# OCA:Banks-I

(Redirected from Banks-I)

Rulestring Banks-I 012-e3-ajk4-akqw5-ajk6-e78/3e4ejr5cinqy6-ei78 B3e4ejr5cinqy6-ei78/S012-e3-ajk4-akqw5-ajk6-e78 Stable B2e3ajk4akqw5ajk6e/S2ei3aejkr4-cny5-e678

Banks-I is an two-state cellular automaton devised by Roger Banks in his 1971 PhD thesis as a simulated model of a universal computer.

## Rule definition and simulation

The original definition of Banks-I by Banks was written in a checkerboard manner and had three rotationally symmetric transition rules, namely:

• Any live cell with two closest orthogonally adjacent live neighbours dies; otherwise it remains alive.
• Any dead cell with three or four orthogonally adjacent live neighbours comes to life; otherwise it remains dead.

In current terminology, what he specified is an isotropic non-totalistic Life-like cellular automaton on the range-1 von Neumann neighbourhood. Since the neighbourhood is a subset of the Moore neighbourhood, the rule can also be written in Hensel notation. However, the resulted rulestring (B3e4ejr5cinqy6-ei78/S012-e3-ajk4-akqw5-ajk6-e78)[1] is very bulky due to reflecting all of the possible configurations of a cell's diagonally adjacent neighbour.

A way to get around the issue is to consider the diagonal neighbours of a cell only and leave orthogonal ones dead, leading to an equivalent rule B3c4c/S01c2n3c4c where patterns are rotated 45 degrees.[2]

Banks-I is a built-in "general binary" rule in Mirek's Cellebration, and also a rule table in Golly that can be loaded via RuleLoader even before supporting non-totalistic rules.

## Constructions

In the PhD thesis, Banks demonstrated how signal circuitry can be built in Banks-I. A "wire" consists of a three-cell-thick block of live cells, and a "signal" is carried by a duoplet-shaped perturbation travelling at lightspeed along the edge of a wire. A five-cell bump extension and an additional cell on the edge can kill a signal.

 Please enable Javascript to view this LifeViewer. The wire, signal and "dead-end" in all orientations (click above to open LifeViewer)

Signals can be emitted at regular intervals with a "clock". These devices are essentially period-2n signal factories driven by Rule 90 replicator loops.

 Please enable Javascript to view this LifeViewer. Clocks at period 8 and 16 respectively (click above to open LifeViewer)

A lone signal can be duplicated when passing a fanout junction, or reflected 90 degrees.

 Please enable Javascript to view this LifeViewer. The signal duplicator (left) and reflector (right) (click above to open LifeViewer)

Following these, Boolean logic elements including ANDNOT, NOT and NOR gates have been built, which are sufficient to construct a general purpose computer based on the universal NOR logic.

With some new observations in 2017, Peter Naszvadi constructed a Rule 110 unit cell in Banks-I, proving the rule Turing-complete.[3]

## Related rules

The black/white reversal of Banks-I is B2e3ajk4akqw5ajk6e/S2ei3aejkr4-cny5-e678, which has a slightly shorter rulestring and corresponds to the diagonal equivalent B2c/S2cn3c4c with patterns rotated 45 degrees. They are related to the Life-like cellular automaton 2×2 due to an infinite family of rectangular oscillators that oscillates according to a block cellular automaton on the Margolus neighbourhood. For any oscillator with a 2×(4n) box of alive cells in B36/S125 (2×2), the corresponding oscillator is an extended barge with 2×(4n) alive cells in B2e3ajk4akqw5ajk6e/S2ei3aejkr4-cny5-e678, or a rectangle of 2×(4n) dots with every coordinate even in B2c/S2cn3c4c.

## References

1. PHPBB12345 (September 20, 2016). "Banks" equivalent (discussion thread) at the ConwayLife.com forums
2. Hunting (March 16, 2020). Re: "Banks" equivalent (discussion thread) at the ConwayLife.com forums
3. Peter Naszvadi (November 1, 2017). Re: List of the Turing-complete totalistic life-like CA (discussion thread) at the ConwayLife.com forums