Questions tagged [circuit-complexity]
Circuit complexity is the study of resource-bounded circuits and the functions computed by such circuits.
349
questions
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Can exponential-size depth-2 $CC^0[m]$ circuits with generalized $MOD_m$ gates compute arbitrary functions from $Z/mZ$ to $Z/2Z$?
Terminology
$CC^0[m]$ is the set of polynomial-sized, constant depth circuits consisting entirely of $MOD_m$ gates for some $m \geq 2$, where a $MOD_m$ gate outputs a 1 if and only if the sum of its ...
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1answer
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Is finding the shortest consistent term to fill a missing line in a truth table still NP-hard?
I understand the logic minimization problem is NP-hard when given the onset, since the last step is equivalent to set cover optimization.
If instead we are given a partial truth table, and we just ...
0
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0answers
71 views
Useful primitives CPUs could provide, from TC0 (or NC1)
I just listened to XYZ talking about "How Universal Is the Idea of Numbers?", and bashing the concept as an accidental historical artifact. He suggested that totally different computational ...
0
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0answers
95 views
Questions about the equivalence between PH and depth-d circuits with respect to an oracle
Consider an oracle $A$ and the language
\begin{equation}
P(A) = \{x\in \{0, 1\}^{n}: \text{the number of strings in }~A~\text{of length}~|x|~\text{is odd}\}.
\end{equation}
I am trying to make sense ...
3
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0answers
103 views
Complete problems for $(NP\cap CoNP)/poly$ class and universal representations
It is conjectured that $NP\cap CoNP$ does not have a complete problem with respect to the polynomial-time many-to-one reductions. I would like to know the current knowledge about the nonuniform ...
4
votes
0answers
189 views
What does width $4$ permutation branching program correspond to?
$L$ can be computed by a family of programs over $S_3$
of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of
polynomial size.
$L$ can be computed by a family ...
3
votes
0answers
82 views
Monotone circuit representations of paths in a graph?
Consider a directed graph $G = (V, E)$ with a source $s \in V$ and sink $t \in V$. From $G$, I can define a monotone Boolean function $\phi_G$ on the set of variables $E$, in the following way: every ...
3
votes
1answer
149 views
$ACC^0$ implementation of a boolean function
Consider the symmetric boolean function
$$F(x_1,\dots,x_n)=1\iff\sum_{i=1}^nx_i\mbox{ is a square}.$$
It is implementable in $TC^0$.
Is there an $ACC^0$ implementation?
The reason I ask is there seems ...
4
votes
0answers
103 views
Trading treewidth for depth in Boolean circuits
We know that languages defined by (poly-sized) Boolean formulae equals $\mathbf{NC}^1$: that Boolean formulae can be simulated in $\mathbf{NC}^1$ was shown by Brent/Spira [B,S], and the converse is ...
0
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73 views
What is the best reduction we know from flavors of $SAT$ to $MCSP$?
Consider $\mathcal P(n,m)$ to be a class from the set $\{k\mbox{-}\mathsf{SAT}(n,m),\mathsf{CIRCUITSAT}(n,m)\}$ where $k$ is fixed, $n$ is number of variables and $m$ is number of clauses.
Denote $\...
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0answers
52 views
Is anything known about NC$^1$ with NP oracle
A few things are known about the class $\textsf{L}$ provided with an $\textsf{NP}$ oracle ($\textsf{L}^\textsf{NP} = \Theta_2^\textsf{P}$ has attracted a bit of attention, for instance [1]) On the ...
1
vote
1answer
63 views
Variable wire weights in DLOGTIME-uniform circuits
The definition of a $DLOGTIME$-uniform circuit family is based on a Turing machine that accepts the language $\langle t, a, b \rangle$, where gate $a$ is of type $t$ and has gate $b$ as a child, ...
7
votes
2answers
636 views
Why should we believe that $NEXP \not \subset P/poly$
I am sorry if this is not an advanced question. Most computer scientists believed that $NEXP \not \subset P/poly$ but they are not even close to this assumption. The main evidence that they are used ...
