We define the symmetric Post Correspondence Problem (PCP) and prove that it is undecidable. As an application we show that the original proof of undecidability of the freeness problem for 3-by-3 integer matrix semigroups works for the symmetric PCP, but not for the PCP in general.

翻译：我们定义了对称后对应问题(PCP ), 并证明它是不可分的。 作为应用程序,我们证明3x3整数矩阵半组自由性问题不可分的原始证据对对对称五氯苯酚有效,但对一般五氯苯酚则不起作用。