Questions tagged [boolean-algebra]

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3
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1answer
88 views

Design an algorithm,have polynomial complexity for deciding satisfiability of a 1-conjective Normal Form boolean formula

I undetstand each part of the word group in this question. I have search for a while but I still can't understand what the whole question want me to do. I will state what I know and give an assumption ...
1
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1answer
108 views

Boolean expression to logic gates

Hello I need help with xy xor z, do I do the logic AND first or the XOR, no parenthesis. Thank You
2
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1answer
25 views

How to fill cell-network partition matrix of a function?

I'm trying to understand a paper (Tandem Networks of Universal Cells, Butler, 1978 1), but I can't make it past the first paragraph: Consider the $x_l … x_{k - 1} | x_k$ partition matrix of a ...
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1answer
193 views

Why is the inner product of -1,+1 binary variables at most $n-2$ and not at most $n-1$?

In short, if $x \neq u_i \in \{\pm1 \}^n$ then why is: $$ \langle x, u_i \rangle \leq n-2 $$ but not: $$ \langle x, u_i \rangle \leq n-1 $$ ? To add context: I was reading understanding machine ...
2
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1answer
161 views

Simplifying this Boolean Expression

I have to simplify A+C'+B'CD but I don't see how. I had to deduce the expression starting from a logic diagram in which two AND gates were used for the B'CD part. Seeing the diagram all I can think ...
2
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3answers
16k 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 ...
4
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1answer
3k views

What is a simple way of explaining what a linear boolean function means in boolean algebra and relating it to the standard definition of linearity?

I was reading notes on computability theory when I came across the term "Linearity" which I was not familiar with, in the context of boolean functions. I am quite comfortable what linear maps mean in ...
2
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0answers
52 views

substituting expressions

I have a set of expressions $E_1 .. E_n$ over boolean variables and I'm looking for an assignment to the variables so that all expressions are satisfied. Normally this would be NP-complete, but I ...
5
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2answers
1k views

Boolean absorption

A and ( A or C ) = A And A or A and C = A How do these identities work? Using the rule A and ( B or C ) = A and B or A and C For the first identity, I get A and A or A and C = A or A ...
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1answer
424 views

What is the simplification of AB + BC + (~B)C?

AB + C is not the answer. The correct answer is AB + BC. HOW?
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2answers
197 views

Is 2QBF in P^NP?

2QBF is the following problem: given a CNF formula $\psi$ on $2n$ variables, determine the truth value of $$\forall x \in \{0,1\}^n . \exists y \in \{0,1\}^n . \psi(x,y).$$ Question: Is 2QBF in $P^{...
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1answer
855 views

How to simplify sum of products boolean expression?

I started with this sum of products: abc’d’ + abc’d + ab’cd’ + a’b’cd’ + a’bc’d + a’bcd + ab’c’d + a’b’c’d I have been able to simplify to this: ...
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1answer
93 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). ...
2
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1answer
258 views

Random forests on monotone training set yields a monotone classifier?

Suppose I train a random forests classifier on a monotone training set. Is the resulting classifier guaranteed to be a monotone function? Suppose I apply the ID3 algorithm (the greedy algorithm) to ...
2
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1answer
92 views

Boolean Algebra: Operation Linkage Laws

Im studying some properties of Boolean Algebra and found the Operation Linkage Laws. Im not able to understand how this laws are possible, and the remark on the proof is not really clear for me. Can ...
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4answers
6k 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?
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1answer
805 views

Extracting Boolean Function using Machine Learning

How can machine learning help extract a Boolean relationship in a given binary input-output data set? Let us assume that the given data set is exhaustive - ie. it cover all possible input ...
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3answers
127 views

How does the following truth table show Y's behaviour? [closed]

a b c Y 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 0 ...
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0answers
163 views

how to evaluate a boolen expression

what is an easy but efficient way to evaluate Boolean Expression. I have been searching a lot in google but not really finding a good book or site which has working algorithm. Input string (A=100) ...
1
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1answer
269 views

Is Functional Complete means Turing Complete?

I noticed that AND, OR, NOT those three logic gates are Functionally Complete, it means I can represent any trues table only by those three gates. A Turing machine may halt or not in a particular ...
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1answer
86 views

How to reduce C′A′B + CAB′ to C′B + CB′?

I faced this Boolean expression: $C'(A'B+A)+C(A'+AB')$ It was solved as follows: $C'(A'B+A)+C(A'+AB')$ $=C'(A+B)+C(A'+B') $ ...by applying absorption laws $(I)$ $=C'A+C'B+CA'+CB'$ $=(C\oplus ...
3
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1answer
609 views

Relation between Lattice and Boolean Algebra

In discrete math, I have read that lattice is a generalized form of boolean lattice. But those places where boolean algebra is mentioned, they don't tell about lattices (digital logic, binary,...). ...
4
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2answers
1k views

Simplest combination of logic gates to produce a given set of outputs

Given a truth table for a truth function that takes n inputs and produces a single output (true or false), what is the fastest way to find the simplest combination of logic gates that will output the ...
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2answers
2k views

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 ...
3
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4answers
1k views

Proving that $A \vee (\neg A \wedge B) \equiv A \vee B$

I'm reading a book at the moment about logic gates and Boolean simplification. There is a part which I can't seem to follow. I can easily work out that $A \vee (\neg A \wedge B) \equiv A \vee B$ ...
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
37 views

How storage affect if we use tertiary language? [duplicate]

as we uses binary language in all the computer systems around us which consist of 0s and 1s (Off and On) bits. Let assume that someone is developing a computer system based on tertiary system -1, 0, 1 ...
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2answers
634 views

How to prove the following xor equation?

