| Summary: | 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.
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