373 lines
12 KiB
TeX
373 lines
12 KiB
TeX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% NOTATION
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%SKETCH
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%
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\newcommand{\sketch}{\mathcal{S}}
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\newcommand{\sketchCols}{B}
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\newcommand{\sketchRows}{M}
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\newcommand{\sketchHash}[1]{h_{#1}}
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\newcommand{\sketchPolar}[1]{s_{#1}}
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\newcommand{\polarFunc}{\sketchPolar(\vec{w})}
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%
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%TIDB
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%
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\newcommand{\pw}{W}
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\newcommand{\numWorlds}{2^N}
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\newcommand{\numTup}{N}
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%\newcommand{\wIndicator}{v_t}
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\newcommand{\wIndicator}[1]{v_{#1}}
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\newcommand{\wVec}{\vec{w}}
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\newcommand{\relation}{R}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\BG}[1]{\todo[inline]{\textbf{Boris says:$\,$} #1}}
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\newcommand{\SF}[1]{\todo[inline]{\textbf{Su says:$\,$} #1}}
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\newcommand{\OK}[1]{\todo[inline]{\textbf{Oliver says:$\,$} #1}}
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\newcommand{\AH}[1]{\todo[inline]{\textbf{Aaron says:$\,$} #1}}
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\newcommand{\AR}[1]{\todo[inline]{\textbf{Atri says:$\,$} #1}}
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%\newcommand{\comment}[1]{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DETAILED PROOFS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newbool{ShowDetailedProofs}
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\newcommand{\detailedproof}[2]{\ifbool{ShowDetailedProofs}{
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\begin{proof}
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#2
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\end{proof}
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}{\noindent\textit{Proof Sketch:}#1\qed}\smallskip}
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\newcommand{\ifnotdetailedproof}[1]{\ifbool{ShowDetailedProofs}{}{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% STRUCTURE
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\mypar}[1]{\smallskip\noindent\textbf{{#1}.}}
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\newcommand{\bfcaption}[1]{\caption{\textbf{#1}}}
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\newcommand{\trimfigurespacing}{\vspace*{-5mm}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% THEOREM LIKE ENVIRONMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
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\newtheorem{Theorem}{Theorem}[section]
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\newtheorem{Definition}{Definition}
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\newtheorem{Lemma}{Lemma}
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\newtheorem{Proposition}{Proposition}
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\newtheorem{Corollary}{Corollary}
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\newtheorem{Example}{Example}
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\newtheorem{Axiom}{Axiom}
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\newcommand{\proofpara}[1]{\medskip\noindent\underline{{#1}:}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% MATH UTILS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\mathtext}[1]{\ensuremath{\,\text{#1}\,}}
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\newcommand{\card}[1]{\vert{#1}\vert}
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% allow for more bb math symbols:
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\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Rel model
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\schema}{\mathbf{D}}
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\newcommand{\relschema}{\mathbf{R}}
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\newcommand{\schemaOf}{\textsc{Sch}}
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\newcommand{\arity}[1]{arity({#1})}
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\newcommand{\db}{D}
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\newcommand{\rel}{R}
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\newcommand{\query}{Q}
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\newcommand{\qClass}{\mathcal{C}}
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% \newcommand{\uDom}{\mathcal{U}}
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\newcommand{\tup}{t}
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\newcommand{\multip}{n}
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\newcommand{\aDom}{\mathbb{D}}
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\newcommand{\aVal}{d}
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\newcommand{\tupDom}{\textsc{TupDom}}
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\newcommand{\tuple}[1]{\left<\;\textsf{\upshape #1}\;\right>}
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\newcommand{\comprehension}[2]{\left\{\;#1\;|\;#2\;\right\}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Algebra
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\selection}{\sigma}
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\newcommand{\projection}{\pi}
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\newcommand{\join}{\Join}
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\newcommand{\union}{\cup}
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\newcommand{\intersection}{\cap}
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\newcommand{\rename}{\rho}
