Rational A utomata a bridge unifying 1d and 2d language theory Marcella Anselmo Dora Giammarresi Maria Madonia Univ of Salerno Univ Roma ID: 198864
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Slide1
Two-dimensional Rational Automata: a bridge unifying 1d and 2dlanguage theory
Marcella Anselmo Dora
Giammarresi
Maria
Madonia
Univ
.
of
Salerno
Univ
. Roma
Tor
Vergata
Univ
.
of
Catania
ITALYSlide2
OverviewTopic: recognizability of 2d languages
Motivation
:
putting
in a
uniform
setting
concepts
and
results
till
now
presented
for
2d
recognizable
languages
Results
:
definition
of
rational
automata
.
They
provide
a
uniform
setting
and
allow
to
obtain
results
in 2d just
using
techniques
and
results
in 1dSlide3
Problem: generalizing the theory of recognizability of formal languages from 1d to 2d
Two-dimensional
string
(or
picture) over a finite alphabet:
finite alphabet ** pictures over L ** 2d language
Two-dimensional (2d) languages
a
b
b
c
c
b
a
a
b
a
a
bSlide4
2d literatureSince ’60 several attempts and different models
4NFA,
OTA,
Grammars
,
Tiling Automata, Wang
Automata, Logic, Operations REC familyMost accreditated generalization:Slide5
REC family is defined in terms of 2d local
languages
It
is
necessary to identify the
boundary of picture p using a boundary symbol
p =
p =
A 2d
language
L
is
local
if
there
exists
a 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
REC family ISlide6
L ** is recognizable
by
tiling
system
if L = (L’)
where L’ G** is a local language and is a mapping from the alphabet of L’ to
the alphabet of L
REC
is
the family
of
two-dimensional
languages
recognizable
by
tiling
system
(
,
,
, )
is
called
tiling
system
REC family IISlide7
Lsq is not local. L
sq
is
recognizable
by tiling system.
Example Lsq = (L’) where L’ is a local
language over G = {0,1,2} and is such that
(0)=
(1)=
(2)
=a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
1
0
0
0
2
1
0
0
2
2
1
0
2
2
2
1
Consider
L
sq
the set
of
all
squares
over
S
=
{a}
L
sq
(p) =
L’
p =Slide8
Why another model? REC family has been deeply studied
Notions
:
unambiguity
,
determinism … Results: equivalences, inclusions, closure properties, decidability properties …
but …ad hoc definitions and techniquesSlide9
This new model of recognition gives:a more natural generalization from 1d to 2da uniform setting for
all
notions
,
results
, techniques presented in the 2d literatureStarting
from Finite Automata for strings we introduce Rational Automata for pictures
From 1d to
2dSlide10
Some techniques can be exported from 1d to 2d (e.g. closure properties) Some results can be exported from 1d to 2d (e.g. classical results on
transducers
)
Some
notions
become more «
natural» (e.g. different forms of determinism)
In this settingSlide11
From Finite Automata to Rational AutomataWe take
inspiration
from
the
geometry
:
Finite sets of symbols are used to define finite automata
that accept rational
sets
of
strings
R
ational
sets
of
strings
are
used
to
define
rational
automata
that
accept
recognizable
sets
of
pictures
Points
Lines
Planes
1
d
2d
Symbols
Strings
Pictures
1
d
2dSlide12
From Finite Automata to Rational AutomataFinite Automaton A = (
S
,
Q
,
q
0, d, F)S finite set of symbolsQ
finite set of statesq0 initial state d finite relation on (Q X S) X 2QF finite set of final statesRational Automaton!!
Symbol String Finite RationalSlide13
Rational automaton H = (AS, SQ, S0, dT, F
Q
)
A
S
= S+ rational set of strings on S
SQ Q+ rational set of statesS0 = q0+ initial statesdT rational relation on (SQ X AS) X 2S
Q computed by transducer TFQ rational
set
of
final
states
A
= (
S
,
Q
,
q
0
,
d
,
F
)
S
finite
set
of
symbols
Q
finite
set
of
states
q
0
initial
state
d
finite
relation on (Q X
S) X 2Q
F
finite set of final
states
Rational
Automata
(RA)
Symbol
String
Finite RationalSlide14
RAH = (AS, SQ, S0, dT, FQ)
d
T
rational
relation on (SQ X AS) X 2SQ computed by transducer
TRational Automata (RA) ctd. If s = s1 s2 … sm
SQ and a = a1 a2 … am
A
S
What
does
it
mean
???
