The Complexity of Iterated Reversible Computation

TL;DR

This paper investigates the complexity of computing iterated polynomial-time bijections, establishing their robustness and characterizing their class as FP^PSPACE, with implications for various computational problems.

Contribution

It introduces a robust definition of iterated reversible computation and characterizes its complexity class as FP^PSPACE, connecting it to natural problems in multiple domains.

Findings

Problems are characterized by FP^PSPACE complexity class.

The definition is robust against variations in reduction types and invertibility requirements.

Includes natural problems in circuit complexity, cellular automata, graph algorithms, and dynamical systems.

Abstract

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.