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 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 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)=aa 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 Lsq the set of all squares over S = {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 2

d

Symbols Strings Pictures1d 2

d

From Finite Automata to Rational Automata

Finite Automaton

A = (S, Q, q0, d, F)

S finite set of symbolsQ finite set of statesq0 initial state

d finite relation on (Q X S) X 2Q

F finite set of final states

Rational Automaton!!Symbol String Finite Rational

Rational automaton H = (AS, SQ, S0, dT, FQ)

AS = S+ rational set of strings on S

SQ Q+ rational set of states

S0 = q0+ initial states

dT rational relation on (SQ X AS) X 2SQ

computed by transducer TFQ rational set of final states

A = (S, Q, q0, d, F)

S finite set of symbolsQ finite set of statesq0 initial state

d finite relation on (Q X S) X 2Q

F finite set of final states

Rational Automata (RA)

Symbol String Finite Rational

RAH = (AS, SQ, S0, dT, FQ)

dT rational relation on (SQ X AS) X 2SQ

computed by transducer T

Rational Automata (RA) ctd.

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

What does it mean???

SQ Q+AS = S+

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 S

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 AS = 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

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 = ( AS, SQ, S0, dT, FQ) with AS = 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 S = {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 a

a a a a

a a a a

a 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 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)

• 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 REC

The 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 given

1. The transduction is a function (i.e. dT on (SQ X AS) 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-UREC

L 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.

Conclusions

Despite 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!