created mirror

Documentation/Math/Makefile

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+all : math.dvi math.ps
+
+include ../lib.mk

Documentation/Math/ajk-scribbles.tex

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+\documentclass{article}
+
+\usepackage{bbm}
+\usepackage{amsmath}
+
+\newcommand{\N}{\mathbbm{N}}
+
+\newtheorem{Def}{Definition}
+\newtheorem{The}{Theorem}[Def]
+
+\DeclareMathOperator{\Dom}{Dom}
+\newcommand{\Dim}[1]{\tilde{#1}}
+
+\begin{document}
+
+Note: we assume here that dimensions are cells instead of strings.
+
+\begin{Def}[space]
+A ZigZag space is the triplet $Z = (X, c, C)$ where $X$ is the set of
+cells, $c : X \rightarrow \Sigma^*$ is the cell content mapping
+and $C \subset X^3$ is the set of connections, if the following hold:
+%
+\begin{enumerate}
+%
+\item If $(d, e, f) \in C$ and $(d, e', f) \in C$ then $e = e'$.
+%
+\item If $(d, e, f) \in C$ and $(d, e, f') \in C$ then $f = f'$.
+%
+\end{enumerate}
+\end{Def}
+
+\begin{Def}[subspace]
+A ZigZag space $Z' = (X', c', C')$ is a subspace of another ZigZag
+space $Z = (X, c, C)$ if the following hold:
+%
+\begin{enumerate}
+%
+\item $X' \subset X$
+%
+\item $c'(x) = c(x)$ for every $x \in X'$.
+%
+\item $C' \subset C$
+%
+\end{enumerate}
+\end{Def}
+
+\begin{The}
+  Let $Z = (X, c, C)$ be a ZigZag space.  For every $d \in X$, there
+  exists a maximal $X' \subset X$ and a unique $\Dim{d} : X'
+  \rightarrow X$ such that for every $x \in X'$ it holds that $(d, x,
+  \Dim{d}(x)) \in C$.  Additionally, $\Dim{d}$ is an injection.
+\end{The}
+
+\begin{Def}[clone dimension]
+  Let $Z = (X, c, C)$ be a ZigZag space.  We say that $d \in X$ is a
+  clone dimension if, for every $x \in \Dom d$ it holds that $c(x) =
+  c(\Dim{d}(x))$.
+\end{Def}
+
+\end{document}

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}
+
+
+