Circuit complexity is the study of resource-bounded circuits and the functions computed by such circuits.
6
votes
0answers
73 views
How small can be a layered boolean circuit for a function with circuit complexity $s$?
Consider a function $f$ computed by a boolean circuit $C$ with $n$ inputs of size $s(n) = \mathsf{poly}(n)$ over the basis $\{\mathsf{XOR},\mathsf{AND},\mathsf{NOT}\}$ (with indegree 2 for the $\...
4
votes
0answers
68 views
Asymptotic complexity of mass production
For a function $f:\{0,1\}^n \rightarrow \{0,1\}^m$, let $C(f)$ be the circuit complexity (for concreteness, constants and NOT gates are free, while 2-input AND gates cost 1).
Let $k{\times}f : \{0,1\}...
0
votes
0answers
47 views
Modular and multiplication operations
Does
modular operation reduce to integer multiplication
integer multiplication reduce to modular operation
in fully linear time and/or in logarithmic time on linear number of processors?
1
vote
0answers
207 views
Impagliazzo lemma, unclear detail in its proof
In Arora-Barak's book on page 378 in the proof of Impagliazzo's Hard Core lemma why did they choose the number 50 in this line: Set $t = \frac{50n}{\epsilon^2}$ ? How this choice then yields the size ...
9
votes
0answers
95 views
Expected value of the evaluation of Boolean circuits of depth $2n$
I am not an expert on circuits and I wonder whether the following problem was already studied (and possibly solved). Any reference or suitable method to solve this question would be welcome.
Let $C_{...
8
votes
1answer
246 views
Nondeterminism is on average useless for circuits?
Savický and Woods (The Number of Boolean Functions Computed by Formulas of a Given Size) prove the following result.
Theorem[SW98]: For every constant $k>1$, almost all boolean functions with ...
2
votes
0answers
39 views
Arithmetic circuits with restrictions on occurrence of pairs of variables
I am curious if the following model was studied or has some obvious lower bounds:
We want to compute a polynomial $P(x_1,x_2, \dots , x_n)$. Suppose we have a graph G on $n$ nodes that we are going ...
6
votes
0answers
116 views
Number of circuits computing a given function
Let's say we have some function that can be computed by a minimal circuit of size $m$ (using some metric, say, the number of gates). Other than this minimal circuit, there will be many other circuits ...
4
votes
0answers
72 views
Circuits computing functions of inputs smaller than $n$
The usual circuit complexity concerns circuits where circuit $C_n$ computes function $f_n$. I am interested in circuits such that $C_n$ can compute $f_i$ for all $i \leq n$. I am assuming that the ...
1
vote
0answers
96 views
Is there any NC-complete problem with respect to logspace reduction?
The question is on the title.
We all know that $\text{NL}$ and $\text{P}$ have such problems. So I wonder the same thing about $\text{NC}$. More interestingly, is there any $k \ge 2$ and any $\text{...
4
votes
0answers
58 views
Can an efficiently computable non-one way permutation be written as the composition of polynomially many easy to compute involutions?
Suppose that $p$ is a polynomial. Then does there exist a polynomial $q$ where if $f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ is a bijection where both $f$ and $f^{-1}$ are computable by circuits with at ...
7
votes
0answers
158 views
Can every efficiently computable permutation be written as the composition of two efficiently computable involutions?
It is well-known that every permutation can be written as the composition of two involutions.
Suppose that $p$ is a polynomial. Then does there exist a polynomial $q$ such that if $f:\{0,1\}^{n}\...
10
votes
3answers
567 views
What are examples of how non-uniformity can be useful?
I'm curious about ways in which you have seen non-uniformity be useful in computation. One way is randomness, as in $BPP \subseteq P/poly$, and another is look-up tables which are used to show that ...
0
votes
0answers
71 views
Why is the circuit class ACC0 important as a mathematical structure? [duplicate]
I agree that ACC0 is a good toy object to study before trying to understand more complicated circuit classes. But here I'm more looking for a mathematical reason, rather than a practical one.
What ...
3
votes
3answers
440 views
Why is the circuit class AC0 unavoidable?
Take AC0.
What is a natural thought process that leads to the definition of AC0?
Does this class arise intrinsically anywhere?
My problem is that in the case of unbounded fan-in, AND and OR gates ...
0
votes
0answers
47 views
Can you find a function which is far easier than it is approximately symmetric?
Define $\mathcal{F}(G) = \{ f : \{0, 1\}^n \rightarrow \{0, 1\} \mid G \leq Aut(f) \}$. Note that all functions can be computed on $n$ bits with circuits of size $2^n/n$. On the other hand, we don't ...
3
votes
1answer
137 views
Proof that all Boolean functions can be computed by $(MOD_2-MOD_3)$ circuit
I was reading "Some properties of MOD m circuits computing simple functions" (Amano & Maruoka, 2003) where the authors prove that every Boolean function can be computed by depth $2$ by $(MOD_2-...
