--- /dev/null
+\usepackage{amsmath}
+\usepackage{soul}
+
+\title{\textit{C\textsubscript{hail}} more: Branching paths}
+\id{004}
+\type{Investigation}
+\version{Under review}
+\exerpt{%
+ Esoteric Licence Section 4a "The word of the Office of the Commissioner (All Hail) is above all [...]"
+}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+ The conceptual expansion of the Commissioner (all-hail) has promted a change to the schema, broadining the $C_{hail}$ scope to all possible outcomes while still containing a trivial path to the identity function.
+
+ The powerfull but fairly loosely defined \textit{Version 3} has also been elaborated with further examples.
+\end{abstract}
+
+\section{Changes}
+
+The previous itteration of \textit{The Schema} will be refered to as \textit{Version 3}, with this version refered to as \textit{Version 4}.
+
+\begin{displaymath}
+\begin{split}
+\textit{Version 3}:\quad & R_n G_i \rightarrow_{\beta} R_{n+1} G_{i+1}\\
+\textit{Version 4}:\quad & R_n G_i \rightarrow_{\beta} R_m G_j
+\end{split}
+\end{displaymath}
+
+In previous versions of the schema, rule $n$ ($R_n$ below) directly evaluated to rule $n+1$ and the changed gamespace. By rules evaluating differently based on the input (the gamespace), $C_{hail}$ contains not only each of the rules that will be applied in the current game (as in \textit{version 3}), but every rule that could ever be applied in the context of the initial gamespace $G_0$.
+
+\section{Structure}
+
+Rules are defined by the following.
+
+\begin{displaymath}
+\begin{split}
+R_n & = (\lambda x.(\texttt{COND}\ x)(\lambda y.R_a (\texttt{<expr>}\ y))(\lambda y.R_b x)) x
+\end{split}
+\end{displaymath}
+
+\texttt{COND} evaulates to either \texttt{TRUE} or \texttt{FALSE}.
+
+\begin{displaymath}
+\begin{split}
+R_n[G_i := x] & \rightarrow_{\beta} (\texttt{COND}\ G_i)(\lambda y.R_a (\texttt{<expr>}\ y))(\lambda y.R_b G_i)G_i\\
+& \rightarrow_{\beta} \texttt{TRUE}(\lambda y.R_a (\texttt{<expr>}\ y))(\lambda y.R_b G_i)G_i\\
+& \rightarrow_{\beta} \lambda y.R_a (\texttt{<expr>}\ y)G_i\\
+& \rightarrow_{\beta} R_a (\texttt{<expr>}\ G_i)\\
+R_n G_i \rightarrow x[G_i := x] & = R_{m} G_j
+\end{split}
+\end{displaymath}
+
+The gamespace could be structured in any way and this could change during the evaluation of $C_{hail}$.
+
+$t$ is the terminal state. $R_t$ evaluates to the identity function and Church Numberal 0.
+
+\begin{displaymath}
+\begin{split}
+R_t G_t & \rightarrow_{\beta} \lambda x.(\lambda y.y) = 0
+\end{split}
+\end{displaymath}
+
+The Commissioner (all hail) always exaluates to the identity function.
+
+\begin{displaymath}
+\begin{split}
+C_{hail} & = \lambda s.R_0 s\\
+C_{hail} G_0 \rightarrow s[G_0 := s] & = R_t(...(R_n(R_0 G_0)))\\
+& = R_t G_t = 0
+\end{split}
+\end{displaymath}
+
+\section{Pick-up game}
+
+Let's consider an example where the gamespace represents the current number of cards in the single players hand, we will disregard the value of each card for the moment.
+
+Some conventional definitions have been used where trivial. The number $n$ is represented as the function $f$ composed with itself $n$ times with the shorthand $f^{(n)}$ and the function incrementing a numberal is represented as \texttt{HIT} (commonly named \texttt{SUCC}). See Appendix section \ref{sec:defs} for further used definitions.
+
+\begin{displaymath}
+\begin{split}
+\texttt{BUST} & = R_t = \lambda x.x\\
+\texttt{HIT} & = \lambda x.\lambda y.\lambda z.y (xyz)\\
+\texttt{PLAYER} & = (\lambda x.(\texttt{FREEWILL}\ x)(\texttt{CHECK} (\texttt{HIT}\ x))(\texttt{PLAYER}\ x)) x\\
+\texttt{CHECK} & = (\lambda x.(\texttt{ISZERO}(\texttt{SUB}\ x f^{(6)}))(\texttt{PLAYER}\ x)(\texttt{BUST}\ f^{(0)})) x\\
+G_0 & = f^{(2)}\\
+C_{hail} & = R_0 = \texttt{PLAYER}
+\end{split}
+\end{displaymath}
+
+This example is a very simple, but playable, game. The player starts with 2 cards and has the choice to hit or stick. When the player has more than 5 cards, the player is bust and the game is over. Other than the function \texttt{FREEWILL}, representing the choice of the player, the game is trivially implemented.
