The Complexity of Iterated Reversible Computation

TL;DR

The paper addresses the complexity of computing iterated applications $f^{(n)}(x)$ where $f$ is a polynomial-time bijection. It introduces a family of nine related functional problems $ extsf{IB}_{x,y}$ based on how $f$ is provided and which reductions are allowed, then proves these variants all collapse to the same class, namely $FP^{PSPACE}$. It provides multiple $FP^{PSPACE}$-complete instances across domains such as circuit complexity, reversible cellular automata (notably via a dimension-reducing 1D construction for Margolus neighborhoods), implicit graphs (second-leaf problems), and piecewise-linear bijections; it also identifies a non-obvious polynomial-time subroutine for iterated integer interval exchange transformations. The work thus unifies diverse problems under a single complexity class and highlights both the richness of reversible computation as a source of complete problems and the nuanced behavior of certain restricted iterates (like IETs) that admit efficient solutions. These results advance our understanding of how reversibility and iteration interplay with space-bounded computation and have implications for both theoretical complexity and models of computation that emphasize reversibility and low-energy dynamics.

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.

Figures (8)

• Figure 4: State space and transitions for Thomason's lollipop algorithm. From any Hamiltonian path (center) with a fixed starting vertex and edge (green), extending the other end of the path by one more edge (blue) can either produce a Hamiltonian cycle (left) or a "lollipop", a shorter cycle with a dangling path (right). Removing one edge from the cycle in a lollipop (red X) produces another Hamiltonian path with the same fixed starting vertex and edge.
• Figure 5: The billiard-ball model. Left: the Margolus neighborhood breaks up the square grid of cells into $2\times 2$ square blocks in two alternating ways, as shown by the blue and red blocks. Right: Blocks with one live cell, or with two diagonal live cells, change in the ways shown; all other blocks remain unchanged.
• Figure 6: A two-track automaton in which the update rule for the top track copies the left neighbor and the update rule for the bottom track copies the right neighbor.
• Figure 7: Helical boundary conditions for the two-dimensional Margolus neighborhood (shown here with circumference 32 in an exploded view with spacing between rows of squares) transform its behavior for a single time step into that of a two-track one-dimensional cellular automaton.
• Figure 8: Left: The interval exchange transformation $x\mapsto (x+\theta)\bmod 1$. Right: More complicated interval exchange transformations can be used to model reflections in mirrored polygons.

Theorems & Definitions (22)

• Lemma 2.2: Jacopini, Mentrasti, and Sontacchi JacMenSon-SIDMA-90
• definition 3.1
• Lemma 3.3
• Lemma 3.5
• Theorem 3.6
• definition 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 5.1
• Lemma 5.2: Imai and Morita ImaMor-TCS-96