A Characterization of Recognizable Picture Languages by Tilings by Finite Sets

As extension of the Kleene star to pictures we introduce the operation of tiling We give a characterization of recognizable picture languages by intersection of tilings by nite sets of pictures Introduction In this paper we considerer picture languages as sets of rectangular arrays of symbols Many formalisms have been introduced to generalize the notion of recognizable languages to the case of picture languages In D Giammarresi and A Restivo introduce the notion of local picture languages which is direct extension of local string languages They then de ne recognizable picture lan guages by projection of local picture languages that is a well known property in string languages This class is very interesting since it corresponds to the class of picture languages recognizable by a particular class of cellular automata called on line tessellation automata and to the class of picture languages de nable by existential expressions in monadic second order logic A survey of the topic is given in the Handbook of Formal Languages In the string language theory it is well known that any recog nizable string language can be obtained by projection of the intersection of the star of two nite languages Formally if a string language R over is recogniz able there exist two nite string languages A and B over an alphabet and a letter to letter morphism such that This work was partially supported by PRC GDR AMI Mod eles et Outils pour les SYs t emes DIStribu es R A B It turns out that for any R R is recognizable if and only if there exist a letter to letter morphism and two nite sets A B f g such that R f j A B g We are interested in nding a similar characterization of recognizable picture languages To do that we need to extend the Kleene star to pictures We propose a extension in terms of tilings Given a picture language L the Kleene star of L denoted by L of this language is the set of all pictures that can be tiled by pictures of L Preliminaries and notations We assume the reader to be familiar with basic formal language theory see for more precisions For picture languages we recall some de nitions from Let be a nite alphabet A picture over is a two dimensional rectangular array of letters of We denote the set of all pictures over by For a picture p of size n m where n is the number of rows and m the number of columns of p we denote by p i j the letter of which occurs in i row and j column starting in the left top corner The set of all the pictures over of size n m is denoted by n m We denote by p also called bounded picture of p the n m picture over f g where is a special letter which does not belong to de ned by i n p i p i n j m p j p m j i n j m p i j p i j For instance if we consider the alphabet fa b cg we have for the picture p p a b a b c c p a b a b c c Let p be a picture of size n m over an alphabet For r n and s m we denote by Tr s p the set of the r s sub pictures of p Tr s p q r s j x n r y m s i r j s q i j p x i y j With pictures we have two concatenation products Let p be a n m picture and p be a n m picture The row concatenation of p with p that is denoted by p p is de ned if and only ifm m and is the n n m picture satisfying i n j m p p i j p i j i n j m p p n i j p i j In the same way the column concatenation of p with p that is denoted by p p is de ned if and only if n n and is the n m m picture satisfying i n j m p p i j p i j i n j m p p i m j p i j We also de ne a clock wise rotation over pictures Let p m n be a picture The rotation of p denoted by P is a picture over of size n m such that