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Documentation/Math/math.ptex

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+% Emacs, note: this is a -*- LaTeX -*- file.
+% MAKE SURE YOU EDIT THE RIGHT FILE, math.ptex and not the math.tex
+% file where gpic has already expanded things.
+\documentclass{article}
+
+\usepackage{rcs}
+\RCS $Date: 2000/07/18 05:21:50 $
+\RCS $Revision: 1.3 $
+\date{Rev.\RCSRevision~~\RCSDate}
+
+\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}
+
+
+