### Two-dimensional rational automata: a bridge unifying one and two

```Two-dimensional Rational
Automata:
a bridge unifying 1d and 2d
language theory
Marcella Anselmo
Univ. of Salerno
Dora Giammarresi
Univ. Roma Tor Vergata
Univ. of Catania
ITALY
Overview
• Topic: 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 1d
Two-dimensional (2d) languages
Two-dimensional string (or picture) over a finite
alphabet:
a b b c
c b a a
b a a b
•  finite alphabet
•  ** pictures over 
• L   ** 2d language
Problem: generalizing the theory of recognizability
of formal languages from 1d to 2d
2d literature
Since ’60 several attempts and different models
• 4NFA,
OTA, Grammars, Tiling Automata, Wang
Automata, Logic, Operations
Most accreditated generalization:
REC family
REC family I
• 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 II
• L  ** is recognizable by tiling system if L =
(L’) where L’  G** is a local language and  is a
mapping from the alphabet G of L’ to the
alphabet  of L
• (, G, , ) is called tiling system
• REC is the family of two-dimensional languages
recognizable by tiling system
Example
Consider Lsq the set of all squares over  = {a}
• Lsq is not local. Lsq is recognizable by tiling system.
• Lsq = (L’) where L’ is a local language over G = {0,1,2}
and  is such that (0)=(1)=(2)=a
p=
1
0 0 0
2
1
2 2
0 0
1
0
2 2 2
1
 L’

(p) =
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
 Lsq
Why another model?
REC family has been deeply studied
• Notions: unambiguity, determinism …
• Results: equivalences, inclusions, closure properties,
decidability properties …
but …
From 1d to 2d
This new model of recognition gives:
• a more natural generalization from 1d to 2d
• a uniform setting for all notions, results,
techniques presented in the 2d literature
Starting from Finite Automata for strings we
introduce Rational Automata for pictures
In this setting
• Some notions become more «natural» (e.g.
different forms of determinism)
• 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)
From Finite Automata to Rational Automata
We take inspiration from the geometry:
Points
Symbols
1d
1d
Lines
Strings
2d
2d
Planes
Pictures
• Finite sets of symbols are used to define finite automata
that accept rational sets of strings
• Rational sets of strings are used to define rational
automata that accept recognizable sets of pictures
From Finite Automata to Rational Automata
Finite Automaton
A = (, Q, q0, d, F)
 finite set of symbols
Q finite set of states
q0 initial state
d finite relation on (Q X ) X 2Q
F finite set of final states
Symbol
String
Finite
Rational
Rational Automaton!!
Rational Automata (RA)
A = (, Q, q0, d, F)
 finite set of symbols
Q finite set of states
q0 initial state
d finite relation on (Q X ) X 2Q
F finite set of final states
Symbol
String
Finite
Rational
Rational automaton H = (A, SQ, S0, dT, FQ)
A = + rational set of strings on 
SQ  Q+ rational set of states
S0 = q0+ initial states
dT rational relation on (SQ X A) X 2SQ
computed by transducer T
FQ rational set of final states
Rational Automata
(RA)
ctd.
RA
H = (A, SQ, S0, dT, FQ)
dT rational relation on (SQ X A) X 2SQ
What does it mean???
computed by transducer T
SQ  Q+
A =  +
If s = s1 s2 … sm SQ and a = a1 a2 … am  A
then q = q1 q2 … qm  dT (s , a)
if q is output of the transducer T
on the string (s1,a1) (s2,a2) … (sm,am) over the alphabet Q X 
Recognition by RA
A computation of a RA on a picture p  ++, p of
size (m,n), is done as in a FA, just considering p as
a string over the alphabet of the columns A = +
i.e. p = p1 p2 … pn with pi  A
Example:
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
p1
p2
p3
p4
picture a a a a  ++
a
a
a
p
a
string
Recognition by RA (ctd.)
The computation of a RA H on a picture p, of
size (m,n), starts from q0m, initial state, and
reads p, as a string, column by column, from
left to right.
FQ is rational
p is recognized by H if, at the end of the
computation, a state qf  FQ is reached.
L(H) = language recognized by H
L(RA) = class of languages recognized by RA
Example 1
RA recognizing Lsq set of all squares over  = {a}
Let Q = {q0,0,1,2} and Hsq = ( A, SQ, S0, dT, FQ) with
A = a+ , SQ = q0+ 0*12*  Q+ , S0 = q0+ , FQ = 0*1,
dT computed by the transducer T
T
L(Hsq) = Lsq
Example 1:computation
a a a a
Computation on p =
T
a a a a
a a a a
a a a a
dT (q04, 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  FQ
p  L(Hsq)=Lsq
RA and REC
This example gives the intuition for the following
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 language
Furthermore
In the previous example the rational
automaton Hsq 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.
Example 2
Consider Lfr=fc the set of all squares over  = {a,b}
with the first row equal to the first column.
• Lfr=fc  L(RA)
• The transition function is realized by a transducer
with states r0, r1, r2, ry, dy for any y  
Similarity with other models
• Rational Graphs
• Iteration of Rational Transducers
• Matz’s Automata for L(m)
Studying REC by RA
• Closure properties
• Determinism: definitions and results
• Decidability results
Closure properties
Proposition L(RA) is closed under union,
intersection, column- and row-concatenation
and stars.
Proof The closure under row-concatenation
follows by properties of transducers.
The other ones can be proved by exporting
FA techniques.
Determinism in REC
The definition of determinism in REC is still
controversial
Different definitions
The “right” one?
Different classes:
DREC, Col-Urec, Snake-Drec
Now, in the RA context, all of them assume a
natural position in a common setting with nondeterminism and unambiguity
Determinism: definition
Two different definitions of determinism can be
given
1. The transduction is a function (i.e. dT on (SQ X A) X SQ)
Deterministic Rational Automaton (DRA)
2. The transduction is left-sequential
Strongly Deterministic Rational Automaton (SDRA)
Col-UREC
DREC
Determinism: results
Theorem
L is in L(DRA) iff L is in Col-UREC
L is in L(SDRA) iff L is in DREC
Remark It was proved Col-UREC=Snake-Drec
In the RA context
Col-UREC=Snake-Drec
follows easily by a classical
result on
transducers Elgot&Mezei1965
Decidability results
Proposition It is decidable whether a RA is
deterministic (strongly deterministic, resp.)
Proof It follows very easily from decidability
results on transducers.
Conclusions
Despite a rational automaton is in principle
more complicated than a tiling system, it has