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：我们研究一类可归约为计算 $f^{(n)}(x)$ 的功能性问题，其中输入为 $n$ 和 $x$，且 $f$ 是多项式时间双射。我们证明，该定义在归约类型的变化以及是否要求 $f$ 具有多项式时间逆或可由可逆逻辑电路计算方面具有鲁棒性。这些问题由复杂性类 $\mathsf{FP}^{\mathsf{PSPACE}}$ 刻画，并包含电路复杂性、细胞自动机、图算法以及分段线性变换描述的动力学系统中的自然 $\mathsf{FP}^{\mathsf{PSPACE}}$ 完全问题。