1
$\begingroup$

Prove that $L\in NC^1$ iff there exists a sequence of poly sized formulas that decides $L$.

I managed to prove the $(\impliedby)$ and I want to prove $(\implies)$.

I feel that we need to take the boolean circuit and duplicate the sub-circuits of $fan-out\gt 1$ into exact circuits but with $fan-out = 1$. I fail to argue why the resulting tree is of polynomial size and not of exponential one.

$\endgroup$
2
$\begingroup$

Let the size of a circuit or formula be the number of leaves. The total number of vertices in a formula is at most twice its size.

You can prove by induction the following claim:

If a function is computed by a fan-in 2 circuit of size $S$ and depth $D$, then it is computed by a formula of size $2^D S$.

If $L$ is in NC1 then its $n$-bit fragments have circuits of size $n^{O(1)}$ and depth $O(\log n)$. The claim shows that there are equivalent formulas of size $n^{O(1)} 2^{O(\log n)} = n^{O(1)}$.

| cite | improve this answer | |
$\endgroup$

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.