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

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 Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 2

Formalizing Star-Shaped Concepts Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 3

Formalizing Star-Shaped Concepts ~ S = S 1.0 ~ S 0.5 ~ S 0.25 Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 4

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 5

Measuring the Size of a Concept  Size of an α-cut:  Original cuboid + Left/right part + Top/bottom part + Corners  2D ellipse + 1D ellipse * height + 1D ellipse * width + 0D ellipse * width * height  Integrate over α-cuts to get overall size of cuboid A B  Use inclusion-exclusion principle on cuboids Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 6

Degree of Subsethood & Implication B B B A 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 7

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 Measuring Relations between Concepts in Conceptual Spaces / Lucas Bechberger 8

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