可识别性的高度不可判定问题
Highly Undecidable Problems about Recognizability by Tiling Systems
摘要：
Altenbernd，Thomas和Wöhrle考虑了有限平铺系统对无限二维词（无限图像）语言的接受，并具有通常的接受条件，如Büchi和Muller的条件[1]。文[9]证明了无限图像的Büchi-可识别语言是否是E-可识别的（或A-可识别的）是不可判定的。我们在这里表明，这两个决策问题实际上是$Π_2^1$ -完整的，因此位于第二层次的分析，和“高度不可判定”。我们给的确切程度的许多其他不可判定的问题Büchi可识别的语言的无限图片。特别地，非空性和无限性问题是$Σ^1_1$ -完全的，普遍性问题、包含问题、等价问题、可确定性问题、可补性问题都是$Π^1_2$ -完全的.它也是$Π^1_2$ -完全的，以确定是否可以接受一个给定的无限图片的Büchi可识别的语言，使用一个自动机模型，对长度为$ω^2$的序数词的行。
英文摘要：
Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Büchi and Muller ones [1]. It was proved in [9] that it is undecidable whether a Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually $Π_2^1$-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 Büchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are $Σ^1_1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all $Π^1_2$-complete. It is also $Π^1_2$-complete to determine whether a given Büchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length $ω^2$.
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