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Spencer Killen 2024-12-09 08:23:54 -07:00
parent 376900f28c
commit e10c2cc2dd
Signed by: sjkillen
GPG Key ID: 3AF3117BA6FBB75B
3 changed files with 52 additions and 13 deletions

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@ -9,8 +9,10 @@
\usepackage[english]{babel} \usepackage[english]{babel}
\usepackage{amsthm} \usepackage{amsthm}
\usepackage{amsmath} \usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools} \usepackage{mathtools}
\usepackage{hyperref} \usepackage{hyperref}
\usepackage[block]{calculation}
\hypersetup{ \hypersetup{
colorlinks=true, colorlinks=true,
linkcolor=black, linkcolor=black,

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@ -56,11 +56,6 @@ So naturally, \Definition{deterministicstablerevisionoperator} applied to \BetaA
For our purposes, we adapt their definition to fit \Approximators and \BetaApproximators. For our purposes, we adapt their definition to fit \Approximators and \BetaApproximators.
\begin{definitionOf}{deterministicbetastablerevisionoperator} \begin{definitionOf}{deterministicbetastablerevisionoperator}
Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$ Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$
\begin{align*}
\Sdetbeta(o)(x, y)_1 \define \minLattice_{\lte} \big( \fixpointsOf(o(\partialApp, y)_1) \setminus ((x \upclosure) \setminus (y \downclosure)) \big)
\\
\Sdetbeta(o)(x, y)_2 \define \minLattice_{\lte} \big( \fixpointsOf(o(x, \partialApp)_2) \setminus ((y \downclosure) \setminus (x \upclosure)) \big)
\end{align*}
\end{definitionOf} \end{definitionOf}
The symmetric version The symmetric version
@ -68,17 +63,56 @@ The symmetric version
\begin{definitionOf}{deterministicbetastablerevisionoperator} \begin{definitionOf}{deterministicbetastablerevisionoperator}
Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$ Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$
\begin{align*} \begin{align*}
\Sdetbeta(o)(x, y)_1 &\define \minLattice_{\lte} \Bigg( \fixpointsOf(o(\partialApp, y)_1) \setminus \bigg( \Sdetbeta(o)(x, y)_1 \define \glb_{\lte} \big( \fixpointsOf(o(\partialApp, y)_1) \setminus ((x \upclosure) \setminus (y \downclosure)) \big)
\Big( ((x \glbBinary y) \upclosure) \union ((x \lubBinary y) \downclosure) \Big)
\setminus
\Big( ((x \glbBinary y) \upclosure) \intersect ((x \lubBinary y) \downclosure) \Big)
\bigg) \Bigg)
\\ \\
\Sdetbeta(o)(x, y)_2 \define \glb_{\lte} \big( \fixpointsOf(o(x, \partialApp)_2) \setminus ((y \downclosure) \setminus (x \upclosure)) \big)
\end{align*}
% This symmetric version likely only works on so-called "A-prudent" pairs
\begin{align*}
\Sdetbeta(o)(x, y)_1 \} &\define \glb (
\fixpointsOf(o(\partialApp, y)_1) \setminus
\Big( ((x \glbBinary y) \upclosure) \symmetricdifference ((x \lubBinary y) \downclosure) \Big)
)\\
\Sdetbeta(o)(x, y)_2 &\define \Sdetbeta(o)(y, x)_1 \Sdetbeta(o)(x, y)_2 &\define \Sdetbeta(o)(y, x)_1
\end{align*} \end{align*}
\end{definitionOf} \end{definitionOf}
\begin{theorem}
For \AnApproximator $o: \LL^2 \rightarrow \LL^2$ s.t.\ $o\image{\LL^c} \subseteq \LL^c$, we have that
$\Sdet(o\restriction_{\LL^c})$ and $\Sdetbeta(o)$ are equivalent.
\end{theorem}
\begin{proof}
First we simplify $\Sdetbeta(o)(x, y)_1$
\begin{calculation}[=]
\minLattice_{\lte} (
\fixpointsOf(o(\partialApp, y)_1) \setminus
\Big( ((x \glbBinary y) \upclosure) \symmetricdifference ((x \lubBinary y) \downclosure) \Big)
\step{inside:
\begin{subcalculation}[=]
((x \glbBinary y) \upclosure) \symmetricdifference ((x \lubBinary y) \downclosure)
\step{Applying $x \lte y$}
(x \upclosure) \symmetricdifference (y \downclosure)
\step{Expand $\symmetricdifference$}
(x \upclosure \union~ y \downclosure) \setminus (x \upclosure \intersect~ y \downclosure)
\step{Using $\fixpointsOf(o(\partialApp, y)_1) \subseteq (y \downclosure)$ we don't quite have this yet tho}
\LL \setminus (x \upclosure \intersect~ y \downclosure)
\end{subcalculation}}
\step[]{Aside \begin{subcalculation}
\step[]{By Assumption}
\forall x \in [\bot, y],~ o(\partialApp, y)_1 \in [\bot, y]
\step[\implies]{Follows}
\fixpointsOf(o)
\end{subcalculation}}
\step{Leverage $\fixpointsOf(o(\partialApp, y)_1) \subseteq [\bot, y]$}
BNy bye
\step*{sdshds}
jdskdjsk
\end{calculation}
\end{proof}
\begin{propositionOf}{deterministicstablerevisionoperatorsymmetry} \begin{propositionOf}{deterministicstablerevisionoperatorsymmetry}
Given an \Approximator $o: \LL^2 \rightarrow \LL^2$ that is \Symmetric, its \BetaStableRevisionOperator $S(o)$ is also \Symmetric. Given an \Approximator $o: \LL^2 \rightarrow \LL^2$ that is \Symmetric, its \BetaStableRevisionOperator $S(o)$ is also \Symmetric.
Under symmetry the two are equivalent for consistent partition Under symmetry the two are equivalent for consistent partition

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@ -4,8 +4,9 @@
\newcommand{\lteSub}[1]{\lteNoLink_{#1}} \newcommand{\lteSub}[1]{\lteNoLink_{#1}}
\newcommand{\image}[1]{[#1]} \newcommand{\image}[1]{[#1]}
\newcommand{\define}{\coloneqq} \newcommand{\define}{\coloneqq}
\newcommand{\union}{\cup} \newcommand{\union}{\mathbin{\cup}}
\newcommand{\intersect}{\cap} \newcommand{\intersect}{\mathbin{\cap}}
\newcommand{\symmetricdifference}{\mathbin{\Delta}}
\newcommand{\glb}{\bigwedge} \newcommand{\glb}{\bigwedge}
\newcommand{\lub}{\bigvee} \newcommand{\lub}{\bigvee}
\newcommand{\glbBinary}{\land} \newcommand{\glbBinary}{\land}
@ -19,4 +20,6 @@
\newcommand{\latticeinterval}[2]{[#1, #2]} \newcommand{\latticeinterval}[2]{[#1, #2]}
\newcommand{\Sdet}{S} \newcommand{\Sdet}{S}
\newcommand{\Sdetbeta}{S^{\beta}} \newcommand{\Sdetbeta}{S^{\beta}}
\let\restrictionWithSpaces\restriction
\renewcommand{\restriction}{{\restrictionWithSpaces}}