User blog:QuasarBooster/Everlasting Egrets

Background
Self-modifying bitwise cyclic tag (SBCT) is a simplification of the Turing complete esoteric language, bitwise cyclic tag, where the program string also functions as its own data string. The following is a summary of the computation rules, referenced from the language's page.

An SBCT program is any finite string of bits (commands), executed as follows: If the program-string is initially empty, execution halts immediately; otherwise, starting at the leftmost bit and halting only when the string becomes empty, the commands are executed in cyclic sequence from left to right (the leftmost bit following next after the rightmost bit). The program pointer advances one bit after each command-execution, and also advances one bit when the goto in a 1-command is executed. Example computation:
 * 0: Delete the leftmost bit.
 * 1: Goto the next command (say x). If the leftmost bit is 1, copy x to the right end of the string.

101 ^

1010 ^

10100   ^

0100 ^

100 ^

1000 ^

000 ^

00 ^

0 ^

(empty string)

The EE function
While BCT is known to be Turing complete (i.e. can stimulate any computable function), the same cannot be said for SBCT yet. It is suspected that the latter is also Turing complete, based on its similarity to the former. For now, if that is assumed to be true, then we can define an uncomputable function, Everlasting egret, as follows: By encoding the output of a computable function as the length of the program-string sequence, ending when it halts, the output of EE would consequently outgrow every computable function.
 * EE(n) = number of steps of the longest-lasting n-bit program in SBCT that halts

Values etc.
The EE function is strictly increasing. Appending 0 to the n-th Everlasting egret bitstring results in an n+1-bit candidate for EE(n+1) that halts in exactly EE(n)+1 steps.

\begin{eqarray} \text{EE}(0)&=&0\\ \text{EE}(1)&=&1\\ \text{EE}(2)&=&4\\ \text{EE}(3)&=&9\\ \text{EE}(4)&\geq&20\\ \text{EE}(5)&\geq&24\\ \text{EE}(6)&\geq&26\\ \text{EE}(7)&\geq&66\\ \text{EE}(8)&\geq&313\\ \text{EE}(9)&\geq&416\\ \text{EE}(10)&\geq&43074\\ \end{eqarray}

The corresponding record bitstrings currently known are, respectively:
 * (empty string)
 * 0
 * 10
 * 101
 * 1110
 * 10111
 * 100111
 * 1111101
 * 10111110
 * 111110011
 * 1011110111