{"DOI":"10.4230/lipics.itcs.2022.70","abstract":"We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c \u2264 n^{1-\u03b5}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-\u03b5}/c must necessarily have depth \u2248 n^c. By setting c = o(n^{1-\u03b5}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems.","author":[{"family":"Fleming","given":"Noah"},{"family":"Pitassi","given":"Toniann"},{"family":"Robere","given":"Robert"},{"family":"Braverman","given":"Mark"}],"id":"unknown","issued":{"date-parts":[[2022,1,25]]},"language":"en","publisher":"Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik","title":"Extremely Deep Proofs","type":"article-journal"}