# Tagged Questions

Circuit complexity is the study of resource-bounded circuits and the functions computed by such circuits.

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### Is length uniform AC0 computable?

Consider the following problem: Input: A binary string $w$. Output: $|w|$ as a binary number. Is it possible to compute this in $\mathsf{DLogTime}$-uniform $\mathsf{AC}^0$ (or equivalently in ...
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### Circuity complexity: monotone circuit of Majority function

As showed in the paper "Monotone Circuits for the Majority Function", is possible to construct a monotone boolean circuit for the majority function on n variables with size O(n^3) and depth 5.3 ...
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### Is unbounded quantum fanout operation experimentally feasible?

It is known that the "unbounded quantum fanout operation" is very powerful: (See, for example, Hoyer et al. : http://theoryofcomputing.org/articles/v001a005/v001a005.pdf). In particular, it is known ...
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### The evaluation problem for AC$^0_d$ formulas is in FO

Let $d \in \mathbb{N}$ be arbitrary. Let $\mathsf{AC^0_d}$-Eval be the following promise problem: Input: A depth $d$ formula $\varphi(x)$ and a binary string $a$. Output: $\varphi(a)$ I am looking ...
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### Circuit complexity lower bounds and uniformity

I have troubles to understand how lower bounds w.r.t. circuit complexity and upper bounds w.r.t. uniform machine models can be used to show completeness results. For example, the word problem for ...
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### Linear Time Hierarchy and Circuit complexity

By Karp-Lipton Theorem we have: $$PH\subseteq P/poly\Rightarrow PH=\Sigma^p_2$$ So this theorem suggest it is unlikely that $PH\subseteq P/poly$. I want to know is there any similar conditional or ...
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### Monotone arithmetic circuit complexity of elementary symmetric polynomials?

The $k$-th elementary symmetric polynomial $S_k^n(x_1,\ldots,x_n)$ is the sum of all $\binom{n}{k}$ products of $k$ distinct variables. I am interested in the monotone arithmetic $(+,\times)$ circuit ...
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### Regular versus TC0

According to the Complexity Zoo, $\mathsf{Reg} \subseteq \mathsf{NC^1}$ and we know that $\mathsf{Reg}$ cannot count so $\mathsf{TC^0} \not\subseteq \mathsf{Reg}$. However it doesn't say if ...
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### When can arguments for $\mathsf{NP}$ be replaced by $\mathsf{NEXP}$ by employing succinct circuits?

According to papers here https://users.cs.duke.edu/~reif/temp/RandomHardProblems/NovaFandina/succinctPermanent.pdf and here http://perso.ens-lyon.fr/natacha.portier/papers/PY86.pdf questions in ...
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### Parallels between communuication complexity and complexity theory

(1) Do we know the polynomial hierarchy $\mathsf{PH^{cc}}$ is infinite in communication complexity? (2) Is there a candidate problem in ...
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### Is Dynamic Programming never weaker than Greedy?

In the circuit complexity, we have separations between powers of various circuit models. In the proof complexity, we have separations between powers of various proof systems. But in the ...
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### Proof that circuit upper bounds for $E$ imply $P \neq NP$

In the official Clay problem description for P versus NP it's stated that $P \neq NP$ would follow from showing that "every language in $E$ [the class of languages recognizable in exponential time ...
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### Lower bounds for nonuniform circuits and oracles separating complexity classes

I have read that Furst, Sax, and Sipser came up with their lower bound for nonuniform AC0 while trying to prove an oracle separation. Can someone explain how proving lower bounds for circuits and ...
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### Why can't we have superlinear bounds on Boolean circuit size for an explicit function?

I am interested about the minimal size (number of gates) of a family of circuits (with negation) over a complete Boolean basis (with fanin 2) that computes some explicit Boolean function. (In other ...
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### Matrix vector multiplication algorithm using minimal number of additions

Consider the following problem: Given a matrix $M$ we want to optimize the number of additions in the multiplication algorithm for computing $v \mapsto Mv$. I find this problem interesting ...
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### What's the “smallest” complexity class for which an $\omega \hspace{.02 in}(n)$ circuit lower bound is known?

I believe the answers to this question give classes such that for all polynomials $p$, there is a problem in the class which does not have circuits of size $p(n)$. However, I'm asking about circuit ...
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### Arithmetic complexity of matrix powering

Assume $M\in\Bbb Z_{\geq0}[x_1,\dots,x_n]^{m\times m}$ be an $m\times m$ matrix in $n$ variables. We know that size of smallest formula that computes $\mathsf{Tr}(M^d)$ where $d\in\Bbb N$ could be ...
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### Consequences of VP = VNP on randomness

According to the answers in posting it is possible that $\mathsf{VP} = \mathsf{VNP}$ and $\mathsf{P} \neq \mathsf{NP}$ are simultaneously correct. $\mathsf{VP} = \mathsf{VNP}$ implies ...
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### How many negations do we need to compute monotone functions?

Razborov proved that the monotone function matching is not in mP. But can we compute matching using a polynomial size circuit with a few negations? Is there a P/poly circuit with $O(n^\epsilon)$ ...
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### smallest circuit size using XOR gates

Suppose we are given a set of n boolean variables x_1,...,x_n and a set of m functions y_1...y_m where each y_i is the XOR of a (given) subset of these variables. The goal is to compute the minimum ...
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### Complexity of multi-linear polynomial computing Boolean function

Let $f:\{0,1\}^{n}\longmapsto\{0,1\}$ be a Boolean function. As usual, let $C(f)$ denote circuit complexity of $f$, i.e, the size of the smallest Boolean circuit computing $f$. As we know that every ...
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### What is largest class of functions $C$ such that we know $\#P$ in not contained in $C$-generated $TC^0$?

