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
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0 0
0 0
1 0
0 0 0 0
0 0 0 0
• 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
0 0 0 0
0 0 1 0
0 0 0 0
Remark: The set of its finite blocks is L01
0 0 0 0
0 0 0 0
0 0 0 0
Overview of the talk
• Topic: Recognizable 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
In this talk
New “unframed” definition for “finite” pictures
• Results of comparison framed vs unframed
with special focus on determinism and unambiguity
M. Anselmo N. Jonoska M. Madonia
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
a
a
a
a
a
a
a
a
1
1
1
1
b
b
b
b
“Unframed” approach
•






Tiling of the (infinite) plane
• No sharp is needed!

a
a
a
a


a
a
a
a


1
1
1
1


b
b
b
b







0 0 0 0
0 0 1 0
0 0 0 0
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  


p=
p=
















• L is local if there exists a finite set  of tiles (i. e. square
pictures of size 22) such that, for any p in L, any subpicture 22 of p is in  (and we write L=L() )
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
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
=
1 0
0 1
0 0
1 0
0 1
0 0
0 0
0 0
 1
 0
 0
 0
0 0
 
0 1
 
0 
1 
0 
0 
 
0 0
 
1 0
 
 1
 
0 
 0
 
1 
 
M. Anselmo N. Jonoska M. Madonia

1 0 0
p= 0 1 0
0 0 1
#
#
p= #
#
#
#
1
0
0
#
#
0
1
0
#
#
0
0
1
#
#
#
#
#
#
Framed vs Unframed 2dim languages
Recognizable languages: REC
• 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
• (, , , ) , 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
M. Anselmo N. Jonoska M. Madonia
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 is factorial local if 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2 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
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
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
0
0
0
0
0
0
0
0
1
0
h
0
0
0
0
0
d
h
0
0
0
0
0
d
h
0
0
0
0
0
e
e
e
c
f
a
a
a
1
b
g
g
g
d
g
g
g
g
g
g
=

e e
e e
e c
e c
c f
c f
f f
f f
e e
a a
e c
a 1
c f
1 b
f f
b b
a a
g g
a 1
g d
1 b
d h
b b
h h
g g
g g
g d
g d
d h
d h
h h
h h
M. Anselmo N. Jonoska M. Madonia
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
(as before)
implies
L REC
• FLOC / LOC
Example: Ld  LOC, not factor-closed
• FREC / REC
(as before)
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim 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

(remove tiles with )
no for F(L)=L
Proposition L  FREC iff L  REC and L=(K)
with K factor-closed local language
M. Anselmo N. Jonoska M. Madonia
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
One possible accepting computation
Remark Usually Determinism implies Unambiguity
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Determinism in REC: DREC
Def. [A, Giammarresi, M 07] A tiling system is tl-brdeterministic if  a,b,c   and s  ,  unique tile a b
c d
such that (d)=s.
s
unique way to fill this
position with a symbol
of  whose projection
matches symbol s
(Analogously tr-bl,bl-tr,br-tl -deterministic tiling system)
DREC languages that admit a tl-br or tr-bl or bl-tr or br-tldeterministic tiling system
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim 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
M. Anselmo N. Jonoska M. Madonia
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 pL has k
pre-images at most (for some k >1).
L is infinitely-ambiguous if it is not finitely ambiguous.
M. Anselmo N. Jonoska M. Madonia
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 ambiguous
factorial recognizable languages
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Example
Recall the example L01
p=
e
e
e
c
f
a
a
a
1
b
0
g
g
g
d
h
0
0
g
g
g
d
h
0
0
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
The unbordered tiling system
for L01 is deterministic but it
is not unambiguous
M. Anselmo N. Jonoska M. Madonia
-1
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
g
g
g
d
g
g
g
d
g
g
g
d
g
g
d
h
g
g
d
h
g
g
d
h
Framed vs Unframed 2dim languages
Example (continued)
p=
e
e
e
c
f
a
a
a
1
b
0
g
g
g
d
h
0
0
g
g
g
d
h
0
0
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
-1
Moreover it can be shown that L01 is an infinitely ambiguous
factorial language.
M. Anselmo N. Jonoska M. Madonia
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
M. Anselmo N. Jonoska M. Madonia
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.
p=
One pre-image at most
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
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Frame unambiguity (II)
In L01
p=
g0
g0
0
g
0
d
g0
0
0
g0
g0
0
g
0
0
0
0
0
e
e
e
c
f
0
0
0
1
0
a
a
a
1
b
0
0
0
0
0
g
g
g
d
h
0
0
0
0
0
g
g
g
d
h
0
0
0
0
0
g
g
g
d
h
-1
g
g
g
d
0
d
g
g
g
d
0
d
g
g
g
d
-1
L01 is frame-unambiguous
Proposition L  DFREC implies L is frame-unambiguous
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Ambiguity in REC vs ambiguity in FREC (I)
In REC
In FREC
Determinism
Determinism
Unambiguity
Frame-Unambiguity
Unambiguity
There are languages
There are languages
•
infinitely ambiguous
•
infinitely ambiguous
•
unambiguous
•
finitely-ambiguous
•
unambiguous
(as far as we know)
M. Anselmo N. Jonoska M. Madonia
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
0a
0a
1
0
b
0
0
0a
0a
1
0
b
0
0
-1
a
a
a
a
b
b
b
b











0a
0a
1
b
0


0a
0a


0a
0a
1
b
0


0a
0a











M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Ambiguity in REC vs ambiguity in FREC (III)
Moreover
• Infinitely factorial ambiguous in FREC and unambiguous in REC







0e
0e
0c
0f


0a
0a
1
b
0


0g
g
0
d
0
h
0







M. Anselmo N. Jonoska M. Madonia






0e
0e
0e


0e
0e
0e






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
M. Anselmo N. Jonoska M. Madonia
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
“shifts of finite type”
Projection
“sofic shifts”
M. Anselmo N. Jonoska M. Madonia
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.
M. Anselmo N. Jonoska M. Madonia
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
Example Consider Ld and the tiling system for it. Teta
contains all the sub-tiles of
#
#
#
#
#
#
1
0
0
#
#
0
1
0
#
#
0
0
1
#
#
#
#
#
#
0 0
1 0
0 1
0 0
T
no
but
0 0 1
1 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
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Finite and infinite ambiguity in FREC
Proposition: For any k >=1, there is a k-factorialambiguous language.
Proposition: Unambiguous-FREC  (Col-UFREC RowUFREC)  Finitely ambiguous FREC
TOGLIERE? SI
Proposition: (Col-UFREC  Row-UFREC)  DFREC 
Frame-unambiguous FREC
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages
Two-dimensional Languages
Picture or 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
Local 2dim languages: first approach
First approach (“framed” one)
Generalization of local 1dim (string) languages
1dim: L= an1bm | n,m>0
 a a a 1 b b 
 a
a a
a 1
2dim:
a
a
a
a
a
a
a
a
1
1
1
1
b
b
b
b
1 b
b 
b b







a
a
a
a


a
a
a
a


1
1
1
1

is finite

b
b
b
b







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
M. Anselmo N. Jonoska M. Madonia
Framed vs Unframed 2dim languages