12
votes
2answers
308 views
Circuit and Formula Lower Bounds for Separating Sparse Sets of Strings
We say that a pair $(P,N)$ of subsets of strings from $\{0,1\}^n$ is an $n$-pair if $|P|=|N|=n$. Intuitively, sucha a pair consists of a set $P$ with $n$ positive $n$-bit strings, and a set $N$ with $...
1
vote
1answer
85 views
Composition theorem for randomized communication complexity
I am currently organizing the literature of composition theorem, and I found the paper by https://www.research.cs.rutgers.edu/~troyjlee/Composition.pdf, in their theorem 5, I find
$$ R_{1/4} (f \circ ...
9
votes
1answer
114 views
Proof of $DLOGTIME-CC^0 = MOD[<,bit]$
Let $CC^0[m]$ be the class of constant-depth, polynomial-sized circuits consisting entirely of $MOD_m$ gates, which put out $1$ iff the sum of their inputs $\equiv 0~(\textrm{mod}~m)$. In the same way ...
4
votes
0answers
138 views
On solving Planar Circuit SAT
This enquiry is three-sided.
Side 1 - State of the art
Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$?
Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$ assuming ...
1
vote
0answers
66 views
Multiplicative-depth 1 composition of arithmetic circuits
I am trying to find information about the following problem. Let $C_1$ and $C_2$ be any two poly-size arithmetic circuits on input vectors x₁ and x₂ correspondingly. Assume that a third arithmetic ...
3
votes
1answer
154 views
What is FO(REGULAR)? (The descriptive complexity equivalent of NC1)
According to Immerman's Descriptive Complexity diagram, there is a logic called $\mathsf{FO(REGULAR)}$ which captures $\mathsf{NC}^1$. However, I can't find the reference where this logic is defined. ...
3
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0answers
101 views
Improving boolean circuits w.r.t. a probability distribution
This is a reference request. Consider the following problem on boolean circuits [ 1 ]:
Given: Boolean circuit $B$ and probability distribution $\mathbb{P}$ on inputs to $B$.
Task: Find one or more ...
14
votes
1answer
360 views
Universal Boolean Formulas
Fix $n\in\mathbb{N}$. Consider a rooted binary tree $T$ in which every non-leaf node contains either AND-gate or OR-gate. Let me say that $T$ is an universal formula if for every $f\colon\{0,1\}^n\to\{...
5
votes
2answers
297 views
Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?
Background
It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$.
Although some natural problems are known to exist, many of them ...
2
votes
0answers
128 views
Matrix multiplication when one matrix is fixed
Let $A$ be a fixed positive entried integer matrix of size $a\times n$ with $\ell$ bits per entry
One is allowed to pre-process this matrix as appropriate.
Given another positive integer entried $B$...
4
votes
0answers
106 views
Is the Complexity Zoo Inclusion Diagram exclusively about classes of decision problems?
The Complexity Zoo includes the class QNC$^0$, which does not seem to be a class of decision problems. When I chase the references of the link provided, they say
“To extend this definition from ...
5
votes
1answer
252 views
What are the consequences of a faster algorithm for $CIRCUIT$-$SAT$?
What is the best algorithm known for $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates?
What is the consequence if there is an $\alpha\in(0,1)$ such that $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates ...
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1answer
178 views
Evidence integer multiplication is in linear time?
After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-...
15
votes
0answers
246 views
Does small circuits for a NP-complete problem contradict ETH?
The remarks of the Theorem 4 in the paper "On the complexity of circuit satisfiability" claims that: if circuit satisfiability (CktSat) problem can be decided by deterministic circuits of $2^{o(n)}$ ...
5
votes
0answers
292 views
Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?
Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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0answers
95 views
Switching lemma for polynomials over $\mathbb{F}_2$
Suppose $f$ is in $\mathbb{F}_2[x_1,...,x_n]$ with total degree $d$.
Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $f$ we can reduce the total ...
2
votes
0answers
112 views
What is the communication complexity of approximating addition?
In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. Specifically, the class of problems I am interested in is parametrized by four ...
2
votes
1answer
109 views
Upper bounds on the circut depth
Suppose $f:\{0,1\}^n \to \{0,1\}$ is a function such that it can be computed by a circuit of size $n^c$ for some constant $c>0$.
