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\BOOKMARK [1][-]{section.1}{\376\377\000P\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 1
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\BOOKMARK [1][-]{section.2}{\376\377\000L\000a\000t\000t\000i\000c\000e\000\040\000T\000h\000e\000o\000r\000y}{}% 2
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\BOOKMARK [1][-]{section.3}{\376\377\000D\000e\000t\000e\000r\000n\000m\000i\000n\000i\000s\000t\000i\000c\000\040\000A\000F\000T}{}% 3
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\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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@ -37,29 +40,15 @@ pdfpagemode=FullScreen,
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\maketitle
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\begin{definition}
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\AnNdaoO is a $\langle \lte, \lte \rangle$-\Monotone function $o: \LL^2 \rightarrow \powersetO(\LL^2)$ s.t.\ for any $x \in \LL^2$
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\begin{itemize}
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\item $o(x, x)_1 = o(x, x)_2$
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\end{itemize}
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\end{definition}
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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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What about the bottom half does it matter?
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\begin{lemma}[From Heyninck]
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\AnNdaoO is consistent, for every $(x, y)$, $o(x, y)_1 \times o(x, y)_2$ contains at least one consistent pair.
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\end{lemma}
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\begin{definition}
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Regular stable revision
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\begin{align*}
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S(o)(x, y)_1 \define \minLattice_{\lte}(\fixpointsOf(o(\cdot, y))) \\
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S(o)(x, y)_2 \define \minLattice_{\lte}(\fixpointsOf(o(x, \cdot)))
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\end{align*}
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\end{definition}
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\begin{definition}
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"beta" stable revision
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\section{Lattice Theory}
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\definitionBody{powerset}{Given a set $S$, we denote its powerset, i.e.\ $\{ x \subseteq S \}$ with $\powerset(S)$. We use $\powersetO(S)$ to denote the powerset of $S$ without the empty set.}
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\definitionBody{poset}{
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We call $\langle S, \lte \rangle$ a {\em poset} (a partially ordered set) if $\lte$ is reflexive, transitive and antisymmetric.
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}
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\definitionBody{lubglb}{An element is an {\em upper or lower bound} of a subset $S$ of a \Poset if it is greater than or equal or less than or equal, respectively, to every element inside $S$.}
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\definitionBody{completelattice}{A {\em complete lattice} is a \Poset $\langle \LL, \lte \rangle$ s.t.\ every subset $S$ of $\LL$ has a unique greatest lower bound $\glb^{\LL} S$ and least upper bound $\lub^{\LL} S$}
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\definitionBody{topbot}{We use $\top^{\LL}$ and $\bot^{\LL}$ to denote $\lub^{\LL} \LL$ and $\glb^{\LL} \LL$ respectively.}
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\definitionBody{monotone}{A function $f: A \rightarrow B$ is {\em monotone} w.r.t. the orderings $\langle A, \lteSub{A} \rangle$ and $\langle B, \lteSub{B} \rangle$ if for all $a_1, a_2 \in A$, $a_1 \lte a_2$ implies $f(a_1) \lte f(a_2)$}
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\definitionBody{image}{Given a function $f: A \rightarrow B$, we use $f[A]$ to denote the {\em image of $f$} w.r.t.\ $A$, i.e.\ the set $\{ f(a) ~|~ a \in A \} \subseteq B$.
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With abuse to notation, when given a set of functions $F$, we write $\bigcup \{ f\image{A} ~|~ f \in F \}$ as $F\image{A}$.}
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\section{Deternministic AFT}
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The original definition of approximators from Denecker et al.\ \cite{denecker2000approximations}, which has been relaxed following Heyninck et al.\ \cite{nondet2}.
