2026-08-17T23:46:55Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15666 2024-03-06T10:55:54Z ddc:004 open_access

Extremely Deep Proofs Fleming, Noah Pitassi, Toniann Robere, Robert Proof Complexity Tradeoffs Resolution Cutting Planes 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 ≤ n^{1-ε}, 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-ε}/c must necessarily have depth ≈ n^c. By setting c = o(n^{1-ε}) 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Noah Fleming and Toniann Pitassi and Robert Robere 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.70 urn:nbn:de:0030-drops-156665 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.70 eng https://creativecommons.org/licenses/by/4.0/legalcode