Q. Is there any nontrivial upper bound on the depth of a circuit ...
7
votes
2answers
350 views
Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power
It seems to be a folklore that we can "handle" $\textrm{ACC}[q]$ circuits not only for prime $q$ but prime power $q$.
For example, authors of this paper say that
... any constant
depth circuit ...
3
votes
2answers
304 views
Is the unbounded fan-in model realistic?
Does the unbounded fan-in circuit model apply in "practical" settings? In other words, are there real-world realisable computers with unbounded fan-in gates?
As I understand, standard silicon ASICs ...
8
votes
1answer
330 views
Is there any quantum analog of the VP vs. VNP problem?
From Wikipedia:
$\mathsf{VP}$: The class VP is the algebraic analog of P; it is the class of polynomials $f$ of polynomial degree that have polynomial size circuits over a fixed field $K$.
$\mathsf{...
5
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0answers
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Is there a universal gate set for classical probabilistic computing?
We know that NAND gates are universal for deterministic classical circuits, Toffoli gates are universal for reversible deterministic classical circuits, and Clifford+T is universal for quantum ...
1
vote
2answers
126 views
When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]
In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
9
votes
1answer
298 views
Depth reduction for Boolean circuits
This result by Tavenas, Koiran and others show that any polynomial computed by a circuit of size $s$ is computed by a depth-4 homogenous circuit of size $s^{\sqrt{d}}$.
Are there any similar results ...
7
votes
1answer
333 views
Can reciprocal inputs speed up monotone computations?
A $(+,\times,1/x_i)$ circuit is a standard monotone arithmetic $(+,\times)$ circuit with the only difference that now besides the input variables $x_1,\ldots,x_n$, also their reciprocals $1/x_1,\...
2
votes
0answers
65 views
How to improve this pseudorandom generator?
Let $f$ be a Boolean function and $\varepsilon > 0$.
There exists a pseudorandom generator $G_f: \{0,1 \}^{n^{\varepsilon}} \to \{0,1 \}^n$ with the following property.
Let $T$ be a set and $p(n)$...
11
votes
1answer
289 views
Reversible polynomial circuit iff polynomial reversible circuit?
My question is about efficiently computable bijective functions. Informally I'm interested in:
If a bijection is computable in polynomial time, can we compute it by a polynomial number of bijective ...
8
votes
0answers
100 views
Does ${\bf CFLPAD}={\bf PPAD}$?
What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a (non-)deterministic finite/push-down automaton encodes the problem?
I asked a similar ...
10
votes
1answer
223 views
Linear circuit complexity classes
The class $\textrm{NC}^i$ is the class functions computable by circuits families of bounded fan-in, $n^{O(1)}$ size and $O(\log^i(n))$ depth.
The $\textrm{NC}$-hierarchy is the union of those classes....
5
votes
1answer
233 views
Boolean circuits which correspond to L/poly
Branching programs are usually used as a computation model for non-uniform logarithmic space $\mathsf{L}/\mathrm{poly}$.
Is there a reference about Boolean circuits corresponding to $\mathsf{L}/\...
7
votes
0answers
265 views
How many different proofs are there of parity is not in AC0?
The theorem that Parity is not in $\mathsf{AC}^0$ is one of the gemstones of complexity theory. I wonder how many different proofs there are of this result? What constitutes "different" is also a part ...
2
votes
1answer
128 views
Circuit complexity of group actions
Suppose that $G$ is a group with $|G|=n$. Suppose that $G$ is generated by elements $g_{1},\dots,g_{k}$. Let $\iota:G\rightarrow S_{2^{N}}$ be an injective group homomorphism such that $\iota(g_{i}):\{...
3
votes
0answers
86 views
How many arithmetic and max operations does it take to compute Dynnikov's action of the braid groups on $\mathbb{Z}^{2n}$?
A function $f:X^{2}\rightarrow X^{2}$ is said to satisfy the Yang-Baxter equation if
$$(f\times\textrm{Id}_{X})\circ(\textrm{Id}_{X}\times f)\circ(f\times\textrm{Id}_{X})=(\textrm{Id}_{X}\times f)\...
11
votes
1answer
330 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
131 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\}...
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vote
0answers
227 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
108 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_{...
