Modal logics of polytopes what we know so far David Gabelaia in - - PowerPoint PPT Presentation

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Modal logics of polytopes what we know so far David Gabelaia in - - PowerPoint PPT Presentation

Jan 24, 2023 223 likes •650 views

Modal logics of polytopes what we know so far David Gabelaia in collaboration with Members of Esakia Seminar Guram Bezhanishvili, Nick Bezhanishvili, Mamuka Jibladze, Evgeny Kuznetsov, Kristina Gogoladze, Maarten Marx, Levan Uridia et

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Modal logics of polytopes – what we know so far

David Gabelaia

in collaboration with Members of Esakia Seminar

Guram Bezhanishvili, Nick Bezhanishvili, Mamuka Jibladze, Evgeny Kuznetsov, Kristina Gogoladze, Maarten Marx, Levan Uridia et alii

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Topology and modal logic

  • McKinsey and Tarski 1944

‒ Interpret propositions as subsets of a topological space ‒ Interpret Boolean operations as their set-theoretic counterparts ‒ Interpret the modal diamond as closure, or as derivative

  • S4 is the modal logic of any crowded, separable,

metrizable space

  • Rasiowa and Sikorski 1963
  • S4 is the modal logic of any crowded, metrizable space
  • So any Rn generates S4
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SLIDE 3

Mapping a map

A B S Map of an Island

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Mapping a map

A B S Map of an Island

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SLIDE 5

Mapping a map

A B S Map of an Island

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SLIDE 6

Mapping a map

A B S Map of an Island

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SLIDE 7

Mapping a map

A B S Map of an Island Mapping f

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Mapping a map

A B S A B S Map of an Island Mapping f A|S A|B B|S

(A|S) | (A|B) | (B|S)

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SLIDE 9

Mapping a map

A B S A B S Map of an Island Mapping f A|S

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SLIDE 10

Mapping a map

A B S A B S Map of an Island Mapping f Kripke frame

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(M)Any subsets – wild logics

  • Any finite connected quasiorder (S4-frame) is an

interior image of Rn

[G. Bezhanishvili and Gehrke, 2002]

  • The subalgebras of the closure algebra (℘(Rn), C)

generate all connected extensions of S4

  • The subalgebras of the closure algebra (℘(Q), C)

generate all normal extensions of S4

[G. Bezhanishvili, DG and Lucero-Bryan, 2015]

  • Too many subsets!
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SLIDE 12

(M)Any subsets – wild logics

  • Any finite connected quasiorder (S4-frame) is an

interior image of Rn

[G. Bezhanishvili and Gehrke, 2002]

  • The subalgebras of the closure algebra (℘(Rn), C)

generate all connected extensions of S4

  • The subalgebras of the closure algebra (℘(Q), C)

generate all normal extensions of S4

[G. Bezhanishvili, DG and Lucero-Bryan, 2015]

  • Too many subsets!
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SLIDE 13

(M)Any subsets – wild logics

  • Any finite connected quasiorder (S4-frame) is an

interior image of Rn

[G. Bezhanishvili and Gehrke, 2002]

  • The subalgebras of the closure algebra (℘(Rn), C)

generate all connected extensions of S4

  • The subalgebras of the closure algebra (℘(Q), C)

generate all normal extensions of S4

[G. Bezhanishvili, DG and Lucero-Bryan, 2015]

  • Too many subsets!
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Nice subsets – tame logics?

  • Piecewise linear subsets = polytopes
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Nice subsets – tame logics?

  • Piecewise linear subsets = polytopes

PCn = C-logic of all polytopal subsets of Rn PDn = d-logic of all polytopal subsets of Rn Our aim is to investigate these modal systems

  • In this talk - PCn
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SLIDE 20

General observations

If A ∩ B = ∅ and A ⊆ CB Then dim(A) < dim(B) Put βA ≡ CA\A (boundary of A) Then βnA = ∅ iff dim(A) < n It follows that each PCn is a logic of finite height.

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Forbidden frames for PCn

. . .

n+1

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Forbidden frames for PCn

. . .

n+1

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Forbidden frames for PCn

. . .

n+1

PCn is an extension of S4.Grzn

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PC1

  • PC1 is the modal logic of a 2-fork

[van Benthem, G. Bezhanishvili and Gehrke, 2003]

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PC2

2 – forbidden frames

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PC2

2 – forbidden frames

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PC2

2 – forbidden frames

Any other forbidden configurations?

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Example

28

ϕ

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Example

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Example

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Lemma: Any crown frame is a partial polygonal interior image of the plane.

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PC2

2 – admitted frames

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PC2 – Axiomatization

Bad, but almost good guys Very nice guys

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PC2

2 – admitted frames

Lemma: Any rooted poset not reducible to any of the forbidden frames is a p-morphic image of a crown frame. Theorem: The logic PC2 is axiomatizable by Jankov- Fine axioms of the five forbidden frames.

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SLIDE 34

PC3

3 – forbidden frames

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PC3

3 – forbidden frames

Any other forbidden configurations?

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SLIDE 36

PC3

3 – Spherical (ope

pen) polyhedra ra

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Planar graphs

  • A graph is planar if it can be drawn on the plane

(=on a surface of a sphere) without intersecting edges

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SLIDE 38

Non-planar graphs

K3,3 K5

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SLIDE 39

PC3

3 – forbidden frames

K5 K3,3

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PC3

3 – forbidden frames

K5 K3,3

Anything else?

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SLIDE 41

Face posets of sphere triangulations