User blog comment:Googleaarex/BMS encoded to 1 symbol system/@comment-30754445-20170720221437

Very nice.

Expanding on this idea, we can do 4 more things:

1. Notate the final [n] in BMS with three commas and then n. For example:

1,1,1,,2,2,1,,3,3,2,,4,4,2,,5,4,2,,,7 would represent (1,1,1)(2,2,1)(3,3,2)(4,4,2)(5,4,2)[7]

2. Fix the generating function at f(n)=n+1.

3. If a binary expression has 3 consecutive ones (representing 3 commas) then all the digits after it are turned to zeros before evaluation.

4. If an expression has a comma in the end, we ignore it.

Armed with these 4 improvements, we now have a function that maps every binary representation to an actual well-defined integer. For example:

88888' → 010101101100111000' ' → 1,1,1,,1,,2,,,3 → 0,0,0,,0,1,,,2 → (0,0,0)(0)(1)[2] → (0)(0)(1)[2] = (0)(0)(0)(0)(0)3 = 5+3 =8  '''

This also gives rise to a fast growing function:

fgf(n) = [the maximum value we can get by plugging in the binary representation of an integer m≤n]

So fgf(88888)≥8.

(It is actually much much larger. As a challange, can anyone here figure out the approximate value of fgf(88888)?)