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\newcommand{\aggregation}[2]{{}_{#1}\gamma_{#2}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Query Languages and Classes
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\raPlus}{\mathcal{RA}^{+}}
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\newcommand{\raFull}{\mathcal{RA}}
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\newcommand{\raAgg}{\mathcal{RA}^{agg}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% ABBREVIATIONS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\abbrPDB}{PDB\xspace}
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\newcommand{\abbrPDBs}{PDBs\xspace}
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\newcommand{\abbrIDB}{IDB\xspace}
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\newcommand{\abbrPQP}{PQP\xspace}
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\newcommand{\abbrPQPs}{PQPs\xspace}
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\newcommand{\abbrUQP}{UQP\xspace}
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\newcommand{\abbrBGQP}{BGQP\xspace}
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\newcommand{\abbrBGQPs}{BGQPs\xspace}
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\newcommand{\abbrCAQP}{CAQP\xspace}
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\newcommand{\abbrCAQPs}{CAQPs\xspace}
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\newcommand{\abbrUADB}{UA-DB\xspace}
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\newcommand{\abbrUADBs}{UA-DBs\xspace}
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\newcommand{\abbrTI}{TI-DB\xspace}
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\newcommand{\abbrTIs}{TI-DBs\xspace}
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\newcommand{\abbrCoddTable}{Codd-table\xspace}
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\newcommand{\abbrCoddTables}{Codd-tables\xspace}
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\newcommand{\abbrVtable}{V-table\xspace}
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\newcommand{\abbrVtables}{V-tables\xspace}
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\newcommand{\abbrCtable}{C-table\xspace}
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\newcommand{\abbrCtables}{C-tables\xspace}
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\newcommand{\abbrPCtable}{PC-table\xspace}
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\newcommand{\abbrPCtables}{PC-tables\xspace}
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\newcommand{\abbrVCtable}{Virtual C-table\xspace}
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\newcommand{\abbrVCtables}{Virtual C-tables\xspace}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMPLEXITY
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\ptime}{\texttt{PTIME}\xspace}
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\newcommand{\npcomplete}{NP-complete\xspace}
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\newcommand{\conpcomplete}{coNP-complete\xspace}
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\newcommand{\conphard}{coNP-hard\xspace}
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\newcommand{\sharpP}{\texttt{\#P}\xspace}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% UNCERTAINTY
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\cSet}{certain}
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\newcommand{\pSet}{possible}
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\newcommand{\cMult}{certainMult}
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\newcommand{\pMult}{possibleMult}
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\newcommand{\cMatches}{certainMatchMult}
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\newcommand{\pMatches}{possibleMatchMult}
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\newcommand{\pdb}{\mathcal{D}}
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\newcommand{\bestG}[1]{BestGuess(#1)}
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\newcommand{\uadb}{\db_{UA}}
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\newcommand{\bgdb}{D_{bg}}
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\newcommand{\prob}{P}
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\newcommand{\probOf}[1]{\prob(#1)}
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\newcommand{\pTupMult}[2]{\prob({#1}, \geq {#2})}
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\newcommand{\rowsmr}{\mathcal{K}_{uncert}}
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\newcommand{\tmatches}{\simeq}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% UNCERTAINTY LABELINGS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\redL}[1]{\textcolor{red}{{#1}}}
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\newcommand{\tcMarker}{T}
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\newcommand{\tuMarker}{\textcolor{red}{F}}
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\newcommand{\uMarker}[1]{\ensuremath{{#1}{}^{\tuMarker}}}
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\newcommand{\cMarker}[1]{\ensuremath{{#1}{}^{\tcMarker}}}
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\newcommand{\notHere}{\ensuremath{\bot}}
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\newcommand{\uLabels}{\mathcal{U}}
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\newcommand{\labelOf}{\textsc{label}}
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\newcommand{\uDom}{{\aDom_{\uLabels}}}
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\newcommand{\typeTI}{TI}
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\newcommand{\typePC}{pc}
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\newcommand{\TUL}{\mathcal{L}}
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\newcommand{\TULDom}{\mathbb{L}}
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\newcommand{\doTUL}{\textsf{label}}
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\newcommand{\TULscheme}{\textsc{Lab}}
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\newcommand{\TULschemeTI}{\TULscheme_{\abbrTI}}
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\newcommand{\TULschemeCtable}{\TULscheme_{\abbrCtable}}
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\newcommand{\TULtype}[1]{\mathcal{L}_{#1}}
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\newcommand{\TULTI}{\TULtype{\typeTI}}
