Questions tagged [circuits]

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

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Connections between circuit complexity and Unique Games Conjecture?

Circuit complexity has connections to many questions in complexity theory. For a couple examples, Ryan Williams shared some in a recent talk and Section 3 of these notes gives simple relations to $\...
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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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Perfect Halver Construction?

A sorting network is a circuit-based approach to sorting, built out of CompareExchange gates, which compute the function: $$\mathsf{CompareExchange}(x,y) = (\min(x,y), \max(x,y))$$ The input to the ...
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195 views

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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331 views

PARITY using depth one TC0 circuit

I need to disprove that a PARITY gate can be simulated using a single MAJORITY gate, or even a ...
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61 views

Circuit complexity of hardest monotone function

Show there exists a monotone function $f\colon \{0,1\}^n \mapsto \{0,1\}$, such that the minimal size of a monotone circuit that computes $f$ is $\Omega(2^n / n^2)$. Use the fact that the number of ...
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34 views

Number of distinct single-assignment forms with $j$ binary function calls?

Given $n$ inputs and $k$ outputs and $j$ identical binary function calls to $g$, how many possible distinct single-assignment forms are there? The only assumption made about $g$ is that if $a = c \...
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228 views

AC0 and first order logic equivalence

The page on descriptive complexity theory in Wikipedia states the following: "First-order logic defines the class FO, corresponding to AC0, the languages recognized by polynomial-size circuits of ...
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28 views

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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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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69 views

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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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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199 views

Example of *small* non monotone circuit such that any equivalent monotone circuit has greater size?

A "general" Boolean (combinatoiral) circuit is a labeled (with the labels: AND, OR, NOT, IN, OUT), directed, acyclic graph, that satisfies: fan-in=2 for the AND and OR nodes fan-n=1 for the NOT ...
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63 views

Are minimum boolean circuit sizes for small problem sizes of an NP-complete problem known?

I think that a table with the following numeric values would be very interesting, but I could not find any table online displaying them: Choose any NP-complete problem (say, clique, but a problem ...
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85 views

Class of languages recognizable by n-bit formulas of size at most $T(n)$

A Boolean (combinatoiral) circuit is a labeled (with the labels: AND, OR, NOT, IN, OUT), directed, acyclic graph, that satisfies: fan-in=2 for the AND and OR nodes fan-n=1 for the NOT nodes fan-...
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49 views

Circuit complexity of random Boolean functions

I just saw a YouTube video where Ryan Williams gives a talk about circuit complexity. He stated that random Boolean functions require exponential size circuits to compute, but I don't understand why ...
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26 views

Simulating Boolean Circuit with RAM

Statement: Every $T(n)$ size bounded Boolean circuit family, could be simulated with $(T(n))^2$ time bounded Random Access Turing Machine (RAM). Could you please supply me with a reference to an ...
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49 views

Does Cook and Ruzzo's result also hold for logspace-uniform AC0?

In Cook's famous paper on $\mathsf{NC}$, he cites the following result: PROPOSITION 4.7 (Cook and Ruzzo, 1983). $\mathsf{AC}^k$ consists of those problems solvable by uniform unbounded fan-in ...
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Implementing depth-3 circuit for XOR

In this set of notes, they claim that there is a size $O(2^{\sqrt n})$ depth-3 circuit (OR -AND -OR) that implements XOR. I tried for a little bit to figure out how to do this, but couldn't find ...
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68 views

Modular reduction in a finite field

Let $\mathbb{F}_p$ be a finite field of prime order $p$. Define $r_q : \mathbb{F}_p \to \mathbb{F}_p$ as $r_q (x) = x \bmod q$ with $q<p$. A tad more formally, treat $x$ as an integer in $[0, p)$ ...
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How to minimize the number of gates of an arithmetic circuit?

A circuit is simply a DAG, with some input wires, some output wires, and some operations on the vertices. Consider an arithmetic circuit where the only operations are addition ($+$) and ...
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2answers
280 views

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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From SETH to circuit lowerbounds

Are there reductions from SETH (Strong Exponential Time Hypothesis) to lowerbounds against threshold circuits? (maybe for computing Boolean functions of the form OR-of-AND-of-OR) In threshold ...
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What are the recent research directions in the topic of circuit lower bounds from derandomization?

