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In [1] it is stated that

"It remains an open question as to whether every function in $\#P$ has $TC^0$ circuits (although it is at least known that not all $\#P$ functions have DLogTime-uniform $TC^0$ circuits)."

$TC^0$ circuits generated by DLogTime functions does not contain $\#P$. We don't know if $TC^0$ circuits generated by arbitrary functions does not contain $\#P$.

Is there anything known about the cases in between these two? E.g. is it known if $TC^0$ circuits generated by $L$ does not contain $\#P$?

  • [1] Agarwal, Allender, and Datta, "On $TC^0$, $AC^0$, and Arithmetic Circuits"
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  • $\begingroup$ @Kaveh You can keep your answer. Maybe you can remark it was for an erroneous version. $\endgroup$ – 1.. Aug 6 '15 at 6:38
  • $\begingroup$ I don't think it answers the question, so it is not really an answer. :) $\endgroup$ – Kaveh Aug 6 '15 at 6:39
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    $\begingroup$ Well it had some nice details. $\endgroup$ – 1.. Aug 6 '15 at 6:42
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This is an (interesting) open problem, as far as I know. Rahul Santhanam and I explicitly mention the problem of proving Permanent is not in LOGSPACE-uniform TC0 in our CCC'13 paper (On Medium-Uniformity and Circuit Lower Bounds).

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    $\begingroup$ More conservatively, one could ask about DTISP $\hspace{-0.04 in}\left(\log(n)\hspace{-0.04 in}\cdot \hspace{-0.03 in}(\log(\log(n)))^{o(1)}\hspace{-0.02 in},\hspace{-0.02 in}O(\log(n))\right)\hspace{-0.02 in}$-uniformity. $\;\;\:\:$ $\endgroup$ – user6973 Aug 12 '15 at 22:24
  • $\begingroup$ @Ricky Demer, that is already known. See for example Chen and Kabanets eccc.hpi-web.de/report/2012/007/download $\endgroup$ – Ryan Williams Aug 13 '15 at 19:56
  • $\begingroup$ Well, by that paper, POLYLOGTIME is a "class of functions C such that we know #P is not contained in" C-uniform TC0. $\:$ Furthermore, by padding, POLYLOGTIME is larger than DLOGTIME. $\endgroup$ – user6973 Aug 13 '15 at 20:43
  • $\begingroup$ Exactly... So, lower bounds for the uniformity you mentioned above are already known. $\endgroup$ – Ryan Williams Aug 14 '15 at 5:34
  • $\begingroup$ ... and, those lower bounds are not mentioned in your answer. $\;$ $\endgroup$ – user6973 Aug 14 '15 at 7:39

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