正则Ω语言中无穷后对应问题的高度不可判定性在有理关系中的三个应用
Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language
摘要：
Harel在[Har 86]中指出，“人们可以定义$Σ_1^1$ -著名的Post Correspondence Problem的完全版本”。我们首先给出了这个结果的一个完整的证明，表明在一个正规的$ω$ -语言的无限后对应问题是$Σ_1^1$ -完全的，因此位于算术层次之外，高度不可判定。我们从这个结果推断，它是$Π_1^1$ -完全的，以确定是否两个给定的无穷有理关系是不相交的。然后证明了有限状态Büchi变换器实现的$ω$ -有理函数的两个判定问题之间存在着惊人的差距。事实上Prieur证明在[Pri 01，Pri 02]，这是决定是否一个给定的$ω$ -有理函数是连续的，而我们在这里表明，它是$Σ_1^1$ -完全确定是否一个给定的$ω$ -有理函数至少有一个点的连续性。其次证明了判定给定$ω$ -有理函数的连续集是否是$ω$ -正则的是$Π_1^1$ -完全的。这给出了[CFS 08]中证明为不可判定的两个问题的精确复杂度。
英文摘要：
It was noticed by Harel in [Har86] that "one can define $Σ_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $ω$-language is $Σ_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $Π_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $ω$-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $ω$-rational function is continuous, while we show here that it is $Σ_1^1$-complete to determine whether a given $ω$-rational function has at least one point of continuity. Next we prove that it is $Π_1^1$-complete to determine whether the continuity set of a given $ω$-rational function is $ω$-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].
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