ECCC-Report TR08-038https://eccc.weizmann.ac.il/report/2008/038Comments and Revisions published for TR08-038en-usMon, 26 Oct 2009 09:55:14 +0200
Revision 2
| Amplifying Lower Bounds by Means of Self-Reducibility |
Eric Allender,
Michal Koucký
https://eccc.weizmann.ac.il/report/2008/038#revision2We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC^1. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC^0 circuits of size n^{1+\epsilon_d}. If one were able to improve this lower bound to show that there is some constant \epsilon>0 such that every TC^0 circuit family recognizing BFE has size n^{1+\epsilon}, then it would follow that TC^0 \not= \NC^1.
We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC^0 and AC^0[6] circuits of size n^{1+c} for some constant c depending on d.
Mon, 26 Oct 2009 09:55:14 +0200https://eccc.weizmann.ac.il/report/2008/038#revision2
Revision 1
| Amplifying Lower Bounds by Means of Self-Reducibility |
Eric Allender,
Michal Koucký
https://eccc.weizmann.ac.il/report/2008/038#revision1We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+epsilon} for every epsilon > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC^1. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC^0 circuits of size n^{1+epsilon_d}. If one were able to improve this lower bound to show that there is some constant epsilon>0 such that every TC^0 circuit family recognizing BFE has size n^{1+epsilon}, then it would follow that TC^0 not= NC^1. We show that proving lower bounds of the form n^{1+epsilon} is not ruled out by the Natural Proof framework of Razborov and Rudich, and thence there is currently no known barrier for separating classes such as ACC^0, TC^0, and NC^1 via existing "natural" approaches to proving circuit lower bounds.
We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC^0 and AC^0[6] circuits of size n^{1+c} for some constant c depending on d.
Tue, 02 Dec 2008 00:00:00 +0200https://eccc.weizmann.ac.il/report/2008/038#revision1
Paper TR08-038
| Amplifying Lower Bounds by Means of Self-Reducibility |
Eric Allender,
Michal Koucky
https://eccc.weizmann.ac.il/report/2008/038We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC^1. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC^0 circuits of size n^{1+\epsilon_d}. If one were able to improve this lower bound to show that there is some constant \epsilon>0 such that every TC^0 circuit family recognizing BFE has size n^{1+\epsilon}, then it would follow that TC^0 \not= \NC^1.
We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC^0 and AC^0[6] circuits of size n^{1+c} for some constant c depending on d.
Fri, 04 Apr 2008 16:26:10 +0300https://eccc.weizmann.ac.il/report/2008/038