Questions tagged [boolean-algebra]

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25 views

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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1answer
92 views

Can minimal CNF contain clause longer than initial CNF?

Let $\Phi$ be a k-CNF and $\Phi_{min}$ be a minimal CNF (one that contains smallest amount of literal occurences) that is equal to $\Phi$. Can $\Phi_{min}$ contain a clause of size $m > k$? What ...
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22 views

Logic minimization via 2 inputs NOR gates: Is it monotone w.r.t to adding a minterm?

notation: $x+y:=\mbox{OR}(x,y)$, $\bar x:=\mbox{NOT}(x)$, $xy:=\mbox{AND}(x,y)$, 1:=TRUE, 0:=FALSE. Let $f$ be a Boolean function of $n$-variables, i.e. $f: \{0,1\}^n \to \{0,1\}$. minterm:= any ...
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1answer
21 views

Relationship between circuit size and formula size in Sipser text

The Sipser text (3rd edition) contains a proof that 3-SAT is NP-Complete based on Boolean circuits. Part of the proof contains the remark that the reduction from the circuit to the Boolean formula can ...
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4answers
5k views

Why is A(A+B) = A [Absorption Law]?

Could you proof it to me that A(A+B) = A? AA + BA [AA = A] A + AB Then what?
3
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0answers
126 views

Boolean function minimization

Does there exist a Boolean function for which no sum-of-products expression that minimizes the number of products also simultaneously minimizes the number of literals (counting repetitions)? ...
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0answers
42 views

Petrick's Méthod With Maxterms

I recently learnt about Quine-McCluskey and Petrick's methods and they are all okay by me using minterms the procedure is as follows : 1- Reduce the prime implicant chart by eliminating the essential ...
2
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0answers
39 views

Boolean matrix / satisfiability problem [duplicate]

Let $M$ be an $m\times n$ matrix with all elements in $\{1,0\}$, $m >> n$. Let $\mathbf{v}_0, \ldots, \mathbf{v}_n$ be the columns of $M$. I want to find all sets of columns $S = \{\mathbf{v}_{...
3
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3answers
2k views

Measuring Complexity of Boolean Satisfiability Problem

How exactly is the complexity of a SAT solver measured? My main concern is that, for $N$ variables, you can have, e.g., an OR of $O(2^N)$ AND terms, which would take at least $O(2^N)$ time to process. ...
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0answers
16 views

Simulating Boolean Circuit with RAM

Statement: Every $T(n)$ size bounded Boolean circuit family, could be simulated with $(T(n))^2$ time bounded Random Access Turing Machine (RAM). Could you please supply me with a reference to an ...
1
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1answer
40 views

Algorithm for idempotent algebra

A boolean algebra expression can be converted into an idempotent algebra using $$\bar a \equiv 1-a, \qquad a \vee b \equiv a+b -ab, \qquad a \wedge b \equiv a \otimes b$$ where $\otimes$ is the ...
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K Maps: Getting simplified boolean form

I have been trying to work out the simplest Boolean form for the following K-map: I get the following expression: (x2 AND X3) OR (x1 AND x2) OR (x0 AND x2) However, via Boolean algebra, I get (x3 OR ...
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0answers
21 views

Which gates are “pre computation” universal?

In the following, by “functions” I will mean 2 input 1 output Boolean logic functions (for conciseness). A function is called “universal” if by using it (sometimes multiple times, chained together), ...
1
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1answer
28 views

Simplification of a multi-index Boolean expression towards computation in fewer steps

Let $x_{ij} \in \{0,1\}$, $1 \leq i \leq M$ (typically, $M = 2000$), $1 \leq j \leq N$ (typically, $N = 10$), be Boolean variables. If possible at all, I would like to simplify the following ...
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0answers
31 views

Why does this finite state machine state transition diagram solution has more states than my solution?

I can't figure out what is wrong with my solution and why does it differ from book's solution. I think the only thing that matters is the previous state of A so that there should be two states, one ...
2
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1answer
54 views

Addition, multiplication, and apostrophe used to represent boolean algebra expressions?

