1. The modal logic of planar polygons
Kristina Gogoladze
Javakhishvili Tbilisi State University
joint work with David Gabelaia, Mamuka Jibladze, Evgeny Kuznetsov and Maarten Marx
June 26, 2014
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The modal logic of planar polygons

2. The modal logic of planar polygons
Introduction
We study the modal logic of the closure algebra P2, generated by the set of all polygons of the Euclidean plane R2. We show that:
The logic is finitely axiomatizable
It is complete with respect to the class of all finite "crown" frames we define
It does not have the Craig interpolation property
Its validity problem is PSpace-complete
TOLO IV
The modal logic of planar polygons

3. The modal logic of planar polygons
Preliminaries
p,q,…     
Modal Language
Topological space X
P, Q,…   \ C I
pp
CCA  CA
  
p  p
(p  q)  p  q
p  p
C() = 
A  C(A)
C(AB) = C(A)  C(B)
C(C(A)) = C(A)
Note that quasiorders can be viewed as (Alexandroff) topologies [Joel’s talk], link to the next slide to show that the modal logic of all topological spaces is S4
As is well known, logic S4 is characterized by reflexive-transitive Kripke frames.
TOLO IV
The modal logic of planar polygons

4. Preliminaries p,q,…      P, Q,…   \ C I
The modal logic of the class of all topological spaces is S4. Moreover, for any Euclidean space Rn, we have Log(Rn) = S4.
[McKinsey and Tarski in 1944]
We study the topological semantics, according to which
modal formulas denote regions in a topological space.
((R2), C)  (A, C)
General spaces
Topological spaces together with a fixed collection of subsets that is closed under set-theoretic operations as well as under the topological closure operator.
General models
Valuations are restricted to modal subalgebras of the powerset.
p,q,…     
Modal Language
Topological space X
P, Q,…   \ C I
Main idea: restrict interpretations of propositional letters to planar polygons. Copy animation from slide about interpretations, add indication that P, Q, etc. denote polygons. BUT, polygons are not closed under Boolean operations. What to do? Link to the next slide – denerate the Boolean algebra.
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The modal logic of planar polygons

5. Lets generate a closure algebra by polygons of R2 and denote it P2.
Preliminaries
Lets generate a closure algebra by polygons of R2 and denote it P2.
Define P_2. Give examples. Mention that the generated Closure (algebra will turn out to be a closure subalgebra.) Define the logic PL_2. With examples. (R^2, P2) PL2 Link to the next slide -> our main goal.
The 2-dimensional polytopal modal logic PL2 is defined to be the set of all modal formulas which are valid on (R2,P2).
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The modal logic of planar polygons

6. Preliminaries What is the modal logic of the polygonal plane?
R. Kontchakov, I. Pratt-Hartmann and M. Zakharyaschev, Interpreting Topological Logics Over Euclidean Spaces., in: Proceeding of KR, 2010
Show Zakhar first. Mention that they also generate Boolean algebra from polygons but inside the Boolean algebra of *regular closed* subsets. This differs from our approach (also their formalisms are different). More in the vein of our approach is what Johan, Mai and Guram did. Show their paper, go to next slide.
J. van Benthem, M. Gehrke and G. Bezhanishvili, Euclidean Hierarchy in Modal Logic, Studia Logica (2003), pp
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The modal logic of planar polygons

7. The logic of chequered subsets of R2
Preliminaries
The logic of chequered subsets of R2
Explain chequered (products of intervals). Say that similar frames play a major role in our study. Pass on to the next slide to define crown frames.
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The modal logic of planar polygons

8. Crown frames n Let  be the logic of all “crown” frames.
Theorem:  coincides with PL2.
Explain crown frames. Name the logic of crown frames. Formulate the main completeness theorem that PL_2 is the logic of crown frames. Say that there are two directions to prove and that we will give sketch of the proof of both, but will need some notions first.
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The modal logic of planar polygons

9. Preliminaries
The map  : X1  X2 between topological spaces X1 = (X1, τ1) and
X2 = (X2, τ2) is said to be an interior map, if it is both open and continuous.
Let X and Y be topological spaces and let  : X  Y be an onto partial interior map.
Then for an arbitrary modal formula  we have Y whenever X.
It follows that Log(X)  Log(Y).
Mention that if we are looking at partial interior maps of the polygonal plane, then preimages should be polygons. Link to the next slide – let us look at some finite partial interior images of the polygonal plane.
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The modal logic of planar polygons

10. The modal logic of planar polygons
Example
Make a polygon instead of the circle. Explain what arrows represent. On the whiteboard define “exterior boundary” and mention that for polygons the arrow represents a strict decrease in dimension. Link to next slide – this construction can be generalized in an obvious manner to all crown frames.
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The modal logic of planar polygons