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\newcommand{\TULC}{\TULtype{\typePC}}
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\newcommand{\AUL}{\mathcal{L}_{A}}
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\newcommand{\AULtype}[1]{\mathcal{L}_{#1}}
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\newcommand{\aLabel}{l}
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\newcommand{\lVal}{{d^{\aLabel}}}
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\newcommand{\ldb}{\db_{\aLabel}}
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\newcommand{\ltupDom}{\textsc{TupDom}_{\uLabels}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% SEMIRINGS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\semN}{\mathbb{N}}
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\newcommand{\semB}{\mathbb{B}}
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\newcommand{\semNX}{\mathbb{N}[X]}
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\newcommand{\semK}{\mathcal{K}}
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\newcommand{\onesymbol}{\mathbbold{1}}
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\newcommand{\zerosymbol}{\mathbbold{0}}
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\newcommand{\multsymb}{\otimes}
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\newcommand{\addsymbol}{\oplus}
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\newcommand{\addormultsymbol}{\bigcirc}
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\newcommand{\monsymbol}{\ominus}
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\newcommand{\kDom}{K}
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\newcommand{\addK}{\addsymbol_{\semK}}
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\newcommand{\multK}{\multsymb_{\semK}}
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\newcommand{\genopK}{\odot_{\semK}}
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\newcommand{\monK}{\monsymbol_{\semK}}
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\newcommand{\oneK}{\onesymbol_{\semK}}
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\newcommand{\zeroK}{\zerosymbol_{\semK}}
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\newcommand{\addOf}[1]{\addsymbol_{#1}}
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\newcommand{\multOf}[1]{\multsymb_{#1}}
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\newcommand{\monOf}[1]{\monsymbol_{#1}}
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\newcommand{\oneOf}[1]{\onesymbol_{#1}}
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\newcommand{\zeroOf}[1]{\zerosymbol_{#1}}
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\newcommand{\dbDomK}[1]{\mathcal{DB}_{#1}}
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%%%%% natural order
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\newcommand{\ordersymbol}{\preceq}
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\newcommand{\geqsymbol}{\succeq}
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\newcommand{\ordK}{\ordersymbol_{\semK}}
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\newcommand{\gordK}{\geqsymbol_{\semK}}
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\newcommand{\geqK}{\geqsymbol_{\semK}}
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\newcommand{\ordOf}[1]{\ordersymbol_{#1}}
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\newcommand{\aHom}{h}
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%%%%% natural order lattice
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\newcommand{\lub}{\sqcup}
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\newcommand{\glb}{\sqcap}
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\newcommand{\glbKof}[1]{\sqcap{#1}}
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%\newcommand{\glbKof}[1]{\left(\bigsqcap_{k\in #1}k\right)}
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\newcommand{\glbK}{\glbKof{\semK}}
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\newcommand{\lubKof}[1]{\sqcup_{#1}}
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%\newcommand{\lubKof}[1]{\left(\bigsqcup_{k\in #1}k\right)}
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\newcommand{\lubK}{\lubKof{\semK}}
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\newcommand{\LubK}{\sqcup_{\semK}}
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\newcommand{\GlbK}{\sqcap_{\semK}}
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\newcommand{\minK}{min_{\semK}}
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\newcommand{\maxK}{max_{\semK}}
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\newcommand{\minOf}[1]{min_{#1}}
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\newcommand{\maxOf}[1]{max_{#1}}
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\newcommand{\setK}{\vec k}
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%%%%%Minima Worlds Semiring
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\newcommand{\mwBool}[1]{\mathbb{B}_{#1}}
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\newcommand{\mwIfElse}[1]{\mwBool{#1}?\mu_{#1}}
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\newcommand{\muMult}{\otimes_{\mu}}
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\newcommand{\boolMult}{\otimes_{\mathbb{B}}}
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\newcommand{\zeroMW}{\zerosymbol_{mw}}
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\newcommand{\oneMW}{\onesymbol_{mw}}
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\newcommand{\mwAdd}{\addsymbol_{mw}}
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\newcommand{\mwMult}{\multsymb_{mw}}
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\newcommand{\hgood}{h_{good}}
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%%%%% Sum of Products Proof
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\newcommand{\productClauseMin}[1]{\sqcap_{\otimes}{#1}}
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\newcommand{\productClauseWorlds}[1]{\productClauseMin{#1}_{worlds}}
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%%%%% N-way direct product
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\newcommand{\dpK}[2]{{#1}^{#2}}
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\newcommand{\dpDom}[2]{{#1}^{#2}}
