I'm back from a lemon tea drinking bout and I brought an improved version of the same notation I've been trying to pull off multiple times under multiple names.
Part 0: Base info[]
Recursion levels[]
- Level I: The most primitive recursion. f(n+1) = f(n) + 1
- Level II: Recursions that are built on higher hyperoperations with usage of one entry. f(n+1) = n*f(n) / f(n+1) = n^(f(n+1))
- Level III: Recursions that are built on higher hypeoperations with usage of multiple entries to stack Level II recursions on top of each other or increase their output in any other way. f(a,b) = f(a*a*a... (b) ...*a)
- Level IV: Recursions that are built on repeatedly stacking recursions on top of each other. f(a,b) = f(a, f(a, f(a,...(a) [b times] / f(a,b,c) = f(a, f(a, ... (a) [f(b, f(b, f(b... (b) [c times] times] / etc.
- Level V: Does the same to the amount of stacked recursions. f(a,b,c)#(d) = f(a,b,c,a,b,c,a,b,c...c) [d repetitions] = [an equation of the likes of Level IV example, but with d repititions in the array and thus d nested levels]
- Level VI: Stacking five levels of a Level V recursion ([... [... [... [... [...(a) times] times] times] times] times]) is a single "5" entry in this level.
- Level VII: The gap between L-VI and L-VII is of the same magnitude as the one between L-V and L-VI
- All subsequent numbered levels (X, XXII, CXI, MMMI etc.) apply the same logic to their corresponding previous level (IX, CX, MMM etc.)
- Level A: Recursions that use entries to identify the recursion level number. / Fa(b) = Level a f(N) with all entries being b.
RLIpC (Recursion Level Increase per Calculation) is a sh#tty term made to define the increase rate of PLANET_M(N). It defines the following equation: FF(n) >{R}> F(n) where R is the amount of recursion levels between FF(n) and F(n).
PLANET_M(N)[]
Examples:
- f(n) = f(n-1)+1
- f(n) = PLANET_1(n)
- f(n) ~ PLANET_2(n) = F(n)
- F(n) = n*F(n-1) // or any function of this speed
Level-III, unlike levels II and I, includes arrays, which might become a problem with increasing such functions. To solve that problem it has been decided that:
- f(N) ; N = {n}
- f(N) ~ PLANET_3(n) = F(N)
- F(N) ; N = {n,n,n,n,n...} with n entries.
=[]
- PLANET_a,2(N) defines a function FF_a(N) that goes F_a(N) levels of recursion beyond F_a(N).
- PLANET_a,3(N) defines a function FFF_a(N) that goes [1]FF_a(N) levels of recursion beyond a certain [1]FF_a(N) that is [2]FF_a(N) levels of recursion beyond a certain [2]FF_a(N)... beyond a certain [ [1]FF_a(N) - 1]FF_a(N) that is [ [1]FF_a(N)]FF_a(N) levels of recursion beyond [ [1]FF_a(N)]FF_a(N).
- PLANET_a,4(N) defines a function FFFF_a(N) that does the same, except the amount m in [m]FFF_a(N) is recurring as many times as needed for this nested recursion to reach FFF_a(N)th level.
- PLANET_a,b(N) defines all next increases for any b, if b є N. (for once, an N doesn't stand for an array. Right here it's the natural number set. Everywhere else it's an array.)