A Thorough Formalization of Conceptual Spaces Lucas Bechberger and - PowerPoint PPT Presentation

A Thorough Formalization of Conceptual Spaces Lucas Bechberger and Kai-Uwe Kühnberger

The Different Layers of Representation A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 2

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 3

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics Sensor / Feature Subsymbolic Layer [0.42; -1.337, ...] Values A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 4

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics ? Sensor / Feature Subsymbolic Layer [0.42; -1.337, ...] Values A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 5

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics Geometric Conceptual Layer Representation Sensor / Feature Subsymbolic Layer [0.42; -1.337, ...] Values A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 6

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics Geometric Conceptual Layer Representation Sensor / Feature Subsymbolic Layer [0.42; -1.337, ...] Values A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 7

The Different Layers of Representation ∀ x :apple ( x ) ⇒ red ( x ) Symbolic Layer Formal Logics Geometric Conceptual Layer Representation Sensor / Feature Subsymbolic Layer [0.42; -1.337, ...] Values A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 8

Conceptual Spaces [Gärdenfors2000] [Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press , 2000 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 9

Conceptual Spaces [Gärdenfors2000]  Quality dimensions  Interpretable ways of judging the similarity of two instances  E.g., temperature, weight, brightness, pitch [Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press , 2000 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 10

Conceptual Spaces [Gärdenfors2000]  Quality dimensions  Interpretable ways of judging the similarity of two instances  E.g., temperature, weight, brightness, pitch  Domain  Set of dimensions that inherently belong together  Color: hue, saturation, and brightness [Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press , 2000 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 11

Conceptual Spaces [Gärdenfors2000]  Quality dimensions  Interpretable ways of judging the similarity of two instances  E.g., temperature, weight, brightness, pitch  Domain  Set of dimensions that inherently belong together  Color: hue, saturation, and brightness  Distance in this space is inversely related to similarity  Within a domain: Euclidean distance  Between domains: Manhattan distance [Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press , 2000 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 12

Conceptual Spaces [Gärdenfors2000]  Quality dimensions  Interpretable ways of judging the similarity of two instances  E.g., temperature, weight, brightness, pitch  Domain  Set of dimensions that inherently belong together  Color: hue, saturation, and brightness  Distance in this space is inversely related to similarity  Within a domain: Euclidean distance  Between domains: Manhattan distance  Concepts  Region + correlation information + salience weights [Gärdenfors2000] Gärdenfors, P. Conceptual Spaces: The Geometry of Thought. MIT press , 2000 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 13

Betweenness A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 14

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z) A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 15

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z) https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 16

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z) https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 17

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z) https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 18

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z) https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 19

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)  Convex region C: ∀ x,z ∈ C : ∀ y : B ( x,y,z ) ⇒ y ∈ C https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 20

Betweenness  B(x,y,z) :↔ d(x,y) + d(y,z) = d(x,z)  Convex region C: ∀ x,z ∈ C : ∀ y : B ( x,y,z ) ⇒ y ∈ C  Star-shaped region S w.r.t. p: ∀ z ∈ S : ∀ y : B ( p,y,z ) ⇒ y ∈ S https://en.wikipedia.org/wiki/Taxicab_geometry#/ media/File:Manhattan_distance.svg A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 21

Convexity and Manhattan distance height age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 22

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 23

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 24

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 25

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 26

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 27

Convexity and Manhattan distance adult height child age A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 28

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 29

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 30

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 31

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 32

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 33

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 34

Formalizing Star-Shaped Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 35

Formalizing Star-Shaped Concepts ~ S = S 1.0 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 36

Formalizing Star-Shaped Concepts ~ S = S 1.0 ~ S 0.5 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 37

Formalizing Star-Shaped Concepts ~ S = S 1.0 ~ S 0.5 ~ S 0.25 A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 38

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 39

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 40

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 41

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 42

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 43

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 44

Intersection of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 45

Unification of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 46

Unification of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 47

Unification of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 48

Unification of Two Concepts A Thorough Formalization of Conceptual Spaces / Bechberger and Kühnberger 49