11. The modal logic of planar polygons
The main results
Theorem: Any crown frame is a partial interior image of the polygonal plane.
Make again a polygon. Better picture of a crown frame. Briefly explain how the map goes (each line segment is a boundary of two open triangles, each open triangle has two line segments as a boundary. This could be said already on the previous slide). Say right here the corollary that PL_2 is a subset of the logic of crown frames. Briefly explain why in terms of satisfiability – for each formula refutable on a crown frame, one can “pull back”the valuation along the partial interior map to refute it on the polygonal plane. For the otjer direction we would need to take direct images of valuations, but they might not be constant on a region. What to do?? Pass on to the next slide – take small enough polygonal neighbourhood.
Corollary: PL2  .
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The modal logic of planar polygons

12. The modal logic of planar polygons
The main results
Theorem: Let  be satisfiable on a polygonal plane. Then  is satisfiable on one of the crown frames.
Make a little animation: first show the plane, the refutation point, mention that formula only depends on finitely many variables, represented by regions, then take a small enough polygonal neighbourhood of the refutation point (show the neighbourhood), then build the partial interior map onto the suitable crown frame (show the crown frame), then push the valuation along the map and refute on the crown frame. Show the corollary (actually our main theorem). Say that not the study of PL_2 can be reduced to the study of crown frames. Let us see what are the simplest frames that are *not* subreductions of crown frames?
Corollary:   PL2. Thus, the logic of the polygonal plane is determined by the class of finite crown frames. Hence this logic has FMP.
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The modal logic of planar polygons

13. The modal logic of planar polygons
Forbidden frames
What is a subreduction? Actually the same as partial interior image. Give geometric intuition why these frames are not partial interior images of the plane (and hence cannot be subreductions of crown frames). Link to next slide – let us make this precise!
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The modal logic of planar polygons

14. Axiomatization
We claim that the logic axiomatized by the Jankov-Fine axioms of these five frames coincides with PL2.
 = (B1)  (B2)  (B3)  (B4)  (B5)
What is \xi(B_1) (the Jankov formula of B_1)? It is a formula which is satisfiable on a frame iff that frame is subreducible to B_1. So negation of \xi(B_1) (the Jankov *axiom* for B_1) is valid on exactly those frames which are *not* subreducible to B_1. Take \xi and go ont o the next slide.
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The modal logic of planar polygons

15. The modal logic of planar polygons
Axiomatization
Lemma 1: Each crown frame validates the axiom .
Lemma 2: Each rooted finite frame G with G is a subreduction of some crown frame.
Number the lemmas. Say that the proof of the first is easy and that the second one is more involved, you won’t give details here, but you can mention that it reduces to some graph-theoretic problem known as the Chinese postman problem. Say that this gives us the axiomatization theorem, but shorter explicit axioms would be nice. Go to next slide.
Theorem: The logic PL2 is axiomatized by the formula .
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The modal logic of planar polygons

16. Shorter Axioms (I) p[p(pp)]
(II) [(rq)][(rq)((rq)  p  p)]
Where  is the formula
(pq)  (pq)  (pq).
If time allows, go on to the whiteboard where the exterior boundary formula is waiting for you and mention that the validity of (I) amounts to B^3(A) being empty for any A, so it measures dimension somehow. Also say that the topologically intuitive version of (II) would be nice. Since fmp and axioms are there, the logic is decidable, so natural next question – what is the complexity?
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The modal logic of planar polygons

17. Complexity
Theorem: The satisfiability problem of our logic is PSpace-complete.
Wolter, F. and M. Zakharyaschev, Spatial reasoning in RCC-8 with boolean region terms, in: Proc. ECAI, 2000, pp
Mention the 2000 paper by Zakhar and Wolter here.
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The modal logic of planar polygons

18. The modal logic of planar polygons
Craig Interpolation
(A) (r  (r  p  p))
(C) (r   s   s)  (r   s  s)
A  C is valid in PL2.
Bring a formula, say that there is no interpolant in the common language, but instead of proving it, just show the lack of amalgamation property, which illustrates the point. There is no crown frame that makes this diagram commute.
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The modal logic of planar polygons

19. The modal logic of planar polygons
Further research
Natural generalizations for spaces of higher dimension. PLn.
Also d-logics and stronger languages.
Natural generalizations for spaces of higher dimension. PL_n. The FMP can be proved similarly due to the “local nature of modal logic”. We can say something about the resulting frames (downset of each point is a Boolean algebra), something about axiomatization (B^n(A) is empty), but full description is an ongoing work. Also d-logics and stronger languages (universal modality, nominal) are a possible future lines of work.
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The modal logic of planar polygons

20. The modal logic of planar polygons
Thank You
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The modal logic of planar polygons