Extending Formal Languages Hierarchies to Higher Dimensions Dora Giammarresi Dipartimento di Informatica. Universit` a Ca’ Foscari di Venezia Via Torino 155, 30173 Venezia Mestre, Italy E-mail: dora@dsi.unive.it and Antonio Restivo Dipartimento di Matematica e Applicazioni, Universit` a di Palermo Via Archirafi 34, 90123 Palermo, Italy E-mail: restivo@altair.math.unipa.it Over 30 years ago, it was remarked by M. Minsky ([11]) that pictures could be regarded as sentences in a picture language. This suggested the idea to extend notions and techniques of languages theory to bidimensional (or multidimensional) environments. A generalization of formal languages to two dimensions is possible in different ways, and several formal models to recognize or generate two-dimensional objects have been proposed in the literature. These approaches, initially motivated by problems arising in the framework of pattern recognition and image processing, play also a role in studies concerning cellular automata and other models of parallel computing. A two-dimensional string is called a picture and is defined as a rectangular ar- ray of symbols taken from a finite alphabet Σ. A two-dimensional language (or picture language) is a set of pictures. We first describe two-dimensional languages defined (recognized, generated) by finite-state devices ([2]). We consider definitions that are “natural extensions” from corresponding ones in the string theory, where “natural extension” has the precise meaning that, they reduce to the string case when restricted to pictures of one row (one column) only. The goal is to inherit as many as possible properties from the string theory. A first natural approach is to define picture languages by means of regular expressions. The following (regular) operations are introduced for set of pictures: row and column concatenations, row and column Kleene closures and boolean operations. A regular expression is then a formula expressing how a specific picture language can be obtained from elementary languages by regular operations. Different families of languages can be defined, de- pending on the choice of operations allowed to be used in the expression. Regarding finite-state machines, several models of them have been designed to recognize pic- Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. c 1999 ACM 00360-0300/99/0900