Reversible Turing Machine: Difference between revisions
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(Note Bennett. Briefly for now until I read his paper in more depth.) |
(→Busy Beaver Champions: Added "halt" to bbch-links of champions) |
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== History == | == 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.<ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973</ref> | 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 quadruple RTM.<ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973</ref> He states that they can also be simulated by a 1-tape quadruple RTM, but with quadratic slowdown. It seems likely that a standard TM can also be simulated by a 1-tape quintuple RTM (the type considered in the rest of this article), however, that was not explicitly discussed in Bennett's paper. | ||
== Definition == | == Definition == | ||
For 1-tape TMs, | For 1-tape quintuple TMs, it is reversible if and only if: | ||
For all states, all transitions to that state: | For all states, all transitions to that state: | ||
| Line 12: | Line 12: | ||
# Must write different symbols | # Must write different symbols | ||
Bruce Smith called this "microscopic reversibility"<ref>https://scottaaronson.blog/?p=4916#comment-1851339</ref> | Bruce Smith called this "microscopic reversibility"<ref>https://scottaaronson.blog/?p=4916#comment-1851339</ref> | ||
The reversible Turing machine function is denoted BB<sub>rev</sub> | |||
== Busy Beaver Champions == | == Busy Beaver Champions == | ||
| Line 19: | Line 21: | ||
!Domain | !Domain | ||
!Max Steps | !Max Steps | ||
!TNF Size | |||
!Champion | !Champion | ||
!Reference | !Reference | ||
|- | |- | ||
| [[BB(2)]] | | [[BB(2)]] | ||
| 6 | | style="text-align: right;" | 6 | ||
| {{TM|0RB1RZ_1LA1RB}} | | style="text-align: right;" | 21 | ||
| {{TM|0RB1RZ_1LA1RB|halt}} | |||
| [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki on Discord] | | [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki on Discord] | ||
|- | |- | ||
| [[BB(3)]] | | [[BB(3)]] | ||
| 17 | | style="text-align: right;" | 17 | ||
| {{TM|0RB1RZ_0LC1RA_1RB1LC}} | | style="text-align: right;" | 356 | ||
| {{TM|0RB1RZ_0LC1RA_1RB1LC|halt}} | |||
| [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki on Discord] | | [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki on Discord] | ||
|- | |- | ||
| [[BB(4)]] | | [[BB(4)]] | ||
| 48 | | style="text-align: right;" | 48 | ||
| {{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ}} | | style="text-align: right;" | 9,853 | ||
| {{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}} | |||
| [https://discord.com/channels/960643023006490684/1243312334907375676/1389809599955210362 Matthew House] and [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki] on Discord | | [https://discord.com/channels/960643023006490684/1243312334907375676/1389809599955210362 Matthew House] and [https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756 Shawn Ligocki] on Discord | ||
|- | |- | ||
| [[BB(5)]] | | [[BB(5)]] | ||
| 388 | | style="text-align: right;" | 388 | ||
| {{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA}} | | style="text-align: right;" | 359,852 | ||
| {{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}} | |||
| [https://discord.com/channels/960643023006490684/1243312334907375676/1389817569342652578 Shawn Ligocki] and [https://discord.com/channels/960643023006490684/1243312334907375676/1389820614415618109 Matthew House] on Discord | | [https://discord.com/channels/960643023006490684/1243312334907375676/1389817569342652578 Shawn Ligocki] and [https://discord.com/channels/960643023006490684/1243312334907375676/1389820614415618109 Matthew House] on Discord | ||
|- | |||
|[[BB(6)]] | |||
| style="text-align: right;" | ≥537,556 | |||
| style="text-align: right;" | ≥9,931,603 | |||
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}} | |||
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400255249847025776 Shawn Ligocki on Discord] | |||
|- | |||
|[[BB(7)]] | |||
| style="text-align: center;" | <math>> 10^{19}</math> | |||
| style="text-align: right;" | ≥542,487,066 | |||
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ}} | |||
|(Early result from enumeration) | |||
|} | |} | ||
| Line 49: | Line 68: | ||
== References == | == References == | ||
<references /> | <references /> | ||
[[category:Functions]] | |||
Latest revision as of 19:10, 16 August 2025
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 quadruple RTM.[1] He states that they can also be simulated by a 1-tape quadruple RTM, but with quadratic slowdown. It seems likely that a standard TM can also be simulated by a 1-tape quintuple RTM (the type considered in the rest of this article), however, that was not explicitly discussed in Bennett's paper.
Definition
For 1-tape quintuple TMs, it is reversible if and only if:
For all states, all transitions to that state:
- Must move in the same direction
- Must write different symbols
Bruce Smith called this "microscopic reversibility"[2]
The reversible Turing machine function is denoted BBrev
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 | TNF Size | Champion | Reference |
|---|---|---|---|---|
| BB(2) | 6 | 21 | 0RB1RZ_1LA1RB (bbch)
|
Shawn Ligocki on Discord |
| BB(3) | 17 | 356 | 0RB1RZ_0LC1RA_1RB1LC (bbch)
|
Shawn Ligocki on Discord |
| BB(4) | 48 | 9,853 | 1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
|
Matthew House and Shawn Ligocki on Discord |
| BB(5) | 388 | 359,852 | 1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
|
Shawn Ligocki and Matthew House on Discord |
| BB(6) | ≥537,556 | ≥9,931,603 | 1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)
|
Shawn Ligocki on Discord |
| BB(7) | ≥542,487,066 | 1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)
|
(Early result from enumeration) |
See Also
- Discussion on Discord on 1 July 2025: https://discord.com/channels/960643023006490684/1243312334907375676/1389756466482647142
References
- ↑ C. H. Bennett, "Logical reversibility of computation", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973
- ↑ https://scottaaronson.blog/?p=4916#comment-1851339