Questions tagged [boolean-complexity]
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24 questions
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3-CNF to 2-CNF reduction
I want to start by saying that I've struggled to find any satisfying answer to this question of mine. I did read this question, but it's slightly different.
My idea is simply that every 3-cnf formula ...
- 101
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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 ($\lor$...
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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 $\textbf{P}$-...
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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 ...
user172436
2
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Lower Bound on Parity of Boolean Functions
Let's say we have boolean functions $f_1, \cdots, f_n$, each of which operates on pairwise disjoint variables (i.e. the variables for each function are unique to that function). Then, how can we show ...
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How to construct a carry-lookahead adder of the optimal $O(n)$ size
Problem (TL;DR): I'd like to know how to construct a CLA adder that has $O(n)$ size and $O(\log n)$ depth using only fan-in 2 AND gates and XOR gates, as suggested in this answer and this answer.
...
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Why is End-Of-The-Line defined in terms of "Arithmetic circuits" instead of "Boolean circuits"
The definition of PPAD (Polynomial parity arguments on directed graphs) revolves around the definition of "End-Of-The-Line"
An exponentially large polynomial-depth arithmetic circuit, $f$, ...
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influence of neighourhood points
Im trying to understand the following question. Suppose $h,f:\{-1,1\}^n\rightarrow \{-1,1\}$ satisfy $\sum_x h(x)f(x)\leq 0.5$, then one can rewrite this as $\textsf{Pr}_x [h(x)=f(x)]\leq 3/4$. Can we ...
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Is there a notation for boolean algebra complexity?
To represent complexity of an algorithm, Computer Scientist is used to using big-O notation.
How about complexity of boolean algebra?
Boolean algebra is commonly used in digital circuit design with ...
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How many different boolean functions exist up to permutation of its $n$ variables
i am relatively new here, so if this was asked before, feel free to redirect me. I am searching for an answer in form of a (iterative or recursive) Formula or even better, an algorithm to list them ...
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Shannon's result that some Boolean functions require exponential circuits
In 1949 Shannon proved, using a non-constructive counting argument, that some boolean functions have exponential circuit complexity, see [1] and many texts on computational complexity. This result has ...
- 8,368
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sum of Boolean characters larger degree
I was curious if someone knew the answer/reference for the following. So it is well-known that if $S\in \{0,1\}^n$, then
$$
\frac{1}{2^n}\sum_{x\in \{0,1\}^n} (-1)^{\langle S, x\rangle}=1
$$
if and ...
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