Questions tagged [boolean-complexity]
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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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Algorithm design: Model redundancy in tests
I've run across an interesting problem at work that I'm not quite sure how to grapple.
Broadly, there is a suite of of $n$ tests to ensure the quality of a product. However, the tests are both time-...
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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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Given a boolean circuit that computes a boolean function, can we always find an equivalent circuit with optimal size?
Let's say that we have a decision problem $P$. Let's also say that $I_n$ is the set of all instances of size $n$ that exist for this problem, and that its cardinality is finite.
There is a sequence of ...
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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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How to convert loop into combinational circuit?
In high-level programming languages, such as C and Python, loop statements such as while and ...
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Acyclic boolean circuit (DAG)
If a function f has a while loop or for loop, can I compile this function into an ...
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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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Formula for computing a specific Fourier coefficient of a boolean function
According to O'Donnell's book ``Analysis of Boolean Functions", in order to determine the Fourier coefficient of a boolean function $f$ on a subset $S$, we take an inner product of $\chi_S$ and $...
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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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Time-complexity of evaluating a CNF formula
Given a Boolean formula over $n$ variables in CNF and a partial assignment to it, all the algorithms I can think of to evaluate the assignment run in time $\Theta(n^2)$. Is it possible to do it in $O(...
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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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Can Boolean circuits of polylog depth represent all Boolean functions?
Consider a Boolean circuit using (2-input) logical-and, (2-input) logical-or and logical-not as basic components. The depth of the Boolean circuit is the length of the longest path from the input to ...
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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 ...
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Is $f(X)f^d(X) = 0$ for a Boolean function $f$?
I'm currently trying to understand a step in the proof for in the Crama and Hammer book on Boolean Functions. The proof is Proposition 4.12, which claims that the self-dualization of Boolean $f$ is ...
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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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Boolean function represented as a column vector: easy way to see if it has full degree?
If you have a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ on $n$ variables, you can represent this function as a $2^n$ vector indexed by the input space such that $
f_x = f(x)$. For ...
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