Download Modal logics based on the derivative operation in topological spaces

Modal logics based on the derivative
operation in topological spaces
Philippe Balbiani, Levan Uridia
Toulouse, Madrid
Outline
1. Modal logic and topology: basic results
2. Modal logic and topology: further direction
3. The derivative operation in topological spaces
4. Iterating the derivative operation
5. On the modal logic of iterated derivative
Modal logic and topology: basic results
A topological space is a pair F = (X , τ ) where
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X is a nonempty set
τ is a set of subsets of X satisfying the following
conditions:
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∅ ∈ τ and X ∈ τ
if A, B ∈ τ then A ∩ B ∈ τ
if (Ai )i is a collection
of subsets of X such that Ai ∈ τ for
S
every i then i Ai ∈ τ
Let F = (X , τ ) be a topological space
I F is called an Alexandroff space if it satisfies the following
condition:
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if (Ai )i is a collection
of subsets of X such that Ai ∈ τ for
T
every i then i Ai ∈ τ
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space
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The elements of τ are called open sets
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For A ⊆ X , a point x ∈ X is called a co-limit point of A iff
there exists an open neighborhood B of x such that
B ⊆ A ∪ {x}
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For A ⊆ X , t(A) will denote the set of co-limit points of A
and In(A) will denote A ∩ t(A)
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Remark that In(A) is the greatest open set contained in A
(called the interior of A)
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space
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The complements of open sets are called closed sets
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For A ⊆ X , a point x ∈ X is called a limit point of A iff for all
open neighborhoods B of x, A ∩ (B \ {x}) is nonempty
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For A ⊆ X , d(A) will denote the set of limit points of A and
Cl(A) will denote A ∪ d(A)
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Remark that Cl(A) is the least closed set containing A
(called the closure of A)
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space
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t(A ∩ B) = t(A) ∩ t(B) and d(A ∪ B) = d(A) ∪ d(B)
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A ∩ t(A) ⊆ t(t(A)) and d(d(A)) ⊆ A ∪ d(A)
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t(X \ A) = X \ d(A) and d(X \ A) = X \ t(A)
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space
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In(X ) = X and Cl(∅) = ∅
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In(A ∩ B) = In(A) ∩ In(B) and Cl(A ∪ B) = Cl(A) ∪ Cl(B)
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In(A) = In(In(A)) and Cl(Cl(A)) = Cl(A)
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In(A) ⊆ A and A ⊆ Cl(A)
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In(X \ A) = X \ Cl(A) and Cl(X \ A) = X \ In(A)
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space (separation axioms)
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F is called a T0 -space iff for each pair of different points
there exists an open set containing one and not containing
the other
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F is called a Td -space iff for each x ∈ X there exists an
open neighborhood A of x such that {x} is closed in A
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F is called a T1 -space iff for each pair of different points
there exists an open set containing exactly one of the
points
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F is called a T2 -space iff for each pair x, y ∈ X of different
points there exists disjoint open neighborhoods of x and y
Modal logic and topology: basic results
Syntax of modal logic
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φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ
Abbreviations
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Topological interpretation
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Standard definitions for the remaining Boolean operations
♦φ ::= ¬¬φ
The interpretation of φ is the interior of φ’s interpretation
As a result,
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The interpretation of ♦φ is the closure of φ’s interpretation
Modal logic and topology: basic results
Topological semantics for modal logic
I Let M = (X , τ, V ) where
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Truth of a modal formula φ at a point x in X is defined as
follows:
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F = (X , τ ) is a topological space
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff there exists O ∈ τ such that x ∈ O and for all
y ∈ O, M, y |= φ
As a result,
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M, x |= ♦φ iff for all O ∈ τ , if x ∈ O then there exists y ∈ O
such that M, y |= φ
Modal logic and topology: basic results
Topological semantics for modal logic
I Let M = (X , τ, V ) where
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Truth of a modal formula φ at a point x in X is defined as
follows:
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F = (X , τ ) is a topological space
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff x ∈ In({y ∈ X : M, y |= φ}
As a result,
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M, x |= ♦φ iff x ∈ Cl({y ∈ X : M, y |= φ}
Modal logic and topology: basic results
Neighborhood semantics for modal logic
I Let M = (W , N, V ) where
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Truth of a modal formula φ at a possible world x in W is
defined as follows:
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(W , N) is a neighborhood structure
