On Recognizable Languages of Infinite Pictures

TL;DR

This paper compares acceptance of infinite picture languages by tiling systems and automata over ordinal words, establishing equivalences and undecidability results for recognition problems.

Contribution

It proves that B"uchi-recognizable infinite picture languages accepted row by row are also recognized by tiling systems, and shows undecidability of certain recognition properties.

Findings

All row-accepted B"uchi infinite picture languages are tiling-recognized.

The converse of the above is not true.

Deciding E- or A-recognizability of B"uchi infinite picture languages is undecidable.

Abstract

In a recent paper, Altenbernd, Thomas and W\"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the B\"uchi and Muller ones, firstly used for infinite words. The authors asked for comparing the tiling system acceptance with an acceptance of pictures row by row using an automaton model over ordinal words of length $\omega^2$. We give in this paper a solution to this problem, showing that all languages of infinite pictures which are accepted row by row by B\"uchi or Choueka automata reading words of length $\omega^2$ are B\"uchi recognized by a finite tiling system, but the converse is not true. We give also the answer to two other questions which were raised by Altenbernd, Thomas and W\"ohrle, showing that it is undecidable whether a B\"uchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable).