Parallel generation of infinite images

Abstract

We introduce a syntactic model for generating sets of finite and infinite images where a finite image can be viewed as an array over finite alphabet and an infinite image is an array with finite number of columns and infinite number of rows. This model, called image grammar, can be considered as a generalization of classical Chromsky grammar.

We study various types of infinite image grammars using Nivat's derivation which is an infinite unrestrictive derivation, Bvichi's (respectively Muller's) which is defined as infinite derivations selected by some repetitive set (respectively sets of rewriting rules). Then we study the combinatorial and language theoretical properties such as complexity measure, closure properties and decidability results. In terms of complexity measure we give a strict infinite hierarchy. We have extended Nivat's and Eilenberg's theorems to infinite images. Unfortunately we prove that Biichi's and McNaughton's theorems cannot be extended to infinite images. We also characterize these families in terms of deterministic image co-substitution and ω-languages

Keywords:

• ω-Words
• Infinite Images
• Büchi Automata
• Muller Automata
• and ω-Regular Languages

C.R. Categories:

• F.4.2
• I.5.1