We prove that, for any natural number \(n\ge 1\), we can find a finite alphabet \(\Sigma \) and a finitary language L over \(\Sigma \) accepted by a one-counter automaton, such that the \(\omega \)-power

$$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$

is \({\varvec{\Pi }}^0_n\)-complete. We prove a similar result for the class \({\varvec{\Sigma }}^0_n\).

中文翻译：

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对于任何有限等级的Borel类，它们都具有单反语言的某些完全$$ \ omega $$ω-次幂

我们证明，对于任何自然数\（N \ GE 1 \） ，我们可以发现一个有限字母表\（\西格玛\）和一个有穷的语言大号超过\（\ Sigma公司\）由一计数器自动机接受，这样的那个\（\ omega \）-功率

$$ \ begin {aligned} L ^ \ infty：= \ {w_0w_1 \ ldots \ in \ Sigma ^ \ omega \ mid \ forall i \ in \ omega ~~ w_i \ in L \} \ end {aligned} $$

是\（{\ varvec {\ Pi}} ^ 0_n \）-完成。我们证明了类\（{\ varvec {\ Sigma}} ^ 0_n \）的相似结果。