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff there exists O ⊆ W such that O ∈ N(x) and
for all y ∈ O, M, y |= φ
As a result,
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M, x |= ♦φ iff for all O ⊆ W , if O ∈ N(x) then there exists
y ∈ O such that M, y |= φ
Modal logic and topology: basic results
Let F = (W , R) be an S4-relational structure
I A subset A of W is called an upset of F iff
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A subset A of W is called a downset of F iff
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x ∈ A and xRy imply y ∈ A
x ∈ A and xR −1 y imply y ∈ A
Remark that
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if A is an upset then W \ A is a downset
if A is a downset then W \ A is an upset
Modal logic and topology: basic results
For a given S4-relational structure F = (W , R), we define the
topology τR on W by declaring
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the upsets of F to be open
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the downsets of F to be closed
Remark that
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the topological space ts(F) = (W , τR ) is an Alexandroff
space
the least neighborhood of x ∈ W is
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the interior of a set A ⊆ W is
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R(x) = {y ∈ W : xRy }
{x ∈ W : R(x) ⊆ A}
the closure of a set A ⊆ W is
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{x ∈ W : R(x) ∩ A 6= ∅}
Modal logic and topology: basic results
For a given topological space F = (X , τ ), we define the binary
relation Rτ on W by
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xRτ y iff x ∈ Cl({y })
Remark that
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the relational structure rs(F) = (W , Rτ ) is an S4-structure
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(W , Rτ ) is a partial order iff F is T0
Modal logic and topology: basic results
Let F = (W , R) be an S4-relational structure, ts(F) = (W , τR )
be the associated topological space and rs(ts(F)) = (W , RτR )
be the associated relational structure. Then
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RτR = R
Let F = (W , τ ) be a topological space, rs(F) = (W , Rτ ) be the
associated relational structure and ts(rs(F)) = (W , τRτ ) be the
associated topological space. Then
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τ ⊆ τRτ
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τ = τRτ iff F is Alexandroff
Modal logic and topology: basic results
Topo-bisimulation: Let M = (X , τ, V ) and M0 = (X 0 , τ 0 , V 0 ) be
topo-models
I A topo-bisimulation between M and M0 is a binary relation
Z ⊆ X × X 0 such that if xZx 0 then
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for all Boolean variables p, x ∈ V (p) iff x 0 ∈ V 0 (p)
for all O ∈ τ , if x ∈ O then there exists O 0 ∈ τ 0 such that
x 0 ∈ O 0 and for all y 0 ∈ O 0 , there exists y ∈ O such that yZy 0
for all O 0 ∈ τ 0 , if x 0 ∈ O 0 then there exists O ∈ τ such that
x ∈ O and for all y ∈ O, there exists y 0 ∈ O 0 such that yZy 0
x and x 0 are topo-bisimilar iff there exists a
topo-bisimulation Z between M and M0 such that xZx 0
Modal logic and topology: basic results
Theorem (Aiello and van Benthem): Let M = (X , τ, V ) and
M0 = (X 0 , τ 0 , V 0 ) be topo-models. If x and x 0 are topo-bisimilar
then
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for all formulas φ, M, x |= φ iff M0 , x 0 |= φ.
Theorem (Aiello and van Benthem): Let M = (X , τ, V ) and
M0 = (X 0 , τ 0 , V 0 ) be finite topo-models. If for all formulas φ,
M, x |= φ iff M0 , x 0 |= φ then
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x and x 0 are topo-bisimilar.
Modal logic and topology: basic results
Topological semantics for modal logic
I Let M = (X , τ, V ) be a topological model
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We say that φ is true in M iff φ is true at every point in M
Let F = (X , τ ) be a topological space
We say that φ is valid in F iff φ is true in every model based
on F
Let C be a class of topological spaces
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We say that φ is valid in C iff φ is valid in every topological
space of C
Modal logic and topology: basic results
Topological interpretation of some valid formulas
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> ↔ >: the whole space is open
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♦⊥ ↔ ⊥: the empty set is closed
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(φ ∧ ψ) ↔ φ ∧ ψ: open sets are closed under finite
intersections
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♦(φ ∨ ψ) ↔ ♦φ ∨ ♦ψ: closed sets are closed under finite
unions
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φ ↔ φ: the interior operator is idempotent
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♦φ ↔ ♦♦φ: the closure operator is idempotent
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φ → φ: the interior of any set is contained in the set
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φ → ♦φ: the closure of any set contains the set
Modal logic and topology: basic results
Several standard modal logics and their axiomatizations
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K is axiomatized by
(K ) (p → q) → (p → q)
(N) from φ, infer φ
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T is axiomatized by K +
(T ) p → p
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K 4 is axiomatized by K +
(4) p → p
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wK 4 is axiomatized by K +
(w4) p ∧ p → p
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S4 is axiomatized by K + (T ) + (4)
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S5 is axiomatized by K + (T ) + (4) +
(B) p → ♦p
Modal logic and topology: basic results
Proposition: Let Top be the class of all topological spaces.
