Dan Asimov: Is there an "Adam & Eve pattern" whose future Life-evolution will (eventually, somewhere) yield every pattern that fits in any fixed-size rectangle and which has an infinite sequence of ancestors? Robert Munafo: Claimed proof answer is "no." WDS: Sorry, I do not follow Munafo's "no" proof. This sentence in his proof is false: "If the Adam/Eve starting pattern's evolution includes any of the block predecessors, its subsequent evolution must lead to the block." Asimov's is actually an interesting question and there are a number of related interesting questions. I can argue that the answer to a weaker question is "yes." Conway et al showed that life is Turing-universal and the computer you can build inside Life can also be a "universal assembler" capable of synthesizing anything that gliders can synthesize, anyhow. (Is that right? I do not have Conway's proof in front of me, but definitely something like this, is true.) So, make a Turing-universal computer and program it to do stuff. For one thing, we can program it to count 1,2,3,4,... which in some sense enumerate every integer, i.e. every possible pattern. Whether it has ancestors or not. However, this enumeration is only in an abstract sense as "bits in memory", not explicitly as life alive-death patterns. We could presumably program our machine to spit out every possible glider configuration to synthesize eventually every possible glider-synthesizable pattern... except the problem is those patterns might then rise up and destroy their creator. To save ourselves from that fate, we could have our computer mentally simulate an Life inside its memory, investigating patterns to see which ones eventually died or were eventually killable with gliders. It could then systematically safely synthesize every pattern 1. synthesizable with gliders 2. whose future was killable with gliders. So this is a pretty impressive set of descendant patterns all coming from one fixed "Genesis seed pattern." And note: any glider-synthesizable pattern automatically has an infinite set of ancestors. It is Turing-undecidable whether a given finite CA pattern, has an infinite sequence of ancestors... indeed it is undecidable whether it has a father, both proven here: Jarkko Kari: Reversibility of 2D cellular automata is undecidable, Physica D 45 (1990) 379-385. More simply we know it is undecidable whether a given pattern has an infinite sequence of nonempty descendants (proof: Turing's halting problem). Q1. Do there exist "cancer" patterns that are unkillable with gliders? [A pattern is "killable with gliders" if some set of incoming gliders will cause it to evolve to the empty state.] Q2. Do there exist patterns with infinite ancestors that are not glider-synthesizable? Q3. Does there exist a lifeform, which enjoys "homeostasis"? That is, suppose we use the usual life time-evolution rule, but with "epsilon random noise" added, i.e. each cell disobeys the rule with probability epsilon at any given timestep (all probabilities independent). Does there exist an epsilon>0 and P>0 such that there exists for all sufficiently large N, a pattern of initial diameter N that will evolve for exponential(N^P) timesteps (staying nonempty) in the noisy world, almost the same way as it was supposed to evolve in the non-noisy world? Meaning the Hamming distance between the noiseless and noisy evolution, stays bounded by a constant C times the area of a bounding ball throughout this timespan, with probability-->1 when epsilon-->0+, where C depends on epsilon and C-->0 when epsilon-->0+. I think question Q3 is especially important; if the answer is "no" that "proves that immortality is impossible"; if yes it proves it pretty much proves it is possible. (Exponential lifespan is not infinite, but it clearly is the best that can be hoped for and is pretty damn near infinite.)