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

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

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 cc b a ab a a b

2d literature

Since ’60 several attempts and different models

• 4NFA, OTA, Grammars, Tiling Automata, Wang Automata, Logic, Operations

REC family

Most accreditated generalization:

•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 22) such that, for any p in L, any sub-picture 22 of p is in

REC family I

• 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

• REC is the family of two-dimensional languages recognizable by tiling system

• (, G, , ) is called tiling system

REC family II

• Lsq is not local. Lsq 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 aa a a aa a a aa a a a

1 0 0 02 1 0 02 2 1 02 2 2 1

Consider Lsq the set of all squares over = {a}

Lsq(p) = L’p =

Why another model?

REC family has been deeply studied

• Notions: unambiguity, determinism …

• Results: equivalences, inclusions, closure properties, decidability properties …

but …ad hoc definitions and

techniques

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

From 1d to 2d

• 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 setting

From Finite Automata to Rational Automata

We take inspiration from the geometry:

• 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

Points Lines Planes1d 2d

Symbols Strings Pictures1d 2d

From Finite Automata to Rational Automata

Finite Automaton A = (, Q, q0, d, F) finite set of symbolsQ finite set of statesq0 initial state d finite relation on (Q X ) X 2Q

F finite set of final states

Rational Automaton!!Symbol String Finite Rational

Rational automaton H = (A, SQ, S0, dT, FQ)A = + rational set of strings on SQ Q+ rational set of statesS0 = q0

+ initial statesdT rational relation on (SQ X A) X 2SQ

computed by transducer TFQ rational set of final states

A = (, Q, q0, d, F) finite set of symbolsQ finite set of statesq0 initial state d finite relation on (Q X ) X 2Q

F finite set of final states

Rational Automata (RA)

Symbol String Finite Rational

RAH = (A, SQ, S0, dT, FQ)dT rational relation on (SQ X A) X 2SQ

computed by transducer T

Rational Automata (RA) ctd.

If s = s1 s2 … sm SQ and a = a1 a2 … am A

What does it mean???

SQ Q+ 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

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

Recognition by RA

Example:

picture +

+string

a a a aa a a aa a a aa a a a

aaaa

aaaa

aaaa

aaaa

p1p p2 p3 p4

The computation of a RA H on a picture p, of size (m,n), starts from q0

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 qf FQ is reached.

FQ is rational

L(H) = language recognized by HL(RA) = class of languages recognized by RA

Example 1

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

RA recognizing Lsq set of all squares over = {a}

L(Hsq) = Lsq

T

Computation on p =

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

Example 1:computation

a a a aa a a aa a a aa a a aT

p L(Hsq)=Lsq

This example gives the intuition for the following

RA 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 language

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.

Furthermore

Example 2

Consider Lfr=fc the set of all squares over = {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

• Lfr=fc L(RA)

• Rational Graphs

• Iteration of Rational Transducers

• Matz’s Automata for L(m)

Similarity with other models

Studying REC by RA

• Closure properties

• Determinism: definitions and results

• Decidability results

Proposition L(RA) is closed under union, intersection, column- and row-concatenation and stars.

Closure properties

Proof The closure under row-concatenation follows by properties of transducers. The other ones can be proved by exporting FA techniques.

Now, in the RA context, all of them assume a natural position in a common setting with non-determinism and unambiguity

Determinism in RECThe definition of determinism in REC is still controversial

Different definitions

Different classes:DREC, Col-Urec, Snake-

Drec

The “right” one?

Two different definitions of determinism can be given1. The transduction is a function (i.e. dT on (SQ X A) X SQ)

Deterministic Rational Automaton (DRA)

Determinism: definition

2. The transduction is left-sequential

Strongly Deterministic Rational Automaton (SDRA)

Col-UREC

DREC

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: results

Decidability results

Proposition It is decidable whether a RA is deterministic (strongly deterministic, resp.)

Proof It follows very easily from decidability results on transducers.

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 theoryFurther steps: look for other results on transducers and finite automata to prove new properties of REC.

Grazie per l’attenzione!