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Overlapping tiles Languages of tiles Conclusion
On languages of one-dimensional overlapping tiles
David Janin,LaBRI, Université de Bordeaux
January, SOFSEM 2013
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Overlapping tiles Languages of tiles Conclusion
General context of research : music modeling
Music has a complex structure with sequential, parallel andhierarchical features.
A language theory of overlapping structures is needed forautomated music analysis [4] and/or automated musicproduction [1].Inverse semigroup theory [7] provides the adequatemathematical foundation for such a development.
Overlapping tiles Languages of tiles Conclusion
General context of research : music modeling
Music has a complex structure with sequential, parallel andhierarchical features.
A language theory of overlapping structures is needed forautomated music analysis [4] and/or automated musicproduction [1].Inverse semigroup theory [7] provides the adequatemathematical foundation for such a development.
Overlapping tiles Languages of tiles Conclusion
General context of research : music modeling
Music has a complex structure with sequential, parallel andhierarchical features.
A language theory of overlapping structures is needed forautomated music analysis [4] and/or automated musicproduction [1].Inverse semigroup theory [7] provides the adequatemathematical foundation for such a development.
Overlapping tiles Languages of tiles Conclusion
General context of research : music modeling
Music has a complex structure with sequential, parallel andhierarchical features.
A language theory of overlapping structures is needed forautomated music analysis [4] and/or automated musicproduction [1].Inverse semigroup theory [7] provides the adequatemathematical foundation for such a development.
Overlapping tiles Languages of tiles Conclusion
General context of research : music modeling
Music has a complex structure with sequential, parallel andhierarchical features.
A language theory of overlapping structures is needed forautomated music analysis [4] and/or automated musicproduction [1].Inverse semigroup theory [7] provides the adequatemathematical foundation for such a development.
Overlapping tiles Languages of tiles Conclusion
1. Overlapping tiles
A rich monoid structure, known since the 70’s, thatgeneralizes the free monoid A∗
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a
s =nil.add(a).add(b).add(a).rem(a).rem(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b
s =nil.add(a).add(b).add(a).rem(a).rem(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b a
s =nil.add(a).add(b).add(a).rem(a).rem(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b a
s =nil.add(a).add(b).add(a).rem(a).rem(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b a
s =nil.add(a).add(b).add(a).rem(a).rem(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b ac
s =nil.add(a).add(b).add(a).rem(a).rem(b).add(c) (fail)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b ab
s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b ab a
s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b).add(a)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b ab a c
s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b).add(a).add(c)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Overlapping one-dimensional tiles
An object oriented data-structure
Overlapping tile = string object with history recording capacity
An exemple
a b ab a c
s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b).add(a).add(c)
Idea : model add(a) by the letter a and rem(a) by the lettera hence s = abaabbac .
Overlapping tiles Languages of tiles Conclusion
Models of overlapping tiles
Let A be an alphabet, let A be the dual alphabet:Positive tiles T+
A = {(u, v ,w) : u, v ,w ∈ A∗}
• • • •u v w
Negative tiles: T−A = {(uv , v , vw) : u, v ,w ∈ A∗}
• • • •u v w
Overlapping tiles Languages of tiles Conclusion
Models of overlapping tiles
Let A be an alphabet, let A be the dual alphabet:Positive tiles T+
A = {(u, v ,w) : u, v ,w ∈ A∗}
• • • •u v w
Negative tiles: T−A = {(uv , v , vw) : u, v ,w ∈ A∗}
• • • •u v w
Overlapping tiles Languages of tiles Conclusion
Overlapping tiles productLet TA = T+
A + T−A .Let (u, v ,w) and (u′, v ,′ ,w ,′ ) ∈ TA be two tiles.Tiles product : (u, v ,w) · (u′, v ′,w ′)Synchronization:
• • • •u v w
• • • •u′ v ′ w ′
+ fusion:
• • • •(uv ∨s u′)v v v ′ v ′(v ′w ′ ∨p w)
extended to 0 when fusion fails.
Overlapping tiles Languages of tiles Conclusion
Overlapping tiles productLet TA = T+
A + T−A .Let (u, v ,w) and (u′, v ,′ ,w ,′ ) ∈ TA be two tiles.Tiles product : (u, v ,w) · (u′, v ′,w ′)Synchronization:
• • • •u v w
• • • •u′ v ′ w ′
+ fusion:
• • • •(uv ∨s u′)v v v ′ v ′(v ′w ′ ∨p w)
extended to 0 when fusion fails.
