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Questions tagged [circuits]

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

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An AC$^1$ circuit for 2-SAT

We know that $NC^1 \subseteq NL \subseteq AC^1$ and that 2-SAT is complete for $NL$. How does one construct an $AC^1$ circuit for 2-SAT? Recall that $AC^1$ circuits have $O(\log n)$ depth where $n$ is the number of bits in the input. I have read…
Sameer Gupta
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2-SAT or 3-SAT or k-SAT in AC-0

This may be an elementary question, but I'm new to circuit complexity. Does 2-SAT in CNF form belong to the complexity class AC$^0$? It seems simple enough to construct an AC$^0$ circuit of depth 2 of polynomial size for 2-SAT. $~$Given $n$…
Martin Seymour
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Lower bound on number of (different) circuits of given size?

For circuits with $n$ input bits, we know that, for any function $s$, there are at most $O(s(n)^{s(n)}) = O(2^{s(n) \log s(n)})$ circuits with size at most $s(n)$. Say two circuits $C_1$ and $C_2$ are different if the function they compute is…
dkaeae
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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'm having trouble understanding the specifications…
user3752935
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Number of circuits with at most $m$ logic gates

I'm working on the same exercise as described in this post: How to show that hard-to-compute Boolean functions exist? In the answer there I don't understand how the number of circuits with at most $m$ gates was found to be $O(m^{4m})$. My…
Rourke Sekelsky
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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 how such a circuit should be represented in a text…
danxinnoble
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CNF and small-depth circuits

I'm reading on small-depth circuits. Since every formula can be turned into a CNF formula, which has depth at most 3, why should we study deeper circuits? Is it because convertion to CNF may result in an explosive blow-up? If so, can you give an…
Zirui Wang
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Question on digital DE-MULTIPLEXER!

How does a demultiplexer ignore/discard/block the non-required outputs? A demultiplexer channels the input to one of the outputs, but there are several outputs. When one output is selected (depending on the state of the select line(s)) then rest of…
samm
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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 write it in infix notation, we've suggested a mix.…
Doug Spoonwood
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How to bound the size of Boolean circuits?

Every function $f\colon \{ 0, 1 \}^n \to \{ 0, 1 \}$ can be computed by a circuit over the standard unbounded fan-in basis $\mathcal{B}_1 = \{ \neg, (\vee^n)_{n \in \mathbb{N}}, (\wedge^n)_{n \in \mathbb{N}} \}$ of depth $3$ and size…
user251130
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How do we enumerate every algorithm in the circuit model?

Consider the family of circuits $\{C_n\}_{n\in \mathbb{N}}$ that are big enough to compute every Boolean function for $n$ variables. We can label the nodes in order starting at the inputs and working down the layers, so that $2^{2^n}$ refers to the…
HiddenBabel
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Boolean circuit size bounds on the majority function

I am a bit lost in the literature. Is it known whether there is a $o(n \log n)$ size boolean circuit family for the majority function?
acupoftea
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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 circuit of size $s$ that computes a function $f:…
Gaussbaby
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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 and can someone explain the steps?
Gaussbaby
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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 verifier for instances of size n that is of size…
user129081
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