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Questions tagged [circuit-complexity]

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Citation/ Proof of a theorem in computational complexity about recursive majority

Circuit Depth Lower Bound for Iterated Majority Function Let $k \geq 3$ be a fixed integer. The function computed by a balanced $k$-ary tree of depth $d(n) = \Theta(\log n)$, where each node computes the majority of its $k$ inputs, cannot be…
CuriousScientist
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Convert the language into CNF or DNF form

A boolean circuit C has n inputs and m outputs, and is constructed with AND, OR, and NOT gates. Each gate has fan-in 2 except the NOT gate which has fan-in 1. The out-degree can be any number. A circuit is not allowed to have any cycles. For example…
Monte_carlo
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Are classes $\textbf{NC}$ and uniform $\textbf{NC}$ the same?

On page 117 in Arora and Barak, the definition of class $\textbf{NC}$: For every $d$, a language $L$ is in $\textbf{NC}^d$ if $L$ can be decided by a family of circuits $\{C_n\}$ where $C_n$ has poly(n) size and depth $O(\log^d n)$. The class…
minh quý lê
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Understanding Reversibility of Quantum Computing in Proving Boolean Circuits as A Subcase of Quantum Circuits

I'm reading quantum computation in Arora and Barak, on page 215 they provide the Lemma 10.10 proving that quantum circuits can simulate boolean circuits. Lemma 10.10 (Boolean circuits as a subcase of quantum circuits) If $f…
minh quý lê
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Whether $\textbf{PH}$ collapses to $\Sigma^p_1$ when $\overline{3SAT}\in \textbf{BP$\cdot$NP}$

I'm doing ex. 7.8 in Arora and Barak Show that if $\overline{3SAT}\in \textbf{BP$\cdot$NP}$, then $\textbf{PH}$ collapses to $\Sigma_3^p$. This is definition of $\textbf{NP}/poly$: A nondeterministic circuit has two inputs $x, y$. We say that $C$…
minh quý lê
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Confusing on the proof $\textbf{NC}^1\subseteq \textbf{L}$ using DFS

On page 430 of Sisper's TOC, Theorem 10.39 proves $\textbf{NC}^1\subseteq \textbf{L}$: PROOF: We sketch a log space algorithm to decide a language $A$ in $\textbf{NC}^1$ . On input $w$ of length $n$, the algorithm can construct the description as…
minh quý lê
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Boolean circuits with fan-out of each gate is 2

I am following the book of Arora and Barak book. We consider Boolean circuits as we do, Specifically, inner nodes are either AND, OR (both – fan-in 2), or NOT (fan-in 1) gates. The fan-out of each gate is 2. The size of a circuit is the number of…
user172436
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Reduction from $\texttt{CKT-SAT}$ to $\texttt{3SAT}$

I am following the Barak and Arora book, in circuit chapter, they use direct reduction from $\texttt{CKT-SAT}$ to $\texttt{3SAT}$ directly without any clue. How to construct an explicit reduction from the language $\texttt{CKT-SAT}$ to the language…
user172436
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DLOGTIME sequential access (uniformity conditions for circuits)

Consider the direct connection language of a circuit family, consisting of tuples (a,p,b,y) where a is a gate number p a binary string is an encoding for a predecessor or type b is a gate type or predecessor of a according to p Y is any string of…
emesupap
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What are the shortest circuits to sum bits?

I am interested in circuits that reduce $2^n-1$ input bits into their sum, represented as an $n$ bit integers. For $n=2$, this is a full adder, which can be implemented with 5 gates of 2 inputs in various ways. Alternatively, it can be directly…
fuz
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Open Problem: Structural Learnability of Pseudo-Random Boolean Circuits

I would like to propose an open problem at the intersection of computational complexity, pseudorandomness, and circuit theory. This problem has potential implications for cryptography, AI model analysis, and the theory of explainability in…
Konstant
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Relationship between Formula Complexity and Depth Complexity of A Boolean Function

A (de Morgan) formula $\phi$ is a rooted binary tree, whose leaves are identified with literals of the forms $x_i$ and $\neg x_i$, and whose internal vertices are labeled as AND ($\land$) or OR ($\lor$) gates. Here, the same literal can be…
minh quý lê
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Boolean circuit with true value

Let $$L=\left\{\,\langle\,B_n,\, \,x\,\rangle:\enspace\substack{B_n \text{ is a boolean circuit and } \\x \in \{0, 1\}^n\text{such that }B_n(x) = 1}\right\}$$ I want to prove that $L$ is $\textbf{P}$-complete. We know that a language is…
Rio
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Prove that $\textbf{NC}$ circuit family can compute log space reduction

Most proofs of the problem that I have seen stop at proving $\textbf{NL}\subseteq \textbf{NC}^2$ and not explicitly point out the circuit. I'm trying to show that $\textbf{NC}$ circuit family can compute log space reduction by constructing…
minh quý lê
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Is it possible there exists a reduction that satisfies conditions of reduction from $\texttt{CKT-SAT}$ to $\texttt{3SAT}?$

I am following the Barak and Arora book. I have asked this question regarding reduction from $\texttt{CKT-SAT}$ to $\texttt{3SAT}.$ Is it possible to show that there exists a reduction that satisfies the condition of asked question (in particular,…
user172436