User blog:Syst3ms/Funky Notation

the name is indeed based off Funky Kong

Hey it's me again. CAN was decent, but it was basically a HAN ripoff whose definition didn't even work properly along with some mere recursive extensions added on top of it. I weakly believe that the limit of the notation introduced here grows comparable to \(f_{\psi(\varepsilon_{\Omega+1})}(n)\). Oh, and before you tell me that "comparable" is the vaguest, most meaningless term ever, I'll let any two functions \(f, g: \mathbb N \mapsto \mathbb N\) be comparable iff \(\exists k\forall n(f(n+k)>g(n)>f(n))\).

This definition is highly reminiscent of pDAN, mainly because of the nesting system.

Definition :
Definition of arrays : - \(m\) is an array, where \(m\) is any natural number - \(\#Am\) is an array, where \(m\) is a natural number, \(A\) is a separator and \(\#\) is a string such that # is an array

Definition of separators : - \(\{m\}\) is a separator, where \(m\) is a natural number - \(\{\#Am\}\) is a separator, where \(m\) is a natural number, \(A\) is a separator, and \(#\) is a string such that \(\{\#\}\) is a separator

Note : - The comma is a shorthand for \(\{0\}\)

Rules : - Base rule : \(a[0] = a+1\) - Trailing rule : \(a[\#\,A\,0] = a[\#]\) and \(\{\#\,A\0\} = \{\#\}\) - Sandwich rule : \(a[\#\,A\,0\,B\,\#'] = a[\#\,B\,\#']\) and \(\{\#\,A\,0\,B\,\#'\} = \{\#\,B\,\#'\}\) iff \(L(A) 0\) : a. If \(\text{ad}=\text{rad}\) : - Let \(f(n) = n[B]\) where \(B\) is the same as \(A_0\) except that \(b\) is decremented. - Replace the entire expression with \(f^a(a)\) - The process ends. - Find the smallest strings X, Y, Z, P, Q and \(c\) such that \(A_{\text{ad}-\text{rad}-1} = “X\,0\,\{P\,c\,Q\}\,Y\,b\,Z”\) b. If \(“\{P\,c\,Q\}”\) is the comma (P and Q are empty and \(c=0\)) : - Replace \(A_{\text{ad}-\text{rad}-1}\) with \("X\,S_a\,,Y\,b-1\,Z"\), where \(S_{x+1} = “[X\,S_x\,,Y\,b-1\,Z]”\) and \(S_0 = “0”\) - The process ends. c. If \(c>0\): - Replace \(A_{\text{ad}-\text{rad}-1}\) with \("X\,S_a"\), where \(S_{x+1} = "0\{c-1\,Q\}S_x"\) and \(S_0 = "1\{c\,Q\}\,Y\,b-1\,Z"\) - The process ends. d. Otherwise, replace \(“\{P\,c\,Q\}b”\) with \(“\{P\,c\,Q\}1\{P\,c\,Q\}b-1”\) jump inside it at its first entry.

Separator comparison process (totally not ripped from exAN) : 1. Let \(A = \{a_1A_1a_2A_2\cdots a_{k-1}A_{k-1}a_k\}\) and \(B = \{b_1B_1b_2B_2\cdots b_{l-1}B_{l-1}b_l\}\) 2. If \(k = 1\) and \(l > 1\), then \(L(A) < L(B)\); if \(k > 1\) and \(l = 1\), then \(L(A) > L(B)\); if \(k = l = 1\), follow step 4; if \(k > 1\) and \(l > 1\), follow step 5 ~ 10 3. If \(a_1 < b_1\), then \(L(A) < L(B)\); if \(a_1 > b_1\), then \(L(A) > L(B)\); if \(a_1 = b_1\), then \(L(A) = L(B)\) 4. Let \(M(A)=\{i\in\{1,2,\cdots,k-1\}|\forall j\in\{1,2,\cdots,k-1\}(lv(A_i)\ge lv(A_j))\}\), and \(M(B)=\{i\in\{1,2,\cdots,l-1\}|\forall j\in\{1,2,\cdots,l-1\}(lv(B_i)\ge lv(B_j))\}\). 5. If \(L(A_{\text{maxM}(A)}) < L(B_{\text{maxM}(B)})\), then \(L(A) < L(B)\); if \(L(A_{\text{maxM}(A)}) > L(B_{\text{maxM}(B)})\), then \(L(A) > L(B)\); or else – 6. If \(|M(A)| < |M(B)|\), then \(L(A) < L(B)\); if \(|M(A)| > |M(B)|\), then \(L(A) > L(B)\); or else – 7. Let \(A = \{\#_1\,A_{\text{maxM}(A)}\,\#_2\}\) and \(B = \{\#_3\,B_{\text{maxM}(B)}\,\#_4\}\) 8. If \(L(\{\#_2\}) < L(\{\#_4\})\), then\(L(A) < L(B)\); if \(L(\{\#_2\}) > L(\{\#_4\})\), then \(L(A) > L(B)\); or else – 9. If \(L(\{\#_1\}) < L(\{\#_3\})\), then \(L(A) > L(B)\); if \(L(\{\#_1\}) > L(\{\#_3\})\), then \(L(A) > L(B)\); if \(L(\{\#_1\}) = L(\{\#_3\})\), then \(L(A) = L(B)\)