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Revision #1 to TR21-158 | 25th November 2021 03:01

Extremely Deep Proofs

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Revision #1
Authors: Noah Fleming, Toniann Pitassi, Robert Robere
Accepted on: 25th November 2021 03:01
Downloads: 15
Keywords: 


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-epsilon}, 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-epsilon}/c must necessarily have depth approximately n^c. By setting c = o(n^{1-epsilon}) 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 [Razborov16]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems.



Changes to previous version:

Minor typos, fixed hyperlinks.


Paper:

TR21-158 | 12th November 2021 00:28

Extremely Deep Proofs


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-epsilon}, 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-epsilon}/c must necessarily have depth approximately n^c. By setting c = o(n^{1-epsilon}) 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 [Razborov16]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems.



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