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\usepackage{amsmath}

\title{Mathematics of ZigZag}
\author{Tuomas J.~Lukka}
\begin{document}
\maketitle

\newtheorem{theorem}{Definition}[section]
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

The purpose of this document is to look at ZigZag from a
mathematical perspective.

It is slowly becoming more and more obvious that this type of analysis
can help us understand some of the conceptually less clear parts of ZigZag
and why they are conceptually less clear.

\section{Tumblers}

\section{Definition of a ZigZag space}

\begin{definition}
% A ZigZag space $Z$ is the tuple $(C, d, t)$ of the set of cells $C$,
% a mapping $d$ from strings to bijections between subsets of $C$, 
% and a mapping $t$ from cells to cell contents
% (either permascroll spans as tumbler addresses or strings)
A ZigZag space $Z$ is the tuple $(C, d, t)$ of the set of cells $C$, a
mapping $d$ from strings to the set \( D := \{ \, f\colon $C'$
\longrightarrow $C''$ \mid \text{$f$ is bijective and } C', C''
\subset C \, \} \) and a mapping $t$ from cells to the set of all
possible cell content (permascroll spans as tumbler addresses or
strings).
\end{definition}

Now, special dimensions can be defined simply as restrictions on 
the space; for instance, as predicates:
\begin{definition}
The ZZspace $Z$ has a clone-dimension $d_0$ iff
for all cells $c$ for which $d(d_0)(c)$ exists, $t(d(d_0)(c)) = t(c)$.
\end{definition}

Likewise, we can define a versioning operation:
\begin{definition}
A ZZ operation $o: Z \rightarrow Z'$ is versioning
if ... XXX
\end{definition}


\section{Solving real problems}

\subsection{Slice spaces}

It is possible to highlight some problems related to defining operations
and combinations of spaces with this type of analysis. One observation
made early on in the coding was that encapsulation makes it complicated
to define operations on e.g.~slice spaces (spaces that consist of a 
combination of several spaces). For example, in $C = f(A,B)$ the ZZspace
is $C$ a slice space if it contains cells corresponding to most cells 
of $A$ and $B$ (excluding {\em preflets}, i.e. cells whose meaning is to
specify for $f$ which cells are to be connected between $A$ and $B$).

The forwards transform is simple: $C = f(A,B)$ as above. 
However, problems begin to appear when we consider that normally
performing an operation functionally: $A' = \Omega(A)$ causes $A'$ to be
saved on the disk to replace $A$. $C' = \Omega(C)$ cannot do the same
as simply because we would like to trace the chain to change $A$ and $B$,
obtaining $(A', B') = f^{-1}(\Omega(f(A,B)))$. This can naturally be quite
complicated. Most combination functions $f$ used in reality are nice but still
this causes a pronounced difficulty in coding the usual operations
(new cell, etc) on slice spaces if starting from this perspective.

Thus, the conceptually simplest way forwards might be defining a whole
new kind of mathematical object, a slice space, which has its own 
operations that naturally distribute to the next level.
After this it is simple to define performance enhancements by caching parts
of the next level space but the conceptual simplicity of directly modifying
only the underlying representation (instead of e.g.~the representation {\em and}
a cache) is appealing.

\begin{definition}
A Slice space $S$ is a 
tuple $(Z_0, s)$ where $Z_0$ is the slice 0, i.e.~the root space,
and $s$ is a mapping from strings to ZigZag spaces 
\end{definition}

\begin{definition}
A slice composition function 
$f_c$ is a mapping
$S \rightarrow Z$
from a slice space to a ZigZag space (the composition function).
\end{definition}



\end{document}