Highly Undecidable Problems about Recognizability by Tiling Systems

TL;DR

This paper investigates the high undecidability levels of various decision problems related to B"uchi-recognizable languages of infinite pictures, establishing their precise positions in the analytical hierarchy.

Contribution

It proves that key decision problems about recognizability by tiling systems are highly undecidable, specifically at the second level of the analytical hierarchy, and determines their exact degrees.

Findings

Non-emptiness and infiniteness are $ ext{Sigma}_1^1$-complete.

Universality, inclusion, and equivalence are $ ext{Pi}_2^1$-complete.

Recognition problems over ordinal words are also $ ext{Pi}_2^1$-complete.

Abstract

Altenbernd, Thomas and W\"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B\"uchi and Muller ones [1]. It was proved in [9] that it is undecidable whether a B\"uchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". We give the exact degree of numerous other undecidable problems for B\"uchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are $\Sigma^1_1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all $\Pi^1_2$-complete. It is also $\Pi^1_2$-complete to determine whether a given B\"uchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length $\omega^2$.