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Revision #3 to TR20-122 | 1st October 2020 00:42

Size Bounds on Low Depth Circuits for Promise Majority

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Revision #3
Authors: Joshua Cook
Accepted on: 1st October 2020 00:42
Downloads: 518
Keywords: 


Abstract:

We give two results on the size of AC0 circuits computing promise majority. \epsilon-promise majority is majority promised that either at most an \epsilon fraction of the input bits are 1 or at most \epsilon are 0.

First, we show super-quadratic lower bounds on both monotone and general depth-3 circuits for promise majority.

For any \epsilon \in (0, 1/2), monotone depth-3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon)}{\ln(\epsilon)}}\right)

For any \epsilon \in (0, 1/2), general depth-3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon^2)}{2\ln(\epsilon)}}\right)

These are the first quadratic size lower bounds for depth-3 \epsilon-promise majority circuits for \epsilon < 0.49.

Second, we give both uniform and non-uniform sub-quadratic size constant-depth circuits for promise majority.

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{non} uniform AC0 circuits of depth-(2 + 2 k) computing \epsilon-promise majority with size
\tilde{O}\left(n^{\frac{1}{1 - 2^{-k}}}\right)

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{uniform} AC0 circuit of depth-(2 + 2 k) computing \epsilon-promise majority with size
n^{\frac{1}{1 - \left(\frac{2}{3}\right)^k} + o(1)}

These circuits are based on incremental improvements to existing depth-3 circuits for promise majority given by Ajtai and Viola combined with a divide and conquer strategy.



Changes to previous version:

Fixed Minor Typos


Revision #2 to TR20-122 | 27th September 2020 00:48

Size Bounds on Low Depth Circuits for Promise Majority





Revision #2
Authors: Joshua Cook
Accepted on: 27th September 2020 00:48
Downloads: 426
Keywords: 


Abstract:

We give two results on the size of AC0 circuits computing promise majority. \epsilon-promise majority is majority promised that either at most an \epsilon fraction of the input bits are 1 or at most \epsilon are 0.

First, we show super-quadratic lower bounds on both monotone and general depth-3 circuits for promise majority.

For any \epsilon \in (0, 1/2), monotone depth-3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon)}{\ln(\epsilon)}}\right)

For any \epsilon \in (0, 1/2), general depth-3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon^2)}{2\ln(\epsilon)}}\right)

These are the first quadratic size lower bounds for depth-3 \epsilon-promise majority circuits for \epsilon < 0.49.

Second, we give both uniform and non-uniform sub-quadratic size constant-depth circuits for promise majority.

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{non} uniform AC0 circuits of depth-(2 + 2 k) computing \epsilon-promise majority with size
\tilde{O}\left(n^{\frac{1}{1 - 2^{-k}}}\right)

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{uniform} AC0 circuit of depth-(2 + 2 k) computing \epsilon-promise majority with size
n^{\frac{1}{1 - \left(\frac{2}{3}\right)^k} + o(1)}

These circuits are based on incremental improvements to existing depth-3 circuits for promise majority given by Ajtai and Viola combined with a divide and conquer strategy.



Changes to previous version:

Accidentally used Jaikumar Radhakrishnan first name instead of his last, minor typos.


Revision #1 to TR20-122 | 26th September 2020 08:30

Size Bounds on Low Depth Circuits for Promise Majority





Revision #1
Authors: Joshua Cook
Accepted on: 26th September 2020 08:31
Downloads: 493
Keywords: 


Abstract:

We give two results on the size of AC0 circuits computing promise majority. \epsilon-promise majority is majority promised that either at most an \epsilon fraction of the input bits are 1 or at most \epsilon are 0.

First, we show super-quadratic size lower bounds on both monotone and general depth-3 circuits for promise majority.

For any \epsilon \in (0, 1/2), monotone depth-3 AC0 circuits for \epsilon-promise majority have size
tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon)}{\ln(\epsilon)}}\right).

For any \epsilon \in (0, 1/2), general depth-3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon^2)}{2\ln(\epsilon)}}\right).

These are the first quadratic size lower bounds for depth-3 \epsilon-promise majority circuits for \epsilon < 0.49.

Second, we give both uniform and non-uniform sub-quadratic size constant-depth circuits for promise majority.

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{non} uniform AC0 circuits of depth-(2 + 2 k) computing \epsilon-promise majority with size
\tilde{O}\left(n^{\frac{1}{1 - 2^{-k}}}\right).

For integer k \geq 1 and constant \epsilon \in (0, 1/2), there exists monotone \emph{uniform} AC0 circuit of depth-(2 + 2 k) computing \epsilon-promise majority with size
n^{\frac{1}{1 - \left(\frac{2}{3}\right)^k} + o(1)}.

These circuits are based on incremental improvements to existing depth-3 circuits for promise majority given by Ajtai and Viola combined with a divide and conquer strategy.



Changes to previous version:

Fixed formatting issues.
Added a related work by Chaudhuri and Jaikumar from 96: "Deterministic Restrictions in Circuit Complexity".
Reworded the motivations to be more clear.
Updated the open problems.
Fixed many typos, grammatical mistakes, and minor errors.


Paper:

TR20-122 | 8th August 2020 07:01

Size Bounds on Low Depth Circuits for Promise Majority





TR20-122
Authors: Joshua Cook
Publication: 16th August 2020 17:37
Downloads: 682
Keywords: 


Abstract:

We give two results on the size of AC0 circuits computing promise majority. \epsilon-promise majority is majority promised that either at most an \epsilon fraction of the input bits are 1, or at most \epsilon are 0.

First, we show super quadratic lower bounds on both monotone and general depth 3 circuits for promise majority.

For any \epsilon \in (0, 1/2), monotone depth 3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon)}{\ln(\epsilon)}}\right)

For any \epsilon \in (0, 1/2), general depth 3 AC0 circuits for \epsilon-promise majority have size
\tilde{\Omega}\left(\epsilon^3 n^{2 + \frac{\ln(1 - \epsilon^2)}{2\ln(\epsilon)}}\right)

These are the first nontrivial size lower bounds on depth 3 promise majority circuits for \epsilon < 0.45.

Second, we give both uniform and non-uniform sub-quadratic size constant depth circuits for promise majority.

For integer k \geq 1, constant \epsilon \in (0, 1/2), there exists monotone non uniform AC0 circuits of depth 2 + 2 \cdot k computing \epsilon-promise majority with size
\tilde{O}\left(n^{\frac{1}{1 - 2^{-k}}}\right)

For integer k \geq 1, constant \epsilon \in (0, 1/2), there exists monotone uniform AC0 circuit of depth 2 + 2 \cdot k computing \epsilon-promise majority with size
n^{\frac{1}{1 - \left(\frac{2}{3}\right)^k} + o(1)}

These circuits are based on incremental improvements to existing depth 3 circuits for promise majority given by Ajtai and Viola combined with a divide and conquer strategy.



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