Overlapping tiles Languages of tiles Conclusion
Overlapping tiles productLet TA = T+
A + T−A .Let (u, v ,w) and (u′, v ,′ ,w ,′ ) ∈ TA be two tiles.Tiles product : (u, v ,w) · (u′, v ′,w ′)Synchronization:
• • • •u v w
• • • •u′ v ′ w ′
+ fusion:
• • • •(uv ∨s u′)v v v ′ v ′(v ′w ′ ∨p w)
extended to 0 when fusion fails.
Overlapping tiles Languages of tiles Conclusion
The monoid T 0A
Theorem (Monoid)T0 = TA + 0 with tile product is an inverse monoid withneutral element 1.
Lemma (Inverses)For every x ∈ T 0
A there exists a unique x−1 ∈ T 0A such that
xx−1x = x and x−1xx−1 = x, i.e. T 0A is an inverse monoid.
In particular, 0−1 = 0, 1−1 = 1 and, for every tilex = (u, v ,w), x−1 = (uv , v , vw).
Lemma (Idempotents)Idempotent elements are 0 and 1 and elements of the form(u, 1,w).
Overlapping tiles Languages of tiles Conclusion
The monoid T 0A
Theorem (Monoid)T0 = TA + 0 with tile product is an inverse monoid withneutral element 1.
Lemma (Inverses)For every x ∈ T 0
A there exists a unique x−1 ∈ T 0A such that
xx−1x = x and x−1xx−1 = x, i.e. T 0A is an inverse monoid.
In particular, 0−1 = 0, 1−1 = 1 and, for every tilex = (u, v ,w), x−1 = (uv , v , vw).
Lemma (Idempotents)Idempotent elements are 0 and 1 and elements of the form(u, 1,w).
Overlapping tiles Languages of tiles Conclusion
The monoid T 0A
Theorem (Monoid)T0 = TA + 0 with tile product is an inverse monoid withneutral element 1.
Lemma (Inverses)For every x ∈ T 0
A there exists a unique x−1 ∈ T 0A such that
xx−1x = x and x−1xx−1 = x, i.e. T 0A is an inverse monoid.
In particular, 0−1 = 0, 1−1 = 1 and, for every tilex = (u, v ,w), x−1 = (uv , v , vw).
Lemma (Idempotents)Idempotent elements are 0 and 1 and elements of the form(u, 1,w).
Overlapping tiles Languages of tiles Conclusion
Within inverse semigroup theory
FactThe set T 0
A of overlapping tiles equipped with the tile productis an inverse monoid known as McAlister monoid [8].
A diagramThe following diagram commutes
(A + A)∗ (A + A)∗/⊥
FIM(A) T 0A
θ θ
η
η
with onto monoid morphisms.
Overlapping tiles Languages of tiles Conclusion
Within inverse semigroup theory
FactThe set T 0
A of overlapping tiles equipped with the tile productis an inverse monoid known as McAlister monoid [8].
A diagramThe following diagram commutes
(A + A)∗ (A + A)∗/⊥
FIM(A) T 0A
θ θ
η
η
with onto monoid morphisms.
Overlapping tiles Languages of tiles Conclusion
2. Languages of tiles
Where it leads to real language theoretical developments
Overlapping tiles Languages of tiles Conclusion
Languages of tiles
Let X and Y ⊆ TA two languages of (non zero) tiles:• sum: X + Y = X ∪ Y ,• product : X .Y = {x · y ∈ TA : x ∈ X , y ∈ Y },• star : X ∗ =
∑k∈ω X k with X 0 = {1} and X k+1 = X k · X ,
• idempotent projection: XE = {x ∈ X : xx = x}.
Overlapping tiles Languages of tiles Conclusion
Classical classes of langages
DefinitionFor all langage L ⊆ TA:
• L is REC when L = ϕ−1(ϕ(L)) for some morphismϕ : TA → S and finite monoid S,
• L is RAT when L is finite combination of finite languagewith sum, product and star,
• L is XRAT when L is finite combination of finite languagewith sum, product, star and idempotent projection,
• L is MSO when L is definable by means of an MSOformulae.
TheoremREC ⊂⊂ RAT ⊂ XRAT = MSO
Overlapping tiles Languages of tiles Conclusion
Classical classes of langages
DefinitionFor all langage L ⊆ TA:
• L is REC when L = ϕ−1(ϕ(L)) for some morphismϕ : TA → S and finite monoid S,
• L is RAT when L is finite combination of finite languagewith sum, product and star,
• L is XRAT when L is finite combination of finite languagewith sum, product, star and idempotent projection,
• L is MSO when L is definable by means of an MSOformulae.
