User:Vel!/pu/My BEAF/Expirement

If a string has [] at the end, it is a successor and has no FS.

Strings are ordered in alphabetical order, with [@] being characters and [@] > [#] if @ > #

If a string has [@] where @ is not nothing and does not end in [], the k'th member of the FS of #[@] is #[K] where K is the k'th member of @'s FS.

If a string has [...[]] at the end, go back until you find a lower bracket. Take the portion of the string starting with the lower bracket and ending at the end. The first member of the FS is the original string with the last bracket removed, and each subsequent member adds the shown portion of the string to the end, with the last bracket removed. If no lower bracket exists and the last bracket isn't the only bracket, the first member of the FS is the string with the last bracket removed, and each subsequent member of the FS is the previous one, followed by the original bracket with the [] removed from the end, then the first member again. Finally, if it is the only bracket, the n'th member of the FS is n copies of the original bracket with the [] removed (it is removed from each copy).

What is this useful for? Well, it can be used to define an accurate version of BEAF which complies to Saibian's climbing method. If you define a slow-growing hierarchy for them, you will find that they are exactly equal to tetrational, exponential, additive, and multiplicative functions of n. This allows definition of pentational arrays, where A&n actually does have A (with X's replaced with n's) entries!