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% \begin{definition}
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% \definitionBody{deterministicapproximator}{
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% A {\em deterministic approximator} $o: \LL \rightarrow \LL$ is an \Exact \Monotone function over a \CompleteLattice $\langle \LL, \lte \rangle$
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% }
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% \end{definition}
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\begin{definitionOf}{deterministicapproximator}
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A {\em deterministic approximator} $o: \LL \rightarrow \LL$ is an \Exact \Monotone function over a \CompleteLattice $\langle \LL, \lte \rangle$
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\end{definitionOf}
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\begin{definitionOf}{deterministicstablerevisionoperator}
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Given \AnApproximator $o: \LL \rightarrow \LL$, its {\em stable revision operator $S(o): \LL^2 \rightarrow \LL^2$} as follows:
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\begin{align*}
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S(o)(x, y) \define (\lfp~o(\partialApp, y), \lfp~(x, \partialApp))
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\end{align*}
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\end{definitionOf}
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In later work, Denecker et al.~\cite{DeneckerMT04} establish an alternative theory of AFT based around operators that are restricted to the smaller domain, $\LL^c = \{ (x, y) \in \LL^2 ~|~ x \lte y \}$.
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The set $\L^c$ only contains the pairs that are \ConsistentPair.
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\definitionBody{consistentpair}{A pair $(x, y)$ is {\em consistent} if $x \lte y$.}
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\begin{definitionOf}{deterministicbetaapproximator}
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A {\em $\beta$-approximator} $o \LL^c \rightarrow \LL^c$ is \AnApproximator restricted and closed under $\LL^c$
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\end{definitionOf}
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% Critically, stable revision is defined differently for \BetaApproximators
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In the same work, Denecker et al.~\cite{DeneckerMT04} use a slightly different notion of stable revision.
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Because, for a \BetaApproximator $o$, the function for any $(x, y) \in \LL^c$
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\begin{align*}
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o(\partialApp, y)_1:& \latticeinterval{\bot}{y} \rightarrow \latticeinterval{\bot}{y} \\
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o(x, \partialApp)_2:& \latticeinterval{x}{\top} \rightarrow \latticeinterval{x}{\top}
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\end{align*}
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So naturally, \Definition{deterministicstablerevisionoperator} applied to \BetaApproximators also uses this limited range.
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For our purposes, we adapt their definition to fit \Approximators and \BetaApproximators.
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\begin{definitionOf}{deterministicbetastablerevisionoperator}
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Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$
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\begin{align*}
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\Sdetbeta(o)(x, y)_1 \define \minLattice_{\lte} \big( \fixpointsOf(o(\partialApp, y)_1) \setminus ((x \upclosure) \setminus (y \downclosure)) \big)
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\\
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\Sdetbeta(o)(x, y)_2 \define \minLattice_{\lte} \big( \fixpointsOf(o(x, \partialApp)_2) \setminus ((y \downclosure) \setminus (x \upclosure)) \big)
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\end{align*}
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\end{definitionOf}
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The symmetric version
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\begin{definitionOf}{deterministicbetastablerevisionoperator}
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Given \AnApproximator $o: \LL^2 \rightarrow \LL^2$
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\begin{align*}
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\Sdetbeta(o)(x, y)_1 &\define \minLattice_{\lte} \Bigg( \fixpointsOf(o(\partialApp, y)_1) \setminus \bigg(
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\Big( ((x \glbBinary y) \upclosure) \union ((x \lubBinary y) \downclosure) \Big)
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\setminus
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\Big( ((x \glbBinary y) \upclosure) \intersect ((x \lubBinary y) \downclosure) \Big)
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\bigg) \Bigg)
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\\
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\Sdetbeta(o)(x, y)_2 &\define \Sdetbeta(o)(y, x)_1
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\end{align*}
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\end{definitionOf}
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\begin{propositionOf}{deterministicstablerevisionoperatorsymmetry}
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Given an \Approximator $o: \LL^2 \rightarrow \LL^2$ that is \Symmetric, its \BetaStableRevisionOperator $S(o)$ is also \Symmetric.