TheoremREC ⊂⊂ RAT ⊂ XRAT = MSO
Overlapping tiles Languages of tiles Conclusion
Languages of tiles definable in MSO
Theorem (Simplicity)L ⊆ TA is definable in MSO if and only if
L = Σi∈I(Li × Ci × Ri)δi = Σi∈I((L−1
i Li)ECi(RiR−1
i )E )δi
for some finite I, δi ∈ {1,−1} and Li , Ci and Ri ⊆ A∗ regularfor every i ∈ I .
Theorem (Robustness)The class of languages of tiles definable in MSO is closedunder booleans, projection, product, star, various projectionoperators, left and right residuals etc. . .
RemarkEmbedding words u ∈ A∗ to tiles (1, u, 1), simplicity theoremalso shows ERAT = MSO.
Overlapping tiles Languages of tiles Conclusion
Languages of tiles definable in MSO
Theorem (Simplicity)L ⊆ TA is definable in MSO if and only if
L = Σi∈I(Li × Ci × Ri)δi = Σi∈I((L−1
i Li)ECi(RiR−1
i )E )δi
for some finite I, δi ∈ {1,−1} and Li , Ci and Ri ⊆ A∗ regularfor every i ∈ I .
Theorem (Robustness)The class of languages of tiles definable in MSO is closedunder booleans, projection, product, star, various projectionoperators, left and right residuals etc. . .
RemarkEmbedding words u ∈ A∗ to tiles (1, u, 1), simplicity theoremalso shows ERAT = MSO.
Overlapping tiles Languages of tiles Conclusion
Languages of tiles definable in MSO
Theorem (Simplicity)L ⊆ TA is definable in MSO if and only if
L = Σi∈I(Li × Ci × Ri)δi = Σi∈I((L−1
i Li)ECi(RiR−1
i )E )δi
for some finite I, δi ∈ {1,−1} and Li , Ci and Ri ⊆ A∗ regularfor every i ∈ I .
Theorem (Robustness)The class of languages of tiles definable in MSO is closedunder booleans, projection, product, star, various projectionoperators, left and right residuals etc. . .
RemarkEmbedding words u ∈ A∗ to tiles (1, u, 1), simplicity theoremalso shows ERAT = MSO.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Language of tiles definable by finite monoid
TheoremREC ⊂⊂ MSO
Proof (Element of)Let ϕ : TA → S a morphism and let s ∈ S with ϕ(0) 6= s.Then there is some x ∈ A∗ and y ∈ A+ such that, for everytile (u, v ,w) ∈ ϕ−1(s) is such that
• u ≤sω(xy),
• v = x(yx)ky ,• and w ≤p (xy)ω.
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
Towards a success story ?
Since then, coping with the collapse of REC , we havedeveloped:
• a notion of quasi-algebraic recognizability for languages oftiles [3]
• a related notion of tile automata [2]• the tiled programing paradigm the T-calculus [6]• a relationship with two-way (or walking) automata [5]• and much more to come: extension to languages of trees,languages of infinite trees, etc. . .
Thanks for your attention !
Overlapping tiles Languages of tiles Conclusion
F. Berthaut, D. Janin, and B. Martin.Advanced synchronization of audio or symbolic musicalpatterns.In Sixth IEEE International Conference on SemanticComputing, pages 202–209. IEEE Society Press, 2012.
D. Janin.Overlaping tile automata.Technical Report RR-1465-12, LaBRI, Université deBordeaux, 2012.D. Janin.Quasi-recognizable vs MSO definable languages ofone-dimensional overlaping tiles.In Mathematical Foundations of computer Science(MFCS), volume 7464 of LNCS, pages 516–528, 2012.
D. Janin.
Overlapping tiles Languages of tiles Conclusion
Vers une modélisation combinatoire des structuresrythmiques simples de la musique.Revue Francophone d’Informatique Musicale (RFIM), 2,2012.D. Janin.Walking automata in the free inverse monoid.Technical Report RR-1464-12, LaBRI, Université deBordeaux, 2012.D. Janin, F. Berthaut, M. DeSainte-Catherine, andS. Salvati andY. Orlarey.The t-calculus : towards a structured programming of timeand space.Technical report, LaBRI, Université de Bordeaux, 2013.
Mark V. Lawson.Inverse Semigroups : The theory of partial symmetries.World Scientific, 1998.