Wikipedia:Reference desk/Archives/Mathematics/2018 May 12

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May 12

Conway's Game of Life: What are the known barriers to glider synthesis?

Concerning the existence of "universal" constructors in CGoL, there are some entirely obvious non-glider-constructible patterns, such as (1) Gardens of Eden, (2) any pattern which satisfies the no grandfather or no great^n-grandfather restriction of which a very few examples have been found only recently (given that incoming gliders can converge from arbitrarily large distances, providing arbitrarily large "ancestries" exceeding any given n), or (3) any finite pattern which satisfies the unique father problem of which none are known so far (given that you cannot have a "stable" configuration consisting of gliders only within a finitely-bounded area of the infinite field). My question is if there provably exists any other finite non-glider-constructible patterns which are not examples of any of the three categories above? DWIII (talk) 08:30, 12 May 2018 (UTC)

Since Life is kind of a specialized area, you might try the forum at the Life Wiki. Another possible 'trivial' solution might be an an oscillator of period n whose only nth ancestor is itself (modulo distant die-offs as in the unique father problem. Just off the top of my head, considering that the unique father problem seems much easier and is yet unsolved, this is probably unanswerable. In other words, to paraphrase Erdős, if powerful aliens landed demanding to know the answer or they would destroy the the Earth, we'd be better off fighting the aliens. --RDBury (talk) 21:11, 13 May 2018 (UTC)

Thanks(!); a quick search of Life Wiki's forum dragged up a recent conversation in their "Thread for basic questions" which discusses that very question (indicating that it is very likely an intractable problem). DWIII (talk) 22:43, 13 May 2018 (UTC)

Idempotent in a ring becoming zero

Let R be a ring and e be an idempotent in R. Define R_e to be {x ∈ R|ex = xe = e}. Also, define x +_e y to be x+y−e. Is R_e a ring with addition +_e, the additive inverse of x being 2e−x, zero being e, and the induced multiplication from R? GeoffreyT2000 (talk) 15:52, 12 May 2018 (UTC)

It seems to be the case. Ruslik_Zero 20:24, 13 May 2018 (UTC)