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\newcommand{\doubleK}[1]{\dpK{#1}{2}}
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\newcommand{\doubleDom}[1]{\dpDom{#1}{2}}
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%%%%% UA-semiring
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\newcommand{\uaName}{UA}
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\newcommand{\uaK}[1]{{#1}_{\uaName}}
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\newcommand{\uaDom}[1]{\kDom{#1_{\uaName}}}
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\newcommand{\uaAdd}[1]{\addOf{\uaK{#1}}}
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\newcommand{\uaMult}[1]{\multOf{\uaK{#1}}}
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\newcommand{\uaZero}[1]{\zeroOf{\uaK{#1}}}
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\newcommand{\uaOne}[1]{\oneOf{\uaK{#1}}}
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\newcommand{\uae}[2]{[#1,#2]}
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\newcommand{\uaN}{\uaK{\semN}}
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\newcommand{\uanAdd}{\uaAdd{\semN}}
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\newcommand{\uanMult}{\uaMult{\semN}}
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%%%%% POSSIBLE WORLD SEMIRINGS
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\newcommand{\pwIndicator}{W}
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\newcommand{\pwDom}{W}
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\newcommand{\pwkDom}{\kDom^{\pwDom}}
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\newcommand{\pwK}{\semK_{\pwIndicator}}
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\newcommand{\pwKof}[1]{{#1}_{\pwIndicator}}
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\newcommand{\pwAdd}{{\addOf{\pwK}}}
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\newcommand{\pwMult}{{\multOf{\pwK}}}
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\newcommand{\pwZero}{{\zeroOf{\pwK}}}
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\newcommand{\pwOne}{{\oneOf{\pwK}}}
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\newcommand{\pwWorld}[1]{\textsc{pw}_{{#1}}}
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\newcommand{\pwe}[1]{\vec{{#1}}}
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\newcommand{\pwHaveCoMinima}[2]{#1 \stackrel{\sqcap}{\sim} #2}
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\newcommand{\pwNotHaveCoMinima}[2]{#1 \not \stackrel{\sqcap}{\sim} #2}
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\newcommand{\pwHaveGlobalMinima}[1]{\stackrel{\sqcap}{\sim} #1}
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\newcommand{\pwNotHaveGlobalMinima}[1]{\not \stackrel{\sqcap}{\sim} #1}
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\newcommand{\pwN}{\pwKof{\semN}}
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\newcommand{\pwB}{\pwKof{\semB}}
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\newcommand{\certainName}{\textsc{cert}}
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\newcommand{\possibleName}{\textsc{poss}}
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\newcommand{\uncertainName}{\textsc{uncert}}
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\newcommand{\pwCertain}{{\certainName}_{\semK}}
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\newcommand{\pwUncertain}{{\uncertainName}_{\semK}}
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\newcommand{\pwPossible}{{\possibleName}_{\semK}}
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\newcommand{\pwCertOf}[1]{{\certainName}_{#1}}
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\newcommand{\pwUncertOf}[1]{{\uncertainName}_{#1}}
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\newcommand{\pwPossOf}[1]{{\possibleName}_{#1}}
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\newcommand{\pwCertainN}{{\certainName}_{\semN}}
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\newcommand{\pwUncertainN}{{\uncertainName}_{\semN}}
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\newcommand{\pwPossibleN}{{\possibleName}_{\semN}}
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\newcommand{\pwCertainB}{{\certainName}_{\semB}}
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\newcommand{\pwUncertainB}{{\uncertainName}_{\semB}}
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\newcommand{\pwPossibleB}{{\possibleName}_{\semB}}
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%%%% K-possible labeling
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%\newcommand{\kpossLab}{{\semK}{-}{\Poss}}
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%%%% UA-DBS
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\newcommand{\uIndicator}{U}
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\newcommand{\uK}{\ensuremath{\semK_{\uIndicator}}}
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\newcommand{\uKDom}{\ensuremath{\kDom_{\uIndicator}}}
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\newcommand{\lCert}{certain}
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\newcommand{\Poss}{possible}
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\newcommand{\uOfName}{uncert}
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\newcommand{\uOf}[1]{h_{\uOfName}({#1})}
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\newcommand{\cOfName}{cert}
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\newcommand{\cOf}[1]{h_{\cOfName}({#1})}
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\newcommand{\dOfName}{det}
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\newcommand{\dOf}[1]{h_{\dOfName}({#1})}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% ENCODING
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\newcommand{\bagEnc}{\textsc{Enc}}
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\newcommand{\rewrN}{\textsf{uRewr}}
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\newcommand{\uAttr}{U}
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\DeclareMathSymbol{\mlq}{\mathord}{operators}{``}
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\DeclareMathSymbol{\mrq}{\mathord}{operators}{`'}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "attrprov"
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%%% End:
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