Two-dimensional Rational Automata: a bridge unifying 1d and 2d language 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 abbc cbaa baab

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 2 2) such that, for any p in L, any sub-picture 2 2 of p is in REC family I

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 REC is the family of two-dimensional languages recognizable by tiling system (,,, ) is called tiling system REC family II

L sq is not local. L sq is recognizable by tiling system. Example L sq = (L) where L is a local language over = {0,1,2} and is such that (0)= (1)= (2)=a aaaa aaaa aaaa aaaa Consider L sq the set of all squares over = {a} L sq (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 PointsLinesPlanes 1d2d SymbolsStringsPictures 1d2d

From Finite Automata to Rational Automata Finite Automaton A = (, Q, q 0,, F) finite set of symbols Q finite set of states q 0 initial state finite relation on (Q X ) X 2 Q F finite set of final states Rational Automaton!! Symbol String Finite Rational

Rational automaton H = (A, S Q, S 0, T, F Q ) A = + rational set of strings on S Q Q + rational set of states S 0 = q 0 + initial states T rational relation on (S Q X A ) X 2 S Q computed by transducer T F Q rational set of final states A = (, Q, q 0,, F) finite set of symbols Q finite set of states q 0 initial state finite relation on (Q X ) X 2 Q F finite set of final states Rational Automata (RA) Symbol String Finite Rational

RA H = (A, S Q, S 0, T, F Q ) T rational relation on (S Q X A ) X 2 S Q computed by transducer T Rational Automata (RA) ctd. If s = s 1 s 2 … s m S Q and a = a 1 a 2 … a m A What does it mean??? S Q Q + A = + then q = q 1 q 2 … q m 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

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 = p 1 p 2 … p n with p i A Recognition by RA Example : picture + string aaaa aaaa aaaa aaaa a a a a a a a a a a a a a a a a p1p1 pp2p2 p3p3 p4p4

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 F Q is reached. F Q is rational L(H) = language recognized by H L(RA) = class of languages recognized by RA

Example 1 Let Q = {q 0,0,1,2} and H sq = ( A, S Q, S 0, T, F Q ) with A = a +, S Q = q * 12 * Q +, S 0 = q 0 +, F Q = 0 * 1, T computed by the transducer T RA recognizing L sq set of all squares over = {a} L(H sq ) = L sq T

Computation on p = T (q 0 4, a 4 ) = output of T on (q 0,a) (q 0,a) (q 0,a) (q 0,a) = 1222 T (1222, a 4 ) = 0122 T (0122, a 4 ) = 0012 T (0012, a 4 ) = 0001 F Q Example 1:computation aaaa aaaa aaaa aaaa T p L(H sq )=L sq

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 H sq mimics a tiling system for L sq 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 L fr=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 r 0, r 1, r 2, r y, d y for any y L fr=fc L(RA)

Rational Graphs Iteration of Rational Transducers Matzs 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. T on (S Q X A ) X S Q ) Deterministic Rational Automaton (DRA) Determinism: definition 2. The transduction is left-sequential Strongly Deterministic Rational Automaton (SDRA) Col-URECDREC

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

Grazie per lattenzione!