\relax 
\@writefile{toc}{\contentsline {section}{\numberline {1}Review: The Resolution Proof System}{1}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces \ensuremath  {\mathsf  {Res}}\xspace  resolution (left) and prover-delayer game graph (right) for the unsat CNF formula\let \relax \let \relax \let \reserved@d = *\def \par }}{2}}
\newlabel{fig:godforsaken-tikz}{{1}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Resolution Lower Bounds}{2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Propositional Pigeonhole Principle}{2}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The pigeonhole principle, with corresponding literals in \PHP [5]{4} indicated}}{3}}
\newlabel{fig:pigeonhole}{{2}{3}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Warm-Up: Tree-Like \ensuremath  {\mathsf  {Res}}\xspace  Lower Bounds for \PHP [n+1]{n}}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces (a) Starting clauses of the monotone resolution refutation (pigeon clauses). (b) An example of the monotone resolution applied, in this case on the third row, or hole 3. (c) Example of a \ensuremath  {\mathsf  {Res}}\xspace  step transformed into a monotone resolution step. Note that on the bottom, we are not applying the monotone resolution rule -- just translating the derived clause to a monotone clause.}}{5}}
\newlabel{fig:matrices}{{3}{5}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Resolution Lower Bounds for \PHP [n+1]{n} (The General Case)}{5}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.1}Monotone Transformation of \PHP [n+1]{n}}{5}}
\newlabel{restomonotone}{{14}{6}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2}\PHP {} Lower Bound for Monotone Refutations}{6}}
\newlabel{monotonelb}{{16}{6}}
\newlabel{sizetowidth}{{17}{7}}
\newlabel{wideclauses}{{18}{7}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Resolution Lower Bounds Beyond \PHP {}}{8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}General Size-Width Tradeoff}{8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Width Lower Bounds From (Boundary) Expansion}{8}}
\citation{BussP}
\citation{raz-wphp}
\citation{MPW}
\citation{PWW}
\@writefile{toc}{\contentsline {section}{\numberline {4}Aside: \ensuremath  {\mathsf  {Res}}\xspace  Upper Bounds for \PHP [m]{n}}{9}}
\citation{Haken85}
\citation{raz-wphp}
\citation{Urquhart87}
\citation{CS88}
\citation{BKPS}
\citation{BussP}
\citation{PWW}
\citation{MPW}
\bibdata{proofcomplexity.bib}
\bibcite{BKPS}{BKPS02}
\bibcite{BussP}{BP97}
\bibcite{CS88}{CS88}
\bibcite{Haken85}{Hak85}
\bibcite{MPW}{MPW02}
\bibcite{PWW}{PWW88}
\bibcite{raz-wphp}{Raz04}
\bibcite{Urquhart87}{Urq87}
\bibstyle{alpha}
\@writefile{toc}{\contentsline {section}{\numberline {5}Some Open Problems}{10}}
\@writefile{toc}{\contentsline {section}{\numberline {6}References}{10}}