Then
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(K ) is valid in Top,
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(N) preserves validity in Top,
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(T ) is valid in Top,
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(4) is valid in Top,
i.e. the modal logic S4 is sound with respect to interpreting as interior.
Modal logic and topology: basic results
The canonical topo-model of S4 is the triple Mc = (Xc , τc , Vc )
where
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Xc is the set of all maximal S4-consistent sets of formulas
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τc is the set generated by arbitrary unions of basic sets, i.e.
sets of the form Oφ = {x ∈ Xc : φ ∈ x}
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Vc : p 7−→ Vc (p) ⊆ Xc is the valuation function defined by
x ∈ Vc (p) iff p ∈ x
Truth lemma: For all formulas φ, for all x ∈ Xc ,
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Mc , x |= φ iff φ ∈ x
Modal logic and topology: basic results
Theorem (completeness):
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S4 is the logic of the class of all topological spaces
Theorem (McKinsey and Tarski):
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S4 is complete for any dense-in-itself metric separable
space.
Theorem (completeness in special spaces):
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S4 is the logic of the rational line
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S4 is the logic of the real line
Modal logic and topology: basic results
Thus, such topological properties as
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being dense-in-itself
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being metric
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being separable
are not definable in the basic modal language
Topological definability: Let C be a class of topological spaces
I C is topologically definable iff there exists a set Γ of modal
formulas such that for each topological space F = (X , τ ),
we have
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F ∈ C iff F |= Γ
Modal logic and topology: basic results
Topological completeness of S4 tells us that
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the class Top of all topological spaces is topo-definable by
the formula >
Proposition (Gabelaia):
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Neither compactness nor connectedness is topo-definable
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None of the separation axioms T0 , Td , T1 and T2 is
topo-definable
Proposition (Gabelaia): Let C be a class of topological spaces
closed under formation of Alexandroff extensions
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Then C is modally definable iff it is closed under taking
open subspaces, interior images, topological sums and it
reflects Alexandroff extensions
Modal logic and topology: basic results
Let F = (X , τ ) be a topological space. Then
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F |= p → p iff F is discrete
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F |= p → ♦p iff every closed subset of X is open
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F |= ♦p → ♦p iff F is extremally disconnected
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F |= ♦p → ♦p iff the set of all dense subsets of X is a
filter
Modal logic and topology: further direction
Let F = (X , τ ) be a topological space
I Co-limit points
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For A ⊆ X , a point x ∈ X is a co-limit point of A iff there
exists an open neighborhood B of x such that B ⊆ A ∪ {x}
For A ⊆ X , t(A) is the set of co-limit points of A
Remind that In(A) = A ∩ t(A)
Limit points
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For A ⊆ X , a point x ∈ X is a limit point of A iff for all open
neighborhoods B of x, A ∩ (B \ {x}) is nonempty
For A ⊆ X , d(A) is the set of limit points of A
Remind that Cl(A) = A ∪ d(A)
Modal logic and topology: further direction
Syntax of modal logic
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φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ
Abbreviations
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Topological interpretation
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Standard definitions for the remaining Boolean operations
♦φ ::= ¬¬φ
The interpretation of φ is the set of co-limit points of φ’s
interpretation
As a result,
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The interpretation of ♦φ is the set of limit points of φ’s
interpretation
Modal logic and topology: further direction
Topological semantics for modal logic
I Let M = (X , τ, V ) where
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Truth of a modal formula φ at a point x in X is defined as
follows:
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(X , τ ) is a topological space
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff there exists O ∈ τ such that x ∈ O and for all
y ∈ O, if x 6= y then M, y |= φ
As a result,
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M, x |= ♦φ iff for all O ∈ τ , if x ∈ O then there exists y ∈ O
such that x 6= y and M, y |= φ
Modal logic and topology: further direction
Topological semantics for modal logic
I Let M = (X , τ, V ) where
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Truth of a modal formula φ at a point x in X is defined as
follows:
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(X , τ ) is a topological space
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff x ∈ t({y ∈ X : M, y |= φ})
As a result,
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M, x |= ♦φ iff x ∈ d({y ∈ X : M, y |= φ})
Modal logic and topology: further direction
Topological semantics for modal logic
I Let M = (X , τ, V ) be a topological model
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We say that φ is true in M iff φ is true at every point in M
Let F = (X , τ ) be a topological space
We say that φ is valid in F iff φ is true in every model based
on F
Let C be a class of topological spaces
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We say that φ is valid in C iff φ is valid in every topological
space of C
Modal logic and topology: further direction
Proposition: Let Top be the class of all topological spaces.
