Modelling reliability of reversible circuits with 2D second-order cellular automata

TL;DR

The paper addresses reliability in reversible computing using 2D second-order CA, analyzing how faults propagate in RCA-based circuits and how signal timing and interactions shape damage spread. The method combines constructions of universal gates from CEOT/CET-like RCAs with quantitative measures of damage, including ${\mathfrak D}_\infty$ and ${\mathfrak D}^\diamond_\infty$, and extensive numerical experiments. Key findings show that even a single defect can lead to rapid, sometimes omnidirectional, damage fronts and that improper signal interactions impose stringent timing requirements, challenging reliable operation. The work highlights fundamental reliability challenges in RCA-based reversible circuits and motivates designing more robust local rules or wiring strategies for practical applications.

Abstract

The cellular automaton is a widely known model of both reversible and irreversible computations. The family of reversible second-order cellular automata considered in this work is appropriate both for construction of logic gates and analysis of damage distribution. The quantities such as formal dimension of damage patterns can be used only for rough estimation of consequences of particular faults and numerical experiments are provided for illustration of some subtleties. Such analysis demonstrates high sensitivity to errors from defects, lack of synchronization and too short intervals between signals.

Figures (19)

• Figure 1: Swap gates for $C_{0}E_{1\dots}^{\,I\!I}$: 'wide' (left) and 'narrow' (right)
• Figure 1: Collisions with different delays in $C_{0}E_{1}^{\,I\!I}$
• Figure 1: Interactions with different delays in $C_{0}E_{1}^{\,I\!I}$
• Figure 2: Signals routing and collisions in a model of switch gate for $C_{0}E_{1\dots}^{\,I\!I}$
• Figure 2: Collisions with different delays in $C_{0}E_{1,3}^{\,I\!I}$