the schützenberger product for syntactic spaces reggio - the... · the schutzenb erger product for...
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![Page 1: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/1.jpg)
The Schutzenberger product for Syntactic Spaces
Mai Gehrke, Daniela Petrisan and Luca Reggio
IRIF, Universite Paris Diderot
ICALP 2016, 12− 15 July 2016
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Motivation and context
Existential quantification in the regular case
The Schutzenberger product for Syntactic Spaces
Conclusion
![Page 3: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/3.jpg)
MSO !Recognition byfinite monoids
!Regular
languages
Varieties of finitemonoids
Varieties of regularlanguages Profinite identities
Eilenberg Theorem Reiterman Theorem
E.g. profinite equations for the regular fragment of AC 0 = FO(N )(Barrington, Compton, Straubing and Therien 1990)
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MSO !Recognition byfinite monoids
!Regular
languages
Varieties of finitemonoids
Varieties of regularlanguages Profinite identities
Eilenberg Theorem Reiterman Theorem
E.g. profinite equations for the regular fragment of AC 0 = FO(N )(Barrington, Compton, Straubing and Therien 1990)
![Page 5: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/5.jpg)
MSO !Recognition byfinite monoids
!Regular
languages
Varieties of finitemonoids
Varieties of regularlanguages Profinite identities
Eilenberg Theorem Reiterman Theorem
E.g. profinite equations for the regular fragment of AC 0 = FO(N )(Barrington, Compton, Straubing and Therien 1990)
![Page 6: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/6.jpg)
What if we are interested in possibly non-regular languages?
I A∗ is the dual Stone space of Reg(A∗)(Birkhoff 1937, Almeida 1994, Pippenger 1997, ...);
I The combination of Eilenberg’s and Reiterman’s theorems canbe seen as an instance of Stone duality (Gehrke, Grigorieff andPin 2008);
![Page 7: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/7.jpg)
What if we are interested in possibly non-regular languages?
I A∗ is the dual Stone space of Reg(A∗)(Birkhoff 1937, Almeida 1994, Pippenger 1997, ...);
I The combination of Eilenberg’s and Reiterman’s theorems canbe seen as an instance of Stone duality (Gehrke, Grigorieff andPin 2008);
![Page 8: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/8.jpg)
Varieties of finite monoids
Varieties of regularlanguages
Lattices of regularlanguages
Lattices of arbitrarylanguages
Profinite identities
Profinite equations
beta-equations
Stone duality
Eilenberg Theorem Reiterman Theorem
![Page 9: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/9.jpg)
What if we are interested in possibly non-regular languages?
I A∗ is the dual Stone space of Reg(A∗)(Birkhoff 1937, Pippenger 1997, Almeida, ...);
I The combination of Eilenberg’s and Reiterman’s theorems canbe seen as an instance of Stone duality (Gehrke, Grigorieff andPin 2008). Here ultrafilter equations are crucial;
I Understanding the correspondence between classes oflanguages defined by some logic (e.g. FO(N )) and topologicalrecognizers, by identifying the construction dual to applying alayer of (existential) quantifier.
![Page 10: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/10.jpg)
What if we are interested in possibly non-regular languages?
I A∗ is the dual Stone space of Reg(A∗)(Birkhoff 1937, Pippenger 1997, Almeida, ...);
I The combination of Eilenberg’s and Reiterman’s theorems canbe seen as an instance of Stone duality (Gehrke, Grigorieff andPin 2008). Here ultrafilter equations are crucial;
I Understanding the correspondence between classes oflanguages defined by some logic (e.g. FO(N )) and topologicalrecognizers, by identifying the construction dual to applying alayer of (existential) quantifier.
![Page 11: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/11.jpg)
Motivation and context
Existential quantification in the regular case
The Schutzenberger product for Syntactic Spaces
Conclusion
![Page 12: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/12.jpg)
Let A be a finite alphabet, and φ(x) be an MSO formula with afree first-order variable x . Assume τ : (A× 2)∗ → M recognisesLΦ(x).
A∗ ⊗ N
A∗ (A× 2)∗
M
π γ1
τ
whereA∗ ⊗ N := {(w , i) ∈ A∗ × N | i < |w |}
is the set of words with a marked spot.
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Let A be a finite alphabet, and φ(x) be an MSO formula with afree first-order variable x . Assume τ : (A× 2)∗ → M recognisesLΦ(x).
A∗ ⊗ N
A∗ (A× 2)∗
M
π γ1
τ
L∃x .Φ(x) = π[γ−11 (LΦ(x))]
![Page 14: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/14.jpg)
Let A be a finite alphabet, and φ(x) be an MSO formula with afree first-order variable x . Assume τ : (A× 2)∗ → M recognisesLΦ(x).
