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cc-preamble.tex
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%--------------------------------------------------------------
% Computational Complexity Preamble
% Author: Antonis Antonopoulos
%--------------------------------------------------------------
%--------------------------------------------------------------
% COMPLEXITY COMMANDS DEFS
%--------------------------------------------------------------
% General Notations
\newcommand{\bO}[1]{\mathcal{O}\left({#1}\right)}
\newcommand{\C}{\mathcal{C}}
\newcommand{\cC}{co\mathcal{C}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\cF}{co\mathcal{F}}
\newcommand{\zo}{\{0,1\}}
\newcommand{\zon}{\{0,1\}^{n}}
\newcommand{\zos}{\{0,1\}^{*}}
\newcommand{\Ss}{\Sigma^{*}}
\newcommand{\Cirfam}{\{C_{n}\}_{n\in\mathbb{N}}}
\newcommand{\Prb}[2]{\Pr_{#1}\left[{#2}\right]}
%---------------------------------------------------------
% Deterministic Classes
\newcommand{\DTIME}[1]{\mathbf{DTIME}[#1]}
\newcommand{\NTIME}[1]{\mathbf{NTIME}[#1]}
\newcommand{\DSPACE}[1]{\mathbf{DSPACE}[#1]}
\newcommand{\NSPACE}[1]{\mathbf{NSPACE}[#1]}
\newcommand{\cP}{\mathbf{P}}
\newcommand{\NP}{\mathbf{NP}}
\newcommand{\cNP}{co\mathbf{NP}}
\newcommand{\EXP}{\mathbf{EXP}}
\newcommand{\E}{\mathbf{E}}
\newcommand{\EE}{\mathbf{EE}}
\newcommand{\NEXP}{\mathbf{NEXP}}
\newcommand{\EXPSPACE}{\mathbf{EXPSPACE}}
\newcommand{\NEXPSPACE}{\mathbf{NEXPSPACE}}
\newcommand{\cL}{\mathbf{L}}
\newcommand{\NL}{\mathbf{NL}}
\newcommand{\cNL}{co\mathbf{NL}}
\newcommand{\SL}{\mathbf{SL}}
\newcommand{\PSPACE}{\mathbf{PSPACE}}
\newcommand{\NPSPACE}{\mathbf{NPSPACE}}
\newcommand{\Stwo}{\Sigma_{2}^{p}}
\newcommand{\Ptwo}{\Pi_{2}^{p}}
\newcommand{\Ttwo}{\Theta_{2}^{p}}
\newcommand{\Ttwod}{\cP^{\NP[\log{n}]}}
\newcommand{\Sp}[1]{\Sigma_{#1}^{p}}
\newcommand{\Pp}[1]{\Pi_{#1}^{p}}
\newcommand{\Dp}[1]{\Delta_{#1}^{p}}
\newcommand{\Syp}[1]{S_{#1}^{p}}
\newcommand{\Sytwo}{S_{2}^{p}}
\newcommand{\PH}{\mathbf{PH}}
\newcommand{\SUBEXP}{\mathbf{SUBEXP}}
\newcommand{\QuasiP}{\mathbf{QuasiP}}
\newcommand{\FP}{\mathbf{FP}}
\newcommand{\FNP}{\mathbf{FNP}}
\newcommand{\TFNP}{\mathbf{TFNP}}
\newcommand{\FTtwod}{\FP^{\NP[\log{n}]}}
%---------------------------------------------------------
% Randomized Classes
\newcommand{\BPTIME}[1]{\mathbf{BPTIME}[#1]}
\newcommand{\RTIME}[1]{\mathbf{RTIME}[#1]}
\newcommand{\BPP}{\mathbf{BPP}}
\newcommand{\RP}{\mathbf{RP}}
\newcommand{\RL}{\mathbf{RL}}
\newcommand{\cRP}{co\mathbf{RP}}
\newcommand{\ZPP}{\mathbf{ZPP}}
\newcommand{\PP}{\mathbf{PP}}
%---------------------------------------------------------
% Counting Classes
\newcommand{\sP}{{\#\mathbf{P}}}
