Classical Reversible Computation with Zero Lyapunov Exponent

| In 1982, Edward Fredkin invented a way to build a time-reversible universal computer using fric-tionless \billiard balls" rolling on a table top, and in which ball \bounces" corresponded to Boolean logical operations. This model made it seem possible to perform unboundedly long computations while consuming zero power. But, achieving that feat seemed to require perfect precision and perfectly precise billiard balls making perfectly precise bounces forever. Actually , we would expect an exponential buildup of impre-cision with time t. If uncorrected, this error buildup would destroy the computer; while on the other hand, continually correcting the errors seems to require power dissipation > 1:34kB T per bounce. We show how to make a variant of Fredkin's computer with zero Lyapunov exponent , and in which we expect errors to grow at worst proportionally to t 3=2. This would seem to allow us to consume kBT per every P bounces, where P can be made to grow as a power law in the mass, precision, velocity, strength and rigidity of the billiard balls. E.g. with perfectly rigid billiard balls of constant size moving at constant velocity, P grows at least proportionally to the sixth root of their masses. But we still need to assume zero friction and we still are subject to the usual limitations of reversible circuitry. The new scheme's \zero-power" advantage seems to come at the cost of requiring more hardware than Fredkin's original scheme: emulating a (reversible with bounded tape) Turing machine for N steps takes hardware growing proportionally to N in my scheme, but not growing with N in Fredkin's scheme. Considering that and some other people's results suggests the following tentative conjectural \no free lunch" principle: \Zero power" computing can be done, but it comes at a price. That price is: performing N computational steps requires a factor of N extra hardwaredelay product, if the energy consumption is reduced from order kBT N to order kBT N 1? (for any 0 1).