Questions tagged [boolean-algebra]
The boolean-algebra tag has no usage guidance.
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Polynomial representations of Boolean functions
The AND boolean function $AND(x)$ can be represented using the polynomial $P(x) = x_1x_2\cdots x_n$. I have a few questions:
Is there a similar polynomial for the PARITY boolean function?
Is there a ...
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Why do Karnaugh maps work?
The question is quite straightforward: Why do Karnaugh maps work? What was the reasoning that led Maurice Karnaugh to come up with these maps? At first glance, it doesn't seem a natural approach, ...
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Equivalent-symmetric (E-symmetric) variables check for Boolean functions
In [1], it states that checking $f(..., x_i, ..., \bar{x}_j, ...) = f(..., \bar{x}_j, ..., x_i, ...)$ (variables $x_i$ and $x_j$ are E-symmetric) is equivalent to checking $f_{x_ix_j} = f_{\bar{x}_i\...
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Why is conjunctive normal form (CNF) "better" for SAT than disjunctive normal form (DNF)?
When hand-manipulating algebra DNF (sum of products) is easier than
CNF (product of sums). Possibly because factoring is more difficult
than expanding. So why is it the opposite for computational ...
CommunityBot
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Complexity Class of the Problem: Existence of Unsatisfying Interpretations in Boolean Formulas
What is the complexity class of the problem if there exist two different interpretations that do not satisfy a given Boolean formula? I believe the problem of existence of an interpretation that does ...
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SOP Boolean expression of state variable and output of a Moore FSM
Disclaimer: I'm studying for an exam, but I don't have to hand in anything.
I have the state transition diagram of the FSM and its states encoding:
I want to find the sum-of-products (SOP) Boolean ...
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Boolean Integer Linear Optimization/Programming
Trying to solve an ILP optimization problem with a number of potential boolean variables and then express constraints on these variables based on those boolean results.
Let's say I am doing 5 coin ...
CommunityBot
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Why do simple Logical Gates have an Irrational amount of Bits?
Suppose $2$ bits are used to encode a message, A and B.
If you know $A$ is $1$, you have one bit of information.
If you know $A\land B$ is $1$, you have two bits of information.
If you know $A\land B$...
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!(ab)*(!a+b)(!b+b)=!a Boolean Algebra
How can I prove that !(ab)(!a+b)(!b+b)=!a* in boolean algebra? This is an exercise from a past paper of my teacher but I can't really find the way to the answer. I would appreciate some help.
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How to solve boolean SAT with equality constraints
Say I have boolean formula in form of a CNF(x1,x2,...) with $x_i$ being boolean variables.
Testing the satisfiability of the CNF is the SAT problem, i.e. determine ...
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Prove HAKMEM Item 23: connection between arithmetic operations and bitwise operations on integers
Prove that for $A, B \in \mathbb{Z}$, $A + B = (A \operatorname{\&} B) + (A \mid B) = (A \oplus B) + 2(A \operatorname{\&} B)$ where $\&$ is bitwise AND, $|$ is bitwise OR and $\oplus$ is ...
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Is this a valid method to merge clauses in CNF formulas?
Assume $$C_1 = (x_1 \lor x_2 \lor \lnot x_3)\hspace{0.2cm} C_2 = (x_4 \lor x_5 \lor x_3) \hspace{0.2cm} C_3 = (x_3 \lor x_5 \lor x_6)$$ Let
$$ \phi_1 = C_1 \land C_2 \land C_3$$ and
$$ \phi_2 = (x_1 \...
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Is it possible to write an AND gate using XOR gates?
How could I express an AND gate using only XOR gates ?
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Is this language PSPACE complete
Prove $PSPACE$-completeness of the language $READALLEXACT$ = $\{$$(M, x, 1^ s , t)$ | $A$ deterministic Turing machine $M$ on input $x$ reads all bits of the input in exactly $t$ steps and using no ...
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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 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 ...
CommunityBot
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7 segment decoder Combinational Logic Circuits using Logic Gates
How can a 7 segment decoder operation be implemented using boolean NOR gate ONLY?
Question:
Here's my truth table and k-map:
Normal circuit diagram:
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Two's complement using logic gates
How can a 4-bit two's complement operation be implemented using boolean NOR gate?
I search lots of 4-bit two's complement videos and articals, but most of them are using XOR gate.
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Reed-Muller Form
I'm having trouble with the Reed-Muller Form, I'm using this notation to make it easier: + = or, * = and, ^ = xor.
This is the given expression (a + !b) * (b + !c).
a + !b = a + (b ^ 1) = a ^ b ^ 1 ^ ...
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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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Method for simplifying complex logic tables
Not sure if this is the right StackExchange site, but back in college (20 years ago) I took a Digital Systems Design class where we learned how to reverse engineer a boolean function to meet the ...
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Boolean Logic when one component switches from 0 to 1
I recently was constructing boolean logic for all sorts of examples from Morris Mano's "Digital Logic and Circuit Design".
I noticed that it is possible to construct a boolean logic wrt the ...
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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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Why can't 3-SAT be solved efficiently if you convert all clauses (x ∨ y ∨ z) into (u ∨ z) by introducing a variable?
Let $a_i$, $b_i$, etc., be a literal, i.e., a variable or the negation of a variable.
3-SAT concerns formulas in CNF form: $(a_1 \vee a_2 \vee a_3) \wedge \dots \wedge (b_1 \vee b_2 \vee b_3)$ (3-CNF)....
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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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Combinational logic check if bits is prime
I wonder if there's Digital Logic Circuit (using combinatorial logic gates) that check if number is prime or not.
For example given input fixed 8-bit that will produce 1-bit output.
00000101 will ...
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Time complexity of negating a CNF formula
Suppose i have a CNF formula. If i negate the CNF formula, then i obtain DNF formula. However, i can't find anywhere on internet that mention the time complexity. What is the time complexity of ...
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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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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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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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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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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
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FALSE
I'm told that this is short ...
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How do i simplify this SOP expression?
Hi i have derived the following SoP (Sum of Products) expression , by analyzing the truth table of a 3 bit , binary to gray code converter. I ask for verification, because i feel as though this answer ...
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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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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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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 $x\in{0,1}$.
...
D.W.♦
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Why can't we reduce register automata to symbolic automata?
When considering automata, they are normally considered over finite alphabets. The following are two types of automata that can handle words over infinite alphabets (definitions included for clarity).
...
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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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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 ...
D.W.♦
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What does it mean to "show algebraically" in propositional logic?
The biconditional operator $\iff$ of Propositional Logic can be defined by the identity
$p \iff q \equiv (\lnot p \lor q) \land (\lnot q \lor p) \quad (1.1)$
Use the identity $(1.1)$ and identities ...
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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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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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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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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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