I am thinking of the classical paper, https://www.cs.sfu.ca/~kabanets/Research/poly.html Can someome link to some papers/reviews that give a sampling of what are the recent thoughts in this direction?...
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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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28 views

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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28 views

Why mod $p$ gates cannot be computed by $ACC^0[q]$ circuits, $p$ and $q$ prime

I am going through Computational Complexity by Arora and Barak, and there I came across the proof of why mod $p$ gates cannot be computed by $ACC^0[q]$ circuits, where $p$ and $q$ are distinct primes. ...
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116 views

Examples of relatively complex truth tables/logic gates in real life?

I'm researching truth tables, logical gates, and boolean algebra expressions. I'm trying to find specific real-life examples of logic gates and/or truth tables used in algorithm or circuit design in ...
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16 views

Boolean circuit multigraph

Let us say that our definition of a circuit is the one of a boolean circuit from [Vollmer]. He uses directed acyclic graphs to represent circuits where the computation nodes are labeled with some ...
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1answer
50 views

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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51 views

Equality checking mod $10$ via arithmetic circuits

I'm interested in implementing equality checking mod 10 in an arithmetic circuit. Is this possible? Preliminary evidence points towards "no", but I thought it best to ask before completely writing it ...
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39 views

projection of arithmetic formulas to determinant

I am looking for a direct proof (i.e. without going through ABPs) that if $f(\bar{x})$ has an arithmetic formula of size $s$ then it is a projection of an $O(s)\times O(s)$ determinant. It seems ...
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What standardized formats (if any) exist for boolean circuits?

Being able to represent a Boolean circuit is useful in a number of areas of Computer Science, such as Circuit Satisfiability, Zero-Knowledge Proofs and Garbled Circuits. Are there any standards for ...
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39 views

About sign-rank of Boolean functions

Do we know of any necessary condition for a Boolean function or say a depth $2$ LTF circuit to have a low (~poly(dim)) sign-rank?
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How many bits we can negate using two/three NOT gates?

How many bits we can negate using two/three NOT gates ? I am newbie at this subject so I ask for help. It is about circuits. Edit After reading link given in comments by @D.W I think that I can ...
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700 views

3 bit binary multiplier?

I have the following 2-bit binary multiplier How can I modify this 2-bit binary multiplier to make it a 3-bit binary multiplier? I notice that there are 2 half-adders, and there are a bunch of ANDs ...
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169 views

On power of $P/poly$

(1) We know that $EXP ⊄ P/poly ⇒ BPP$ is in $SUBEXP$. Does $SUBEXP ⊄ P/poly$ mean $P=BPP$ or anything close? (2) We know that if $NP$ is in $P/poly$ then $PH$ collapses to second level. What is the ...
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Arithmetic problems known to be in TC^{i+1} but not known to be in TC^i

Is there an arithmetic problem that is known to be in $TC^{i+1}$ but not known in $TC^i$ for any $i\geq0$? Concrete examples for $i=0$ would be of most utility however any arithmetic example is fine.
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Does an Expression in RPN Give us a Linear Way of Writing What Happens in a Circuit?

I mean, say we want to show how we can implement an OR gate in terms of a NAND gate. If we write in Polish notation, then we've suggested that the circuit takes the gates before the inputs. If we ...
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How do I triangularise a netlist?

I have a circuit that is represented as a netlist (specifically, an and-inverter graph). The desired outputs of this circuit are known. We can assume that some combination of the primary inputs will ...
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126 views

Simplifying circuits using boolean algebra

I am having a lot of trouble simplifying my circuit using boolean algebra. I am very new to this and any explanation would be greatly appreciated. I have y'+z+w'x+wx' I feel like I could use DeMorgan'...
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Is VHDL a description language for a Boolean circuit or are both concepts unrelated

I am looking for a way to translate basic c programs (subset of c or java or some declarative programming language) to a Boolean circuit. I know that Turing machines are reducible to Boolean circuits ...
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146 views

Functional Unit and Micro-operations Schematics

I'm sitting an exam on Computer Architecture in a few days and i'm stuck on a particular type of question. I'm asked to: Provide a detailed schematic for a functional unti that implements the ...
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1answer
241 views

Show that problem is PSPACE-complete - path in directed graph

I have a following problem: Given $n$ and graph of size $2^n$, and circuit with $2n$ input gates. Directed edge between $k$ and $l$ exists iff only and only we encode $k$ and $l$ as bits and launch ...
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1answer
28 views

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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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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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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Mechanism of Howard's algorithm

How does Howard's algorithm avoids re-mapping of the non-critical nodes ?
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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^...