Then
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(K ) is valid in Top,
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(N) preserves validity in Top,
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(w4) is valid in Top,
i.e. the modal logic wK 4 is sound with respect to interpreting as set of co-limit points.
Modal logic and topology: further direction
For a given wK 4-relational structure F = (W , R), we define the
topology τR on W by declaring
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F = (W , R) to be the reflexive closure of F
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F = (W , R) to be the irreflexive fragment of F
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dR to be the limit operator in F
Remark that
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the topological space ts(F) = (W , τR ) is an Alexandroff
space
the least neighborhood of x ∈ W is
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the interior of a set A ⊆ W is
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R(x) = {y ∈ W : xRy}
{x ∈ W : R(x) ⊆ A}
the closure of a set A ⊆ W is
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{x ∈ W : R(x) ∩ A 6= ∅}
Modal logic and topology: further direction
For a given topological space F = (X , τ ), we define the binary
relation Rτ on W by
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xRτ y iff x ∈ dτ ({y})
Remark that
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the relational structure rs(F) = (W , Rτ ) is an irreflexive
wK 4-structure
Modal logic and topology: further direction
Lemma:
1. If F = (W , R) is a wK 4-frame then RτR ⊆ R.
2. If F = (W , R) is an irreflexive wK 4-frame then RτR = R.
3. If F = (X , τ ) is a topological space then Rτ−1 (A) ⊆ d(A).
4. If F = (X , τ ) is an Alexandroff space then Rτ−1 (A) = d(A).
Corollary: For a nonempty set X , there is a 1-1 correspondence
between
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Alexandroff topologies on X
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reflexive and transitive binary relations on X
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irreflexive and weakly transitive binary relations on X
Modal logic and topology: further direction
Theorem (Esakia):
1. wK 4 is the logic of all topological spaces.
2. wK 4 is the logic of all finite topological spaces.
3. wK 4 has the effective finite model property with respect to
the class of all topological spaces.
The derivative operation in topological spaces
Let F = (X , τ ) be a topological space
I Co-limit points
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Limit points
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For A ⊆ X , a point x ∈ X is a co-limit point of A iff there
exists an open neighborhood B of x such that B ⊆ A ∪ {x}
For A ⊆ X , t(A) is the set of co-limit points of A
Remind that In(A) = A ∩ t(A)
For A ⊆ X , a point x ∈ X is a limit point of A iff for all open
neighborhoods B of x, A ∩ (B \ {x}) is nonempty
For A ⊆ X , d(A) is the set of limit points of A
Remind that Cl(A) = A ∪ d(A)
Remind that
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t(X \ A) = X \ d(A)
d(X \ A) = X \ t(A)
The derivative operation in topological spaces
In words
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x belongs to the set of co-limit points of A iff some open
set B around x is contained in A ∪ {x}
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x belongs to the set of limit points of A iff every open set B
around x intersects A \ {x}
This set d(A) of limit points of A has many other names
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derivative
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derived set
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Cantor-Bendixson derivative
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set of limit points
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set of accumulation points
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set of cluster points
The derivative operation in topological spaces
Example: Let IR be the reals with the usual topology and
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A = {m
: m ≥ 1}. Then
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d(A) = {0}
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d(d(A)) = ∅
Example: Let IR be the reals with the usual topology and
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B = {m
+ n1 : m, n ≥ 1}. Then
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d(B) = A ∪ {0}
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d(d(B)) = {0}
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d(d(d(B))) = ∅
The derivative operation in topological spaces
Example: Let F = (X , τ ) be a topological space such that
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X is an infinite set
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there exists an infinite subset D of X such that τ is the
family of subsets A ⊆ W such that A = ∅ or D \ A is finite
Then
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d(A) = ∅ if D ∩ A is finite
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d(A) = A if D ∩ A is infinite
The derivative operation in topological spaces
Example: If F = (W , R) is an S4-frame then its (Alexandroff)
topology is defined as follows:
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the opens are the R-closed sets
Moreover,
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d(A) = {x ∈ W : there exists y ∈ A such that x 6= y and
xRy}
The derivative operation in topological spaces
Proposition: Concerning the derivative operation d on a
topological space F = (X , τ ),
1. If every set of the form d({x}) is open then F is a
T1 -space.