4
votes
0answers
113 views
Is $L\subset NC^1$
Arora and Barak's online book claims in exercise 6.11 that $NC^1=L$. While the $NC^1\subset L$ direction is relatively straightforward and explained in many other texts, I couldn't prove or find the $...
6
votes
0answers
127 views
Has what I am calling “helpfulness” here been studied?
We say that a Boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$ is helpful for another Boolean function $g$ if $f(x)$ can be computed with a smaller circuit given $g(x)$ as an extra input bit. I'...
1
vote
2answers
207 views
Efficiently computable by a “simple” algorithm?
I am interested in the relation between "program complexity" and "computational complexity".
In particular, I was wondering
What is known about the minimal length a program must have to solve a ...
4
votes
1answer
116 views
Which $SIZE$-$DEPTH(s, d)$ classes with $log(s(n))^{d(n) - 1} = o(n)$ can we not separate by known methods?
Define $SIZE$-$DEPTH(s, d)$ to be the functions which are computed by circuit families of size $O(s(n))$ and less than depth $d(n)$.
We know from Boppana's 1997 paper on the average sensitivity of ...
2
votes
0answers
48 views
Which invertible linear transformations can be computed reversibly without ancilla/garbage bits just as easily as they can be computed irreversibly?
Suppose that $L:F_{2}^{n}\rightarrow F_{2}^{n}$ is an invertible linear transformation. Then define $w(L)$ to be the gate count of the smallest reversible circuit on $n$ bits without ancilla/garbage ...
2
votes
0answers
72 views
Scaled down and scaled up versions of Impagliazzo-Wigderson Therem
A famous theorem due to Impagliazzo and Wigderson states that if some function in $E=DTIME[2^{O(n)}]$ requires circuits of size $2^{\Omega(n)}$ then P=BPP.
When can we change $P$ with some ...
1
vote
0answers
102 views
Solving 0/1 integer programming and solving ACC-of-SYM circuits
I am referring to the proof of Theorem 1.4 in this STOC 2014 paper, https://arxiv.org/abs/1401.2444. In particular my question is about the argument that begins in the 8th line of page 9 where the ...
1
vote
0answers
88 views
Possibly small circuit complexity class containing REG?
What is the smallest well-known Boolean-circuit complexity class containing all the regular languages over the binary alphabet {0,1}?
If we believe Theorem 2 in
Koucký, Circuit Complexity of ...
-5
votes
1answer
117 views
Should GCT focus on $PSPACE\not\subseteq P/poly$?
GCT tries to show $P$ is not $NP$ by showing $NP$ is not in $P/poly$.
Could it be useful in showing $\Sigma_{i+1}\not\subseteq P^{\Sigma_i}/Poly$ at every $i>0$?
Suppose if it turns out that $\...
14
votes
1answer
172 views
What are the consequences of $P \subseteq L/poly$?
A language is in $L/poly$ if there exists a logspace Turing machine that decides the language with polynomial amount of advice.
See here for more info: https://en.wikipedia.org/wiki/L/poly
...
2
votes
1answer
94 views
Complexity of maximising weighted sum of and functions on a set of binary variables
Suppose we have a set of binary variables $a_1, ..., a_n$ that $a_i\in\{0,1\}$. Now we define $m$ and functions over a subset of them: $$j\in\{1,...,m\}: f_j=x_1\land x_2\land...\land x_k$$ in which
...
4
votes
0answers
165 views
Is $NEXP^{NP}$ known to not be contained in $NP/poly$?
To the best of my knowledge, it is known that $NEXP^{NP} \nsubseteq P/poly$, but it's still not known if $NEXP \nsubseteq P/poly$.
For more info, see "Superpolynomial circuits, almost sparse oracles ...
14
votes
2answers
422 views
OR-circuit complexity of a dense linear operator
Consider the following simple monotone circuit model: each gate is just a binary OR. What is the complexity of a function $f(x)=Ax$ where $A$ is a Boolean $n \times n$ matrix with $O(n)$ 0's? Can it ...
0
votes
0answers
146 views
On $i.o.P/poly$?
Is $NEXP^{NP}\not\subseteq i.o.P/poly$?
Is there any consequence if $NP$ or $PP$ is in $i.o.P/poly$?
Showing $NEXP^{NP}\not\subseteq P/poly$ needs Karp-Lipton.
What is the best $i.o.P/poly$ ...
5
votes
0answers
93 views
Non-Uniform Classes of Languages not Closed Under Complement
Most "normal" non-uniform circuit classes are closed under complement. Just add a negation gate to the output of a circuit, and if necessary, apply De-Morgan's law.
Now there are some natural non-...
1
vote
1answer
123 views
Is $CAPP \in P$ known to collapse some quantum complexity classes to classical ones?
Lets define the class
$ZBQP = \{ L \mid \exists \textit{P-uniform circuit family } \{C_i\}, \forall n \in \mathbb{N}, |x| = n, |\langle 0|C_n|x \rangle - I(x \in L)| \leq 9/10 \Longleftrightarrow x \...
