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Framed vs Unframed Two-dimensional languages

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Framed vs Unframed

Two-dimensional languages

Marcella Anselmo Natasha Jonoska Maria Madonia

Univ. of Salerno Univ. South Florida Univ. of Catania

ITALY USA ITALY

Two-dimensional(2dim)languages

In the literature two kinds of 2dim languages

- Sets of finite pictures

Ex.L01= the set of finite pictures with one occurrence of symbol “1” at most and symbol “0” in the other positions

- Tilings of the infinite plane

Ex. Tiling of the infinite plane with one occurrence of symbol “1” at most and symbol “0” in the other positions

Remark: The set of its finite blocks is L01

Overview of the talk

- Topic:Recognizable2dim languages

- Motivation:In the literature
- recognizable = (symbol-to-symbol) projection of local
- with two different approaches
- framed for finite pictures and
- unframed for the infinite plan

In this talk

New “unframed” definition for “finite” pictures

- Results of comparison framed vs unframed
- with special focus on determinism and unambiguity

Framed vs Unframed 2dim languages

Local 2dim languages

“Framed” approach

- Generalization of local 1dim (string) languages
- sharp () is needed to test locality conditions on the boundaries

“Unframed” approach

- Tiling of the (infinite) plane

- No sharp is needed!

p =

p =

- L islocalif there exists a finite set of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of is in (and we write L=L() )

p

Local languages: LOC

- finite alphabet, **all pictures over ,
- L ** 2dim language

- To define local languages, identify the boundary of a picture p using a boundary symbol

Framed vs Unframed 2dim languages

0

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p =

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#

Example of local language

Ld = the set of square pictures with symbol “1” in all main diagonal positions and symbol “0” in the other positions

Framed vs Unframed 2dim languages

Recognizable languages: REC

- L is recognizable by tiling systemif L= (L’) where L’ is a local language and is a mapping from the alphabet of L’ to the alphabet of L

- (, , , ) , where L’=L(), is called tiling system

- REC is the family of two-dimensional languages recognizable by tiling system[Giammarresi, Restivo 91]

Example: LSq = all squares over {a}

is recognizable by tiling system.

Set L’=Ld and (1)= (0)= a

Framed vs Unframed 2dim languages

Factorial local/recognizable languages

- Factorial recognizable languages (FREC) are defined in terms of factorial local languages (FLOC)

Do not care about the boundary of a picture!

- L isfactorial localif there exists a finite set of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of p is in (and we write L=Lu() ) (throw away the … hat!!!)

- L is factorial tiling recognizable if L= (L’) where L’ is a factorial local language and is a mapping from the alphabet of L’ to the alphabet of L

(, ,, ) , where L’=Lu(), is called unborderedtiling system

Framed vs Unframed 2dim languages

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Example of L in FREC

L01 = the set of pictures with one occurrence of symbol “1” at most and symbol “0” in the other positions

Framed vs Unframed 2dim languages

LOC and FLOC, REC and FREC

/

/

- L FLOC or L FREC implies L factor-closed
- (i.e. L=F(L) where F(L) is the set of all factors of L)

- L FLOC implies L LOC

(adding everywhere)

- L FREC implies L REC
- (as before)

- FLOC LOC
- Example: Ld LOC, not factor-closed

- FREC REC
- (as before)

Framed vs Unframed 2dim languages

Characterization of FLOC inside LOC and

of FREC inside REC

Proposition FLOC = LOC Factor-closed

Proof LLOC and L factor-closed implies L FLOC. Indeed

nofor F(L)=L

(remove tiles with )

Proposition L FREC iff L REC and L=p(K) with K factor-closed local language

Framed vs Unframed 2dim languages

Determinism and unambiguity

- “Computing”by a tiling system(, , , )
- Given a picture p** looking for p’ ** such that
- (p’)=p (i.e. for a pre-image p’ of p)

- Determinism
- One possible next step

- Unambiguity

Remark Usually Determinism implies Unambiguity

Framed vs Unframed 2dim languages

a

b

c

d

unique way to fill this position with a symbol of whose projection matches symbol s

s

Determinism in REC: DREC

Def. [A, Giammarresi, M 07] A tiling systemis tl-br-deterministicif a,b,c and s , unique tile

such that (d)=s.

(Analogously tr-bl,bl-tr,br-tl -deterministictiling system)

DREC languages that admit a tl-br or tr-bl or bl-tr or br-tl-

deterministic tiling system

Framed vs Unframed 2dim languages

/

/

/

Unambiguity in REC: UREC

Definition [Giammarresi,Restivo 92]A tiling system (, , , ) is unambiguous for L** if for any pL there is a unique p’ L’ such that (p’)=p (p’ pre-image of p).

L ** is unambiguous if it admits an unambiguous tiling system.

