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Complexity of Grammar Induction with Quantum Types
Antonin Delpeuch École Normale Supérieure and University of Oxford June 5, 2014 Quantum Physics and Logic, Kyoto
É C O L E N O R M A L E S U P É R I E U R E
Complexity of Grammar Induction with Quantum Types Antonin Delpeuch - - PowerPoint PPT Presentation
Complexity of Grammar Induction with Quantum Types Antonin Delpeuch - - PowerPoint PPT Presentation
Complexity of Grammar Induction with Quantum Types Antonin Delpeuch cole Normale Suprieure and University of Oxford June 5, 2014 Quantum Physics and Logic, Kyoto C O L E N O R M A L E S U P R I E U R E Syntax-semantics interface
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Syntax-semantics interface
S C Syntactic category Semantic category F M Free monoidal category
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Syntax-semantics interface
S C Syntactic category Semantic category F M Free monoidal category G
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Syntax-semantics interface
S C Syntactic category Semantic category F M Free monoidal category G ∃H ?
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Syntax-semantics interface
S C Syntactic category Semantic category F M Free monoidal category G ∃H ? An object A ∈ S is grammatical ⇔ ∃f ∈ S(A, I). Given G and a finite P ⊂ Ob(M), is there an H such that the diagram commutes and H(P) is grammatical?
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An example
S
F
− − − − − − − − − − → C pivotal compact closed
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An example
S
F
− − − − − − − − − − → C pivotal compact closed x1 x2 x3 x4 A B C B∗ A∗ A C ∗ A∗ A A∗
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An example
S
F
− − − − − − − − − − → C pivotal compact closed x1 x2 x3 x4 A C B B∗ A∗ A C ∗ A∗ A A∗
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An example
S
F
− − − − − − − − − − → C pivotal compact closed x1 x2 x3 x4 C B A A∗ B∗ A A∗ C ∗ A A∗
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An example
S
F
− − − − − − − − − − → C pivotal compact closed x1 x2 x3 x4 C B A A∗ B∗ A A∗ C ∗ A A∗ x1 x5 x6 x7 C B A B∗ C ∗ C A∗ C ∗ C C ∗
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An example
S
F
− − − − − − − − − − → C pivotal compact closed x1 x2 x3 x4 A C B B∗ A∗ A C ∗ A∗ A A∗ x1 x5 x6 x7 A C B B∗ C ∗ C C ∗ A∗ C C ∗
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The betweenness problem
A : finite set C ⊂ A3 : finite set of triples Problem: find a total ordering < of A such that ∀(a, b, c) ∈ C, a < b < c or c < b < a This problem is NP-complete1 and reduces to grammar inference from pivotal to compact closed categories.
1Guttmann and Maucher (2006)
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A hierarchy
Bi-closed Symmetric closed Autonomous Pivotal Self-dual pivotal Self-dual compact closed Compact closed
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A hierarchy
Bi-closed Symmetric closed Autonomous Pivotal Self-dual pivotal Self-dual compact closed Compact closed = NP-complete induction
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A hierarchy
Bi-closed Symmetric closed Autonomous Pivotal Self-dual pivotal Self-dual compact closed Compact closed = NP-complete induction
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A hierarchy
Bi-closed Symmetric closed Autonomous Pivotal Self-dual pivotal Self-dual compact closed Compact closed = NP-complete induction
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A hierarchy
Bi-closed Symmetric closed Autonomous Pivotal Self-dual pivotal Self-dual compact closed Compact closed = NP-complete induction
2 3
2Dudau-Sofronie, Tellier, and Tommasi (2001) 3Béchet, Foret, and Tellier (2007)
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