25 votes

Proof that circuit upper bounds for $E$ imply $P \neq NP$

I don't think the paper in the other answer contains an answer to your question. Indeed I am not really sure a proof has been published, because the result follows from other well known results. The ...
21 votes
Accepted

How many negations do we need to compute monotone functions?

Markov proved that any function of $n$ inputs can be computed with only $\lceil \log (n+1)\rceil$ negations. An efficient constructive version was described by Fisher. See also an exposition of the ...
  • 2,809
19 votes
Accepted

Is BPP vs. P a real problem after we know BPP lies in P/poly?

Not sure how much of an answer this is, I'm just indulging in some rumination. Question 1 could be equally asked about P $\neq$ NP and with a similar answer -- the techniques/ideas used to prove the ...
  • 7,140
18 votes
Accepted

smallest circuit size using XOR gates

This is NP-hard. See: Joan Boyar, Philip Matthews, René Peralta. Logic Minimization Techniques with Applications to Cryptology. http://link.springer.com/article/10.1007/s00145-012-9124-7 The ...
17 votes
Accepted

Regular versus TC0

Take $S_5$ as alphabet and $$L= \{ \sigma_1\cdots \sigma_n \in S_5^*\mid \sigma_1\circ\cdots\circ\sigma_n = \text{Id}\}$$ Barrington proved in [2] that $L$ is $\textrm{NC}^1$-complete for $\textrm{AC}...
  • 992
16 votes
Accepted

For what c is division by c in AC0?

Addition and subtraction of binary numbers are in $\mathsf{AC^0}$. For any constant number $c$, $x \bmod c$ is $\mathsf{AC^0}$ reducible to division by $c$ ($\lfloor x/c \rfloor$): $$x \bmod c = x ...
  • 3,236
16 votes
Accepted

Small circuits for circuit evaluation problem

I have learned from talking to Ryan Williams (who deserves the credit for my being able to post this answer) that it is known from Paul and Pippenger that Circuit Eval can be decided by a quasilinear ...
15 votes
Accepted

What is an equivalent definition of mP/poly in terms of a Turing machine?

There is a notion of a monotone non-deterministic and, more generally, alternating Turing machine in the paper Monotone Complexity by Grigni and Sipser. Since polynomial time is the same as ...
15 votes

Regular versus TC0

Regular languages with unsolvable syntactic monoids are $\mathrm{NC}^1$-complete (due to Barrington; this is the underlying reason behind the more commonly quoted result that $\mathrm{NC}^1$ equals ...
15 votes

Collapses under the assumption that $NEXP\subseteq P/Poly$

A whole lot of fun things happen. Most of the ones I know of start with the IKW paper. There, the collapse $\textrm{NEXP} = \textrm{MA}$ is shown, and (I think) is the strongest literal collapse of ...
15 votes
Accepted

Circuit and Formula Lower Bounds for Separating Sparse Sets of Strings

Yes, any such pair can be separated by a formula of size $O(n)$. More generally, any disjoint pair $P,N\subseteq\{0,1\}^n$ of size $s=|P|+|N|$ can be separated by a decision tree of size $O(s)$, which ...
14 votes

Scope of natural proofs barrier

Let $f: \{0,1\}^* \rightarrow \{0,1\}$ be a function, and let $C$ be a class of algorithms working on finite slices of $f$. Every circuit lower bound whatsoever is a proof that $f \notin C$, for some $...
14 votes
Accepted

Why is HAMILTONIAN CYCLE so different from PERMANENT?

The following is a proof over any ring of characteristic zero that the Hamiltonian cycle polynomial is not a polynomial-size monotone projection of the permanent. The basic idea is that monotone ...
13 votes
Accepted

Collapses under the assumption that $NEXP\subseteq P/Poly$

I believe the strongest is that $NEXP = MA$. This was proved by Impagliazzo Kabanets and Wigderson. See https://scholar.google.com/scholar?cluster=17275091615053693892&hl=en&as_sdt=0,5&...
11 votes

Can one prove $\mathsf{PARITY} \notin \mathsf{AC}^0$ using Linial-Mansour-Nisan theorem and the knowledge of fourier spectrum of $\mathsf{PARITY}$?

