Natural

There are two distinct definitions of natural.

Naturalness

Under the narrower definition, a natural object is one that occurs often in the ash of random patterns. There is no precise measure of naturalness, since the most useful definition of "random" in this context is open to debate; nonetheless, objects such as blocks, blinkers, beehives and gliders are very natural and appear in the majority of soups, while objects such as figure eights and queen bee shuttles are less natural, occurring rarely. Darts, Gosper glider guns, etc., are not natural as of November 2023, having never appeared in an asymmetric soup.

Catagolue

Less strictly, an object is said to occur naturally if it has been found in the ash of an asymmetric random soup at least once; the object in question may then also be said to be natural, e.g. "a natural Coe ship" (even though it forms rarely, and is not very natural in the above sense).

Objects found in the ash of symmetric random soups are said to occur semi-naturally (or sometimes almost naturally). The dart is an example of a semi-natural pattern, having appeared in soups with bilateral symmetry.

Under the influence of Catagolue, the definition of a "random soup" in this context is a 16 × 16 pattern where a SHA-256 preimage string is known.

Particulars and edge cases

GPU censuses are not exhaustive, but simulate soups with a preliminary first pass in the GPU, then repeat with the CPU (adding their results to the haul) if behaviour is deemed interesting. Gliders, xWSSes and oscillators of periods 2, 3 and 6 are not deemed so, but for other objects, rates of occurrence in CPU and GPU censuses should be considered equivalent and combined.^{[citation needed]}

As a cryptographic hash algorithm, SHA-256 may be treated as a pseudo-random number generator. Catagolue generates input strings by concatenating the haul identifier string with the index of the soup within the haul. As such, its input space may be arbitrarily large. If it is truly unbiased, as the input length approaches infinity, every one of the 2²⁵⁶ outputs is mapped to with probability approaching 1, making it surjective. However, the problem of finding an input to SHA-256 that produces a given output is computationally hard, and no faster method of solving it is known than iterating over possible inputs until a match is obtained, so likewise the problem of proving its surjectivity is currently intractible.

If SHA-256 is surjective, every pattern with a predecessor fitting in a 16 × 16 bounding box is natural (by the definition that a preimage string would be known to exist), and if its output is uniform, this would prove that it occurs in a lower bound of 1/2²⁵⁶th of the set of soups. On June 7, 2020, using Logic Life Search, Oscar Cunningham found a 16 × 16 soup from which the 2-engine Cordership emerges, which would make it natural.

Some symmetrical objects have occurred in censuses of a different symmetry than themselves, that are not a superset of their own. For instance, the pseudo-barberpole has occurred in C4_1, but the soup in question is based on a reaction near the centre that changes from C4_1 to D8_1, so it is not considered natural.

xp2_0g4j04g53z32x3 is an unusual case, it has C2_1 symmetry but occurred once in a C2_2-symmetric soup. apgsearch creates C2_2 soups by generating a C1 soup and joining it to a rotated and displaced version of itself, reinterpreting the seed string as a C1 soup will produce a C1 occurrence of it. Cha cha was also subject to this edge case, having a C2_2 occurrence that works asymmetrically, but later a C1 occurrence (with the same seed as the C2_2 occurrence) was uploaded to Catagolue.

See also

• Natural periodic objects (category)
• Semi-natural periodic objects (category)
• List of natural spaceships
• Common
• First natural glider, Second natural glider, First natural block

External links

• Natural at the Life Lexicon
• Small Unnatural Objects Watchlist (discussion thread) at the ConwayLife.com forums
• "Natural Life objects". Game of Life Information. A census of 44,631,748 objects from randomly generated 1024 x 1024 (0.125) patterns.