+
+If the player should keep ``sticking'', the game may not evaluate to $R_t$. This is solved for us cleanly by entropy which ties up loose ends, evaluating the player to $R_t$.
+
+Lets evaluate the example game.
+
+\begin{displaymath}
+\begin{split}
+C_{hail} G_0 & = f^{(2)} \texttt{PLAYER}\\
+& \rightarrow_{\beta} (\lambda x.(\texttt{FREEWILL}\ x)(\texttt{CHECK} (\texttt{HIT}\ x))(\texttt{PLAYER}\ x)) f^{(2)}\\
+& \rightarrow_{\beta} (\texttt{FREEWILL}\ f^{(2)})(\texttt{CHECK} (\texttt{HIT}\ f^{(2)}))(\texttt{PLAYER}\ f^{(2)})
+\end{split}
+\end{displaymath}
+
+Let's say the player decides to hit.
+
+\begin{displaymath}
+\begin{split}
+& \rightarrow_{\beta} \texttt{TRUE}(\texttt{CHECK} (\texttt{HIT}\ f^{(2)}))(\texttt{PLAYER}\ f^{(2)})\\
+& \rightarrow_{\beta} \texttt{CHECK} (\texttt{HIT}\ f^{(2)})\\
+& \rightarrow_{\beta} \texttt{CHECK}\ f^{(3)}
+\end{split}
+\end{displaymath}
+
+Let's check if the player is bust.
+
+\begin{displaymath}
+\begin{split}
+& \rightarrow_{\beta} (\lambda x.(\texttt{ISZERO}(\texttt{SUB}\ x f^{(6)}))(\texttt{PLAYER}\ x)(\texttt{BUST}\ f^{(0)})) f^{(3)}\\
+& \rightarrow_{\beta} (\texttt{ISZERO}(\texttt{SUB}\ f^{(3)} f^{(6)}))(\texttt{PLAYER}\ f^{(3)})(\texttt{BUST}\ f^{(0)})\\
+& \rightarrow_{\beta} (\texttt{ISZERO}\ f^{(0)})(\texttt{PLAYER}\ f^{(3)})(\texttt{BUST}\ f^{(0)})\\
+& \rightarrow_{\beta} \texttt{TRUE}(\texttt{PLAYER}\ f^{(3)})(\texttt{BUST}\ f^{(0)})\\
+& \rightarrow_{\beta} \texttt{PLAYER}\ f^{(3)}\\
+\end{split}
+\end{displaymath}
+
+The player has successfully hit and the move is with the player again.
+
+\appendix
+
+\section*{Appendix}
+
+\section{Common definitions}
+
+\label{sec:defs}
+
+\begin{displaymath}
+\begin{split}
+\texttt{PRED} & = \lambda n.(\lambda f.(\lambda x.n (\lambda g.\lambda h.h (g f)) (\lambda u.x) (\lambda u.u)))\\
+\texttt{SUB} & = \lambda m.(\lambda n.n\ \texttt{PRED}\ m)\\
+\texttt{SUCC} & = \lambda n.(\lambda f.(\lambda x.f (n f x))) \\
+\texttt{TRUE} & = \lambda x.(\lambda y.x) \\
+\texttt{FALSE} & = \lambda x.(\lambda y.y)\\
+\texttt{ISZERO} & = \lambda n.n (\lambda x.\texttt{FALSE}) \texttt{TRUE}
+\end{split}
+\end{displaymath}
+
+\end{document}
Item \@id \textit{(\@version)}\par
\begin{quotation}
\noindent\textcolor{gray}{\textit{%
- "\@exerpt "
+ "\@exerpt{}"
}}
\end{quotation}
\thispagestyle{empty}
\noindent Published by The Office of the Commissioner (all hail).\par
\noindent Designed by Goodnight Publishing.\par
\vspace{5mm}
- \noindent For more information, see: \texttt{https:// all-hail.co.uk}\par
- \noindent Address inquiries to: \texttt{commissionerallhail@gmail.com}\par
+ \noindent For more information, see: \texttt{https://all-hail.co.uk}\par
\section*{Licence}
All works related to Blackjack 2, including those published by the Office of the Commissioner (all hail) are licenced under the \texttt{Blackjack 2 Esoteric Licence}.
- For the latest version, see: \texttt{https://blackjack2.neocities.org/licence.txt }\par
+ For the latest version, see: \texttt{https://all-hail.co.uk/licence.txt }\par
\vspace*{\fill}
\begin{flushright}
\includegraphics[height=3cm]{goodnight}
projects / ootc-docs / commitdiff