Questions tagged [arithmetic-circuits]

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36
votes
5answers
2k views

Integer multiplication when one integer is fixed

$n$ is a parameter in the problem. For every $n$ we pick a random integer $a_n\in\{2^{n-1},2^{n-1}+1,\dots,2^n-1\}$ where $n\in\{1,2,\dots\}$. Problem: Given $n$ what is the complexity of ...
23
votes
1answer
939 views

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 ...
22
votes
4answers
1k views

Monotone arithmetic circuits

The state of our knowledge about general arithmetic circuits seems to be similar to the state of our knowledge about Boolean circuits, i.e. we don't have good lower-bounds. On the other hand we have ...
22
votes
2answers
1k views

Lower bound for determinant and permanent

In light of the recent chasm at depth-3 result (which among other things yields a $2^{\sqrt{n}\log{n}}$ depth-3 arithmetic circuit for the $n \times n $ determinant over $\mathbb{C}$), I have the ...
21
votes
1answer
565 views

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 ...
15
votes
1answer
296 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 ...
15
votes
1answer
529 views

Any polynomial which is hard to count but easy to decide?

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 $f(x_1,\ldots,x_n)$...
14
votes
1answer
298 views

Machine characterization of $SAC^i$

$SAC^i$ is the class of decision problems solvable by a family of $O({\log}^i{n})$ depth circuits with unbounded-fanin OR and bounded-fanin AND gates. Negations are only allowed at the input level. It ...
14
votes
1answer
686 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 ...
14
votes
1answer
2k views

A course for learning algebraic complexity

I want to learn about algebraic algorithms and complexity thoery. In particular, I am interested in PIT. Is there a set of lecture notes, books, papers and surveys for students who have read standard ...
14
votes
1answer
1k views

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 ...
13
votes
1answer
373 views

Adleman's theorem over infinite semirings?

Adleman has shown in 1978 that $\mathrm{BPP}\subseteq \mathrm{P/poly}$: if a boolean function $f$ of $n$ variables can be computed by a probabilistic boolean circuit of size $M$, then $f$ can be also ...
12
votes
3answers
638 views

Arithmetic circuits with $\min$, $\max$, and average over $[0,1]$

Consider a circuit that takes as inputs numbers in $[0,1]$, and has gates that consist of the functions $\max(x, y)$, $\min(x, y)$, $1 - x$, and $\frac{x+y}{2}$. The output of the circuit is then also ...
12
votes
1answer
531 views

Are arithmetic circuits weaker than boolean?

Let $A(f)$ denote the minimum size of a (non-monotone) arithmetic $(+,\times,-)$ circuit computing a given multilinear polynomial $$ f(x_1,\ldots,x_n)=\sum_{e\in E}c_e\prod_{i=1}^n x_i^{e_i}\,, $$ ...
11
votes
2answers
279 views

Straight line complexity of monomials

Let $k$ be some field. As usual, for an $f\in k[x_{1},x_{2},\ldots,x_{n}]$ we define $L(f)$ to be the straight-line complexity of $f$ over $k$. Let $F$ be the set of monomials of $f$, namely the ...
11
votes
2answers
1k views

Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size

I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication. It is known that the determinant of an $n\times n$ matrix can ...
11
votes
0answers
298 views

Generalizations of the determinant/permanent problem?

A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with ...
10
votes
2answers
831 views

Implications of Riemann Hypothesis variants in TCS

The over ~1½ century old Riemann Hypothesis has deep implications in mathematics and a large edifice of math theory is now proved conditionally on it and numerous variants. I recently came across a ...
10
votes
2answers
270 views

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 ...
10
votes
1answer
435 views

Why lower bounds for boolean Circuits does not imply arithmetic circuits lower bounds

My question is why lower bounds for depth 3 Boolean circuits with gates "and" and "xor" for determinant does not imply the same lower bounds for arithmetic circuits over $\mathbb{Z}$? What is wrong ...
10
votes
0answers
154 views

Checking a long product of matrices

Given a set of n-by-n integer matrices $\{A_1, \dots, A_m\}$, for a word $w=w_1 w_2\cdots w_l$ over $\{1, \dots, m\}$, we define $A_w:=A_{w_1}\cdots A_{w_l}$. The question is to decide, given $\{A_1,...
9
votes
2answers
721 views

Cancellation and determinant

Berkowitz algorithm provides a polynomial size circuit with logarithmic depth for determinant of a square matrix using matrix powers. The algorithm implicitly uses cancellation. Is cancellation ...
9
votes
1answer
295 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 ...
9
votes
1answer
232 views

Randomized identity-testing for high degree polynomials?