I'm looking at a worksheet that expresses boolean logic expressions using multiplication, addition, and apostrophes; something I've never seen before. I can make a guess that the apostrophe is ...
1
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1answer
54 views

All 16 Boolean Logic Gates

I was doing some reseach and I came across a website that had listed 16 different boolean logic operators. I was wondering if all of them were real, and if so, what do they do. https://www....
2
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1answer
25 views

Significance of quantifier ordering in quantified boolean formulas (kQBF vs. QBF)

I am studying solvers of quantified boolean formulas (QBF) as a generalization of SAT solving. The standard DIMACS format of SAT specification is extended to QDIMACS, which adds "a ..." and "e ..." ...
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2answers
57 views

Does the naive conversion of a Boolean Formula to CNF have a polynomial or exponential complexity?

I am reading the naive conversion to CNF, this procedure is explaining in this book book, but I have not found a conplexity analysis of this algorithm: elimination of equivalence Elimination of ...
2
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2answers
54 views

Simplifying the Boolean expression $A + \bar{A}\bar{B}$?

So I'm trying to simplify the Boolean expression (1) $A + \bar{A}\bar{B}$. I noticed that by Karnaugh maps this is equivalent to $A+\bar{B}$, and I also noticed that if I take the complement of (1), ...
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1answer
88 views

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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34 views

Can numbers (not sets of numbers) in Linear Integer Arithmetic form a Boolean algebra?

As far as I understood, Boolean algebra is just one of the many first-order theories (1). It has the signature $\{\sqcap, \sqcup, \neg, \bot, \top\}$ and the axioms: associativity, commutativity, ...
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1answer
25 views

Expressing unsigned comparison through signed comparison of 2's complement

Let n > 0 be a natural number and for any two reminders a, b modulo 2^n we have that <...
2
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1answer
72 views

3-CNF to “independent form”

Is it possible to convert all logical formulae into a form such that each variable ends up in exactly 1 "factor" of the and operation? ($\wedge$). Any combination of operations is allowed, though the ...
1
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1answer
44 views

What is the most efficient way to test whether a set $X \subset \{0, 1\}^n$ and its complement $\{0, 1\}^n \setminus X$ are linearly separable?

I am interested in algorithms that have optimal running time, and ideally which are also very easy to implement. If you can also give some tips on how to implement the algorithm(s) you mention in the ...
2
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1answer
111 views

Generate random matrix and its inverse

I want to randomly generate a pair of invertible matrices $A,B$ that are inverses of each other. In other words, I want to sample uniformly at random from the set of pairs $A,B$ of matrices such that ...
2
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1answer
57 views

Conjunctive normal form to simple elementary algebra

I'm curious to know the computational complexity class of each step in this method of converting a CNF formula into simple elementary algebra. An example: $$\phi=\left(x_1 \vee x_2 \right) \wedge \...
3
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1answer
62 views

Polynomial size Boolean circuit for counting number of bits

Given a natural number $n \geq 1$, I am looking for a Boolean circuit over $2n$ variables, $\varphi(x_1, y_1, \dots, x_n, y_n)$, such that the output is true if and only if the assignment that makes ...
3
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1answer
111 views

Is is possible to determine if a given number is xor combination of some numbers?

I have been given a number Y which is ($a$ xor $b$ xor $c$ xor $d$ xor $e$ ) of some numbers ($a$,$b$,$c$,$d$,$e$) and another no X. Now i have to determine if X is a xor combination of ($a$,$b$,$c$,$...
2
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2answers
870 views

Which is the correct XOR Gate Symbol

I'm confused between 2 XOR gate symbols, they have a minor difference but I'd still like to know if they truly are identical. One looks like - The other, like Notice, how for one of them the ...
2
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3answers
15k views

Absorption Law Proof by Algebra

I'm struggling to understand the absorption law proof and I hope maybe you could help me out. The absorption law states that: $X + XY = X$ Which is equivalent to $(X \cdot 1) + (XY) = X$ No problem ...
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0answers
50 views

Iterating over a union of sets denoted by bitmasks

Consider the set $\mathbb{B}^n$ of all $n$-digit binary numbers. Let us define a bitmask as a tuple $M=(m_0,\ldots,m_{n-1})$, where $m_i\in \{0,1,*\}$. Such bitmask denotes a set $S \subset \mathbb{B}^...
3
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2answers
120 views

Is every X3SAT instance with no cycles satisfiable?

