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Questions tagged [boolean-algebra]

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20
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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 ?
14
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7answers
152k views

How to construct XOR gate using only 4 NAND gate?

xor gate, now I need to construct this gate using only 4 nand gate ...
9
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2answers
2k views

Are Boolean functions Turing complete

A Boolean function is a function $f:\{0,1\}^n\rightarrow\{0,1\}$. The boolean basis $(\vee,\wedge)$ is known to be Turing complete as it allows any sequence $s\in\{0,1\}$ to be flipped or to be left ...
6
votes
1answer
193 views

Is there an intuitive proof for the existence of hard functions?

I am referring to the theorem on page 115 of the book by Arora and Barak, which states that, ``For every $n>1$, there exists a function $f:\{0,1\}^n \rightarrow \{0,1\}$ that cannot be computed by ...
6
votes
1answer
55 views

Small world theorem for set constraints

Let $S_1,\dots,S_n$ be variables representing unknown sets. A set expression has the form $S_i$, $\overline{E}$ (the complement of $E$), or $E \cap E'$, where $E,E'$ are set expressions. A ...
6
votes
0answers
118 views

Boolean formula that agrees with most truth assignments

Let $X_1,\dots,X_n$ be $n$ boolean variables. I have an unknown predicate $P(X_1,\dots,X_n)$ on these boolean variables. Of course, I can view the predicate as a function $f_P : \{0,1\}^n \to \{0,1\}...
5
votes
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 ...
5
votes
2answers
318 views

Sensitivity and Block sensitivity

May be this question is really silly and obvious but I am missing something subtle. I am reading on Sensitivity and Block sensitivity. Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a Boolean function. Let ...
5
votes
1answer
665 views

Which CNF boolean formulas blow up exponentially at conversion to DNF?

If I'm correct, some boolean formulas in CNF require exponential size when being converted to an equivalent DNF version (and vice versa). But what is an example of such a formula (and is there a ...
5
votes
2answers
1k 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 ...
5
votes
1answer
188 views

Finding a graph-theoretic representation of expressions in Boole's algebra

I just read "Boole's Algebra Isn't Boolean Algebra" by Theodore Halperin (behind a paywall here). I don't have a strong background in abstract algebra, so, frankly, the paper is a bit over my head but ...
4
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1answer
73 views

A universal operator necessarily generates $\neg x$ for input $x,…,x$

I originally posted this on math.stackexchange, but then deleted it and moved it here since I think it would fit this site more. I saw a claim in a slideshow from a basic computer architecture course ...
4
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1answer
315 views

Issue understanding the reduction of SAT to 3-SAT in poly time

Reading this http://classes.soe.ucsc.edu/cmps102/Spring10/lect/17/SAT-3SAT-and-other-red.pdf, I came to know that reducing a clause $C_i$ from a $SAT$ instance containing more than 3 literals to a $3-...
4
votes
1answer
75 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 ...
4
votes
1answer
368 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 ...
4
votes
1answer
246 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 ...
4
votes
1answer
214 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 ...
4
votes
1answer
59 views

Common parse tree for several formulas

I have a large (~1k) number of boolean formulas like: f1(x) = p1 AND p2 f2(x) = (p1 AND p2) OR p3 f3(x) = p4 OR !p5 The argument x is a set, and the predicates (...
4
votes
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 ...
4
votes
1answer
61 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 ...
3
votes
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
votes
2answers
1k 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. ...
3
votes
2answers
118 views

Classical Computation without NOT

Is it possible to do universal classical computation using bits and 2-bit gates when you cannot perform a NOT operation on a single bit (hence cant do CNOT and so on). If yes, what are the possible ...
3
votes
2answers
81 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 ...
3
votes
1answer
524 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,...). ...
3
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2answers
4k views

Boolean algebraic expression vs Propositional logic expression

There is a lot of similarity between Propositional logic and Boolean algebraic expressions. Similar aspects : 1) Both has variables of two states. 2) Operations of Boolean algebra and ...
3
votes
1answer
81 views

Programmatically checking equivalence of statements

So as part of a theorem-prover/checker, I'm using Prolog to try to determine the equivalence of statements that have been parsed into tree form, e.g. $x=2$ is represented as ...
3
votes
1answer
31 views

