Questions tagged [circuit-complexity]

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

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7
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1answer
237 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,\...
12
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0answers
184 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 $2^{o(n)}$ ...
16
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1answer
2k views

Boolean Functions Where Sensitivity Equals Block Sensitivity

Some of the work on sensitivity vs. block sensitivity has been aimed at examining functions with as large a gap as possible between $s(f)$ and $bs(f)$ in order to resolve the conjecture that $bs(f)$ ...
5
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182 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 ...
2
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1answer
144 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 ...
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80 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
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0answers
68 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 ...
14
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1answer
544 views

VC dimension of polynomials over tropical semirings?

As in this question, I am interested the $\mathbf{BPP}$ vs. $\mathbf{P}$/$\mathrm{poly}$ problem for tropical $(\max,+)$ and $(\min,+)$ circuits. This question reduces to showing upper bounds for the ...
2
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1answer
84 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 ...
6
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2answers
224 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 ...
4
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3answers
1k views

Are Boolean circuits 'universal'

I have a question, but I don't seem to know enough computer science terminology in order to look up an answer. So I wonder if you guys could help a poor physicist like me. I would like to know if ...
8
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1answer
206 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
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1answer
258 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 $K$. $\...
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54 views

Which research fields deal with this variant definition of Boolean circuit depth?

Disclaimer: I admit that the question is not very clear. I think it cannot be helped because the question is very open-ended. First of all, I present the interested type of circuits. We only consider ...
29
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1answer
2k views

Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
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86 views

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
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2answers
104 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, ...
8
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1answer
219 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 ...
11
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1answer
201 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 $\...
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53 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 $...
9
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225 views

Complexity of a problem over acyclic context-free grammars

Let $G$ be an acyclic, context-free grammar over a fixed alphabet $\Sigma=\{a_1,\dots,a_k\}$ with the restriction (without loss of generality) that $|w|=2$ for each rule $A\to w$ in the grammar. ...
16
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3answers
737 views

Which $AC^0$ problems are not “truly feasible”?

Neil Immerman's famous Picture of The World is the following (click to enlarge):                    &...
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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 ...
13
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2answers
337 views

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

It is known that if $NP\subseteq P/Poly$ then the polynomial hierarchy collapses to $\Sigma_2^{P}$ and $MA = AM$. What are the strongest collapses known to happen if $NEXP\subseteq P/Poly$?
13
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1answer
224 views

What are bounded-treewidth circuits good for?

One can talk of the treewidth of a Boolean circuit, defining it as the treewidth of the "moralized" graph on wires (vertices) obtained as follows: connect wires $a$ and $b$ whenever $b$ is the output ...
10
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1answer
173 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....
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181 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 ...
5
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1answer
188 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}/\...
2
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1answer
112 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
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0answers
79 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)\...
4
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0answers
109 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\}...
17
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1answer
632 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 ...
11
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1answer
319 views

Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
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0answers
220 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 ...
4
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1answer
123 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 ...
1
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0answers
93 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 ...
9
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0answers
100 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
281 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
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0answers
42 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 ...
7
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0answers
140 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
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0answers
76 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
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0answers
157 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
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0answers
62 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 ...
5
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0answers
246 views

What precisely is the extra power afforded by using deeper nets?

For any choice of activation function (fix the choice for all the hidden nodes for both the following DNNs) do we know of functions which some $k$ (hidden layer) DNN can compute but a $(k-1)-$DNN can'...
7
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0answers
180 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}\...
9
votes
3answers
583 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 ...
3
votes
3answers
585 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 ...
3
votes
1answer
166 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
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0answers
133 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 $...
1
vote
2answers
219 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 ...