Two-Dimensional Languages

  title={Two-Dimensional Languages},
  author={Dora Giammarresi and Antonio Restivo},
  booktitle={Handbook of Formal Languages},
The aim of this chapter is to generalize concepts and techniques of formal language theory to two dimensions. Informally, a two-dimensional string is called a picture and is defined as a rectangular array of symbols taken from a finite alphabet. A two-dimensional language (or picture language) is a set of pictures. 

Two-Dimensional Languages

A two-dimensional (2-d) language is a set of 2-d arrays of symbols, whose theories give the mathematical foundation for 2-d information processing. There are various formal methods for treating 2-d

Two-Dimensional Rational Automata: A Bridge Unifying One- and Two-Dimensional Language Theory

The model provides a uniform setting for the most important notions, techniques and results presented in the last decades for recognizable two-dimensional languages, and establishes new connections between one- and two- dimensional language theory.

A note on identities of two-dimensional languages

Two-dimensional pattern languages

Several classes of array languages obtained by generalising Angluin's pattern languages to the two-dimensional case are introduced and compared with respect to their expressive power and their closure properties are investigated.

Recognizable vs. Regular Picture Languages

  • O. Matz
  • Linguistics, Computer Science
  • 2007
The class of regular word languages plays a central role in formal language theory. Considerable effort has been made to transfer definitions and applications from word languages to their

Two-Dimensional Languages and Cellular Automata

It is shown that cellular automata whose two-dimensional languages are factorial-local have the same characterization as one-sided Cellular automata with SFT traces.

Extending formal language hierarchies to higher dimensions

Two-dimensional languages defined (recognized, generated) by finite-state devices are described and definitions that are “natural extensions” from corresponding ones in the string theory are considered, where “ natural extension” has the precise meaning that, they reduce to the string case when restricted to pictures of one row (one column) only.

On Two-Dimensional Pattern Matching by Finite Automata

The first of these two models presents an automata based version of the Bird and Baker approach with lower space complexity than the original algorithm, and a new model for two- dimensional approximate pattern matching using the two-dimensional Hamming distance is introduced.

Higher Dimensional Automata

It is provided the basics of a 2-dimensional theory of automata on series-parallel biposets and relates these classes to languages of series-Parallel bipOSets definable in monadic second-order logic.

Regular Expressions and Context-Free Grammars for Picture Languages

A new concept of regular expression and context-free grammar for picture languages (sets of matrices over a finite alphabet) is introduced and these two formalisms are compared and connected.



A survey of two-dimensional automata theory

Parallel Generation of Finite Images

A syntactic model for generating sets of images, where an image can be viewed as an array over finite alphabet is defined, called image grammar, which can be considered as a generalization of classical Chomsky grammar.

Recognizable Picture Languages

A new notion of recognizability for picture (two-dimensional) languages is proposed extending the characterization of one-dimensional recognizable languages in terms of local languages and alphabetic mappings and the undecidability of the emptiness problem is proved.

On Logics, Tilings, and Automata

  • W. Thomas
  • Computer Science, Mathematics
  • 1991
A notion of “graph acceptor” is introduced which can specify monadic second-order properties and allows to treat known types of finite automata in a common framework.

Two-dimensional finite automata and unacceptable functions

In this note, we first introduce the concept of “functions acceptable by two-dimensional finite automata” and then give several functions unacceptable by two-dimensional finite automata.

Two-Dimensional Finite State Recognizability

It is shown that some natural families of two-dimensional languages (finite languages, regular languages, locally testable languages) are recognizable and that the emptyness problem is undecidable for this family of languages.

Cyclic closure properties of automata on a two-dimensional tape