2. d(d(A)) ⊆ A ∪ d(A).
3. If F is a Td -space then d(d(A)) ⊆ d(A).
A dense-in-itself topological space F = (X , τ ) is a DSO space
iff
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every derived set d(A) is open
The derivative operation in topological spaces
Topological semantics for modal logic
I Let M = (X , τ, V ) where
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Truth of a modal formula φ at a point x in X is defined as
follows:
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(X , τ ) is a topological space
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff x ∈ t({y ∈ X : M, y |= φ})
As a result,
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M, x |= ♦φ iff x ∈ d({y ∈ X : M, y |= φ})
The derivative operation in topological spaces
Theorem: Let F = (X , τ ) be a topological space. Then
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F |= p → ♦p iff F is dense-in-itself
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F |= p → p iff F is a Td -space
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F |= ♦p → ♦p iff all derived sets are open
Theorem: The logic KD45 is sound and complete for DSO
topological models.
Iterating the derivative operation
Let F = (X , τ ) be a topological space
I Limit points
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For A ⊆ X , a point x ∈ X is a limit point of A iff for all open
neighborhoods B of x, A ∩ (B \ {x}) is nonempty
For A ⊆ X , dτ (A) is the set of limit points of A
Remind that Clτ (A) = A ∪ dτ (A)
In general, dτ (dτ (A)) ⊆ A ∪ dτ (A)
If F is a Td -space, dτ (dτ (A)) ⊆ dτ (A)
Sometimes, (X , τ ) will be denoted by (X , dτ ) or by (X , d)
Iterating the derivative operation
Let F = (X , d) be a topological space
I Iterated limit points
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Let Ld be the set of all functions d 0 : 2X 7−→ 2X such that
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Let ≤d be the binary relation on Ld defined by
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for all A ⊆ X , d 0 (A) ⊆ d(A)
d 0 ≤d d 00 iff for all A ⊆ X , d 0 (A) ⊆ d 00 (A)
Obviously, (Ld , ≤d ) is a complete lattice
Iterating the derivative operation
Let F = (X , d) be a topological space
I Iterated limit points
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If F is a Td -space, let θ be the following operator:
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Ld −→ Ld
d 0 7−→ θ(d 0 ) = d ◦ d 0
Obviously, θ is a ≤d -preserving operator
Thus,
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d∗ =
W
{d 0 ∈ Ld : d 0 ≤d θ(d 0 )}
is the greatest fixpoint of θ
Iterating the derivative operation
Proposition: Let F = (X , d) be a topological space. If F is a
Td -space then for all A ⊆ X ,
1. d ∗ (A) ⊆ d(A).
2. d(d(A)) ⊆ d(A).
3. d(d ∗ (A)) ⊆ d ∗ (A).
4. d ∗ (d(A)) ⊆ d ∗ (A).
5. d ∗ (d ∗ (A)) ⊆ d ∗ (A).
6. d ∗ (A) ⊆ d(d ∗ (A)).