In a test paper, a question is given as : ...
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3answers
4k views

Sum of 3 integers with full adder

1)Is it possible for a full adder to add three e.g 4 bit numbers? I mean I know the full adder has 3 inputs and two outputs but the second bit of C comes from the previous block (as shown in the image ...
2
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1answer
567 views

Algorithm for simplifying ANF or polynomials?

I have some digital logic circuits in Algebraic Normal Form, and am limited to using XOR and AND logic gates. For instance: $B_{out} = B_1 B_2 \oplus B_1 B_3$ I was wondering, are there any ...
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2answers
92 views

How do I simplify this boolean expression?

I have constructed truth tables to prove that: $ABC + ABC'+ AB'C +A'BC = AB+AC+BC$ How do I prove it by simplifying the expression? I know that I can simplify: $ABC + ABC' = AB(C+C')=AB$. However I ...
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1answer
121 views

Boolean expression logic law confusion

I've been trying to attempt a particular question that I need to translate truth table into boolean expression but I'm completely stuck on one point now. First, I worked it out by using Sums of ...
1
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1answer
57 views

Counterexample for equality in Quantified Boolean Formula

I'm looking for a counterexample for the following (false) equality. It should exist. $n$ is an odd integer. $f(x_1, x_2 .. x_n)$ is a boolean expression in 3 CNF form with boolean variables x. $A =...
0
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1answer
598 views

Question about idempotent and Dominance Laws in Boolean Algebra [closed]

If I have the following statements: For Idempotent: Since X * X = X, would that imply that ~X * ~X = ~X For Dominance: Since X + 1 = 1 would that imply that ~X + 1 = 1
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2answers
50 views

Boolean algebra truths with more than one digit

I understand that when you 0 AND 0 it will result in 0 and that 1 AND 1 will result in 1 etc but what I don't understand is if the question was 01 AND 11. How would I work this out?
2
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1answer
52 views

About a universal (functionally complete) function producing a constant

I read in a note the following: Suppose we have a boolean function $f(x,y,z,w)$, if $f(a,a,a,a)=1$ then $f$ can't be functionally complete. Why is that? how does it imply that $f$ can't produce ...
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1answer
94 views

Question simplifying boolean expression

This is what I have so far, so I just want to know if I am doing everything right or wrong. I am novice in the topic. I have been given: $(A \oplus B) \land \neg A \lor A$ XOR = $\oplus$ By order ...
2
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1answer
142 views

Help me understand the logic behind x - y in binary by boolean?

x and y are 4 bit signed numbers (2s complement..) x - y can be obtained by: !(!x + y) I know that in 2s complement ...
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1answer
384 views

Simplifying Boolean Expression

I am designing a 4-bit comparator with a look ahead unit using a bit slice approach. I have to break the propagation of the Logical expressions for (A<B)i and <...
4
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1answer
442 views

What happens to uninterpreted predicates in Ackermann's reduction?

I know the procedure to apply the Ackermann's reduction to a formula that doesn't involve uninterpreted predicates. But, how do we treat the uninterpreted boolean predicates? Nearly all the examples I ...
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2answers
252 views

Which formula corresponds to this K-Map?

I have a K-Map and I need to figure out which expression isn't equivalent to the provided K-Map. $f(w,x,y,z) = \sum(3,7,9,11,13,15) + \Phi(4,5,6)$ We know that both options (a) and (b) are ...
4
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1answer
389 views

What is the XNOR of 3 or more inputs?

We know that for 3 variables (A=0,B=1,C=1), f$_1$ = (A XNOR B XNOR C) = 1, since the input has even number of 1's. But if we were to do this step by step, f$_2$ = (A XNOR (B XNOR C)) = (A XNOR (1 ...
2
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1answer
428 views

Result of ANDing & ORing functions expressed as CDNF & CCNF

I came across following fact: If we AND two functions expressed as CDNF (Canonical Disjunctive Normal Form), then the result contains sum of commont minterms. For example consider two functions $...
2
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1answer
32k views

Two's complement Using ONLY Logic Gates

How can a 4-bit two's complement operation be implemented using only boolean logic gates (AND, OR, NOR, NOT, NAND, XOR, and XNOR)? (This question was redirected to CS from Stack Overflow)
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2answers
1k views

How to simplify boolean expression? [closed]

I am having trouble simplifying logical expressions to a much simpler form, can someone provide me some insight on how to approach the problem? Let's assume i have the following expression: $ABCD + A\...
4
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1answer
316 views

Number of solutions to linear system of equations over GF(2)

Linear systems of equations over the reals have either 0, 1 or infinitely many solutions. However, when applied to finite fields (specifically GF(2)), infinitely many is not an option. Is there a ...
2
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2answers
146 views

Converting a digital circuit to two layers of OR and AND gates

The other day someone mentioned to me that you could take an arbitrary digital circuit which mapped N input bits to M output bits, and replace it with a layer of OR gates and a layer of AND gates. I ...
1
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1answer
85 views

How to Apply Binary Transformations?

One of the steps in Binary Index Tree algorithm is to find a node's parent which is done by un-setting the rightmost SET bit. For example: if a node has index 1010 than it's parent is 1000. To apply ...
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2answers
5k views

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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1answer
1k views

Implementing a Boolean function with NOR gates

Implement f(a, b, c, d) = Σ m(3, 4, 5, 6, 7, 11, 15) as a 2-level gate circuit (a) Using OR gates and NOR gates. (b) Using NOR gates only. I have found that F=ab+d using Karnaugh map. I have also ...