8
$\begingroup$

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"
$\endgroup$
  • $\begingroup$ @Kaveh You can keep your answer. Maybe you can remark it was for an erroneous version. $\endgroup$ – Turbo 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
  • 1
    $\begingroup$ Well it had some nice details. $\endgroup$ – Turbo Aug 6 '15 at 6:42
6
$\begingroup$

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).

$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.