Iterating the derivative operation
Let F = (W , R) be a relational structure
I Iterated relation
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Let LR be the set of all binary relations R 0 on W such that
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Let ≤R be the binary relation on LR defined by
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R0 ⊆ R
R 0 ≤R R 00 iff R 0 ⊆ R 00
Obviously, (LR , ≤R ) is a complete lattice
Iterating the derivative operation
Let F = (W , R) be a relational structure
I Iterated relation
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If F is a K 4-relational structure, let f be the following
operator:
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LR −→ LR
R 0 7−→ f (R 0 ) = R ◦ R 0
Obviously, f is a ≤R -preserving operator
Thus,
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R∗ =
W
{R 0 ∈ LR : R 0 ≤R f (R 0 )}
is the greatest fixpoint of f
Iterating the derivative operation
Proposition: Let F = (W , R) be a relational structure. If F is a
K 4-relational structure then for all x, y ∈ W ,
1. If xR ∗ y then xRy .
2. If x(R ◦ R)y then xRy.
3. If x(R ◦ R ∗ )y then xR ∗ y .
4. If x(R ∗ ◦ R)y then xR ∗ y .
5. If x(R ∗ ◦ R ∗ )y then xR ∗ y.
6. If xR ∗ y then x(R ◦ R ∗ )y .
Iterating the derivative operation
For a given Td topological space F = (X , d), we define the
binary relation Rd on W by
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xRd y iff x ∈ d({y})
Remark that in all cases
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Rd∗ = Rd ∗
For a given K 4-relational structure F = (W , R), we define the
function dR : 2W 7−→ 2W by
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dR (A) = R −1 (A)
Remark that in some cases
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dR∗ 6= dR ∗
Iterating the derivative operation
An iterated relational structure is a triple F = (W , R, S) where
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W is a nonempty set
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R is a transitive binary relation on W
S is the greatest fixpoint of the following operator:
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LR −→ LR
S 7−→ f (S) = R ◦ S
where LR is the complete lattice of the set of all binary
relations on W contained in R
Sometimes, S will be denoted by R ∗
Iterating the derivative operation
Example: Let F = (W , R) where W = IN. Then
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R∗ = ∅
Example: Let F = (W , R) where W = IN ∪ {ω}. Then
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R ∗ = IN × {ω}
Example: Let F = (W , R) where W = IN ⊕ IN0 . Then
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R ∗ = {(m, m0 ): m ∈ IN and m0 ∈ IN0 }
Iterating the derivative operation
Proposition: Let F = (W , R, S) be an iterated relational
structure. Then
1. S ⊆ R.
2. R ◦ R ⊆ R.
3. R ◦ S ⊆ S.
4. S ◦ R ⊆ S.
5. S ◦ S ⊆ S.
6. S ⊆ R ◦ S.
Iterating the derivative operation
Proposition: There exists no first-order formula φ(u, v ) in
Lan(=, R) with free variables u, v such that for all iterated
relational structures F = (W , R, S) and for all possible worlds
x, y in W ,
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xSy iff F |= φ[x, y]
Iterating the derivative operation
Proof:
1. Assume such φ(u, v ) exists and let r be its quantifier rank.
2. Let F1 = (ZZ, <) and F2 = (ZZ ⊕ ZZ0 , < ⊕ <0 ).
3. Let x1 = −2r and y1 = +2r . Remark that not x1 S1 y1 .
4. Let x2 = 0 and y2 = 00 . Remark that x2 S2 y2 .
5. Lemma: The Duplicator wins the r -move
Ehrenfeucht-Fraı̈ssé game on (F1 , x1 , y1 ) and (F2 , x2 , y2 ).
6. By 5, for all first-order formulas ϕ(u, v ) of quantifier rank r
in Lan(=, R) with free variables u, v , F1 |= ϕ[x1 , y1 ] iff
F2 |= ϕ[x2 , y2 ].
