118 lines
3.8 KiB
TeX
118 lines
3.8 KiB
TeX
\documentclass{article}
|
|
|
|
\usepackage{xspace}
|
|
\usepackage{hyperref}
|
|
\usepackage{bm}
|
|
\usepackage[english]{babel}
|
|
\usepackage{amsthm}
|
|
\usepackage{amsmath}
|
|
\usepackage{mathtools}
|
|
\usepackage{hyperref}
|
|
\hypersetup{
|
|
colorlinks=true,
|
|
linkcolor=black,
|
|
filecolor=black,
|
|
urlcolor=black,
|
|
pdfpagemode=FullScreen,
|
|
}
|
|
|
|
\input{notation.tex}
|
|
\input{glossary.tex}
|
|
\pagenumbering{arabic}
|
|
\pagestyle{plain}
|
|
|
|
\newtheorem{theorem}{Theorem}
|
|
\newtheorem{corollary}{Corollary}
|
|
\newtheorem{lemma}{Lemma}
|
|
\newtheorem{proposition}{Proposition}
|
|
|
|
\begin{document}
|
|
|
|
% \maketitle
|
|
|
|
|
|
Hello world\cite{tarskilatticetheoretical1955}
|
|
First, we generalize Knaster-Tarski Fixpoint Theorem.
|
|
|
|
|
|
|
|
\section{Background}
|
|
\input{sections/background.tex}
|
|
|
|
\section{Content}
|
|
|
|
\begin{theorem}[Tarski-Knaster Fixpoint Theorem~\cite{tarskilatticetheoretical1955}]\label{tarskitheorem}
|
|
For a \Monotone function $o$ over a \CompleteLattice $\langle \L, \lte \rangle$, we have that $\langle \fixpointsOf(o), \lte \rangle$ is a \CompleteLattice.
|
|
\end{theorem}
|
|
|
|
\begin{theorem}
|
|
Trivial attempt using set preimage / image to attain LUB / GLB in image lattice does not work.
|
|
|
|
Consider antichain of 2 that map to a chain in image.
|
|
The antichain's LUB does maps to a third, distinct element which extends the chain to three in the image
|
|
|
|
If we attain LUB of preimage, then its not the least element :)
|
|
\end{theorem}
|
|
|
|
\begin{theorem}\label{imagelattice}
|
|
For a \Monotone function $o$ over a \CompleteLattice $\langle \L, \lte \rangle$, we have that $\langle o\imageNoLink{\L}, \lte \rangle$ is a \CompleteLattice.
|
|
\end{theorem}
|
|
\begin{proof}
|
|
Consider $\langle \L', \lte \rangle$ where
|
|
\begin{align*}
|
|
\L' \define \{ x' ~|~ x \in \L \} \\
|
|
x' \lte y' \iff x \lte y \textrm{ where } x, y \in \L
|
|
\end{align*}
|
|
Clearly, $\langle \L', \lte' \rangle$ is a \CompleteLattice.
|
|
Combining $\L$ and $\L'$, we formulate $\L^*$:
|
|
\begin{align*}
|
|
\L^* &\define \{ \top^*, \bot^* \} \union \L \union \L' \\
|
|
\forall x^*, y^* \in \L^*, x^* \lte y^* &\iff \begin{cases}
|
|
(x^* = \bot^*) \lor (y^* = \top^*), \\
|
|
\textrm{or } x, y \in \L, \\
|
|
\textrm{or } x', y' \in \L'
|
|
\end{cases}
|
|
\end{align*}
|
|
It is easy to show that $\langle \L^*, \lte \rangle$ is a \CompleteLattice.
|
|
Consider the operator $o^*$ that maps elements from $\L$ to $\L'$ and every element in $\L'$ to itself.
|
|
\begin{align*}
|
|
o^*(x) \define \begin{cases}
|
|
o(x)' & \textrm{if } x \in \L \\
|
|
x & \textrm{otherwise}
|
|
\end{cases}
|
|
\end{align*}
|
|
We have $o^*\image{\L \union \L'} \subseteq L'$ and $\fixpointsOf(o^*) = o\image{\L}' \union \{ \bot^*, \top^* \}$.
|
|
By Tarski-Knaster Fixpoint Theorem (Theorem \ref{tarskitheorem}), $\fixpointsOf(o^*)$ is a \CompleteLattice.
|
|
Thus, $o\image{\L} \union \{ \bot^*, \top^* \}$ is a \CompleteLattice. For any $S \subseteq \L$, we have $\glb S \not\in \{ \bot^*, \top^* \}$ and $\lub S \not\in \{ \bot^*, \top^* \}$, thus $\langle o\image{\L}, \preceq \rangle$ is a \CompleteLattice.
|
|
\end{proof}
|
|
|
|
The following follows directly from the contrapositive of Theorem \ref{imagelattice}.
|
|
\begin{corollary}
|
|
If $\langle o\image{\L}, \lte \rangle$ is not a \CompleteLattice, then $o$ is not \Monotone.
|
|
\end{corollary}
|
|
|
|
|
|
\begin{definition}
|
|
An approximator set $H$ captures a nondeterministic approximator $o$ if for each consistent pair $(x, y)$
|
|
\begin{align*}
|
|
(H\image{(x, y)}_1, H\image{(x, y)}_2) = o(x, y)
|
|
\end{align*}
|
|
\end{definition}
|
|
|
|
\begin{theorem}
|
|
A nondeterministic approximator $o$ has a set of
|
|
\end{theorem}
|
|
|
|
\begin{theorem}
|
|
For any nondeterministic approximator $o$, there exists an approximator set $H$ s.t.\ gamma stable models correspond to n-stable models
|
|
\end{theorem}
|
|
\begin{proof}
|
|
foo
|
|
\end{proof}
|
|
|
|
|
|
\bibliographystyle{plain}
|
|
\bibliography{references}
|
|
|
|
\end{document}
|