Questions tagged [circuits]

A computation model in which the computation is described via circuits of various logic gates.

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Prove $\text{CorrectSuccintSolver} \in \mathbf{coNP}$

Define the following languages: $$ \text{SUCC-CVAL}=\{(S,x,i) : \substack{S \text{ is a succint representation for circuit } C \\ \text{ and } C_i(x)=1 \text{ where } C_i \text{ is the i'th gate in }...
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Show that any monotone Boolean function is computable by a circuit containing only AND and OR gates

A Boolean function $f : \{0, 1\}^n → \{0, 1\}$ is called monotone if changing any of the $n$ input bits $x_1, \ldots , x_n$ from $0$ to $1$ can only ever change the output $f(x_1, \ldots ,x_n)$ from $...
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Circuits for Modular Arithmetic

I've read this which describes how to do do integer arithmetic in circuits. The one thing that it does not describe is how to do these operations with a modulus. How can modular arithmetic be done in ...
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Algorithm to reduce a Circuit-SAT to NAND-SAT

I am trying to construct an algorithm to reduce OR, AND and NOT gates into NAND-SAT. Can someone give me a hint as to where to start?
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Smallest Circuit for Square of Sparse Symmetric Matrix

I have an n by n symmetric matrix, and I would like to compute its square in as small a circuit complexity as possible. It's sparse: there are sqrt(n) nonzero entries in each row/column, so the input ...
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What class is the language $(C,(v_i)_{i=1}^m,x)$ complete to s.t. $C(x)$ is a boolean circuit with $m$ gates with values $\{v_i\}_{i=1}^m$

Given the following language: $$ L=\left\{\,(\,C,\,\{v_i\}_{i=1}^m, \,x\,) \enspace :\enspace \substack{C(x) \text{ is a boolean circuit with } m \text{ gates} \\i\text{'th gate value is } v_i \text{...
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Decidable languages unconditionally not in P/poly

What are some nice/natural examples of languages not contained in $P/\mathit{poly}$, preferably decidable ones? I'm interested in unconditional results rather than examples such as the Karp–Lipton ...
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Nandgame--I am not sure I understand the Data Flip-Flop specifications

Nandgame (nandgame.com) has you solve puzzles of increasing complexity which culminate in constructing a simple CPU. You start at the level of nand gates, and build everything else up out of those. I'...
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Make the forward counter go down

Suppose, I have a 4-bit binary Incrementor that uses XOR gates to increment the inputted number by the value of 1 (b0001, to be precise). Suppose, we connect it to 4 D-Flip-Flops (DFF) to create a ...
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Problem with understanding Multi-party security circuit for secure stable matching

I am reading the following paper: MPCircuits: Optimized Circuit Generation for Secure Multi-Party Computation Paper Link I have following question: We have two groups shown in the circuit. Why we ...
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Why aren't P and P/poly trivially the same?

The definition of P is a language that can be decided by a polynomial time algorithm. The definition of P/poly can be taken to mean a language that can be decided by a polynomial-size circuit (see ...
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Number of possible boolean functions in a DAG of lookup tables?

A K-input lookup table (K-LUT) can represent any function with K boolean inputs and a single boolean output. The number of possible functions represented by this LUT is $2^{2^K}$ according to this ...
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How does fan-out change circuit complexity?

Edit: Here's maybe a clearer presentation of my question. In a Boolean formula, all the gates have fan-out 1, and the graph representing the formula is a tree. In a Boolean circuit, the gates can have ...
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Details wanted on the reduction from Circuit Value to CFG Membership

Consider a Boolean Circuit $C$ which takes $n$ inputs and has one output. Notation: Let $\textit{size}(C)$ be the size of circuit $C$: the total number of gates in $C$. Let $G = (V,\Sigma,R,S)$ be a ...
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Circuit satisfiability problem : SAT-C to SAT-2C

I have the following language : $L=\{\langle C_1,C_2\rangle \text{ | } C_1 \text{ and } C_2 \text{ are two circuits that calculate different function}\}$. We can call this language SAT-2C. Prove that ...
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Mechanism of Howard's algorithm

How does Howard's algorithm avoids re-mapping of the non-critical nodes ?
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What was the original paper that showed a simulation of turing machines via circuits?