Let $f$ be an $n$-variate polynomial given as an arithmetic circuit of size poly$(n)$, and let $p = 2^{\Omega(n)}$ be a prime. Can you test if $f$ is identically zero over $\mathbb{Z}_p$, with time ...
9
votes
1answer
217 views

Checking if a polynomial factors into linear factors

Let $f\in\mathbb{Q}[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial given by an arithmetic circuit $C$ of size $s$. Given $C$ as the input, is there a deterministic algorithm to check whether all the ...
8
votes
1answer
185 views

Is there an alternate proof or an exposition of: Exponential lower bound for $\Sigma\Pi\Sigma$ circuits [Grigoriev-Karpinski(1998)]?

Is there an alternate proof or an exposition of the result of Grigoriev and Karpinski (STOC 1998, doi:10.1145/276698.276872) on the exponential lower bounds for Depth 3 arithmetic circuits computing $\...
8
votes
0answers
875 views

Complexity of Polynomial Division

Let $P(x)$ be a univariate polynomial with integer coefficients where both coefficients and degrees are in binary and let $q(x)$ be another polynomial also with integer coefficients where the degrees ...
8
votes
0answers
290 views

Grigoriev-Karpinski for the permanent

Grigoriev and Karpinski (ps.Z) showed that any depth-3 circuit over a fixed finite field computing $\mathrm{Det}_n$ requires $2^{\Omega(n)}$ size. I had the misconception(?) until recently that the ...
7
votes
4answers
352 views

Reference request: Arithmetic circuit complexity

I am completely new to this field. I want to read the prelims and unfortunately, I don't see any book written for a beginner in this area. Can anyone give me some basic starting points/references(text/...
7
votes
2answers
138 views

White-box sparse interpolation

Let $C$ be an arithmetic circuit that represents a polynomial $f\in\mathbb K[x_1,\dotsc,x_n]$, with the promise that $f$ has at most $k$ nonzero terms. What is (known about) the complexity of ...
7
votes
1answer
377 views

Uniformity vs. nonuniformity in algebraic complexity theory

I understand that the study of Boolean circuits and nonuniform complexity classes was introduced to (hopefully) prove separation of uniform complexity classes. In this sense, nonuniform computational ...
7
votes
1answer
142 views

Lower bounds for noncommutative arithmetic circuits with exact division?

It is known that for general arithmetic circuits there is not much of a difference between standard model and one with division: any circuit with divisions that computes a polynomial can be simulated ...
7
votes
1answer
330 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,\...
7
votes
2answers
261 views

How efficiently can circuits over sets of naturals be transformed to boolean circuits?

I am interested in reducing a circuit over sets of naturals (see here for some basic notions about this type of circuits) to a boolean circuit computing the same output. A very basic circuit of this ...
7
votes
0answers
141 views

Computing the Fourier-Walsh coefficients of an arithmetic circuit

Given an $f$-fan in arithmetic circuit of depth $d$ over $GF(q)$ (for some prime $q$) and the set of variables $(x_1,\ldots,x_n)$, what is the state of the art upper bound for calculating the Fourier-...
6
votes
1answer
368 views

Degree restriction for polynomials in $\mathsf{VP}$

why do we need the restriction on the degree for a polynomial to be in VP due to which $(\prod_{i=0}^{n} x_i)^{2^n}$ $\notin$ VP even though it has a poly-size circuit?
6
votes
1answer
206 views

arithmetic circuits for polynomials (updated)

I am concerned about the following question, consider $P_n(x)= \sum_{i=0}^n \frac{x^n}{n!}$ Is there a straight line program (or arithmetic circuits) of polynomial size (wrt $n$) for the polynomial ...
6
votes
1answer
230 views

Has there been a study of circuits operating on arrays?