S
Q
Q
+
A
S
=
S
+
t
hen
q
=
q
1
q
2
…
q
m
d
T
(
s
,
a
)
if
q
is
output
of the
transducer
T on the string
(s
1,a
1) (s
2,a
2) … (s
m,a
m) over
the alphabet
Q
X S
Slide15
A computation of a RA on a picture p S++, p of size
(m,n)
,
is
done as in a FA, just considering p as a string over the alphabet of the columns A
S = S+ i.e. p = p1 p2 … pn with pi AS
Recognition by RA
Example
:
picture
S
++
string
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
p
1
p
p
2
p
3
p
4Slide16
The computation of a RA H on a picture p, of size (m,n), starts
from
q
0
m, initial state, and reads p, as a string, column by column, from left to right.
Recognition by RA (ctd.) p is recognized by H if, at the end of the computation, a state q
f FQ is reached.
F
Q
is
rational
L(
H
)
=
language
recognized
by
H
L(
RA
)
=
class
of
languages
recognized
by
RASlide17
Example 1 Let Q = {q
0
,
0,1,2
}
and
Hsq = ( AS, S
Q, S0, dT, FQ) with AS
= a+ , SQ = q0+
0
*
12
*
Q
+
,
S
0
=
q
0
+
,
F
Q
=
0
*
1,
d
T
computed
by
the
transducer
T
RA
recognizing
L
sq
set
of
all
squares
over
S
=
{a}
L(
H
sq
) =
L
sq
TSlide18
Computation on p =dT
(
q
0
4
, a4) = output of T on (q0,a) (q0,a) (
q0,a) (q0,a) = 1222 dT (1222, a4) = 0122 dT (0122, a4) = 0012 dT (0012, a4) = 0001 F
QExample 1:computation
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
T
p
L(
H
sq
)
=
L
sqSlide19
This example gives the intuition for the followingRA and REC
Theorem
A
picture
language
is
recognized by a Rational Automaton iff it is tiling recognizable
Remark This
theorem
is
a
2d
version
of
a
classical
(
string
)
theorem
Medvedev
’64
:
Theorem
A
string
language
is
recognized
by
a Finite Automaton
iff it
is the projection
of a local
languageSlide20
In the previous example the rational automaton H
sq
mimics
a
tiling
system for Lsq
but …in general the rational automata can exploit the extra memory of
the states of the transducers
as
in the
following
example
.
FurthermoreSlide21
Example 2 Consider
L
fr=fc
the set
of
all squares over S = {a,b} with the first row equal to the first column
. The transition function is realized by a transducer with states r0, r1, r2, ry, dy for any y
S Lfr=fc L(RA)Slide22
Rational GraphsIteration of Rational TransducersMatz’s Automata for L(m)
S
imilarity
with
other modelsSlide23
Studying REC by RAClosure
properties
Determinism
:
d
efinitions and resultsDecidability resultsSlide24
Proposition L(RA) is closed under
union
,
intersection
,
column- and
row-concatenation and stars.Closure propertiesProof The
closure under row-concatenation follows by properties
of
transducers
.
The
other
ones
can
be
proved
by
exporting
FA
techniques
. Slide25
Now, in the RA context, all of them assume a natural position in a common setting with non-determinism and unambiguity
Determinism
in REC
The
definition
of
determinism in REC is still controversialDifferent definitions
Different classes:DREC, Col-Urec
,
Snake-Drec
The “right”
one
?Slide26
Two different
definitions
of
determinism can be givenThe transduction is a function (i.e. dT on (
SQ X AS) X SQ)Deterministic Rational Automaton (DRA)Determinism: definition
The transduction is left-sequentialStrongly Deterministic Rational Automaton (
SDRA
)
Col-UREC
DRECSlide27
Remark It was proved Col-UREC=Snake-Drec with ad hoc techniques Lonati&Pradella2004.In the RA
context
Col-UREC
=
Snake-Drec
follows easily by a classical result on transducers Elgot&Mezei1965
Theorem L is in L(DRA) iff L is in Col-URECL is in L(SDRA) iff L is
in DREC
Determinism
:
resultsSlide28
Decidability resultsProposition It
is
decidable
whether
a RA is
deterministic (strongly deterministic, resp.)Proof It follows very easily from decidability
results on transducers.Slide29
ConclusionsDespite a rational automaton is in principle more complicated than a tiling system, it
has
some major
advantages
:
It
unifies concepts coming from different motivations It allows to use results of the string language theory
Further steps: look for other results on transducers and finite automata to prove new properties of REC.Slide30
Grazie per l’attenzione!