So, this is some sort of E system.
P is for penguiN
Ex is for extending
\(ExP\{m+1\}n = Exp\{m\}^{n}n\) If \(m > 0\) (Means recurse \(Exp\{m\}\) on n , n times)
\(ExP\{m+1\}n = 10\uparrow^{n}n\uparrow n\) if m = 0
\(ExP\{m\}n\,S_{p+1}\,q = ExP\{m\}q\,\underbrace{S_{p}(q\,S_{p}(q\cdots}_{ExP\{m\}n \,S_{p}s }(q)\)
\(ExP\{m\}n\,S_{p+1}\,q = ExP\{m\}n\uparrow q\) when p is zero
Ok ok I don’t know how to not use ellipses, so just bear with me.
\(Pn = ExP\{n\}n\,S_{n}^n\,n\)
\(Pn\wedge^{m}q = Pn\underbrace{\wedge q\wedge q\wedge q\cdots\wedge q}_{\text{Pn q’s}}\)
Where \(\wedge\) operates as follows:
\(Pa_1\wedge a_2\wedge \cdots\wedge a_n = P^{Pa_n}a_1\wedge a_2\wedge\cdots\wedge a_n-1\)
\(Pa_1\wedge a_2\cdots\wedge 1\wedge a_n = Pa_1 \wedge a_2\cdots \wedge a_n\) when a part of the array is 1, delete it.
E.g. \(P2\wedge2 = P^{P2}2\wedge1 = P^{P2}2 = \underbrace{PP\cdots P}_{P2\,Ps}2\)
Note that superscripts denote recursion.
Extended P system
\(xPn = P^{n}n\wedge^{n}n\)
\(xPn\#^{m}o = xPn\#n\underbrace{\#o\#o\cdots\#o}_{xPm\,o’s}\)
\(xPa_1\#\cdots\#a_n-1\#a_n = xPa_1\#\cdots\#xPa_n-1\) So, we remove the last part of the array and replace the second to last with \(xPa_n\), which is the original last.
\(xPa\#\#b = xP\underbrace{a\#a\#a \cdots \#a}_{b}\)
\(xPa\underbrace{\#\cdots\#}_{n}b = xP\underbrace{a\underbrace{\#\cdots\#}_{n-1}a\cdots a\underbrace{\#\cdots\#}_{n-1} a}_{b}\) there are b sets of n-1 octothropes.
The next Hyperion is ^
\(\left. \begin{matrix} &&xPa\underbrace{\#\cdots \#}b\\ & &xPa\underbrace{\#\cdots \#}b \\ & & \;\;\;\;\underbrace{\quad\;\; \vdots \quad \;\;}\\ & &xPa\underbrace{\# \cdots \#}b \\ & & b\quad \#'s \end{matrix} \right \} xPa\#b\)