It is a very standard construction in most complexity theory courses to turn a turing machine into a circuit. I thought this was due to Cook, but it looks like he did the reduction to SAT not through ...
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Simple example of exponential gap between monotone and non-monotone circuits

Is there a simple example of a monotone Boolean function $f:\{0,1\}^m \to \{0,1\}$ that we know can be computed by a polynomial-size circuit, but cannot be computed by any polynomial-size monotone ...
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Combinational Logic Circuits and Theory of Computation

I'm trying to link Combinational Logic Circuits ( computers based on logical gates only ) with everything I have learned recently in Theory of Computation. I was wondering whether combinational ...
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1answer
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What is the comparator circuit?

The standard circuits $AC^i$, $NC^i$ are constructed using $AND$, $OR$ and $NOT$ of various fan-ins, fan-outs and depths. What is the comparator gate constituted from? Structurally why is it believed $...
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What is the depth of comparator circuit required in Gale Shapely and STCONN?

Stable matching problem and $STCONN$ can be solved using comparator circuits (refer https://arxiv.org/abs/1208.2721). What is the depth of the $CC$ circuit necessary for stable matching? Is it in $CC^...
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Comparing PRAM and Circuit Complexity, $NC^i$

I wondered about the following quote from NC (Wikipedia): $NC^i$ is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates of at most two inputs and ...
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Is an AND gate which is noisy 1/3 of the time on only one of its inputs universal?

Imagine you have a noise-free NOT gate, and an AND gate with the usual truth table 00 0 01 0 10 0 (*) 11 1 but such that the case (*) is wrong 1/3 of the time, ...
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Show that a circuit of size $s$ can be converted to a DeMorgan circuit computing the same function of size at most $2s$

I am trying to prove the above statement. A DeMorgan circuit is a circuit that has only $\{ \wedge, \vee, \neg \}$ gates, and the negation is applied only to input variables. So, assuming we have a ...
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Is there distinction between $(C/poly)\cap(coC/poly)$ and $(C\cap coC)/poly$?

Let $C$ be an uniform complexity class for example $NL$ or $NP$. Is there distinction between $(C/poly)\cap(coC/poly)$ and $(C\cap coC)/poly$?
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Construct a Circuit computing all boolean functions over n bits

Let $ n∈N $ . Construct a circuit with $ C_n(x_1,\dots,x_n) $ with $ 2^{2^n} $ outputs $ y_1,\dots,y_{2^{2^n}} $ which computes all distinct boolean functions $ f_i:\{0,1\}^n→\{0,1\}$ such that $ ...
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How can you convert the depth of a boolean circuit to its size?

I know that the depth of a circuit is the maximal length from an input gate to the output gate of the circuit and its size is its number of gates. Is there a formula that you can go from depth to size ...
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Problems that are easy on boolean formulas but become NP-hard on circuits?

Many problems that take a boolean circuit as input are NP hard to compute. Do we have examples of such problems that become polynomial time computable when only boolean formulas are allowed as input? ...
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Is it possible to construct a C^5(U) with V^2=U and no work qubits (Nielsen & Chuang Exercise 4.28)

My question is related to the exercise 4.28 in the book of Nielsen and Chuang (Quantum Computation and Quantum Information). Here is the exercise For $U=V^2$ with $V$ unitary, construct a $C^5(U)$ ...
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Relevance of depth for $NP$-completeness of fan-in $2$ and fan-out $1$ modest depth circuits?

Let $\mathcal C$ be a circuit of $m=f(n)$ input wires where every input is taken in the set $\{x_1,x_1',\dots,x_n,x_n'\}$ where $x_n\in\{0,1\}$ and $x_n+x_n'=1$ holds (not all $x_i,x_i'$ necessarily ...
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Prove lower bound on boolean circuit

Given matrix $A \in \{0,1\}^{n \times m}$ with $n$ rows and $m = 2^n - 1$ columns. Where $j$-th column is binary decomposition of $j$ ($j = 1 \dots 2^n - 1$). For example, if $n = 3$: $ A = \begin{...
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Iterated multiplication of permutation matrices

Given $m$ matrices of size $n\times n$ each of which is promised to be a permutation is it in $\mathit{quasiAC}^0$ or $\mathit{AC}^0$ to multiply the permutations where $m=\mathit{poly}(n)$ $m=\...
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What is uniformity in Boolean circuits exactly?