Much ink has been spilled studying the theory surrounding computation by combinatorial circuits operating on bits or boolean values - with AND, OR and NOT gates (as those are enough to implement any ...
6
votes
1answer
201 views

VNP is closed under taking coefficients using Valiant's criterion

We consider the family of polynomials $$\{f_n(x_1,\dots,x_n,y_1,\dots,y_n)\}\in\mathsf{VNP}$$ We want to show that the family $$\{h_n(x_1,\dots,x_n)\}$$ is in $\mathsf{VNP}$. Where $h_n(x_1,\dots,...
6
votes
0answers
103 views

Practical algorithms for finding small arithmetic circuits

I have a multivariate integer polynomial $f : \mathbb{Z}^n \to \mathbb{Z}$ given as either as a circuit or as a list of monomials. I am interested in practical (though obviously exponential time) ...
5
votes
1answer
188 views

Can a polynomial sized arithmetic ciruits perform integer division?

Can we perform integer division with a polynomial size arithmetic circuit over $\mathbb{Q}$ that takes as input the numerator and denominator?
5
votes
1answer
92 views

How quickly can we decompose a number into a set of residues?

Basically, if we are given a natural $x$, in binary, with $x < m_0 \cdot m_1 \cdot m_2 \cdot \dots \cdot m_n$, how quickly can we find $x \bmod m_0$, $x \bmod m_1$,..., and $x \bmod m_n$? In other ...
5
votes
1answer
604 views

Lower bounds for Polynomials computing the boolean functions

Expressing a boolean function $f$ $:\{ 0,1 \}^{n} \rightarrow \{0,1 \}$ using a polynomial $P(x_{1},...,x_{n})$, where $x_{1},...,x_{n}$ may be integer, finite fields, or other fields. One of the most ...
5
votes
1answer
232 views

Evaluation of an arithmetic formula where the time depends on the length of the arguments of gates

Let $(X,+,\cdot)$ be a commutative ring. Let $|\cdot|\colon X\to \mathbb{N}$ be a function that satisfies $|x+y|\leq |x|+|y|$ and $|xy|\leq |x|+|y|$. We call the function length, and length is always ...
5
votes
0answers
386 views

About the ``recent" paper by Razborov in the Annals of Mathematics

Recently this paper on complexity theory was published at the Annals of Mathematics by Razborov, http://annals.math.princeton.edu/2015/181-2/p01. Curiously this seems to have been submitted to the ...
4
votes
1answer
405 views

Analogies between VNP and NP

Valiant introduced the class VNP with respect to "arithmetic circuits" over 35 years ago in a "rough" analogy to NP. Recently, there have been major advances in the area of arithmetic circuits eg as ...
4
votes
1answer
185 views

Computing every boolean function with a polynomial over $\mathbb{F}_3$?

The following paper briefly mentions the power of $MOD_6$ gates (page 3), and relies on the unstated fact that every boolean function can be computed with an arithmetic circuit of depth 2 over $\...
4
votes
0answers
103 views

Complexity of equivalence testing of arithmetic circuits

I have a very specific problem that appears to be close to equivalence testing for arithmetic circuits (checking whether the computed functions are the same). Since I'm completely new to the field, ...
4
votes
0answers
102 views

Size depth tradeoffs for monotone arithmetic circuits

Are there any size depth trade-offs known for monotone arithmetic circuits that compute permanent and determinant?
3
votes
1answer
166 views

Arithmetic circuits over $GF(p)$ for computing $x \bmod 2$ given input $x \in GF(p)$

Let $p$ be a prime. Suppose we consider the function $f: GF\left(p \right) \rightarrow \left\{0, 1 \right\}$ where $f(x) = x \bmod 2$ for all. The question is the following: are there any known ...