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Questions tagged [boolean-algebra]

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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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1 vote
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Why do we use {+1, -1} in place of {0, 1} for the Fourier analysis of boolean functions?

I want to know what will change if we keep on using {0,1} for our Fourier analysis of boolean functions? What are the things, which can not be performed with {0,1} and can be done with {+1, -1}?
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Fourier Dimension of Boolean functions

I was recently reading about Fourier dimension of Boolean functions. What I understand is that if we take the Fourier expansion of $f: \{\pm1\}^n \to \{\pm1\}$ and consider the monomials with non zero ...
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Boolean Expression and Logic Circuit

In a fish tank, the health of goldfish depends on the pH value of the water, temperature and oxygen level of the water in the tank. A sensor based control system is to be developed to make a healthy ...
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1 vote
2 answers
152 views

Clarification regarding linear boolean functions!

I am a little confused when it comes to linear boolean functions. According to this post: What is a simple way of explaining what a linear boolean function means in boolean algebra and relating it to ...
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simplification of boolean expression using k-map

I have the following boolean expression which has to be solved using K-map: ...
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1 vote
1 answer
19 views

How to come up with combination a short-circuit evaluation table?

(a || b) || (c && d)) Given the above, how do I derive the table below: a b c d output T - - - TRUE F T - - TRUE F F T T TRUE F F T F FALSE F F F - FALSE I'm told that this is short ...
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1 answer
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Influence of a variable in composition of Boolean functions

Suppose $f$ and $g$ are Boolean functions without a constant term, and where every variable has the same influence. How to show every variable will have the same influence in $f \circ g$? To me it ...
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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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Is [F(a, b, c) = a' + b] functionally or logically complete?

I'm having a problem determining whether [F(a, b, c) = a' + b] is functionally(logically) complete or not. I would really appreciate it if you could help me. P.S: I can't have 1 or 0 as input.
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Can element occur in a CNF formula?

For example, is $(X \vee 1)$ a valid formula in conjunctive normal form (CNF)? If yes, then I would have to consider such formulas when trying to prove a statement about all CNF formulas.
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Is there an algorithm for generating non comparable boolean vectors?

First some context: A Boolean Network of $n$ components is a function $f$ from the set $\{0,1\}^n$ (set of vectors of $n$ components whose values are 0 or 1) to itself. The dynamical behavior of a ...
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2 votes
1 answer
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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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Is there an efficient way to generate a pseudo-boolean function from a linear constraint?

I would like to define a pseudo-boolean function $f$ such that $f(x) = 0$ for all logically valid combinations of $x\in{0,1}$ and $f(x) > 0$ for all logically invalid combinations of $x\in{0,1}$. ...
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graph representation of a Boolean function

I'm trying to classify a certain family of Boolean functions, and need to represent the function as a graph. Is there any well-known graph representation for a Boolean function that captures the ...
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3 votes
1 answer
42 views

Stalmarck's method: x ≡ x → z, does z have to be true?

I have been researching Ståmarck's method 1. In the paper cited here, some rules are given. Rules are made of triplets (x, y, z) such that: y $\to$ z $\equiv$ x where x, y and z are booleans which ...
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Is there a known relationship between Kolmogorov Complexity of a binary string and the logic optimization of the corresponding Boolean function?

I haven't thought about how to go about proving it or finding a counterexample (I probably don't have the right background), but it seems intuitive to me that, given some representation of a Boolean ...
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-1 votes
1 answer
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Can anyone solve this? Is the answer 4 or 7. I'm confused

I'm trying to solve this but I'm confused with different answers. I'm getting 4 but the answer written is 7. Please guide me.
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1 vote
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Efficient bit-level implementation of Boolean polynomials with few variables

A Boolean polynomial in $n$ variables $x_1, \dots, x_n$ is an expression of the form $$\sum_{\mathbf{s} \in \{0,1\}^n} c_{\mathbf{s}} x_1^{s_1} \cdots x_n^{s_n}, \quad \text{ where } c_s \in \{0,1\} .$...
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What boolean algebra simplification is applied in this Shannon's expansion?