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Under symmetry the two are equivalent for consistent partition
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\end{propositionOf}
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\begin{proof}
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By \Symmetry, we have $\fixpointsOf(o(\partialApp, y)_1) = \fixpointsOf(o(x, \partialApp)_2)$
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\end{proof}
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\Proposition{deterministicstablerevisionoperatorsymmetry}
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\begin{definition}
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\AnNdaoO is a $\langle \lte, \lte \rangle$-\Monotone function $o: \LL^2 \rightarrow \powersetO(\LL^2)$ s.t.\ for any $x \in \LL^2$
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\begin{itemize}
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\item $o(x, x)_1 = o(x, x)_2$
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\end{itemize}
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\end{definition}
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What about the bottom half does it matter?
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\begin{lemma}[From Heyninck]
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\AnNdaoO is consistent, for every $(x, y)$, $o(x, y)_1 \times o(x, y)_2$ contains at least one consistent pair.
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\end{lemma}
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\begin{definition}
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Regular stable revision
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\begin{align*}
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S(o)(x, y)_1 \define \minLattice_{\lte}(\fixpointsOf(o(\cdot, y))) \\
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S(o)(x, y)_2 \define \minLattice_{\lte}(\fixpointsOf(o(x, \cdot)))
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\end{align*}
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\end{definition}
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30
glossary.tex
30
glossary.tex
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\newcommand{\definitionBody}[2]{\hypertarget{glossary:#1}{#2}}
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\newcommand{\definitionLink}[2]{\hyperlink{glossary:#1}{#2}\xspace}
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\newcommand{\Definition}[1]{\definitionLink{#1}{Definition~\ref{definition:#1}}}
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\NewEnviron{definitionOf}[1]{
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\begin{definition}\label{definition:#1}
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\definitionBody{#1}{\BODY}
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\end{definition}
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}
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\NewEnviron{propositionOf}[1]{
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\begin{proposition}\label{proposition:#1}
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\BODY
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\end{proposition}
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}
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\newcommand{\Proposition}[1]{\hyperref[proposition:#1]{Proposition~\ref{proposition:#1}}}
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\newcommand{\LeastFixpoint}{\definitionLink{leastfixpoint}{least fixpoint}}
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\newcommand{\ConsistentPair}{\definitionLink{consistentpair}{consistent}}
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\newcommand{\Monotone}{\definitionLink{monotone}{monotone}}
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\newcommand{\Exact}{\definitionLink{deterministicexactapproximator}{exact}}
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\newcommand{\Symmetric}{\definitionLink{deterministicsymmetricapproximator}{symmetric}}
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\newcommand{\Symmetry}{\definitionLink{deterministicsymmetricapproximator}{symmetry}}
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\newcommand{\BetaApproximator}{\definitionLink{deterministicbetaapproximator}{$\beta$-approximator}}
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\newcommand{\BetaStableRevisionOperator}{\definitionLink{deterministicbetastablerevisionoperator}{$\beta$-stable revision operator}}
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\newcommand{\BetaApproximators}{\definitionLink{deterministicbetaapproximator}{$\beta$-approximators}}
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\newcommand{\Approximator}{\definitionLink{deterministicapproximator}{approximator}}
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\newcommand{\Approximators}{\definitionLink{deterministicapproximator}{approximators}}
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\newcommand{\AnApproximator}{\definitionLink{deterministicapproximator}{an approximator}}
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\let\SdetNoLink\Sdet
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\renewcommand{\Sdet}{\definitionLink{deterministicstablerevisionoperator}{\SdetNoLink}}
