We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its definition, and in whether we require $f$ to have a polynomial-time inverse or to be computible by a reversible logic circuit. These problems are characterized by the complexity class $\mathsf{FP}^{\mathsf{PSPACE}}$, and include natural $\mathsf{FP}^{\mathsf{PSPACE}}$-complete problems in circuit complexity, cellular automata, graph algorithms, and the dynamical systems described by piecewise-linear transformations.

翻译：我们研究了一类函数问题，这类问题可归约为对于输入$n$和$x$计算$f^{(n)}(x)$，其中$f$是一个多项式时间双射。我们证明，该定义在归约类型（用于其定义）以及是否要求$f$具有多项式时间逆或由可逆逻辑电路计算方面是稳健的。这些问题以复杂性类$\mathsf{FP}^{\mathsf{PSPACE}}$为特征，并包括电路复杂性、细胞自动机、图算法以及由分段线性变换描述的动态系统中的自然$\mathsf{FP}^{\mathsf{PSPACE}}$完全问题。