Codd's cellular automaton

A simple configuration in Codd's cellular automaton. Signals pass along wire made of cells in state 1 (blue) sheathed by cells in state 2 (red). Two signal trains circulate around a loop and are duplicated at a T-junction onto an open-ended section of wire. The first (7-0) causes the sheathed end of the wire to become exposed. The second (6-0) re-sheathes the exposed end, leaving the wire one cell longer than before.

Codd's cellular automaton is a cellular automaton (CA) devised by the British computer scientist Edgar F. Codd in 1968. It was designed to recreate the computation- and construction-universality of von Neumann's CA but with fewer states: 8 instead of 29. Codd showed that it was possible to make a self-reproducing machine in his CA, in a similar way to von Neumann's universal constructor but never gave a complete implementation.

Historical Context

In the 1940s and 50's, John von Neumann posed the following problem^[1]:

• What kind of logical organization is sufficient for an automaton to be able to reproduce itself?

He was able to construct a Universal Constructor with a square grid and 29 states. E.F. Codd found a simpler machine with only eight states^[2]. Therefore von Neumann's question had to be modified:

• What kind of logical organization is necessary for an automaton to be able to reproduce itself?

Three years after Codd's work, Edwin Roger Banks showed a 4-state CA (appendix IV in his thesis) that was also computation- and construction-universal but again didn't implement a self-reproducing machine^[3].

John Devore, in his 1973 master's thesis, was able to greatly reduce the size of Codd's machine design while using almost the same 8-state CA. More than just a theoretical device, Devore's machine was possibly the first implementation of a self-reproducing CA. The machine itself could fit inside computer memory but the data tape stretched for thousands of cells, so simulation of the entire self-reproduction cycle was not possible. Decades later, Devore's original design would complete self-reproduction in simulation using the Golly program.

In 1984, Christopher Langton extended Codd's cellular automaton to create Langton's loops^[4], which also exhibits reproductive behavior but uses only a very small number of cells compared to the hundreds of thousands needed for self-reproduction in von Neumann, Codd or Banks' cellular automata.

Comparison of CA rulesets

CA	number of states	symmetries	computation- and construction-universal	size of self-reproducing machine
von Neumann	29	none	yes	130,622 cells
Codd	8	rotations	yes	283,126,588 cells[5]
Devore	8	rotations	yes	94,794 cells
Banks-IV	4	rotations and reflections	yes	unknown
Langton's loops	8	rotations	no	86 cells

Specification

The construction arm in Codd's CA can be moved into position using the commands listed in the text. Here the arm turns left, then right, then writes a cell before retracting along the same path.

Codd's CA has 8-states and the von Neumann neighborhood with rotational symmetry.

The table below shows the signal-trains needed to accomplish different tasks. Some of the signal trains need to be separated by two blanks (state 1) on the wire to avoid interference, so the 'extend' signal-train used in the image at the top appears here as '70116011'.

purpose	signal train
extend	70116011
extend_left	4011401150116011
extend_right	5011501140116011
retract	4011501160116011
retract_left	5011601160116011
retract_right	4011601160116011
mark	701160114011501170116011
erase	601170114011501160116011
sense	70117011
cap	40116011
inject_sheath	701150116011
inject_trigger	60117011701160116011

Universal computer-constructor

Codd designed a self-replicating computer in the cellular automaton, based on Wang's W-machine. However, the design was so colossal that it evaded implementation until 2009, when Tim Hutton constructed an explicit configuration^[5]. There were some minor errors in Codd's design, so Hutton's implementation differs slightly, in both the configuration and the ruleset.

See also

• Artificial life
• Cellular automaton
• Conway's game of life
• Langton's loops
• von Neumann cellular automaton
• Wireworld

References

1. ^ von Neumann, John; Burks, Arthur W. (1966). "Theory of Self-Reproducing Automata." (Scanned book online). www.walenz.org. Archived from the original on 2008-01-05. http://web.archive.org/web/20080105213853/http://www.walenz.org/vonNeumann/index.html. Retrieved 2008-02-29. 
2. ^ Codd, Edgar F. (1968). Cellular Automata. Academic Press, New York. 
3. ^ Banks, Edwin (1971). Information Processing and Transmission in Cellular Automata. PhD thesis, MIT, Department of Mechanical Engineering. http://www.bottomlayer.com/bottom/banks/banks_commentary.htm. 
4. ^ Langton, C. G. (1984). "Self-Reproduction in Cellular Automata". Physica D: Nonlinear Phenomena 10 (1-2): 135–144. doi:10.1016/0167-2789(84)90256-2. 
5. ^ ^{a b} Hutton, Tim J. (2010). "Codd's self-replicating computer". Artificial Life 16 (2): 99–117. doi:10.1162/artl.2010.16.2.16200. PMID 20067401. http://www.sq3.org.uk/papers/Hutton_CoddsSelfReplicatingComputer_2010.pdf. 

External links

• The Rule Table Repository has the transition table for Codd's CA.
• Golly - supports Codd's CA along with the Game of Life, and other rulesets.
• Download the complete machine (13MB) and more details.