0
votes
1answer
184 views
What do stronger circuit lower bounds give in terms of derandomization?
We have $EXP\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\epsilon})$ at every $\epsilon>0$.
This is essentially $DTIME(2^{O(n)})\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\...
27
votes
0answers
605 views
Is BPP= P known for ANY uniform model of computation?
Many believe that BPP $=$ P "should" hold for Turing machines. We even have some "witnesses" for this: otherwise some "strange" things would happen; see e.g. this paper by Implagliazzo and Wigderson.
...
2
votes
1answer
95 views
Non-Uniform Lower Bounds for NSPACE
If I'm not mistaken it is not known whether $E^{NP} \subseteq {\rm SIZE}(n)$
where $E^{NP}$ is the class of problems solvable by a TM which works in time $2^{O(n)}$ and is allowed to make queries of ...
1
vote
0answers
52 views
What is the average sensitivity of a quantum circuit with depth $d$ and size $s$?
We have some quantum circuit $C$ with $k$ ancillae and $n$ input bits of depth $d$ and size $s$, and we can define a function $f$ which, for any $x \in \{0, 1\}^n$, is the random variable which is the ...
0
votes
0answers
19 views
On $PP$, $MP$ and $\oplus P$ classes?
In here $PP^{\oplus P}\subseteq MP$ is shown and it is left as an open problem on whether $MP=PP^{\oplus P}$.
$MP\subseteq P/poly$ collapses the polynomial hierarchy and $PH\subseteq P/poly^{\oplus P}...
12
votes
1answer
461 views
Is BPP vs. P a real problem after we know BPP lies in P/poly?
We know (for now about 40 years, thank Adleman, Bennet and Gill) that the inclusion BPP $\subseteq$ P/poly, and an even stronger BPP/poly $\subseteq$ P/poly hold.
The "/poly" means that we work non-...
-3
votes
1answer
182 views
Some questions about the depth hierarchy for threshold circuits [closed]
(I am hugely editing the question. My initial question was if lowerbounds on threshold circuits say anything about P/NP and it seems that they dont. Irrespective of P/NP its an independently true fact ...
8
votes
0answers
97 views
Reference for a circuit lower bound for slightly superexponential time
It is known that $EXP$ doesn't have circuits of size $n^k$. On the other hand proving $10 n$ lower bound on circuit size for $E$, $NE$ or even $E^{NP}$ is a known open problem.
My question is ...
3
votes
1answer
125 views
Are there any parameterized problems in non-uniform FPT that are suspected (but not proven) to be in uniform-FPT?
Getting Started
Consider a parameterized problem $F$. We use $n$ to denote the input size and $k$ to denote the parameter. Consider the fixed levels of $F$ which we denote by $\{F_k\}_{k\in\mathbb{...
7
votes
0answers
147 views
Complexity Class Equalities on the Edge of Inconsistency
What are some of the most extreme potential equalities between computational complexity classes (especially if there is a barrier to refuting them)? These may give us an opportunity to prove better ...
3
votes
0answers
159 views
Circuit's with gate computing only “simple” functions
I was curious if something like that is known or was studied.
Let's call a function simple if it is computable by $AC_0$ circuit of depth $\leq d$ and size $\leq n^k$ for fixed $k,d$.
Now let's ...
6
votes
0answers
111 views
Are there generalizations for Chow's theorem?
The Chow's theorem as it stands holds only for a single linear threshold gate. That these gates are uniquely determined by their first $n+1$ Fourier coefficients.
Are there other circuits for which ...
17
votes
1answer
529 views
Example demonstrating the power of non-deterministic circuits
A non-deterministic Boolean circuit has, in addition to the ordinary inputs $x = (x_1,\dots,x_n)$, a set of "non-deterministic" inputs $y=(y_1,\dots,y_m)$. A non-deterministic circuit $C$ accepts ...
1
vote
1answer
77 views
Some consequences of the Roychowdhury-Orlitsky-Siu result from 1994
This pertains to the proof of theorem 1.1 in this paper, http://dl.acm.org/citation.cfm?id=2897636
So Roychowdhury-Orlitsky-Siu had shown that the number of depth $2$ linear threshold gate circuits ...
10
votes
1answer
428 views
Deciding the Most Significant Bit of Binary Multiplication
I am interested in determining the complexity of the following decision problem: Given two integers $l_1$ and $l_2$ (each with at most m bits), decide whether the most significant bit of the ...
3
votes
1answer
59 views
parallel algorithms for the determinant of the Hessenberg matrix
I am interested in highly parallel algorithms for computing the determinant of matrices of a special form (over finite fields).
It is known that computing determinant of general matrices over finite ...
10
votes
2answers
400 views
High-level overview of Razborov's method of approximations
What is Razborov's method of approximations? Can someone give a high-level overview and the intuition behind it?