\newcommand{\oddP}{{\oplus\mathbf{P}}}
\newcommand{\modP}{{\mathbf{Mod}_{k}\mathbf{P}}}
\newcommand{\UP}{\mathbf{UP}}
\newcommand{\FewP}{\text{\textbf{Few}}\mathbf{P}}
%---------------------------------------------------------
% Interactive Proof Classes
\newcommand{\IP}{\mathbf{IP}}
\newcommand{\MA}{\mathbf{MA}}
\newcommand{\cMA}{co\mathbf{MA}}
\newcommand{\AM}{\mathbf{AM}}
\newcommand{\cAM}{co\mathbf{AM}}
\newcommand{\PCP}[2]{\mathbf{PCP}[#1,#2]}
%---------------------------------------------------------
% Non-Uniform Classes
\newcommand{\SIZE}[1]{\mathbf{SIZE}[#1]}
\newcommand{\Ppoly}{\mathbf{P}_{\mathbf{/poly}}}
\newcommand{\ACz}[1]{\mathbf{AC}^{0}[#1]}
\newcommand{\ACCz}{\mathbf{ACC}^{0}}
\newcommand{\NC}[1]{\mathbf{NC}^{#1}}
\newcommand{\AC}[1]{\mathbf{AC}^{#1}}
\newcommand{\TC}[1]{\mathbf{TC}^{#1}}
\newcommand{\SC}[1]{\mathbf{SC}^{#1}}
\newcommand{\RNC}{\mathbf{RNC}}
\newcommand{\EXPpoly}{\mathbf{EXP}_{\mathbf{/poly}}}
%---------------------------------------------------------
% Subclasses of TFNP
\newcommand{\PLS}{\mathbf{PLS}}
\newcommand{\PPA}{\mathbf{PPA}}
\newcommand{\PPAD}{\mathbf{PPAD}}
\newcommand{\PPP}{\mathbf{PPP}}
%---------------------------------------------------------
% Pseudo Classes
\newcommand{\pseudo}[2]{\texttt{pseudo}_{#1}{#2}}
%---------------------------------------------------------
% Relations & Reductions
\newcommand{\cs}{\subseteq}
\newcommand{\cns}{\nsubseteq}
\newcommand{\cps}{\subsetneq}
\newcommand{\kred}{\le^{p}_{m}} %Karp Reduction
\newcommand{\tred}{\le^{p}_{T}} %Cook/Turing Reduction
\newcommand{\ttred}{\le^{p}_{tt}} %Truth-Table Reduction
\newcommand{\lred}{\le_{l}} %Log-space reduction
%---------------------------------------------------------
% Common Languages/Problems
\newcommand{\SAT}{\mathtt{SAT}}
\newcommand{\sSAT}{\mathtt{\#SAT}}
\newcommand{\PERM}{\mathtt{PERMANENT}}
\newcommand{\TSP}{\mathtt{TSP}}
\newcommand{\CVP}{\mathtt{CVP}}
\newcommand{\GI}{\mathtt{GI}}
\newcommand{\GNI}{\mathtt{GNI}}
\newcommand{\FACT}{\mathtt{FACTORING}}
\newcommand{\HP}{\mathtt{HALTING\ PROBLEM}}
\newcommand{\TQBF}{\mathtt{TQBF}}
%---------------------------------------------------------
% Quantifier Characterizations
\newcommand{\explus}{\exists^{+}}
\newcommand{\qcc}[2]{(#1/#2)}
%---------------------------------------------------------
% Operators on Complexity Classes
\newcommand{\opN}[1]{\mathcal{N}\cdotp{#1}}
\newcommand{\opD}[1]{\Delta\cdotp{#1}}
\newcommand{\BP}[1]{\mathcal{BP}\cdotp{#1}}
\newcommand{\opP}[1]{\mathcal{P}\cdotp{#1}}
\newcommand{\opR}[1]{\mathcal{R}\cdotp{#1}}
\newcommand{\Alm}[1]{\text{Almost}{#1}}
%---------------------------------------------------------