# Questions tagged [circuit-complexity]

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

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### (0,1)-vector XOR problem

this is a rewrite of another recent question of mine  that wasnt stated well (it had a semi obvious simplification, mea culpa) but I think theres still a nontrivial question at the heart of it. ...
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### Hierarchy theorem for circuit size

I think that a size hierarchy theorem for circuit complexity can be a major breakthrough in the area. Is it an interesting approach to class separation? The motivation for the question is that we ...
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### Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
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### Computational Power of Neural Networks?

Let's say we have a single-layer feed forward neural network with k inputs and one output. It calculates a function from $\lbrace 0,1\rbrace ^{n}\rightarrow\lbrace 0,1\rbrace$, it's fairly easy to ...
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### k-CNF ←→ k-DNF conversion to minimize errors

the following problem/question seems fundamental/hard. it appears in some circuit theory proofs, graph theory, and maybe elsewhere. looking for any nontrivial insight. will add various known/nearby ...
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### Formula size lower bounds for AC0 functions

Question: What is the best known formula size lower bound for an explicit function in AC0? Is there an explicit function with an $\Omega(n^2)$ lower bound? Background: Like most lower bounds, ...
764 views

### Is there a better than linear lower bound for factoring and discrete log?

Are there any references that provide details about circuit lower bounds for specific hard problems arising in cryptography such as integer factoring, prime/composite discrete logarithm problem and ...
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### Parity and $AC^0$

Parity and $AC^0$ are like inseparable twins. Or so it has seemed for the last 30 years. In the light of Ryan's result, there will be renewed interest in the small classes. Furst Saxe Sipser to Yao ...
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### Barriers and Monotone Circuit Complexity

Natural proofs is a barrier towards proving lower bounds on the circuit complexity of boolean functions. They do not directly imply any such barrier in proving lower bounds on the $monotone$ circuit ...
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### What is the big version of NC?

$\mathsf{NC}$ captures the idea of efficiently parallelizable, and one interpretation of it is problems that are solvable in time $O(\log^c n)$ using $O(n^k)$ parallel processors for some constants $c$...
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### 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 ...
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### 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 ...
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### What's the “smallest” complexity class for which a superlinear circuit bound is known?

Apologies for asking a question that must surely be in a lot of standard references. I'm curious about exactly the question in the title, in particular I am thinking of Boolean circuits, no depth ...
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### Circuit lower bounds over arbitrary sets of gates

In the 1980s, Razborov famously showed that there are explicit monotone Boolean functions (such as the CLIQUE function) that require exponentially many AND and OR gates to compute. However, the basis ...
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### Random monotone function

In Razborov-Rudich's Natural Proofs paper, page 6, in the part they discuss that there are "strong lowerbounds proofs against monotone circuit models" and how they fit into the picture, there are the ...
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### References on Circuit Lower Bounds

Preamble Interactive proof systems and Arthur-Merlin protocols were introduced by Goldwasser, Micali and Rackoff and Babai back in 1985. At first, it was thought that the former is more powerful than ...
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### Integer multiplication when one integer is fixed

Let $A$ be a fixed positive integer of size $n$ bits. One is allowed to pre-process this integer as appropriate. Given another positive integer $B$ of size $m$ bits, what is the complexity of ...
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### Deciding if a given $\mathsf{NC}^0$ circuit computes a permutation

What is the complexity of deciding whether an $\mathsf{NC}^0$ circuit with $n$ input bits and $n$ output bits computes a permutation of $\{0,1\}^n$? in the other words, whether every bit strings in ...
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### Status on circuit lower bounds for polylog-bounded depth circuits

Bounded depth circuit complexity is one of the main areas of research within circuit complexity theory. This topic has origins in results like "the parity function is not in $AC^{0}$" and "the mod $p$ ...
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### Stronger notions of uniformizations?

One gap that I was always aware that I don't really understand is between non uniform and uniform computational complexity where the circuit complexity represents the non uniform version and Turing ...
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### Circuit complexity of Majority function

Let $f: \{0,1\}^n \to \{0,1\}$ be the majority function, i.e. $f(x) = 1$ if and only if $\sum_{i = 1}^n x_i > n/2$. I was wondering if there was a simple proof of the following fact (by "simple" I ...
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### Constructivity in Natural Proof and Geometric Complexity

Recently, Ryan Willams proved that Constructivity in Natural Proof is unavoidable to derive a separation of complexity classes : $\mathsf{NEXP}$ and $\mathsf{TC}^{0}$. Constructivity in Natural ...
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### Is $AC^0$ with bounded fanout weaker than $AC^0$?

In the survey "Small Depth Quantum Circuits" by D. Bera, F. Green and S. Homer (p. 36 of ACM SIGACT News, June 2007 vol. 38, no. 2), I read the following sentence: The classical version of $QAC^0$ (...
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### PARITY $\notin$ $AC^0$ with bounded fanout: easy proof?

$AC^0$ is the class of constant-depth polynomial-size circuits with NOT gates and unbounded fan-in AND and OR gates, where inputs and gates also have unbounded fanout. Now consider a new class, call ...
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### Lower bounds for the size of nondeterministic circuits

It is known that the minimum size of $U_2$-circuits computing the parity function exactly equals $3(n-1)$. The lower bound proof is based on the gate elimination method. Recently, I noticed that the ...
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### Do we have any nontrivial uniform circuits?

Given an algorithm running in time $t(n)$, we can convert it into a "trivial" uniform circuit family for the same problem of size at most $\approx t(n)\log t(n)$. On the other hand, it might be that ...
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### Practical consequences of $Parity \notin AC^0$

Background Circuit complexity $AC^0$ is defined as the set of circuit families (i.e. sequences of circuits, one for each input size) of bounded depth and polynomial size built using unbounded fan-in ...
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### 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 ...
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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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### Can addition be carried out in less than depth 5?

Using carry look ahead algorithm we can compute addition using a polynomial size depth 5 (or 4?) $AC^0$ circuit family. Is it possible to reduce the depth? Can we compute the addition of two binary ...
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### Are all the functions whose fourier weight is concentrated on the small sized sets computed by AC0 circuits?

Are all the functions whose fourier weight is concentrated on the small sized sets(or terms with low degree) computed by $\mathsf{AC}^0$ circuits ?
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### Better lower bounds than 3n for non-boolean functions?

Blum's $3n-o(n)$ lower bound is the best known circuit lower bound over the complete basis for an explicit function $f : \{0,1\}^n \to \{0,1\}$, cf. Jukna's answer to this question for related results....
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### 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 ...
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### Randomness and small circuits complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined as $\textrm{BPP}$ with respect to $\textrm{P}$. More formally we provide ...
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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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### 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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### Circuit Complexity Charaterization for DLogTime and NLogTime

$\mathsf{DLogTime}$ and $\mathsf{NLogTime}$ are two of the smallest complexity classes we have. (Note that logarithmic time hierarchy $\mathsf{LH}$ is equal to $\mathsf{AC}^0$ and these are the first ...
### Does the $\mathsf{TC^0}$ hierarchy collapse?
Do we know that the $\mathsf{TC^0}$ hierarchy does not collapse ($\mathsf{TC^0_d} \subsetneq \mathsf{TC^0_{d+1}}$ for all $d$)? The Zoo entry for $\mathsf{TC^0}$ only mentions the separation between ...
The following question is related to the optimality of the Bellman-Ford $s$-$t$ shortest path dynamic programming algorithm (see this post for a connection). Also, a positive answer would imply that ...