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.

Comment: 35 pages, 8 figures

• 68-XX - Computer science
• 68Q06 - Networks and circuits as models of computation; circuit complexity
• 68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)
• 68Q80 - Cellular automata (computational aspects)