On Recognizable Languages of Infinite Pictures

In a recent paper [1], Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for acceptance of infinite words. This way they extended the classical theory of recognizable languages of finite pictures, [13], to the case of infinite pictures. On the other hand automata reading ordinal words have been first considered by Büchi in order to study the decidability of the monadic second order theory on countable ordinals. In particular he defined automata reading words of length ω 2 , [5,6]. Another model of automaton reading words of length ω 2 has been studied by Choueka in [7] and it has been shown by Bedon that these two models are equivalent [2,3]. They accept the so called regular ω 2 -languages which can also be defined by generalized regular expressions, see also the work of Wojciechowski [26,27]. In [1] 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 ω 2 . We give in this paper a solution to this problem, showing that the class of languages of infinite pictures which are accepted by Büchi automata reading words of length ω 2 is strictly included in the class of languages of infinite pictures which are Büchirecognized by some finite tiling system. Another way to compare these two classes is to compare the topological complexity of languages in each of them, with regard to the Borel and projective hierarchies. We then determine the topological complexity of Büchi-recognized languages of infinite pictures. This way we show that Büchi tiling systems have a much greater accepting power than automata over ordinal words of length ω 2 . Using topological arguments, we give also the answer to two questions raised in [1], showing that it is undecidable whether a Büchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). For that purpose we use a very similar technique as in a recent paper where we have proved several undecidability results for infinitary rational relations [12]. The paper is organized as follows. In section 2 we recall basic definitions for pictures and tiling systems. Büchi automata reading words of length ω or ω 2 are introduced in section 3. We compare the two modes of acceptance in section 4. Undecidability results are proved in section 5.

Let Σ be a finite alphabet and # be a letter not in Σ and let Σ = Σ ∪ {#}. If m and n are two integers > 0 or if m = n = 0, a picture of size (m, n) over Σ is a function p from {0, 1, . . . , m + 1} × {0, 1, . . . , n + 1} into Σ such that p(0, i) = p(m + 1, i) = # for all integers i ∈ {0, 1, . . . , n + 1} and p(i, 0) = p(i, n + 1) = # for all integers i ∈ {0, 1, . . . , m + 1} and p(i, j) ∈ Σ if i / ∈ {0, m + 1} and j / ∈ {0, n + 1}. The empty picture is the only picture of size (0, 0) and is denoted by λ. Pictures of size (n, 0) or (0, n), for n > 0, are not defined. Σ ⋆,⋆ is the set of pictures over Σ. A picture language L is a subset of Σ ⋆,⋆ .

An ω-picture over Σ is a function p from ω × ω into Σ such that p(i, 0) = p(0, i) = # for all i ≥ 0 and p(i, j) ∈ Σ for i, j > 0. Σ ω,ω is the set of ω-pictures over Σ. An ω-picture language L is a subset of Σ ω,ω . For Σ a finite alphabet we call Σ ω 2 the set of functions from ω × ω into Σ. So the set Σ ω,ω of ω-pictures over Σ is a strict subset of Σω 2 .

As usual, one can imagine that, for integers j > k ≥ 1, the j th column of p is on the right of the k th column of p and that the j th row of p is “above” the k th row of p. This representation will be used in the sequel.

We introduce now tiling systems as in the paper [1]. A tiling system is a tuple A=(Q, Σ, ∆), where Q is a finite set of states, Σ is a finite alphabet, ∆ ⊆ ( Σ × Q) 4 is a finite set of tiles. A Büchi tiling system is a pair (A,F ) where A=(Q, Σ, ∆) is a tiling system and F ⊆ Q is the set of accepting states. A Muller tiling system is a pair (A, F ) where A=(Q, Σ, ∆) is a tiling system and F ⊆ 2 Q is the set of accepting sets of states.

Tiles are denoted by (a 3 , q 3 ) (a 4 , q 4 ) (a 1 , q 1 ) (a 2 , q 2 ) with a i ∈ Σ and q i ∈ Q, and in general, over an alphabet Γ, by

We will indicate a combination of tiles by:

A run of a tiling system A=(Q, Σ, ∆) over a (finite) picture p of size (m, n) over Σ is a mapping ρ from {0, 1, . . . , m + 1} × {0, 1, . . . , n + 1} into Q such that for all (i, j) ∈ {0, 1, . . . , m} × {0, 1, . . . , n} with p(i, j) = a i,j and ρ(i, j) = q i,j we have a i,j+1 a i+1,j+1 a i,j a i+1,j • q i,j+1 q i+1,j+1 q i,j q i+1,j ∈ ∆.

A run of a tiling system A=(Q, Σ, ∆) over an ω-picture p ∈ Σ ω,ω is a mapping ρ from ω × ω into Q such that for all (i, j) ∈ ω × ω with p(i, j) = a i,j and ρ(i, j) = q i,j we have a i,j+1 a i+1,j+1 a i,j a i+1,j • q i,j+1 q i+1,j+1 q i,j q i+1,j ∈ ∆.

We now reca

…(Full text truncated)…

This content is AI-processed based on ArXiv data.