7. By 6, F1 |= φ[x1 , y1 ] iff F2 |= φ[x2 , y2 ].
8. By 1, x1 S1 y1 iff x2 S2 y2 : a contradiction with 3 and 4.
Iterating the derivative operation
Proposition: There exists no first-order sentence φ in
Lan(=, R, S) such that for all relational structures
F = (W , R, S),
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F is an iterated relational structure iff F |= φ
On the modal logic of iterated derivative
Syntax of modal logic
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φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ | ∗ φ
Abbreviations
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Topological interpretation
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Standard definitions for the remaining Boolean operations
♦φ ::= ¬¬φ
♦∗ φ ::= ¬∗ ¬φ
The interpretation of φ is the set of co-limit points of φ’s
interpretation
The interpretation of ∗ φ is the set of iterated co-limit
points of φ’s interpretation
As a result,
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The interpretation of ♦φ is the set of limit points of φ’s
interpretation
The interpretation of ♦∗ φ is the set of iterated limit points of
φ’s interpretation
On the modal logic of iterated derivative
Relational semantics for modal logic
I Let M = (W , R, S, V ) where
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Truth of a modal formula φ at a possible world x in W is
defined as follows:
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(W , R, S) is an iterated relational structure
V is a valuation function: p 7−→ V (p) ⊆ X
M, x |= p iff x ∈ V (p)
M, x |= φ iff R(x) ⊆ {y ∈ X : M, y |= φ}
M, x |= ∗ φ iff S(x) ⊆ {y ∈ X : M, y |= φ}
As a result,
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M, x |= ♦φ iff R(x) ∩ {y ∈ X : M, y |= φ} =
6 ∅
M, x |= ♦∗ φ iff S(x) ∩ {y ∈ X : M, y |= φ} =
6 ∅
On the modal logic of iterated derivative
Relational semantics for modal logic
I Let M = (W , R, S, V ) be an iterated relational model
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Let F = (W , R, S) be an iterated relational structure
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We say that φ is true in M iff φ is true at every possible
world in M
We say that φ is valid in F iff φ is true in every model based
on F
Let C be a class of iterated relational structures
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We say that φ is valid in C iff φ is valid in every iterated
relational structure of C
On the modal logic of iterated derivative
Proposition: There exists no modal formula φ in Lan() such
that for all iterated relational structures F = (W , R, S),
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F |= ∗ p ↔ φ
On the modal logic of iterated derivative
Proof:
1. Assume such φ exists.
2. Let F = (IN, <, ∅). Remark that F 6|= ¬∗ ⊥.
3. Let F 0 = (W 0 , R 0 , S 0 ) where (W 0 , R 0 ) is the ultrafilter
extension of (IN, <) and S 0 = R 0 ∗ . Remark that
F 0 |= ¬∗ ⊥.
4. By 1, F 0 |= ¬φ(p := ⊥).
5. Thus, F |= ¬φ(p := ⊥).
6. By 1, F |= ¬∗ ⊥: a contradiction with 2.
On the modal logic of iterated derivative
Proposition: The modal logic of the class of all iterated
relational structures is the least normal logic Log(, ∗ )
containing the following axioms:
(Ax1 ) p → ∗ p
(Ax2 ) p → p
(Ax3 ) ∗ p → ∗ p
(Ax4 ) ∗ p → ∗ p
(Ax5 ) ∗ p → ∗ ∗ p
(Ax6 ) ∗ p → ∗ p
Proof: Step-by-step construction.
On the modal logic of iterated derivative
Proposition: Log(, ∗ ) has not the finite model property with
respect to the class of all irreflexive iterated relational
structures.
Proof:
1. Take φ = ♦∗ >.
2. Remark that φ is valid in the irreflexive iterated relational
structure (IR, <, <).
3. For all finite irreflexive iterated relational structures
(W , R, S), S = ∅.
4. Thus, φ is satisfiable in no finite irreflexive iterated
relational structure.
On the modal logic of iterated derivative
Open problems:
1. Philosophical interpretation of ∗ in terms of beliefs ?
2. Finite model property of Log(, ∗ ) with respect to the
class of all relational structures validating it ?
3. Modal definability in Lan(, ∗ ) of the class of all iterated
relational structures ?
4. Decidability/complexity of the membership problem in
Log(, ∗ ) ?
5. What is the modal logic of ∗ alone ? K 4 ?
6. Do the class of all Td -spaces of the form (X , d, d ∗ ) and the
class of all K 4-relational structures of the form (W , R, R ∗ )
validate the same formulas in Lan(, ∗ ) ?
Bibliography
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Van Benthem, J., Bezhanishvili, G.: Modal logics of space.
In Aiello, M., Pratt-Hartmann, I., van Benthem, J.
(Editeurs) : Handbook of Spatial Logics. Springer (2007)
217–298.
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Bezhanishvili, G., Esakia, L., Gabelaia, D.: Some results
on modal axiomatization and definability for topological
spaces. Studia Logica 81 (2005) 325–355.
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Parikh, R., Moss, L., Steinsvold, C.: Topology and
epistemic logic. In Aiello, M., Pratt-Hartmann, I., van
Benthem, J. (Editeurs) : Handbook of Spatial Logics.
Springer (2007) 299–341.
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Shehtman, V.: Derived sets in Euclidean spaces and
modal logic. ITLI Prepublication Series X-90-05 (1990).