Difference between revisions of "Unit cell"
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A '''unit cell''' is a subset (usually rectangular or square) of the Life plane that tiles over the plane, along with a fixed number of distinct patterns, with each tile assuming one of the patterns, such that it simulates a [[cellular automaton]], possibly itself. A '''unit Life cell''' is a unit cell that simulates the [[Conway's Game of Life|Game of Life]]. To avoid single [[cell]]s themselves being considered unit cells, the size of a unit cell must be greater than 1x1. It is also a restriction that only finite sized patterns are accepted as unit cells excluding infinite one-cell thick bars in [[HighLife]] for example, which simulates Rule-54<ref>https://en.wikipedia.org/wiki/Rule_54</ref>; many [[ | A '''unit cell''' is a subset (usually rectangular or square) of the Life plane that tiles over the plane, along with a fixed number of distinct patterns, with each tile assuming one of the patterns, such that it simulates a [[cellular automaton]], possibly itself. A '''unit Life cell''' is a unit cell that simulates the [[Conway's Game of Life|Game of Life]]. To avoid single [[cell]]s themselves being considered unit cells, the size of a unit cell must be greater than 1x1. It is also a restriction that only finite sized patterns are accepted as unit cells excluding infinite one-cell thick bars in [[HighLife]] for example, which simulates Rule-54<ref>https://en.wikipedia.org/wiki/Rule_54</ref>; many [[amphichiral]] 1D elementary automata can be embedded in other rules via this method. | ||
== Simulating a 2D cellular automaton == | == Simulating a 2D cellular automaton == | ||
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=== Constructions === | === Constructions === | ||
Complex one-dimensional cellular automata usually can be simulated via constructed unit cells. Purpose of this is basically the containing automaton inherits some useful properties of the embedded, which is usually not trivial to prove. Such properties are logic universality, Turing-completeness etc. It is known that many examples are constructed in order to simulate Rule-110<ref>https://en.wikipedia.org/wiki/Rule_110</ref> and allegedly Rule-30<ref>https://en.wikipedia.org/wiki/Rule_30</ref> in one case. One is constructed in CGoL by [[Jason Summers]]<ref>http://pentadecathlon.com/lifenews/2005/12/rule_110_unit_cell.html</ref> and later it was trivially shown using [[Golly]] that it is a | Complex one-dimensional cellular automata usually can be simulated via constructed unit cells. Purpose of this is basically the containing automaton inherits some useful properties of the embedded, which is usually not trivial to prove. Such properties are logic universality, Turing-completeness etc. It is known that many examples are constructed in order to simulate Rule-110<ref>https://en.wikipedia.org/wiki/Rule_110</ref> and allegedly Rule-30<ref>https://en.wikipedia.org/wiki/Rule_30</ref> in one case. One is constructed in CGoL by [[Jason Summers]]<ref>http://pentadecathlon.com/lifenews/2005/12/rule_110_unit_cell.html</ref> and later it was trivially shown using [[Golly]] that it is a polyglot (works in [[EightLife]], too) | ||
==== Selection of life-like rules with constructed W110 unit cells ==== | ==== Selection of life-like rules with constructed W110 unit cells ==== | ||
* B3[8]/S23[8] | * B3[8]/S23[8] polyglot<ref>http://www.conwaylife.com/forums/viewtopic.php?f=2&t=3154</ref>: works in [[Life]], [[EightLife]], [[Pedestrian Life]] and [[HoneyLife]] | ||
* Banks-I<ref>http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52584</ref>: proves logic universality using Cook's results, simplifying Banks' proofs from 1971 | * Banks-I<ref>http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52584</ref>: proves logic universality using Cook's results, simplifying Banks' proofs from 1971 | ||
* B35/S236<ref>http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52043</ref><ref>https://plus.google.com/100003628603413742554/posts/AmrSkLDLbNG</ref> | * B35/S236<ref>http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52043</ref><ref>https://plus.google.com/100003628603413742554/posts/AmrSkLDLbNG</ref> | ||
Revision as of 10:00, 14 April 2018
A unit cell is a subset (usually rectangular or square) of the Life plane that tiles over the plane, along with a fixed number of distinct patterns, with each tile assuming one of the patterns, such that it simulates a cellular automaton, possibly itself. A unit Life cell is a unit cell that simulates the Game of Life. To avoid single cells themselves being considered unit cells, the size of a unit cell must be greater than 1x1. It is also a restriction that only finite sized patterns are accepted as unit cells excluding infinite one-cell thick bars in HighLife for example, which simulates Rule-54[1]; many amphichiral 1D elementary automata can be embedded in other rules via this method.
Simulating a 2D cellular automaton
The first unit Life cell was constructed by David Bell in 1996.[2] It employs standard glider logic to determine whether or not a glider should be present. The two states differ by a single glider. In 2004, Jared Prince modified David Bell's unit Life cell to support two (and therefore multiple) layers of Life universes, coined "deep cell".[3]
More recently OTCA metapixel was constructed that simulates any Life-like cellular automaton.[4] Designed to run quickly in HashLife, it has the advantage of having two states that are clearly distinct when zoomed out.
The P1 megacell is currently the largest unit cell. Like the OTCA metapixel, it has clearly visible states. It is capable of simulating any rule, including non-totalistic and asymmetric rules, that uses the standard eight-cell neighborhood. It also has unusual positioning, being a square with diagonal edges. This allows much of its information to be transmitted with gliders.
Simulating a 1D cellular automaton
Natural occurrences
Many replicators also act as a one-dimensional cellular automaton, for example the replicator in HighLife.
Constructions
Complex one-dimensional cellular automata usually can be simulated via constructed unit cells. Purpose of this is basically the containing automaton inherits some useful properties of the embedded, which is usually not trivial to prove. Such properties are logic universality, Turing-completeness etc. It is known that many examples are constructed in order to simulate Rule-110[5] and allegedly Rule-30[6] in one case. One is constructed in CGoL by Jason Summers[7] and later it was trivially shown using Golly that it is a polyglot (works in EightLife, too)
Selection of life-like rules with constructed W110 unit cells
- B3[8]/S23[8] polyglot[8]: works in Life, EightLife, Pedestrian Life and HoneyLife
- Banks-I[9]: proves logic universality using Cook's results, simplifying Banks' proofs from 1971
- B35/S236[10][11]
- DryLife and 3 more[12]
References
- ↑ https://en.wikipedia.org/wiki/Rule_54
- ↑ Paul Callahan (March 1, 1996). "The Unit Life Cell".
- ↑ Jared Prince (September 27, 2004). "Game of Life Deep Cell".
- ↑ "OTCAmetapixel".
- ↑ https://en.wikipedia.org/wiki/Rule_110
- ↑ https://en.wikipedia.org/wiki/Rule_30
- ↑ http://pentadecathlon.com/lifenews/2005/12/rule_110_unit_cell.html
- ↑ http://www.conwaylife.com/forums/viewtopic.php?f=2&t=3154
- ↑ http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52584
- ↑ http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52043
- ↑ https://plus.google.com/100003628603413742554/posts/AmrSkLDLbNG
- ↑ http://conwaylife.com/forums/viewtopic.php?f=11&t=2597&start=25#p58579
External Links
- Unit cell at the Life Lexicon