UREC= all unambiguous recognizable 2dim languages

Proposition [A, Giammarresi, M 07]

LOC DREC UREC REC

Framed vs Unframed 2dim languages

Ambiguity in REC

Definition L ** is finitely-ambiguous if there exists a tiling system for L such that every picture pL has k pre-images at most (for some k >1).

L is infinitely-ambiguous if it is not finitely ambiguous.

Framed vs Unframed 2dim languages

Determinism and unambiguity in FREC

- DFREC = languages that admit a deterministic unbordered tiling system

- UFREC = languages that admit an unambiguous unbordered tiling system

- Finitely-ambiguous and infinitely ambiguousfactorial recognizable languages

Framed vs Unframed 2dim languages

Example

Recall the example L01

-1

p =

-1

The unbordered tiling system for L01 is deterministic but it is not unambiguous

-1

Framed vs Unframed 2dim languages

Example (continued)

-1

p =

Moreover it can be shown that L01 is an infinitely ambiguous factorial language.

Framed vs Unframed 2dim languages

Unambiguity in FREC

Proposition.UFREC = FLOC

Proof. If L FLOC then is the identity.

If L UFREC any symbol in has an unique pre-image and then is a one-to-one mapping

- Remarks.
- UFREC is a very limited notion

- DFREC does not imply UFREC

A better suited definition of unambiguity is necessary

Framed vs Unframed 2dim languages

Frame-unambiguity (I)

Definition An unbordered tiling system for L is frame-unambiguos at p L if, once we fix a frame of local symbols in p, p has at most one pre-image.

One pre-image at most

p =

Definition LFREC is frame-unambiguous if it admits a frame-unambiguous unbordered tiling system.

Remark The frame of boundary symbols in UREC is replaced by a frame of local symbols

Framed vs Unframed 2dim languages

Frame unambiguity (II)

-1

-1

In L01

p =

L01 is frame-unambiguous

Proposition L DFREC implies L is frame-unambiguous

Framed vs Unframed 2dim languages

Ambiguity in REC vs ambiguity in FREC (I)

Determinism

Frame-Unambiguity

Determinism

Unambiguity

In REC

In FREC

Determinism

Unambiguity

- There are languages
- infinitely ambiguous
- finitely-ambiguous
- unambiguous

- There are languages
- infinitely ambiguous
- unambiguous
- (as far as we know)

Framed vs Unframed 2dim languages

Ambiguity in REC vs ambiguity in FREC (II)

Remark Frame reduces the ambiguity degree

- Finitely-ambiguous factorial in FREC and unambiguous in REC

-1

Framed vs Unframed 2dim languages

Ambiguity in REC vs ambiguity in FREC (III)

Moreover

- Infinitely factorial ambiguous in FREC and unambiguous in REC

Framed vs Unframed 2dim languages

Conclusions

- Frame can enforce size and content of recognized pictures

- Frame can reduce ambiguity degree

Additional memory

Factorial recognizable 2dim symbolic dynamical systems

analogies and interpretations in symbolic dynamics

Framed vs Unframed 2dim languages

Grazie

Conclusions

Frame can enforce size and content of recognized pictures

Frame can reduce ambiguity degree

Additional memory

Tilings of the plane 2dim symbolic dynamical systems

analogies and interpretations in symbolic dynamics

Note

When sets of tilings are invariant under translations, in symbolic dynamics:

Local

Projection

“shifts of finite type”

“sofic shifts”

Framed vs Unframed 2dim languages

Decidability properties

Proposition: It is decidable whether a given unbordered tiling system is unambiguous and whether it is deterministic.

Proposition: It is undecidable whether a given unbordered tiling system is frame-unambiguous.

Framed vs Unframed 2dim languages

Removing tiles with # does not always work …

- Given a tiling system for L REC, this does not allow to recognize F(L) as element of FREC

ExampleConsider Ldand the tiling system for it. Teta contains all the sub-tiles of

T no

F(L)

but

- Given a tiling system for L=F(L) REC, we cannot prove that this allow to recognize L as subset of FREC

Framed vs Unframed 2dim languages

Finite and infinite ambiguity in FREC

Proposition: For any k >=1, there is a k-factorial-ambiguous language.

Proposition: Unambiguous-FREC (Col-UFREC Row-UFREC) Finitely ambiguous FREC

TOGLIERE? SI

Proposition: (Col-UFREC Row-UFREC) DFREC Frame-unambiguous FREC

Framed vs Unframed 2dim languages

Pictureor two-dimensional string over a finite alphabet:

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- finite alphabet
- ** all 2dim rectangular words (pictures) over
- L **2dim language

Two-dimensional Languages

Local 2dim languages: first approach

First approach (“framed” one)

Generalization of local 1dim (string) languages

1dim: L= an1bm | n,m>0

is finite

2dim:

Unambiguity in FREC (II)

One pre-image

UFREC

Fix no local symbol

Fix first column or first row of local symbols

Fix two consecutive sides of local symbols

DFREC

Fix the frame

New definition

Framed vs Unframed 2dim languages