Reversible Turing Machine

A Reversible Turing Machine (RTM) is a Turing machine for which the computation can always be run backwards from any step back to the previous configuration (and so forth all the way to the start of the computation). This property (called logical reversibility) has theoretical implications for the limits of computation. Specifically, non-reversible computation cannot scale beyond some limit due to the inherent energy cost whereas reversible computations may be able to.

History

Charles Bennett described Reversible Turing Machines in a 1973 paper in which he proves that any standard TM can be simulated by a 3-tape RTM. This demonstrates that RTMs with 3+ tapes are universal. It is not entirely clear if 1-tape RTMs (considered in the remainder of this article) are universal.

Definition

For 1-tape TMs, they are reversible if and only if:

For all states, all transitions to that state:

1. Must move in the same direction
2. Must write different symbols

Bruce Smith called this "microscopic reversibility"

Busy Beaver Champions

We can restrict the Busy Beaver competition to only (1-tape) RTMs when doing that we get the following champions:

Domain	Max Steps	Champion	Reference
BB(2)	6	0RB1RZ_1LA1RB (bbch)	Shawn Ligocki on Discord
BB(3)	17	0RB1RZ_0LC1RA_1RB1LC (bbch)	Shawn Ligocki on Discord
BB(4)	48	1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)	Matthew House and Shawn Ligocki on Discord
BB(5)	388	1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)	Shawn Ligocki and Matthew House on Discord

See Also

• Discussion on Discord on 1 July 2025: https://discord.com/channels/960643023006490684/1243312334907375676/1389756466482647142