I have two questions on Kaveh's answer to Definition of uniform boolean circuit : Kaveh mentions that the input is in unary encoding. In the definition it says the input is $1^n$, afaik $1^n$ is a ...
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How to find the reverse of a binary string with simple binary operators?

I was wondering is it possible to create a simple circuit that detects if an input (a binary string) is a palindrome? So my approach is to feed the input to a circuit that reverses the input, ie if ...
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Newbie 2s complement to binary circuits help

I'm creating a circuit with in3, in2, in1, and in0, and outputs F11-F0. The circuit has to take the input in 2s complement and output the number in binary with the sign. F11 represents the sign bit (...
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Circuits and Closure Under Reductions

Suppose that $A$ and $B$ are languages such that $A\leq_P B$ (many-to-one Karp reduction), and $B\in \mathbf{P/poly}$. How do we prove that $A\in\mathbf{P/poly}$? Using similar ideas like Cook-Levin (...
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Symmetric functions in NC¹

A boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ is symmetric if $f(x)$ depends only on the number of $1$s in $x$. It is known that every boolean function is in $\mathrm{NC}^1$, i.e. there ...
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Were boolean logic used in the analog computers?

I began a simple collection of the events behind todays computers. My knowledege in these fields is so limited, and I read: "In the 1930s and working independently, American electronic engineer ...
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Generalizing Quantum Computation

When you first learn more about computation you can imagine it in terms of boolean circuits. That is you get a boolean vector $v \in \lbrace 0,1\rbrace ^n$ which you can then apply a circuit $C$ to ...
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Closure properties of Alternating Circuit 1 level

Recall that $\mathsf{AC^1}$ is the class of circuits with unbounded fan-in, polynomial size, and logarithmic depth. Is this class closed under Kleene star? I thought it would be simple since it is ...
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Converse of Impagliazzo, Kabanets, Wigderson

I am trying to prove that $\text{NEXP} = \text{MA} \Rightarrow \text{NEXP} \subseteq P/\text{Poly}$. I tried to approach the result via trying out the contrapositive, that $\text{NEXP} \nsubseteq P/\...
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Can you build a solver circuit from a verifier circuit?

Can you build a solver from a verifier? I see that if you start with an NP-verifier TM the answer is yes, you can build a solver TM. How about for circuits? Can you go from a circuit that implements a ...
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Functions with small support have small circuits

I have been trying to understand the use of circuit models for boolean functions, and came across this question, that I am trying to struggle to understand: Show that if a function $f\colon \{0,1\}^n→\...
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Is NP in NP/Poly?

The answer is yes, NP/poly is defined as the class of problems solvable in polynomial time by a non-deterministic Turing machine that has access to a polynomial-bounded advice function--the advice ...
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single circuit simulating multiple Turing machines

You can simulate polynomial time Turing machines with polynomial size circuits, can you simulate multiple poly time TMs with a single poly size circuit?
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Simulation of circuits with circuits

From classical results of universal simulation of Turing machines there exists a Universal Turing machine simulating any Turing machine with time complexity 𝑇(𝑛) in time 𝑇(𝑛)log𝑇(𝑛). Is there is ...
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Hardness of boolean functions

For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$, $H_{avg}(f)$ is a function from $\mathbb{N}\longrightarrow \mathbb{N}$, termed as the average case hardness, if $\forall$ circuit $C_n$ of ...
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Universality of the Toffoli gate

Regarding the quantum Toffoli gate: is it classicaly universal, and if so, why? is it quantumly universal, and why?
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Connection between Pseudo random generators and hardness

For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$ $H_{avg}(f)$ is defined as the largest $S(n)$ s.t. for all circuit $C_n$ of size $S(n)$, $\Pr_{x\in U_n}[C_n(x)=f(x)]<1/2+1/S(n)$. Here $...
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Existence of boolean function with exponential average case hardness

Show that for every large enough $n$, there is a boolean function $f\colon \{0,1\}^n\longrightarrow\{0,1\}$, whose average case hardness is exponential. The question is taken from Arora Barak ...

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