无限图像可识别语言的判定问题
Decision Problems for Recognizable Languages of Infinite Pictures
摘要：
Altenbernd，Thomas和Wöhrle在[ATW 02]中考虑了有限镶嵌系统对无限二维词（无限图像）语言的接受，通常的接受条件，如Büchi和Muller的，首先用于无限词。许多经典的判定问题都是在形式语言理论和自动机理论中研究的，现在自然地出现在无限图像的可识别语言中。本文首先回顾了[Fin 09 b]的一些最新结果，其中我们给出了无限图像的Büchi可识别语言的许多不可判定问题的确切程度，这些问题实际上位于分析层次的第一或第二层，并且“高度不可判定”。然后，我们在这里证明了一些更（高）的不可判定性结果。我们首先证明了它是$Π_2^1$ -完全的，以确定是否一个给定的Büchi可识别的语言的无限图片是明确的。然后我们研究基数问题。利用[FL 09]的结果，证明了判定一个Büchi可识别的无限图像语言是否可数无限是$D_2(Σ_1^1)$ -完全的，判定一个Büchi可识别的无限图像语言是否不可数是$Σ_1^1$ -完全的.接下来，我们考虑无限图像的可识别语言的补语。利用集合论的一些结果，我们证明了无限图像的Büchi可识别语言的补集的基数可能依赖于公理系统ZFC的模型.我们证明了问题，以确定是否补一个给定的Büchi可识别的语言的无限图片是可数的（分别，不可数）是在类$Σ_3^1 \setminus (Π_2^1 \cup Σ_2^1)$（分别在类$Π_3^1 \setminus (Π_2^1 \cup Σ_2^1)$）。
英文摘要：
Altenbernd, Thomas and Wöhrle have considered in [ATW02] 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 infinite words. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We first review in this paper some recent results of [Fin09b] where we gave the exact degree of numerous undecidable problems for Büchi-recognizable languages of infinite pictures, which are actually located at the first or at the second level of the analytical hierarchy, and "highly undecidable". Then we prove here some more (high) undecidability results. We first show that it is $Π_2^1$-complete to determine whether a given Büchi-recognizable languages of infinite pictures is unambiguous. Then we investigate cardinality problems. Using recent results of [FL09], we prove that it is $D_2(Σ_1^1)$-complete to determine whether a given Büchi-recognizable language of infinite pictures is countably infinite, and that it is $Σ_1^1$-complete to determine whether a given Büchi-recognizable language of infinite pictures is uncountable. Next we consider complements of recognizable languages of infinite pictures. Using some results of Set Theory, we show that the cardinality of the complement of a Büchi-recognizable language of infinite pictures may depend on the model of the axiomatic system ZFC. We prove that the problem to determine whether the complement of a given Büchi-recognizable language of infinite pictures is countable (respectively, uncountable) is in the class $Σ_3^1 \setminus (Π_2^1 \cup Σ_2^1)$ (respectively, in the class $Π_3^1 \setminus (Π_2^1 \cup Σ_2^1)$).
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