LMN theorem shows that if f is a boolean function$(f:\{-1,1\}^n \rightarrow \{-1,1\})$ computable by an $\text{AC}^0$ circuit of size M, $$\sum_{S:|S|> k} \hat f(S)^{2} \leq 2^{-\Omega(k/(\log M)^{...
  • 317
11 votes

Better lower bounds than 3n for non-boolean functions?

here are new results on this said to be the 1st in ~3 decades and some brief commentary A better-than-3n lower bound for the circuit complexity of an explicit function / Find, Golovnev, Hirsch, ...
  • 10.9k
11 votes

What are examples of how non-uniformity can be useful?

An example I like is the argument that $\mathrm{NE\subseteq coNE}/(n+1)$ by counting strings in the language (see e.g. https://blog.computationalcomplexity.org/2004/01/little-theorem.html).
10 votes
Accepted

What is the minimum size of a circuit that computes PARITY?

It is possible to compute parity using only 2.33n + C gates. The construction is quite simple and is given in this article. http://link.springer.com/article/10.3103/S0027132215050083 Here is an ...
10 votes
Accepted

Consequences of $NP\subseteq P/poly$ to $BQP$

I'm not aware of any direct consequence of $NP\subset P/poly$ for $BQP$. Of course it might lessen the interest in quantum computing, since it would mean that we could do something far more ...
10 votes
Accepted

Evaluate boolean circuit on batch of similar inputs

I'd consider it unlikely that such a trick is easy to find and/or will give you significant gains, as it would give nontrivial satisfiability algorithms. Here's how: First of all, while ostensibly ...
10 votes
Accepted

Randomness and small circuits complexity classes

Most nonuniform complexity classes—$\mathrm{NC^1}$ included—are closed under the $\mathrm{BP}$ operator by the same argument as $\mathrm{BPP\subseteq P/poly}$: using the Chernoff–Hoeffding bound, the ...
10 votes
Accepted

Why is the circuit class AC0 unavoidable?

The class $\mathbf{AC}^0$ arises very naturally as the circuit characterization of problems definable by formulas of first-order logic. A language over the alphabet $\{ 0,1 \}$ is in $\mathbf{AC}^0$ ...
10 votes

What are examples of how non-uniformity can be useful?

One example is $\mathbf{NL} \subseteq \mathbf{UL}/\text{poly}$. This theorem was proven by Reinhardt and Allender in their paper "Making Nondeterminism Unambiguous". Without going into the details, ...
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10 votes
Accepted

Linear circuit complexity classes

It follows from work of Valiant [1, 2] that linear-size $\textrm{NC}^1$ can be simulated by $2^{O(n / \log \log n)}$-size circuits of depth three and unbounded fan-in. For a nice exposition of this ...
10 votes
Accepted

$AC^0$[subexp] vs. NC

No, $\mathrm{AC}^0[2^{n^\delta}]$ is not included in NC; it is not even included in $\mathrm{SIZE}[2^{n^\epsilon}]$ for $\epsilon<\delta$. Indeed, any Boolean function on $n^\delta$ inputs, padded ...
9 votes
Accepted

How expensive may it be to destroy all long s-t paths in a DAG?

[Self answer; this is a shortened version, the old one can be found here] We realized with Georg Schnitger that the answer to my question is strongly negative: there are DAGs (even of constant degree)...
  • 6,645
9 votes
Accepted

What are bounded-treewidth circuits good for?

We now understand that for any fixed bound $k \in \mathbb{N}$ on the treewidth, we can convert any Boolean circuit of treewidth less than $k$ to a so-called d-SDNNF circuit, in linear time and with ...
  • 7,942
9 votes

Integer multiplication when one integer is fixed

I am not sure whether this is directly relevant to the question, but the following elementary result might be of interest. Given a fixed natural number $k$, the operation $n \to kn$ can be realized by ...
  • 4,761
9 votes

What's the “smallest” complexity class for which an $\omega \hspace{.02 in}(n)$ circuit lower bound is known?

$S^p_2$ and $PP$ are both known not to have $n^k$-circuits for any fixed k and there is no known containment between them. Details in my blog post. Update: As Rickey Demer points out, these results ...
9 votes
Accepted

Matrix vector multiplication algorithm using minimal number of additions

This is essentially the problem that motivated Valiant to introduce matrix rigidity into complexity (as far as I understand the history). A linear circuit is an algebraic circuit whose only gates ...

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