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Questions tagged [boolean-complexity]

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Communication complexity of testing balancedness of a Boolean polynomial

The problem I consider is the following: given the $2^n$ coefficients of a Boolean polynomial $f : \{0, 1\}^n \rightarrow \{0, 1\}$, determine if $f$ is balanced namely if the truth table of $f$ contains exactly $2^{n-1}$ zeroes and $2^{n-1}$ ones.…
Charles Bouillaguet
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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 been strengthened later, showing that, in the…
Martin Berger
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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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Approximate the parity function in L1-norm

Consider the parity function $MOD_2(x) = x_1 \oplus \cdots \oplus x_n$ for $x \in \mathbb{F}_2^n$. I am concerned about the degree bounds for a real polynomial $f$ which approximates $MOD_2$ well in the L1-norm i.e. $\frac{1}{2^n}\sum_{x \in…
TheGuy
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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 that $L(f_1 \oplus \cdots \oplus f_n) \geq…
dino-t
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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 bijective. The specific statement that I am…
Berk U.
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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 only if $S=0^n$ and is $0$ otherwise. Suppose I…
postasguest
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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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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-intensive and frequently run so it's necessary to…
lyberius
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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. Settings and motivation: I am doing a research…
AXX
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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, $f$, specifies a graph where every node has an in-degree and…
Andrew Baker
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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 acyclic boolean circuit that ensures that each gate is topologically ordered, i.e. there is no feedback? If so, how exactly is that done, by loop expansion? And if…
Emison Lu
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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 $f$ and then divide the result by $2^n$. In the paper…
user154975
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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 $O(n)$? I'd say no, because otherwise SAT would be…
Noel Arteche
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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 all. But if you know the name of what i am…
vreithinger
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