On Recognizable Languages of Infinite Pictures

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).

💡 Research Summary

The paper investigates two fundamental models for recognizing languages of infinite two‑dimensional words (infinite pictures). The first model is the finite tiling system, a well‑studied extension of finite picture recognizers to the infinite case, equipped with Büchi or Muller acceptance conditions. The second model reads pictures “row by row” using ordinal automata that process words of length ω² (the first uncountable ordinal after ω). The authors compare the expressive power of these models, determine their topological complexity, and settle several decision‑theoretic questions left open by Altenbernd, Thomas and Wöhrle.

Key definitions.
An infinite picture over a finite alphabet Σ is a function p: ω×ω → Σ∪{#} with a border of the special symbol #. The j‑th row of p is the ω‑word p(1,j)p(2,j)… and the j‑th column is p(j,1)p(j,2)…. A tiling system A = (Q, Σ, Δ) assigns a state from Q to each cell, subject to a finite set Δ of admissible 2×2 tiles. A run is accepting if the set of states occurring infinitely often satisfies a Büchi set F⊆Q (or a Muller set F⊆2^Q). Such systems define the class TS(Σ^{ω,ω}).

An ordinal Büchi automaton for words of length α (here α = ω²) is a sextuple (Σ, Q, q₀, Δ, γ, F). Transitions for successor positions are given by Δ ⊆ Q×Σ×Q, while limit‑step transitions use γ ⊆ P(Q)×Q: the automaton looks at the set of states visited co‑finally before the limit and moves accordingly. Acceptance requires that the final state after reading the whole word belongs to F. For ω²‑words this model is equivalent to the Choueka automata studied by Bedon; both accept exactly the regular ω²‑languages, i.e., those definable by ω²‑regular expressions.

Row‑by‑row acceptance.
Given an infinite picture p, the authors encode it as an ω²‑word (\bar p) by interleaving rows: (\bar p(ω·n+m) = p(m+1, n+1)) for n,m≥0. A picture language L is said to be accepted “row by row” if the set (\bar L = {\bar p \mid p∈L}) is a regular ω²‑language. This class is denoted BA(Σ^{ω,ω}).

Main result (Theorem 4.1).
Every language in BA(Σ^{ω,ω}) is also Büchi‑recognizable by some finite tiling system, but the converse fails. The inclusion BA ⊆ TS is proved by constructing, from a regular ω²‑language (\bar L), a tiling system that simulates the underlying ω‑automata for each row and simultaneously checks that the sequence of row‑labels belongs to the ω‑language R that generated (\bar L) via substitution (Proposition 3.5). The tiling system’s state at a cell contains five components: (i) the guessed index i_j indicating which component language R_{i_j} the current row belongs to, (ii) the current state of the corresponding row‑automaton, (iii) a “ant” direction marker (right, down, or idle) that traverses the picture, (iv) a flag for whether the ant is moving vertically, and (v) a marker for the Muller acceptance condition. Horizontal propagation checks each row against its R_{i_j}; vertical propagation checks that the ω‑sequence i₁i₂… belongs to R. A Muller condition guarantees that infinitely many rows satisfy their respective Büchi conditions, yielding a Büchi‑accepting tiling system.

To show the inclusion is strict, the authors use topological arguments. Languages recognized by Büchi tiling systems can be Σ₁¹‑complete (analytic‑complete), for example by encoding the non‑emptiness of a recursively enumerable set of infinite trees. Such languages lie beyond the Borel hierarchy. In contrast, every regular ω²‑language is Borel (indeed Π₂⁰ or Σ₂⁰). Hence there exist tiling‑system languages that are not row‑by‑row regular, proving TS ⊄ BA.

Topological complexity.
The paper establishes that TS(Σ^{ω,ω}) contains languages of arbitrary analytic complexity, while BA(Σ^{ω,ω}) is confined to Borel sets of low rank. This separation provides a clean topological characterization of the expressive gap between the two models.

Undecidability results (Section 5).
Two decision problems posed by Altenbernd et al. are resolved negatively. For a given Büchi tiling system, it is undecidable whether its language is E‑recognizable (every row eventually visits an accepting state) or A‑recognizable (every row visits accepting states infinitely often). The proof reduces the classic emptiness problem for Büchi tiling systems (known to be Π₁⁰‑complete) to these properties by constructing a modified tiling system that forces a rejection precisely when a particular row fails to satisfy the E or A condition. Consequently, deciding E‑ or A‑recognizability is at least as hard as the emptiness problem and therefore undecidable.

Conclusion.
The work clarifies the relationship between two natural acceptance mechanisms for infinite pictures. Row‑by‑row ω²‑automata are strictly weaker than finite tiling systems, a fact reflected both in their Borel rank and in the existence of Σ₁¹‑complete tiling‑system languages. Moreover, fundamental verification questions such as E‑ and A‑recognizability are shown to be undecidable. These findings deepen our understanding of infinite two‑dimensional automata, provide a bridge between automata theory and descriptive set theory, and suggest further exploration of higher‑dimensional infinite structures and their logical properties.