Measuring Relations between Concepts in Conceptual Spaces Lucas - - PowerPoint PPT Presentation

Measuring Relations between Concepts in Conceptual Spaces

Lucas Bechberger and Kai-Uwe Kühnberger https://www.lucas-bechberger.de

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 2

The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337] ∀ x:apple (x ) ⇒red(x ) Formal Logics Sensor / Feature Values

?

Conceptual Layer Geometric Representation

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 3

Formalizing Star-Shaped Concepts

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 4

Formalizing Star-Shaped Concepts

S = S1.0 ~ S0.5 S0.25 ~ ~

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 5

Formalizing Star-Shaped Concepts

• We can already create new concepts from old ones
• Intersection ( green ∩ banana = green banana )
• Unification ( citrus fruit = orange lemon grapefruit … )
• Projection onto a subspace ( apple↓(size) = apple’s typical size )
• We also want to talk about relations between concepts
• Compare their sizes ( |apple| < |fruit| )
• Subsethood ( apple fruit ↔ apple is-a fruit )
• Implication ( apple → red )
• (Similarity & Betweenness)

∩ ∩ ∩ ∩

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 6

Measuring the Size of a Concept

• Size of an α-cut:
• Original cuboid
• 2D ellipse
• Integrate over α-cuts to get overall size of cuboid
• Use inclusion-exclusion principle on cuboids

A B

+ Left/right part + Top/bottom part + Corners + 0D ellipse * width * height + 1D ellipse * width + 1D ellipse * height

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 7

Degree of Subsethood & Implication

A B B B A A

Kosko, B. Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence Prentice Hall, 1992

Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 8

Research Contributions

• Formalized ways of measuring relations between concepts
• Size
• Subsethood
• Implication
• All of this has been implemented:

https://github.com/lbechberger/ConceptualSpaces

• Future work: use this formalization in practice

Thank you for your attention!

Questions? Comments? Discussions? https://www.lucas-bechberger.de @LucasBechberger