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Extremely Deep Proofs

Authors Noah Fleming, Toniann Pitassi, Robert Robere

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Author Details

Noah Fleming
  • University of California, San Diego, CA, USA
  • Memorial University, Canada
Toniann Pitassi
  • University of Toronto, Canada
  • Columbia University, New York, NY, USA
  • IAS, Princeton, NJ, USA
Robert Robere
  • McGill University, Montreal, Canada

Funding

  • Fleming, Noah: Research supported by NSERC.
  • Pitassi, Toniann: Research supported by NSERC, the IAS School of Mathematics and NSF Grant No. CCF-1900460.
  • Robere, Robert: Research supported by NSERC.

Cite AsGet BibTex

Noah Fleming, Toniann Pitassi, and Robert Robere. Extremely Deep Proofs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 70:1-70:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/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 ≤ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof Complexity
  • Tradeoffs
  • Resolution
  • Cutting Planes

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