Framed vs Unframed Two-dimensional languages Marcella Anselmo Natasha Jonoska Maria Madonia Univ. of Salerno Univ. South Florida Univ. of Catania ITALY.

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

• Sets of finite pictures

• Tilings of the infinite plane

Remark: The set of its finite blocks is L01

In the literature two kinds of 2dim languages

0 0 0 00 0 1 00 0 0 00 0 0 0

0 00 01 0

0 0 0 00 0 0 0

0 0 0 00 0 1 00 0 0 0

0 0 0 00 0 0 00 0 0 0

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

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

M. Anselmo N. Jonoska M. Madonia Framed vs Unframed 2dim 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

Overview of the talk

New “unframed” definition for “finite” pictures

In this talk

• Results of comparison framed vs unframed

with special focus on determinism and unambiguity

• Topic: Recognizable 2dim languages

Local 2dim languages

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

a a 1 ba a 1 ba a 1 ba a 1 b

a a 1 b a a 1 b a a 1 b a a 1 b

“Framed” approach

• Tiling of the (infinite) plane 0 0 0 00 0 1 00 0 0 0

• No sharp is needed!

“Unframed” approach


• finite alphabet, ** all pictures over ,

L ** 2dim language

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

Local languages: LOC

p =

p =

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


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

Example of local language

1001

10

0010

0000

01

1

10

00

00

01

0

00

00

0

1

=0100

100

010

001p =

#####

#100#

#010#

#001#

#####

p =


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

Recognizable languages: REC

Example: LSq = all squares over {a}

is recognizable by tiling system.

Set L’=Ld and (1)= (0)= a

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

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


Do not care about the boundary of a picture!

Factorial local/recognizable languages

• L is factorial local if there exists a finite set of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22 of 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 unbordered tiling system

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


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

Example of L in FREC

eeee

bbff

fcfc

ffff

aaee

gggg

ggaa

dg1a

hdb1

hhbb

dgdg

b1fc

1ace

hdhd

hhhh

=cece

e e e c f

a a a 1 b

g g g d h

g g g d h

g g g d h

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0


LOC and FLOC, REC and FREC

• L FLOC implies L LOC

• 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 FREC implies L REC (as before)

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

• FREC REC (as before)

(adding everywhere)

/

/


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

(remove tiles with )no for F(L)=L

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


Determinism and unambiguity

Remark Usually Determinism implies 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)

• Unambiguity

One possible accepting computation

• Determinism

One possible next step


Def. [A, Giammarresi, M 07] A tiling system is tl-br-deterministic if a,b,c and s , unique tile

a b

c d

(Analogously tr-bl,bl-tr,br-tl -deterministic tiling system)

s

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

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

deterministic tiling system

Determinism in REC: DREC

such that (d)=s.


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/ //


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.


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 ambiguous factorial recognizable languages


Example

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

p =

e e e c f

a a a 1 b

g g g d h

g g g d h

g g g d h

Recall the example L01

0 0 0 0

0 0 0 0

0 0 0 0

g g g d

g g g d

g g g d

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

-1

g g d h

g g d h

g g d h

-1

-1


Example (continued)

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

p =

e e e c f

a a a 1 b

g g g d h

g g g d h

g g g d h

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

-1


Unambiguity in FREC

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

Proposition. UFREC = FLOC

Remarks.

• UFREC is a very limited notion

• DFREC does not imply UFREC

A better suited definition of unambiguity is necessary


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 mostp =

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


Frame unambiguity (II)

Proposition L DFREC implies L is frame-unambiguous

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

p =

e e e c f

a a a 1 b

g g g d h

g g g d h

g g g d h

In L01

0 0 0 0

0 0 0 0

0 0 0 0

g g g d

g 0 0 d

g g g d

g g g d

g g g d

g g g d

-1

-1

L01 is frame-unambiguous


Ambiguity in REC vs ambiguity in FREC (I)

In REC

Determinism

Unambiguity

There are languages

• infinitely ambiguous

• finitely-ambiguous

• unambiguous

In FREC

There are languages

• infinitely ambiguous

• unambiguous

(as far as we know)

Determinism

Unambiguity

Determinism

Frame-Unambiguity


Ambiguity in REC vs ambiguity in FREC (II)

• Finitely-ambiguous factorial in FREC and unambiguous in REC

Remark Frame reduces the ambiguity degree

0 0 1 0

0 0 1 0

a a 1 b

a a 1 b

a a 1 b

a a 1 b

0 0

0 0

a a

a a

b b

b b

0 0 1 0

0 0 1 0

0 0

0 0

a a

a a

-1


Ambiguity in REC vs ambiguity in FREC (III)

• Infinitely factorial ambiguous in FREC and unambiguous in REC

e e c f

a a 1 b

g g d h

Moreover

0 0 0 0

0 0 1 0

0 0 0 0

e e e

e e e

0 0 0

0 0 0


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

Grazie


Conclusions

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

Local

Projection

“shifts of finite type” “sofic shifts”

Frame can enforce size and content of recognized picturesFrame can reduce ambiguity degree

Additional memory

Tilings of the plane 2dim symbolic dynamical systems

analogies and interpretations in symbolic dynamics


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.


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

Removing tiles with # does not always work …

Example Consider Ld and the tiling system for it. Teta contains all the sub-tiles of

T no

# # # # ## 1 0 0 ## 0 1 0 ## 0 0 1 ## # # # #

0 01 0

0 10 0 but

0 0 11 0 0

F(L)

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


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

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

TOGLIERE? SI

Two-dimensional Languages

Picture or two-dimensional string over a finite alphabet:

abaab

bcabc

acbba

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

Local 2dim languages: first approach

Generalization of local 1dim (string) languages

a a a 1 b b a a a 1 b b

2dim:

a a 1 ba a 1 ba a 1 ba a 1 b

a a 1 b a a 1 b a a 1 b a a 1 b

is finitea a 1 b b b a a 1 b

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

First approach (“framed” one)


Unambiguity in FREC (II)

New definition

One pre-image

Fix no local symbolUFREC

Fix first column or first row of local symbols

Fix two consecutive sides of local symbols

DFREC

Fix the frame

Framed vs Unframed Two-dimensional languages Marcella Anselmo Natasha Jonoska Maria Madonia Univ. of Salerno Univ. South Florida Univ. of Catania ITALY.

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