Learning a small disjunction using an input distribution of our choice

I have a boolean function $f: \{0,1\}^n \to \{0,1\}$ that I know takes the form $$f(x_1,\dots,x_n) = x_{i_1} \lor x_{i_2} \lor \dots \lor x_{i_k}.$$ I don't know the values of $i_1,\dots,i_k$, but I ...
3
votes
1answer
132 views

Couting Self dual functions

The Dual of a Boolean function $F(x_1, x_2, ..., x_n)$, written as $F^D$ is the same expression as that of $F$ with $+$ and $.$ swapped. $F$ is said to be self dual if $$F=F^D$$ How can we count ...
3
votes
1answer
639 views

Karnaugh map with don't care: increasing the number of groups instead of simplifying

AB 00 01 11 10 00 | x | 1 | 0 | 1 | CD 01 | 0 | 1 | x | 0 | 11 | 1 | x | x | 0 | 10 | x | 0 | 0 | x | The answer to the ...
3
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0answers
34 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
votes
0answers
104 views

Simple example of exponential gap between monotone and non-monotone circuits

Is there a simple example of a Boolean function $f:\{0,1\}^m \to \{0,1\}$ that we know can be computed by a polynomial-size circuit, but cannot be computed by any polynomial-size monotone circuit? ...
2
votes
2answers
8k 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 ...
2
votes
2answers
237 views

Is my simplified explaination of the XOR swap correct?

The XOR swap is a well-known in-place algorithm to swap two values, by XOR:ing them bitwise. It goes as follows: a = a ^ b b = a ^ b a = a ^ b Now, I was ...
2
votes
1answer
138 views

Can the Euclidean distance function be computed using only XOR's

The Eulidean distance function $d$ of $x$ and $y$ is given by: $ d(x,y)=\sqrt{x^2-y^2} $ Let us assume that $x$ and $y$ are fixed-point numbers, or $x,y$ are element of some subfield $f_n$ of $F_p$. ...
2
votes
3answers
106 views

Is $A\odot B\odot C = A\oplus B\oplus C$?

(The notations used: $\oplus$ is XOR operator $\odot$ is XNOR operator) I was solving a problem, where they asked which of the given options give equation for the difference of full subtractor. ...
2
votes
1answer
21 views

Emulating equal operator using multiplication

I have two values $A$ and $B$, I want to know if I can implement the equals $=$ operation as the product of the two values. I can apply any function to $A$ and any function to $B$, but I need to use ...
2
votes
1answer
55 views

Are there any techniques for checking whether a clause is subsumed by another clause when adding it to a cnf formula?

When doing variable elimination on a formula in cnf form, there is created a lot of new clauses. Is there any efficient way to check if these are subsumed by other, already existing clauses?
2
votes
2answers
110 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 ...
2
votes
2answers
461 views

Karnaugh map simplification

I'm working through an example that looks like a fairly simple Karnaugh map and simplification, but I feel stupid that I can't seem to understand the correct answer. This is the map: My groupings: ...
2
votes
1answer
45 views

Intuition for maxterms

I understand that in terms of minterms, F (Boolean Function) = Sum of Products and thus will yield true when either of the products is true. But I am unable to develop any intuition for maxterms, ...
2
votes
1answer
174 views

How to find minimum number of k-input LUTs needed to express a n variable boolean function?

An k-input LUT (look up table) takes in atmost k-inputs and gives 1 output (which is a function of the k inputs). I need to devise an algorithm to find the minimum number of k-input LUT's required to ...
2
votes
1answer
471 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 ...
2
votes
2answers
43 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 ...
2
votes
1answer
36 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
votes
1answer
30 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 ...
2
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1answer
642 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 ...
2
votes
1answer
187 views

Infinite Boolean circuits as a model of computation

Boolean circuits are non-uniform models of computation in that they require a different circuit for each length of input. The typical way of uniformizing a family of Boolean circuits is to define a ...
2
votes
1answer
31 views

2CNF with 3 variable occurences

If we do not allow unit clauses, can 2CNF containing 3 occurences per variable be unsatisfiable? Every time I try to connect opposing variables to a cycle, I need to draw 4 edges for at least one ...
2
votes
1answer
50 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 ...