Found this question about Shannon's expansion. While I am trying to follow its logic, found one super convenience simplification used. Can we do this in general while dealing with boolean algebra? or ...
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Boolean formula for graph 3COL

For a given undirected graph $G=(V,E)$ I'm trying to construct a boolean polynomially computable formula $\varphi$ with the following property: $\varphi$ is satisfiable $\iff$ vertices of $G$ can be ...
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3 answers
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Why is SAT based on the CNF?

I have been reading up on Boolean logic and, specifically, the Boolean satisfiability problem. I have seen several people mention that the expression must be converted to conjunctive normal form (CNF) ...
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1 answer
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Sorting by boolean algebra (hardware) instead of algorithm (software)

Consider there's an 5 elements list that foreach element are 2-bits. Forexample [01,00,10,00,11], if the list is sorted, we hope the output like this [00,00,01,10,11] Maybe that case seems complicated,...
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Functions expressible in conjunctive normal form, but with XOR replacing OR

What are all the functions $f:\{0,1\}^n\rightarrow\{0,1\}$ that can be expressed as a product of affine Boolean functions? For example, if $x_1,x_2,x_3\in\{0,1\}$ then $x_1x_2x_3\oplus x_2x_3 \oplus ...
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Doubt regarding the physical folding of a two dimensional K-map

While grouping terms in a k-map, if we pair terms on the first row with the ones on the last, it can be interpreted as the folding the 2-D map in the form of a cylinder, along the horizontal axis. ...
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-1 votes
1 answer
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Four Queens Problem to a Conjunctive Normal Form

Given a chessboard with 4 rows and 4 columns (4x4) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Assign a Boolean variable to each cell of the board as below (1, 2, 3, etc. are variable names) If a ...
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4 votes
2 answers
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Compact representation for quantified boolean formula

I got black-box (too big to analyze) boolean formula f(...) with 3 sets of input arguments: $x_1... x_i, y_1... y_j, z_1... z_k$. And I want to find such values for x-arguments that for every y-...
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2 votes
1 answer
101 views

What are the limits of Boolean Algebra?

Any decision problem algorithm can be represented as a boolean expression. The rules of boolean algebra (De Morgan's law, distributivity, etc.) can be used to manipulate and simplify that expression, ...
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1 vote
1 answer
30 views

Definition and use of a Boolean equation system

According to google, a system of boolean equations is defined as follows Essentially, a Boolean equation system is an ordered sequence of fixed point equations over Boolean variables, with associated ...
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Minimization of an expression through K-map, in which there are more chances of errors and with Quine-McCluskey Method there are less chance of errors

Minimization of any expression of 4 variables through K-map, in which there are more chances of errors (like error in grouping) and the same expression with Quine-McCluskey Method there are less ...
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-2 votes
1 answer
60 views

Simplify $X'(X+Y) + (Y+X.X) ( X + Y') + Z + X.Z$

I wanna know if $X'(X+Y)$ means $X'.X+Y.X'$? Does it have an AND gate after $X'$? Notation: $X'$ : NOT $X$ $X + Y$: $X$ OR $Y$ (OR gate) $X.Y$ : $X$ AND $Y$ (AND gate) New to boolean, can't seem ...
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1 vote
2 answers
52 views

Simplifying the boolean expression xy + (!x)z + yz with boolean algebra?

So using K-maps I was able to simplify xy + (!x)z + yz to xy + (!x)z, and I double-checked that the truth tables are the same. I'm having trouble understanding how I would have used boolean algebra to ...
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1 answer
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How to perform AND on binary "recursive repeating sequences"?

Suppose, we have a two binary sequences, encoded as "recursive repeating sequences" (I don't know exactly how to name them). Each sequence can contain other sequences and has number related ...
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-1 votes
1 answer
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Prove x . y' . z' is y' . z'

Please, how can I prove that x . y' . z' simplifies to y' . z'? I have tried without success. Below is the context of my question; I am taking a course on Coursera
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2 votes
0 answers
71 views

Prove lower bound on boolean circuit

Given matrix $A \in \{0,1\}^{n \times m}$ with $n$ rows and $m = 2^n - 1$ columns. Where $j$-th column is binary decomposition of $j$ ($j = 1 \dots 2^n - 1$). For example, if $n = 3$: $ A = \begin{...
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1 vote
1 answer
63 views

How to find the reverse of a binary string with simple binary operators?

