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\newcommand{\nb}[1]{{\color{violet} NB: #1}}
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\newcommand{\todo}[1]{{\color{red} TODO: #1}}
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\newcommand{\anupam}[1]{{\color{orange} Anupam: #1}}
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\newcommand{\patrick}[1]{{\color{blue} Patrick: #1}}
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%\newtheorem{theorem}{Theorem}
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\newtheorem{proposition}[theorem]{Proposition}
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%\newtheorem{lemma}[theorem]{Lemma}
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%
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%
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%\theoremstyle{definition}
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%\newtheorem{definition}[theorem]{Definition}
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\newcommand{\Word}{\mathbb{W}}
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\newcommand{\Nat}{\mathbb{N}}
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\newcommand{\arith}{B}
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\newcommand{\basic}{\mathit{BASIC}}
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\newcommand{\ind}{\mathit{IND}}
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\newcommand{\pind}{\mathit{PIND}}
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\newcommand{\cind}[1]{#1\text{-}\ind}
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\newcommand{\cpind}[1]{#1\text{-}\pind}
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\renewcommand{\min}{\mathit{MIN}}
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\newcommand{\cmin}[1]{#1\text{-}\min}
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\newcommand{\prefix}{\preccurlyeq}
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\newcommand{\wit}[2]{\mathit{wit}^{#1}_{#2}}
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\newcommand{\Wit}[2]{\mathit{Wit}^{#1}_{#2}}
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\newcommand{\dfn}{:=}
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\newcommand{\seqar}{\rightarrow}
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\newcommand{\proves}{\vdash}
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\renewcommand{\epsilon}{\varepsilon}
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\newcommand{\pred}{p}
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\renewcommand{\succ}[1]{s_{#1}}
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\newcommand{\cond}{C}
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\newcommand{\smsh}{\#}
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\newcommand{\pair}[3]{\langle ; #1,#2 , #3 \rangle}
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\newcommand{\safe}{{S}}
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\newcommand{\normal}{N}
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\newcommand{\pv}{\mathit{PV}}
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\newcommand{\pvbci}[1]{\pv^{#1}_{\mathrm{BC}}}
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\newcommand{\mubci}[1]{\mu\mathrm{BC}^{#1}}
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\newcommand{\mubc}{\mu\mathrm{BC}}
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\newcommand{\bc}{\mathrm{BC}}
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\newcommand{\sigp}[1]{\Sigma^p_{#1}}
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\newcommand{\pip}[1]{\Pi^p_{#1}}
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\newcommand{\fphi}[1]{\Box^p_{#1}}
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\newcommand{\fph}{\Box^p}
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	\newcommand{\ph}{\mathbf{PH}}
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	\newcommand{\pspace}{\mathbf{PSPACE}}
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	\newcommand{\fpspace}{\mathbf{FPSPACE}}
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	\newcommand{\ptime}{\mathbf{P}}
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	\newcommand{\fptime}{\mathbf{FP}}
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	\newcommand{\nc}{\mathbf{NC}}
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	\newcommand{\ac}{\mathbf{AC}}
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	\newcommand{\exptime}{\mathbf{EXP}}
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	\newcommand{\np}{\mathbf{NP}}
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	\newcommand{\conp}{\mathbf{coNP}}
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	\newcommand{\poly}{\mathrm{poly}}
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\newcommand{\cnot}{\neg}
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\newcommand{\cimp}{\supset}
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\newcommand{\cor}{\vee}
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\newcommand{\cand}{\wedge}
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\newcommand{\ciff}{\equiv}
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\newcommand{\size}[1]{|#1|}                        %% length of a word
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\newcommand{\mode}{\; \underline{\mbox{mod}}\;}   %% mod 2^{|x|}