Framed vs Unframed
This presentation is the property of its rightful owner.
Sponsored Links
1 / 33

Framed vs Unframed Two-dimensional languages PowerPoint PPT Presentation


  • 52 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Framed vs Unframed Two-dimensional languages

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Framed vs unframed two dimensional languages

Framed vs Unframed

Two-dimensional languages

Marcella Anselmo Natasha Jonoska Maria Madonia

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

ITALY USA ITALY


Framed vs unframed two dimensional languages

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


Framed vs unframed two dimensional languages

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


Framed vs unframed two dimensional 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!


Framed vs unframed two dimensional languages

p =

p =

  • L islocalif there exists a finite set  of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of 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


Framed vs unframed two dimensional languages

0

0

0

1

0

0

0

1

0

1

0

1

0

0

1

0

0

0

0

0

 =

0

0

0

0

1

0

0

0

0

1

1

0

1

0

1

0

p =

#

#

#

#

#

1

0

0

#

1

0

0

#

0

1

0

#

0

1

0

#

p =

0

0

1

#

0

0

1

#

#

#

#

#

#

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


Framed vs unframed two dimensional 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


Framed vs unframed two dimensional 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 22) such that, for any p in L, any sub-picture 22of 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


Framed vs unframed two dimensional languages

e

c

c

f

e

e

f

f

e

e

c

f

f

f

a

1

1

b

e

e

f

f

a

a

c

f

b

b

 =

a

1

1

b

a

a

b

b

g

g

g

d

d

h

e

h

h

c

g

d

d

h

g

g

h

h

g

g

g

d

d

h

h

e

h

c

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


Framed vs unframed two dimensional 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


Framed vs unframed two dimensional languages

Characterization of FLOC inside LOC and

of FREC inside REC

Proposition FLOC = LOC  Factor-closed

Proof LLOC and L factor-closed implies L FLOC. Indeed

nofor 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


Framed vs unframed two dimensional 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

  • One possible accepting computation

  • Remark Usually Determinism implies Unambiguity

    Framed vs Unframed 2dim languages


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional languages

    /

    /

    /

    Unambiguity in REC: UREC

    Definition [Giammarresi,Restivo 92]A tiling system (, , , ) is unambiguous for L** if for any pL 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


    Framed vs unframed two dimensional languages

    Ambiguity in REC

    Definition L ** is finitely-ambiguous if there exists a tiling system for L such that every picture pL 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional languages

    Example (continued)

    -1

    p =

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

    Framed vs Unframed 2dim languages


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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 LFREC 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional languages

    Ambiguity in REC vs ambiguity in FREC (III)

    Moreover

    • Infinitely factorial ambiguous in FREC and unambiguous in REC

    Framed vs Unframed 2dim languages


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional languages

    Grazie


    Framed vs unframed two dimensional languages

    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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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


    Framed vs unframed two dimensional 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


    Two dimensional languages

    Pictureor two-dimensional string over a finite alphabet:

    a

    b

    b

    c

    a

    c

    b

    a

    c

    b

    b

    a

    a

    b

    a

    • finite alphabet

    •  ** all 2dim rectangular words (pictures) over 

    • L **2dim language

    Two-dimensional Languages


    Framed vs unframed 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:


    Framed vs unframed two dimensional languages

    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


  • Login