Exactly 1 in 3 SAT (X3SAT) is a variation of the Boolean Satisfiability problem. Given a set of clauses, where each clause has three literals, is there an assignment such that in each clause exactly ...
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2answers
399 views

how do we demonstrate using Boolean algebra that these NAND and NOR gate combinations are XOR gates

How do we demonstrate using Boolean algebra that these NAND and NOR gate combinations are XOR gates?
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2answers
104 views

Prime Implicants in Boolean Function

A Boolean function $F(X_1, X_2, X_3, X_4, X_5, X_6)$ of six variables is defined as $F = 1$, when three or more input variables are at logic 1. otherwise 0. How many essential prime implicants does F ...
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2answers
2k views

Converting truth table to algebraic normal form

Is there any efficient algorithm to convert a given truth table of a Boolean function to its equivalent algebraic normal form (ANF)? I have seen that Sage has one implementation (official ...
1
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1answer
59 views

Modeling equality in an ILP

Lets say we have integer variables $a \in\mathbb{Z}^n$ and $M \in\{0,1\}^{n\times L}$. I am promised $a_i \leq L$, for some fixed constant $L$. I want to model the constraint $$M_{i,j} \iff (a_i=j)$...
2
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1answer
51 views

Logic of multiple variables in ILP

Is there a better way to represent an AND of $n$ variables together other than creating $O(n)$ new variables and constraints?
2
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1answer
74 views

how many boolean functions exist that satisfy the condition

How many boolean functions exist that satisfy the following condition? $$\neg f(x_1,x_2,x_3,....,x_n) = f(\neg x_1, \neg x_2,...,\neg x_n)$$
2
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1answer
266 views

State Transition Diagrams for XOR

I have found two different versions for State Transition Diagrams for XOR. I'm confused as to why one has 2 states and the other has 3. What are the implied states in each please? Are they both ...
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2answers
87 views

Why is Boolean satisfiability such a rare case?

In the space of all K-sat formulas, True and False should have an equal set size. For every un-Satisfiable formula (F), there will an F' (or F-prime) which will be Satisfiable by definition. I cannot ...
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1answer
2k views

Boolean Algebra : Using identities, prove x(x + y) = x

Example: Prove the ABSORPTION LAW: $$ x(x + y) = x $$ $ Solution: \\ x(x + y) \\ = (x + 0)(x + y) \;\;\;\;\; Identity \;Law \\ = x + (0 · y) \;\;\;\;\;\;\;\;\;\;\; Distributive \;Law \\ = x + y · 0 ...
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0answers
191 views

How do I get the NAND gate configuration for full adder from the logic table?

I'm self-studying, but I've gotten stuck already. If I'm given the logic table for a full-adder or any two-output table, how do I figure out the NAND-gate configuration, preferably methodically? ...
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1answer
44 views

Truth table and logic circuit problem

Does anyone know how to make this into a truth table and a logic circuit? (x•y)+x If so please send me the answer.
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0answers
89 views

Can I use the Quine-McCluskey to simplify a CNF which is not a product of maxterms?

As I understand it the Quine-McCluskey method allows you to start with a set of maxterms (or minterms), and combine them pairwise in a systematic way into a smaller set of clauses with a smaller set ...
2
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2answers
562 views

Absorption rule in Boolean algebra

I am confused regarding the absorption rule which states: A OR (A AND B) = A. I do not completely understand how the expression simplifies to A and while i have seen proofs for this question, i still ...
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1answer
171 views

Simplifying SOP: implementing OR with NAND

I am learning how to implement basic logic gates using NAND. I have learnt that you can use De Morgan's theorem as such: $a+b = \bar{\bar a} + \bar{\bar b} = \overline{(\bar a *\bar b)}$ In other ...
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1answer
52 views

How to “logically” solve boolean logic

I came across of one excellent book Elements of Computing Systems and in chapter 1 we need to implement the "primitive" logic gates (for example: Not, And, Or, Xor, Mux, etc) based on a Nand gate. ...
4
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1answer
90 views

Why are there two not operators in lambda calculus?

From Wikipedia: $\mathrm{true} = \lambda a. \lambda b. a$ $\mathrm{false} = \lambda a. \lambda b. b$ Because true and false choose the first or second parameter they may be combined to ...
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1answer
1k views

Number of literals in the given boolean expression

Count the number of literals in the following expression : F = AB' + BC' + CD' + DE' According to me, the answer should be 8. But my solution suggests that the answer should be 6. Can anyone help me ...