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[Challenge] A monospecific rule with an endemic object

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[Challenge] A monospecific rule with an endemic object

Postby Rhombic » January 6th, 2019, 9:58 am

Building from this post, which I recommend reading for some background, shows the possibility of there being a monospecific rule containing an endemic object as its only object.
Defining "monospecific rule" as a CA rule in which there only exists one stable object. "Endemic" was coined a while back: an object which can only exist in one rule*.

* For a given rule domain! It's obvious that the Demonoid works in all 8 MAP variations of CGoL, but unless we are referring to MAP rules, pointing that out is beyond pedantic.


So anyway, how to approach this:
  • The juicy rule to be found is within the isotropic non-totalistic domain. It would be stunning.
  • I profoundly doubt such a situation would happen within totalistic rules, so we should prove that this is impossible.
  • As a more approachable problem, it would be useful as a first step to find such a case in the two-state anisotropic rule domain (i.e. MAP rules). I think this should be easy enough to attempt!
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Re: [Challenge] A monospecific rule with an endemic object

Postby Rhombic » January 7th, 2019, 6:02 am

As a first reduction (generic for all rule domains), if a pattern is endemic, all birth and survival conditions are used as well as the "no-birth" and death counterparts. For MAP rules, this seems like an absolute headache to analyse but there probably are some clever tricks.

For many-state rules, which I did not contemplate, the number of possible combinations grows exponentially[citation needed] with the number of states, so even though there will certainly be a case where this is possible, the search space is extremely huge... or is it?
I remember the Mona Lisa UC from one cell, and if a new state was added to this rule (let's call it state X) we might have a plausible solution. The UC will build a Mona Lisa when in empty space. If any of the cells of any live state encounter an unexpected environment (i.e. there must have been at least 2 live state 1 cells at the beginning), then the cell changes state to state X, which spreads through all adjacent live cells in a B123/S fashion (all state X cells die after they are created). This "virus" will destroy the entire pattern if it was interacting with anything else but what it's meant to be. Therefore, only full copies of the pattern will remain as all interacting ones will have died.

The only problem might come with the endemic constraint: how do we make sure that the pattern only appears in a single many state rule table? This is a big problem for the current UC as the more states that you have, the worse it gets. However, for a simpler (say 7x7) pattern, with the least possible number of states and maximising the neighbourhoods, this should be possible... hopefully?

Do I have any problems with my reasoning?
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