A∗ ⊗ N
A∗ (A× 2)∗
M
π γ1
τ
A∗ −→ Pfin(M)
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DefinitionIf M is a monoid, then ♦M is a bilateral semidirect product: it hasPfin(M)×M as underlying set and the multiplication is given by
(S ,m) ∗ (T , n) := (S · n ∪m · T ,m · n).
LemmaIf M recognises LΦ(x), then ♦M recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
M ♦M
An alternative to ♦M is the block product U1�M.
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DefinitionIf M is a monoid, then ♦M is a bilateral semidirect product: it hasPfin(M)×M as underlying set and the multiplication is given by
(S ,m) ∗ (T , n) := (S · n ∪m · T ,m · n).
LemmaIf M recognises LΦ(x), then ♦M recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
M ♦M
An alternative to ♦M is the block product U1�M.
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DefinitionIf M is a monoid, then ♦M is a bilateral semidirect product: it hasPfin(M)×M as underlying set and the multiplication is given by
(S ,m) ∗ (T , n) := (S · n ∪m · T ,m · n).
LemmaIf M recognises LΦ(x), then ♦M recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
M ♦M
An alternative to ♦M is the block product U1�M.
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Motivation and context
Existential quantification in the regular case
The Schutzenberger product for Syntactic Spaces
Conclusion
![Page 19: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/19.jpg)
Consider a finite monoid M and a morphism µ:A∗ � M. Then theBoolean algebra of A∗-languages recognised by µ is isomorphic tothe power-set algebra P(M).
µ:A∗ � M
P(A∗)←↩ P(M):µ−1
There is a correspondence between monoid quotients andembeddings as power-set subalgebras closed under quotients.This is a special case of a more general phenomenon, namelyStone duality.
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Stone duality for Boolean algebras
Boolean Algebras Stone Spacesop
Ultrafilters
Clopens
Stone spaces = Zero-dimensional compact Hausdorff spaces
An example is
P(A∗) β(A∗) =Stone-Cech
compactification of A∗
Ultrafilters
Clopens
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Stone duality for Boolean algebras
Boolean Algebras Stone Spacesop
Ultrafilters
Clopens
Stone spaces = Zero-dimensional compact Hausdorff spaces
An example is
P(A∗) β(A∗) =Stone-Cech
compactification of A∗
Ultrafilters
Clopens
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The recognising objects
Let B ↪→ P(A∗) be any Boolean algebra of languages closed underquotients. Dually, we have a continuous surjection τ :β(A∗)� XB.
β(A∗) XB
A∗ M
τ
τ
NOTE: A∗ is a monoid which acts continuously (on the left and onthe right) on β(A∗). Further, A∗ is dense in β(A∗).
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The recognising objects
Let B ↪→ P(A∗) be any Boolean algebra of languages closed underquotients. Dually, we have a continuous surjection τ :β(A∗)� XB.
β(A∗) XB
A∗ M
τ
τ
NOTE: A∗ is a monoid which acts continuously (on the left and onthe right) on β(A∗). Further, A∗ is dense in β(A∗).
![Page 24: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/24.jpg)
The recognising objects
Let B ↪→ P(A∗) be any Boolean algebra of languages closed underquotients. Dually, we have a continuous surjection τ :β(A∗)� XB.
β(A∗) XB
A∗ M
τ
τ
NOTE: A∗ is a monoid which acts continuously (on the left and onthe right) on β(A∗). Further, A∗ is dense in β(A∗).
![Page 25: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/25.jpg)
The recognising objects
Let B ↪→ P(A∗) be any Boolean algebra of languages closed underquotients. Dually, we have a continuous surjection τ :β(A∗)� XB.
β(A∗) XB
A∗ M
τ
τ
NOTE: M is a monoid which acts continuously (on the left and onthe right) on X . Further, M is dense in X .
![Page 26: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/26.jpg)
The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
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The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
![Page 28: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/28.jpg)
The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
![Page 29: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/29.jpg)
The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
![Page 30: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/30.jpg)
The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
![Page 31: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/31.jpg)
The recognising objects
DefinitionA Stone space with an internal monoid is a pair (X ,M) where
I X is a Stone space
I M is a dense subspace of X equipped with a monoid structure
I the biaction of M on itself extends to a biaction of M on Xwith continuous components
τ−1(C ) β(A∗) X C
τ−1(C ) ∩ A∗ A∗ M C ∩M
τ
τ
![Page 32: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/32.jpg)
Now, let A be a finite alphabet and φ(x) be any (!) formula with afree first-order variable x . Assume τ recognises LΦ(x).
β(A∗ ⊗ N)
β(A∗) β(A× 2)∗
X
βπ βγ1
τ
There is a continuous map
β(A∗) −→ V(X )
into the Vietoris hyperspace of X .
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Now, let A be a finite alphabet and φ(x) be any (!) formula with afree first-order variable x . Assume τ recognises LΦ(x).