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 ...
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### Analogues of different complexity classes in various models

We suspect following relation: $$TC^0\subsetneq NC^1\subsetneq L\subsetneq NL\subsetneq AC^1\subsetneq NC^2\subsetneq P\subsetneq NP\subsetneq PH\subsetneq PSPACE$$ in Turing/boolean circuit ...
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### What is the power of general poly-size permutation branching programs?

Call $\mathsf{PPBP}$ the class of languages decided by poly-size families of permutation branching programs, which are layered branching programs (i.e., the ones defined here) whose transitions ...
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### Computed circuit complexity of decision problems

Has anyone explored what is the circuit complexity of classic decision problems such as Primes or Graph-Isomorphism for small input size $N$? While most people are interested in the how the scaling ...
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### Symbolic Execution of the Quine-McCluskey Algorithim

If I understand correctly, the Quine–McCluskey algorithm will find the minimum boolean formula size for given boolean function. Has there been any attempts to (for lack of a better term) symbolically ...
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### How expensive may it be to destroy all long s-t paths in a DAG?

We consider DAGs (directed acyclic graphs) with one source node $s$ and one target node $t$; parallel edges joining the same pair of vertices are allowed. A $k$-cut is a set of edges whose removal ...
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### What is an equivalent definition of mP/poly in terms of a Turing machine?

P/poly is the class of decision problems solvable by a family of polynomial-size Boolean circuits. It can alternatively be defined as a polynomial-time Turing machine that receives an advice string ...
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### Are monotone Boolean functions in P well-approximated by monotone polynomial-size circuits?

Question 1: Is it true that for every polynomial $p(n)$ and $\epsilon >0$ there is a polynomial $q(n)$ such that every monotone Boolean function on $n$ variables that can be expressed by a Boolean ...
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### A trivial question on hierarchy [closed]

According to wiki we know that $\mathsf{ACC^0\subseteq TC^0\subseteq NC^1\subseteq L\subseteq P\subseteq NP\subsetneq NEXPTIME}.$ Class $\mathsf{ACC^0}$ is included in $\mathsf{TC^0}$ is in ...
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### What would signify hierarchy collapse to first level?

We know that $$\mathsf{NP\subseteq P/Poly \iff coNP\subseteq P/Poly\implies PH=\Sigma_2^P}=\mathsf{\Pi_2^P}$$ $$\mathsf{NP\subseteq P/Log\iff coNP\subseteq P/Log\implies PH=\Sigma_0^P=\Pi_0^P}$$ ...
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### Simple candidates for pseudorandom permutations?

Even though it is not known whether one-way functions exist, there are several candidate functions used in practice for cryptographic applications that are efficiently computable but are conjectured ...
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### Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
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### Is there a counting complexity class for succint problems?

Encoding NP-complete problems succintly often makes them NEXP-complete. I am wondering if counting the number of solutions to such a problem with a succint encoding would be any harder than solving ...
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### Scope of natural proofs barrier

The natural proofs barrier of Razborov and Rudich states that under credible cryptographic assumptions one cannot hope to separate NP from P/poly by finding combinatorial properties of functions that ...
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### Computing $\operatorname{MAJ}_n$ by $\operatorname{MAJ}_m$ in depth 2

Can the majority of $n$ bits be computed by a depth 2 formula all of whose gates compute the majority of $m$ bits where $m=O(n^c)$ for a constant $c<1$? Such a formula contains $m+1$ gates and ...
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### Idea for a white-box PIT deterministic algorithm in polynomial time

Just a warning, I am an amateur and this algorithm probably doesn't work. A high level description of the algorithm: Be $f(x_1,...,x_n)$ a polynomial, $n$ the number of variables and $d$ is the ...
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### Complexity of reversible Circuit Value

I am wondering what is known about the complexity of the reversible Circuit Value Problem (rCVP) and the corresponding reversible Satisfiability problem (rSAT). More precisely: a circuit ...
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### size bounds for circuits recognizing graph properties (reachability, cyclicity, …)

I am interested in the following. Let the inputs of the circuit correspond to the arcs of a directed graph. The circuit has to output 1 iff there is a directed path from a given node S to a given node ...
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### For what c is division by c in AC0?

Suppose that our input is a binary $x$ and we have to output $\lfloor x/c \rfloor$, where $c$ is some constant integer. This is just a shift if $c$ is a power of two, but what about other numbers? Can ...
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### Why is HAMILTONIAN CYCLE so different from PERMANENT?

A polynomial $f(x_1,\ldots,x_n)$ is a monotone projection of a polynomial $g(y_1,\ldots,y_m)$ if $m$ = poly$(n)$, and there is an assignment $\pi:\{y_1,\ldots,y_m\}\to\{x_1,\ldots,x_n, 0,1\}$ such ...
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### What is the strongest known lower bound against SIZE(n)?

What is the best known lower bound against (nonuniform) circuits of size $O(n)$? I understand that we don't know of any explicit functions that need circuits of size more than something like $5n$. But ...
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### Arithmetic circuits with just one threshold gate

When restricted to $0$-$1$ inputs, every $\{+,\times\}$-circuit $F(x_1,\ldots,x_n)$ computes some function $F:\{0,1\}^n\to \mathbb{N}$. To obtain a boolean function, we can just add one fanin-1 ...
Green [1] showed that $PP^{PH}$ is properly contained in $PSPACE$ relative to some oracle. Around the same time, in the famous "voting polynomials" paper [2], it was shown that $PP$ is properly ...
Every monotone arithmetic circuit, i.e. a $\{+,\times\}$-circuit, computes some multivariate polynomial $F(x_1,\ldots,x_n)$ with nonnegative integer coefficients. Given a polynomial ...