### Amplifying lower bounds by means of self-reducibilityAmplifying lower bounds by means of self-reducibility

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 Author Allender, Eric ♦ Kouck, Michal Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2010 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Circuit complexity ♦ Lower bounds ♦ Natural proofs ♦ Self-reducibility ♦ Time-space tradeoffs Abstract We observe that many important computational problems in $NC^{1}$ share a simple self-reducibility property. We then show that, for any problem $\textit{A}$ having this self-reducibility property, $\textit{A}$ has polynomial-size $TC^{0}$ circuits if and only if it has $TC^{0}$ circuits of size $n^{1+&epsis;}$ for every &epsis;> 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 $NC^{1}$ and has the self-reducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth $\textit{d}$ $TC^{0}$ circuits of size $n^{1+&epsis;_{d}}.$ If one were able to improve this lower bound to show that there is some constant &epsis;> 0 (independent of the depth $\textit{d})$ such that every $TC^{0}$ circuit family recognizing BFE has size at least $n^{1+&epsis;},$ then it would follow that $TC^{0}$ ≠ $NC^{1}.$ We show that proving lower bounds of the form $n^{1+&epsis;}$ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence 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 $\textit{d}$ $TC^{0}$ and $AC^{0}[6]$ circuits of size $n^{1+c}$ for some constant $\textit{c}$ depending on $\textit{d}.$ ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2010-03-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 57 Issue Number 3 Page Count 36 Starting Page 1 Ending Page 36

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Source: ACM Digital Library