ECCC-Report TR19-120https://eccc.weizmann.ac.il/report/2019/120Comments and Revisions published for TR19-120en-usSun, 26 Mar 2023 15:28:37 +0300
Revision 2
| Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation |
Or Meir
https://eccc.weizmann.ac.il/report/2019/120#revision2One of the major open problems in complexity theory is proving super-logarithmic
lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5, 3/4) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions $f \diamond g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:
- The multiplexor relation is "complete" for the approach of Karchmer et. al.
in the following sense: if we could prove (a variant of) their conjecture
for the composition $f \diamond MUX$ for every function $f$, then this would
imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
- A simpler proof of a lower bound for the multiplexor relation due
to Edmonds et. al. Our proof has the additional benefit of fitting
better with the machinery used in previous works on the subject.Sun, 26 Mar 2023 15:28:37 +0300https://eccc.weizmann.ac.il/report/2019/120#revision2
Revision 1
| Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation |
Or Meir
https://eccc.weizmann.ac.il/report/2019/120#revision1One of the major open problems in complexity theory is proving super-logarithmic
lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5, 3/4) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions $f \diamond g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:
- The multiplexor relation is "complete" for the approach of Karchmer et. al.
in the following sense: if we could prove (a variant of) their conjecture
for the composition $f \diamond MUX$ for every function $f$, then this would
imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
- A simpler proof of a lower bound for the multiplexor relation due
to Edmonds et. al. Our proof has the additional benefit of fitting
better with the machinery used in previous works on the subject.Tue, 25 Feb 2020 14:23:43 +0200https://eccc.weizmann.ac.il/report/2019/120#revision1
Paper TR19-120
| Toward Better Depth Lower Bounds: Two Results on the Multiplexor Relation |
Or Meir
https://eccc.weizmann.ac.il/report/2019/120One of the major open problems in complexity theory is proving super-logarithmic
lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5, 3/4) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions $f \diamond g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
As a way to realize this program, Edmonds et. al. (Computational Complexity 10, 3) suggested to study the "multiplexor relation" $MUX$, which is a simplification of functions. In this note, we present two results regarding this relation:
- The multiplexor relation is "complete" for the approach of Karchmer et. al.
in the following sense: if we could prove (a variant of) their conjecture
for the composition $f \diamond MUX$ for every function $f$, then this would
imply $\mathbf{P}\not\subseteq\mathbf{NC}^1$.
- A simpler proof of a lower bound for the multiplexor relation due
to Edmonds et. al. Our proof has the additional benefit of fitting
better with the machinery used in previous works on the subject.Sun, 15 Sep 2019 17:52:07 +0300https://eccc.weizmann.ac.il/report/2019/120