Stacked up-arrow notation

This notation is in the form $a[\uparrow\boldsymbol{\uparrow}^b\uparrow] c$ or $a[\uparrow\boldsymbol{\uparrow}^b\uparrow] ^cd$

I define it recursively as follows:

$a[\uparrow\boldsymbol{\uparrow}^{0}\uparrow] b$ = $a^b$

$a[\uparrow\boldsymbol{\uparrow}^{b+1}\uparrow] 0$ = $a[\uparrow\boldsymbol{\uparrow}^{b}\uparrow] a$

$a[\uparrow\boldsymbol{\uparrow}^b\uparrow] ^1c$ = $a[\uparrow\boldsymbol{\uparrow}^b\uparrow] c$

$a[\uparrow\boldsymbol{\uparrow}^b\uparrow] ^{c+1}d$ = $a[\uparrow\boldsymbol{\uparrow}^b\uparrow] (a[\uparrow\boldsymbol{\uparrow}^b\uparrow]^cd)$

$a[\uparrow\boldsymbol{\uparrow}^{b+1}\uparrow] (c+1)$ = $a[\uparrow\boldsymbol{\uparrow}^{b}\uparrow] ^{a[\uparrow\boldsymbol{\uparrow}^{b+1}\uparrow] c}a$

$a[\uparrow^\uparrow]b$

Maybe unfinished