express a similar structure about another dimension
Metaphors transfer knowledge about one conceptual dimension to another
E.g. space mapped to time
In case of contrast classes, one set of dimensions is not really mapped onto another set but the conceptual space is mapped onto a subspace of itself retaining the same topological structure.

10. Primary and secondary properties
Predicates are assigned regions of space (red)
Secondary properties (tall)
“Parasitical” on other properties
“Big chihuahua”

11. Marr (1982) Cylinders Prototypical vector for an object – image schema
Length
Width
Angle between the dominating and the other one
Position of the added cylinder
Prototypical vector for an object – image schema
Subordinate cat. – subregions of the convex region
on each level of the hierarchy an object is described by a comparatively
small number of coordinates based on lengths and angles.
Thus the object can be
identified as a hierarchially structured vector in a (higher order) conceptual space.
Furthermore, a ’prototypical’ vector for an object category like ’bird’ identifies a
spatial structure that can serve as an image schema for that category. Such an
image schema represents a basic level category (cf. Section 3), while subordinate
categories like ’ostrich’ are represented by subregions of the convex region
associated with the prototypical object. Superordinate categories like ’animal’ do
not have any associated image schemas.

12. Action space
Spatio-temporal patterns of forces that generate the movement

13. Functional concepts Function of an object can be analysed
Actions it affords
Functional concept = convex region in action space

14. Conceptual spaces Ideal to represent
Concepts on basic level of conceptualization
Spatial-relations concepts
Rules follow from the topological structure
For example:
A point in a conceptual space will always have an internally consistent set of properties
Something cannot be blue and yellow at the same time
Everything that is green is also colored
Nothing is in the same place in the same time
Transitivity – as in “earlier than”

15. Questions?