I was wondering is it possible to create a simple circuit that detects if an input (a binary string) is a palindrome? So my approach is to feed the input to a circuit that reverses the input, ie if ...
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2 votes
1 answer
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What can be said about the Fourier expansion of compositions of Boolean functions, if anything?

Let $\chi_S: \mathbb{F}_2^n \to \{\pm 1\}, x \mapsto (-1)^{\sum_{i \in S}x_i}$ for every $S \subseteq [n]$ denote the parity functions from $\mathbb{F}_2^n$ to $\{\pm 1\}$. Then, of course, every $f: \...
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1 vote
0 answers
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Intuition for a 2-Monotone Boolean Function

I am currently studying Chapter 8 of Krama and Hammer's textbook on Boolean Functions, and having a hard time understanding what it means for a function to be "k-monotone." I am currently ...
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3 votes
1 answer
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What's the average number of transistor switches needed to do an N-bit x N-bit multiply?

I want to know how switch-efficient a multiplier can be. If I need to do many $N$-bit by $N$-bit multiplies, and each bit is determined by flipping a coin, what's the average number of transistor ...
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1 vote
0 answers
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What is the intuitive logic behind the working of the Variable Entered K-Map (VEM)?

In this site here they have just said how to minimize a function using VEM. But no intuitive logic behind the same has been stated, making it too mechanical. And I am very bad at memorizing things, so ...
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2 votes
0 answers
34 views

closure property violated by palindrome language

It is well established that palindrome language is non-regular. The one way to prove it is by means of pumping lemma. The other way is violating the closure properties of regular language. The ...
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1 answer
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Functions with small support have small circuits

I have been trying to understand the use of circuit models for boolean functions, and came across this question, that I am trying to struggle to understand: Show that if a function $f\colon \{0,1\}^n→\...
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0 votes
1 answer
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Simplify Boolean expression

The following Boolean expression is simplified into its minimal number of literals: $$x'y' + yz +x'yz' \implies x'+yz.$$ How do you logically conclude this using the Boolean Laws?
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2 answers
70 views

is duality principle in boolean algebra is true for every expression

Let say A = 1 and B = 1 and then A+B = 1 now by using duality(replacing or gate by and gate and 1 by 0) we can say that, A.B = 0 but this is not 0, because 1.1 = 1, so please anyone clear my ...
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3 votes
0 answers
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Circuit complexity of hardest monotone function

Show there exists a monotone function $f\colon \{0,1\}^n \mapsto \{0,1\}$, such that the minimal size of a monotone circuit that computes $f$ is $\Omega(2^n / n^2)$. Use the fact that the number of ...
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Karnaugh Map: does maximal overlap always produce simplest boolean expression?

Suppose I have a 4x4 Karnaugh map with a few cells that are don't cares, and there are two ways of producing 3 groups of 4 cells. One of these ways overlaps groupings more that the other. Is one way ...
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1 vote
1 answer
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Distribution of random Fourier coefficients

Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by $$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
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4 votes
0 answers
241 views

Can every sentence of first-order logic be converted into an equisatisfiable equation in Boolean algebra?

There may be some theoretical literature, unknown to me, that addresses this question. If possible, I would like a practical approach to this problem. My attempt involves the use of an equational ...
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0 votes
1 answer
92 views

For a logic gate to be universal, must it necessarily be able to perform duplication?

It is said that a gate that can simulate AND and NOT is universal and able to recreate any classical circuit. I was looking at some of the circuits simulated by NAND, and for some of them, we need to ...
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Computational complexity in Boolean network

An Boolean control networks can be expressed as \begin{equation} \label{ControlBN} \left\{\begin{array}{l}{x_{1}(t+1)=f_{1}\left(x_{1}(t), \cdots, x_{n}(t), u_{1}(t), \cdots, u_{m}(t)\right),} \\ {x_{...
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