davidjanin, labri,universitédebordeaux france janin@labri fileoverlapping tileslanguages of...

Overlapping tiles Languages of tiles Conclusion

On languages of one-dimensional overlapping tiles

David Janin,LaBRI, Université de Bordeaux

[email protected]

January, SOFSEM 2013

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.

1. Overlapping tiles

A rich monoid structure, known since the 70’s, thatgeneralizes the free monoid A∗

Overlapping one-dimensional tiles

An object oriented data-structure

Overlapping tile = string object with history recording capacity

An exemple

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 .

An exemple

a b ac

s =nil.add(a).add(b).add(a).rem(a).rem(b).add(c) (fail)

An exemple

a b ab

s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b)

An exemple

a b ab a

s =nil.add(a).add(b).add(a).rem(a).rem(b) .add(b).add(a)

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)

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)

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

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 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.

• • • •u v w

• • • •u′ v ′ w ′

+ fusion:

• • • •(uv ∨s u′)v v v ′ v ′(v ′w ′ ∨p w)

• • • •u v w

• • • •u′ v ′ w ′

+ fusion:

• • • •(uv ∨s u′)v v v ′ v ′(v ′w ′ ∨p w)

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).

The monoid T 0A

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

with onto monoid morphisms.

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

with onto monoid morphisms.

2. Languages of tiles

Where it leads to real language theoretical developments

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}.

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

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

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.

i Li)ECi(RiR−1

i )E )δi

i Li)ECi(RiR−1

i )E )δi

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.

• u ≤sω(xy),

3. 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 !

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.

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.

Mark V. Lawson.McAlister semigroups.Journal of Algebra, 202(1):276 – 294, 1998.

davidjanin, labri,universitédebordeaux france janin@labri fileoverlapping tileslanguages of...

Documents