In the bi-infinite Post Correspondence Problem ($\Z$PCP), it is asked whether the same bi-infinite word can be constructed correspondingly from a given finite set of pairs of words. In this article, we study its complexity with respect to the arithmetical hierarchy and prove that it is in $\Si^0_2 \setminus (Π^0_1 \cup \Si^0_1)$ and, therefore, at the level 2 of the arithmetical hierarchy. For the proof, we present a sequence of reductions starting from the nonhalting of the Turing machine all the way to $\Z$PCP via infinite PCP, an $s$-shift infinite PCP and $s$-shift $\Z$PCP for all natural numbers $s$. In the process, we prove that the infinite PCP is undecidable for injective morphisms, and that the infinite injective PCP, $s$-shift infinite PCP, $s$-shift $\Z$PCP and the non-termination problem for (deterministic and reversible) semi-Thue systems are all $Π^0_1$-complete.

翻译：在双无穷波斯特对应问题（$\Z$PCP）中，我们询问是否可以从给定的有限对词集合中对应地构造出同一个双无穷词。在本文中，我们研究了它在算术层级中的复杂度，并证明它属于$\Si^0_2 \setminus (Π^0_1 \cup \Si^0_1)$，因此位于算术层级的第二层。为了证明这一点，我们提出了一系列归约，从图灵机的不停机问题出发，通过无穷PCP、对所有自然数$s$的$s$-移位无穷PCP和$s$-移位$\Z$PCP，最终归结到$\Z$PCP。在此过程中，我们证明了对于单射态射的无穷PCP是不可判定的，并且无穷单射PCP、$s$-移位无穷PCP、$s$-移位$\Z$PCP以及（确定性和可逆）半Thue系统的不终止问题都是$Π^0_1$-完全的。