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\let\SdetbetaNoLink\Sdetbeta
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\renewcommand{\Sdetbeta}{\definitionLink{deterministicbetastablerevisionoperator}{\SdetbetaNoLink}}
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\newcommand{\Image}{\definitionLink{image}{monotone}}
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\let\imageNoLink\image{}
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\renewcommand{\image}[1]{\imageNoLink{#1}}
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\newcommand{\image}[1]{[#1]}
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\newcommand{\define}{\coloneqq}
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\newcommand{\union}{\cup}
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\newcommand{\intersect}{\cap}
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\newcommand{\glb}{\bigwedge}
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\newcommand{\lub}{\bigvee}
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\newcommand{\glbBinary}{\land}
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\newcommand{\lubBinary}{\lor}
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\newcommand{\powerset}{\wp}
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\newcommand{\powersetO}{\wp^o}
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\newcommand{\upclosure}{\uparrow}
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\newcommand{\downclosure}{\downarrow}
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\newcommand{\partialApp}{\cdot}
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\newcommand{\lfp}{\textbf{lfp}}
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\newcommand{\latticeinterval}[2]{[#1, #2]}
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\newcommand{\Sdet}{S}
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\newcommand{\Sdetbeta}{S^{\beta}}
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\definition{powerset}{Given a set $S$, we denote its powerset, i.e.\ $\{ x \subseteq S \}$ with $\powerset(S)$. We use $\powersetO(S)$ to denote the powerset of $S$ without the empty set.}
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\definition{poset}{
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\definitionBody{powerset}{Given a set $S$, we denote its powerset, i.e.\ $\{ x \subseteq S \}$ with $\powerset(S)$. We use $\powersetO(S)$ to denote the powerset of $S$ without the empty set.}
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\definitionBody{poset}{
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We call $\langle S, \lte \rangle$ a {\em poset} (a partially ordered set) if $\lte$ is reflexive, transitive and antisymmetric.
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}
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\definition{lubglb}{An element is an {\em upper or lower bound} of a subset $S$ of a \Poset if it is greater than or equal or less than or equal, respectively, to every element inside $S$.}
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\definition{completelattice}{A {\em complete lattice} is a \Poset $\langle \LL, \lte \rangle$ s.t.\ every subset $S$ of $\LL$ has a unique greatest lower bound $\glb^{\LL} S$ and least upper bound $\lub^{\LL} S$}
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\definition{topbot}{We use $\top^{\LL}$ and $\bot^{\LL}$ to denote $\lub^{\LL} \LL$ and $\glb^{\LL} \LL$ respectively.}
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\definition{monotone}{A function $f: A \rightarrow B$ is {\em monotone} w.r.t. the orderings $\langle A, \lteSub{A} \rangle$ and $\langle B, \lteSub{B} \rangle$ if for all $a_1, a_2 \in A$, $a_1 \lte a_2$ implies $f(a_1) \lte f(a_2)$}
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\definition{image}{Given a function $f: A \rightarrow B$, we use $f[A]$ to denote the {\em image of $f$} w.r.t.\ $A$, i.e.\ the set $\{ f(a) ~|~ a \in A \} \subseteq B$.
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\definitionBody{lubglb}{An element is an {\em upper or lower bound} of a subset $S$ of a \Poset if it is greater than or equal or less than or equal, respectively, to every element inside $S$.}
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\definitionBody{completelattice}{A {\em complete lattice} is a \Poset $\langle \LL, \lte \rangle$ s.t.\ every subset $S$ of $\LL$ has a unique greatest lower bound $\glb^{\LL} S$ and least upper bound $\lub^{\LL} S$}
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\definitionBody{topbot}{We use $\top^{\LL}$ and $\bot^{\LL}$ to denote $\lub^{\LL} \LL$ and $\glb^{\LL} \LL$ respectively.}
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\definitionBody{monotone}{A function $f: A \rightarrow B$ is {\em monotone} w.r.t. the orderings $\langle A, \lteSub{A} \rangle$ and $\langle B, \lteSub{B} \rangle$ if for all $a_1, a_2 \in A$, $a_1 \lte a_2$ implies $f(a_1) \lte f(a_2)$}
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\definitionBody{image}{Given a function $f: A \rightarrow B$, we use $f[A]$ to denote the {\em image of $f$} w.r.t.\ $A$, i.e.\ the set $\{ f(a) ~|~ a \in A \} \subseteq B$.
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With abuse to notation, when given a set of functions $F$, we write $\bigcup \{ f\image{A} ~|~ f \in F \}$ as $F\image{A}$.}
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