121 lines
3.3 KiB
TeX
121 lines
3.3 KiB
TeX
\documentclass{article}
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\usepackage{natbib}
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\usepackage{xspace}
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\usepackage{xparse}
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\usepackage{environ}
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\usepackage{hyperref}
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\usepackage{bm}
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\usepackage[english]{babel}
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\usepackage{amsthm}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{mathtools}
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\usepackage{hyperref}
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\usepackage[block]{calculation}
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\hypersetup{
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colorlinks=true,
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linkcolor=black,
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filecolor=black,
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urlcolor=black,
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pdfpagemode=FullScreen,
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}
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\input{../notation.tex}
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\input{../glossary.tex}
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\pagenumbering{arabic}
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\pagestyle{plain}
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\newtheorem{theorem}{Theorem}
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\newtheorem{corollary}{Corollary}
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\newtheorem{lemma}{Lemma}
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\newtheorem{proposition}{Proposition}
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\newtheorem{definition}{Definition}
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\newtheorem{example}{Example}
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\title{Consistent n-Approximators ($\beta$)}
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\author{}
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\date{}
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\begin{document}
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\maketitle
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\section{Preliminaries}
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\input{sections/preliminaries.tex}
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\begin{definition}
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\AnNdaoO is {\em extra consistent} if for every $\lte$-\Prefixpoint $y$ of $o(x, \cdot)_2$, $x \lte y$.
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\end{definition}
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\begin{definition}
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"beta" stable revision
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\begin{align*}
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S(o)(x, y)_2 \define \minLattice_{\lte}(\fixpointsOf(o(x, \cdot)) \setminus ((y \downclosure) \setminus (x \upclosure)))
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\end{align*}
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\end{definition}
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\begin{theorem}
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For an extra consistent \Ndao $o$, regular stable revision is equivalent to beta stable revision
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\end{theorem}
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\begin{example}
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SHowing without ultra consistency
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\begin{align*}
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o(x, y) \define (\{ \bot \}, \{ \bot \})
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\end{align*}
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below properties don't hold
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\end{example}
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\begin{proposition}\label{prop:ultra-consistency}
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Works fo rdouble sided ordering
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Given \AnNdaoO $o: \LL^2 \rightarrow \powersetO(\L)^2$ that is \Monotone from $\lte_p^2$ to $<_p^2$,\ we have for any consistent pair $(x, y) \in \LL^2$,
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\begin{align*}
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o(x, y) \lte_p^2 (o(x, y)_2, o(x, y)_1)
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\end{align*}
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\end{proposition}
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\begin{lemma}
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This probably needs double sides :()
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Given an \Ndao $o$, if $y$ is a \Prefixpoint of $o(x, \cdot)_2$, then for some $y' \in o(x, \cdot)_2$, we have
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$x \lte y' \lte y$.
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\end{lemma}
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\begin{proof}
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begin
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\end{proof}
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% iven an \gls{ndao} $o$, its {\em $\beta$-n-stable fixpoints} are fixpoints of the following
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% \begin{align*}
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% B^o_{high}(x) &\define \{ a ~|~ a \in o({x}, a), \neg \exists a',
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% \\ &\hspace{1.5cm}(\boxed{x \preceq_{}}~ a' \prec_{} a) \land (a' \in o({x}, a))) \}\\
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% S(o)(x, y) &\define (C^{o}_{low}(y), B^{o}_{high}(x))
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% \end{align*}}
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% \newcommand{\betastablefixpoint}{\hyperlink{glossary:betastablefixpoint}{$\beta$-stable fixpoint}}
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% \newglossaryentry{betastablefixpoint}{
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% name={$\beta$-stable fixpoint},
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% description={
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% An \gls{interpretation} $(T, P)$ is a {\em $\alpha$-stable fixpoint} (or a $\beta$-stable fixpoint) if it is a \gls{fixpoint} of some $h \in H$ and for each $h' \in H$, none of the following hold
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% \begin{enumerate}[(i.)]
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% \item $\stablerevisionoperator(h')(T, P)_1 \prec_{} T$,
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% \item ($\alpha$-stable only)~$\stablerevisionoperator(h')(T, P)_2 \prec_{} P$, nor
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% \item ($\beta$-stable only) $\exists Z \in \L, T \preceq_{} (h'(T, Z)_2 = Z) \prec_{} P$
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% \end{enumerate}
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% }}
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\bibliographystyle{plain}
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\bibliography{../references}
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\end{document}
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