β(A∗ ⊗ N)
β(A∗) β(A× 2)∗
X
βπ βγ1
τ
There is a continuous map
β(A∗) −→ V(X )
into the Vietoris hyperspace of X .
![Page 34: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/34.jpg)
DefinitionLet X be any topological space. Then its Vietoris hyperspace hasV(X ) = {K ⊆ X | K is closed} as underlying set, and its topologyis the generated by the sets of the form
�U = {K ∈ V(X ) | K ⊆ U}, ♦U := {K ∈ V(X ) | K ∩ U 6= ∅}
for U an open subset of X .
Theorem
X Stone space ⇒ V(X ) Stone space,
and Pfin(X ) is dense in V(X ).
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DefinitionLet X be any topological space. Then its Vietoris hyperspace hasV(X ) = {K ⊆ X | K is closed} as underlying set, and its topologyis the generated by the sets of the form
�U = {K ∈ V(X ) | K ⊆ U}, ♦U := {K ∈ V(X ) | K ∩ U 6= ∅}
for U an open subset of X .
Theorem
X Stone space ⇒ V(X ) Stone space,
and Pfin(X ) is dense in V(X ).
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DefinitionLet X be any topological space. Then its Vietoris hyperspace hasV(X ) = {K ⊆ X | K is closed} as underlying set, and its topologyis the generated by the sets of the form
�U = {K ∈ V(X ) | K ⊆ U}, ♦U := {K ∈ V(X ) | K ∩ U 6= ∅}
for U an open subset of X .
Theorem
X Stone space ⇒ V(X ) Stone space,
and Pfin(X ) is dense in V(X ).
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DefinitionIf X is a Stone space, ♦X :=V(X )× X with the product topology.Then (♦X ,♦M) is a Stone space with an internal monoid.
Proposition
If (X ,M) recognises LΦ(x), then (♦X ,♦M) recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
(X ,M) (♦X ,♦M)
Theorem
B(♦X ,A) = 〈B(X ,A) ∪ B(X ,A× 2)∃〉,
where B(X ,A× 2)∃ is the Boolean algebra closed under quotientsgenerated by {L∃ | L ∈ B(X ,A× 2)}.
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DefinitionIf X is a Stone space, ♦X :=V(X )× X with the product topology.Then (♦X ,♦M) is a Stone space with an internal monoid.
Proposition
If (X ,M) recognises LΦ(x), then (♦X ,♦M) recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
(X ,M) (♦X ,♦M)
Theorem
B(♦X ,A) = 〈B(X ,A) ∪ B(X ,A× 2)∃〉,
where B(X ,A× 2)∃ is the Boolean algebra closed under quotientsgenerated by {L∃ | L ∈ B(X ,A× 2)}.
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DefinitionIf X is a Stone space, ♦X :=V(X )× X with the product topology.Then (♦X ,♦M) is a Stone space with an internal monoid.
Proposition
If (X ,M) recognises LΦ(x), then (♦X ,♦M) recognises L∃x .Φ(x).
LΦ(x) L∃x .Φ(x)
(X ,M) (♦X ,♦M)
Theorem
B(♦X ,A) = 〈B(X ,A) ∪ B(X ,A× 2)∃〉,
where B(X ,A× 2)∃ is the Boolean algebra closed under quotientsgenerated by {L∃ | L ∈ B(X ,A× 2)}.
![Page 40: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/40.jpg)
Motivation and context
Existential quantification in the regular case
The Schutzenberger product for Syntactic Spaces
Conclusion
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What I have presented:
I Notion of recognition in the topological setting
I ♦ as dual construction to ∃
What I have not presented (look up in the paper!):
I The binary product
I Ultrafilter equations
Future directions:
I “Good” basis of equations
I Connections with generalization of block product(Krebs et al.)
![Page 42: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/42.jpg)
What I have presented:
I Notion of recognition in the topological setting
I ♦ as dual construction to ∃
What I have not presented (look up in the paper!):
I The binary product
I Ultrafilter equations
Future directions:
I “Good” basis of equations
I Connections with generalization of block product(Krebs et al.)
![Page 43: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/43.jpg)
What I have presented:
I Notion of recognition in the topological setting
I ♦ as dual construction to ∃
What I have not presented (look up in the paper!):
I The binary product
I Ultrafilter equations
Future directions:
I “Good” basis of equations
I Connections with generalization of block product(Krebs et al.)
![Page 44: The Schützenberger product for Syntactic Spaces Reggio - The... · The Schutzenb erger product for Syntactic Spaces Mai Gehrke, Daniela Petri˘san and Luca Reggio IRIF, Universit](https://reader033.vdocuments.net/reader033/viewer/2022042913/5f49353929fbd84dfa182ce1/html5/thumbnails/44.jpg)
The Schutzenberger product for Syntactic Spaces
Mai Gehrke, Daniela Petrisan and Luca Reggio
IRIF, Universite Paris Diderot
ICALP 2016, 12− 15 July 2016