Coherence, Similarity, and Concept Generalisation Roberto - - PowerPoint PPT Presentation

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Coherence, Similarity, and Concept Generalisation Roberto - - PowerPoint PPT Presentation

Oct 31, 2022 271 likes •443 views

Coherence, Similarity, and Concept Generalisation Roberto Confalonieri 1 , Oliver Kutz 1 , Nicolas Troquard 1 , Pietro Galliani 1 , Daniele Porello 1 , Rafael Pe naloza 1 , Marco Schorlemmer 2 1 Free University of Bozen-Bolzano, Italy 2 Artificial

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Coherence, Similarity, and Concept Generalisation

Roberto Confalonieri1, Oliver Kutz1, Nicolas Troquard1, Pietro Galliani1, Daniele Porello1, Rafael Pe˜ naloza1, Marco Schorlemmer2

1 Free University of Bozen-Bolzano, Italy 2 Artificial Intelligence Research Institute, IIIA-CSIC, Spain

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What is this about?

◮ Joint coherence of concepts (wrt background ontology) ◮ Thagard-style coherence maximising partition(s) ◮ Common Generalisation of maximising partition(s)

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What is this about?

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SLIDE 4

What is this about?

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SLIDE 5

What is this about?

White Cats Black Cats

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Prime Numbers − − +

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SLIDE 6

What is this about?

Greyscale Cats

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Prime Numbers

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Main Procedure

Input:

◮ ALC TBox (background ontology) T ◮ Concepts C1 . . . Cn

Output:

◮ Coherence Maximising Partitions A ⊆ {C1 . . . Cn} ◮ Common Generalisations (as justifications)

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Main Procedure

Input:

◮ ALC TBox (background ontology) T ◮ Concepts C1 . . . Cn

Output:

◮ Coherence Maximising Partitions A ⊆ {C1 . . . Cn} ◮ Common Generalisations (as justifications)

  • 1. Define generalisation refinement operator from T ;
  • 2. Compute similarity between pairs (Ci, Cj);
  • 3. Evaluate coherence or incoherence between pairs;
  • 4. Find coherence maximising partitions;
  • 5. Compute respective generalisations.
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Generalisation Refinement Operator

Given a TBox T , and a concept C, the generalisation operator γ(C) is defined along the lines of (Confalonieri et al, 2016). Main Idea: we generalise a concept C either by finding a concept D ∈ sub(T ) which is immediately more general than C or by generalising one of the subformulas inside C. γ(C) is the (finite) set of all such generalisations.

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Generalisation Refinement Operator

Given a TBox T , and a concept C, the generalisation operator γ(C) is defined along the lines of (Confalonieri et al, 2016). Main Idea: we generalise a concept C either by finding a concept D ∈ sub(T ) which is immediately more general than C or by generalising one of the subformulas inside C. γ(C) is the (finite) set of all such generalisations.

◮ γ∗(C) is the set of all concepts which can be obtained from C

through repeated generalisations;

◮ For D ∈ γ∗(C), λ(C γ

− → D) = smallest number of steps from C to D.

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Common Generalisation(s)

CD = G ∈ γ∗(C) ∩ γ∗(D) s.t. ∀G ′ ∈ γ∗(C) ∩ γ∗(D),

◮ λ(C γ

− → G) + λ(D

γ

− → G) < λ(C

γ

− → G ′)+λ(D

γ

− → G ′), or

◮ λ(C γ

− → G) + λ(D

γ

− → G) = λ(C

γ

− → G ′) + λ(D

γ

− → G ′) and λ(G

γ

− → ⊤) ≥ λ(G ′ γ − → ⊤) Not Unique! C D CD ⊤

λ(C

γ

− → CD) λ(D

γ

− → CD) λ(CD

γ

− → ⊤)

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Concept Similarity

Sλ(C, D) =      λ(CD

γ

− → ⊤) λ(CD

γ

− → ⊤) + λ(C

γ

− → CD) + λ(D

γ

− → CD) if C or D = ⊤ 1

  • therwise.

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⊤

2 2 13

Sλ(White Cats, Black Cats) = 13 17

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Coherence and Incoherence

◮ C coheres with D (+) if Sλ(C, D) > 1 − δ ◮ C incoheres with D (−) if Sλ(C, D) ≤ 1 − δ

where δ = λ(CD

γ

− → ⊤) max{λ(C

γ

− → ⊤), λ(D

γ

− → ⊤)}

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⊤

2 2 13 15 15

Sλ(White Cats, Black Cats)= 13 17 δ = 13 15 1 − δ < 13 17 : +

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Coherence Maximising Partitions

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Prime Numbers − − +

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Conclusions

What we have:

◮ Theoretical framework for joint coherence of concepts ◮ Common generalisation as justification

What we need:

◮ Implementation ◮ Real-world (not toy) tests ◮ Graded coherence/incoherence ◮ Exploration of different similarity measures (e.g. Resnik 1995)

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References

◮ Confalonieri, R., Eppe, M., Schorlemmer, M., Kutz, O.,

Pe˜ naloza, R., Plaza, E.: Upward Refinement Operators for Conceptual Blending in EL++. Annals of Mathematics and Artificial Intelligence (2016)

◮ Thagard, P.: Coherent and creative conceptual combinations.

In: Creative thought: An investigation of conceptual structures and processes, pp. 129–141. American Psychological Association (1997)

◮ Resnik, Philip. Using Information Content to Evaluate

Semantic Similarity in a Taxonomy. (1995)

◮ Schorlemmer, M., Confalonieri, R., Plaza, E.: Coherent

concept invention. In: Proceedings of the Workshop on Computational Creativity, Concept Invention, and General Intelligence (C3GI 2016) (2016)