Share Your Experience: Take the 2024 Developer Survey

# Questions tagged [boolean-algebra]

The tag has no usage guidance.

282 questions
Filter by
Sorted by
Tagged with
1 vote
50 views

### Is there a linear programming method that is polynomial in the number of variables, constraints and bitlength of numbers?

AFAIK, Interior Point method for solving a system of linear inequations is polynomial in the number of variables and constraints. Probably there are others. I don't need to optimize any function (...
83 views

• 1,664
65 views

### Prove or disprove that the Quine-McCluskey method generates the circuit with the minimum inputs and minimum gates?

Recently, when I self-learnt Discrete Mathematics and Its Applications 8th by Kenneth Rosen, it says in 12.4 Minimization of Circuits which uses the Karnaugh Map or the Quine-McCluskey method: ...
• 203
82 views

### Prove that it is impossible to construct the toffoli gate using only CNOT gates

Show that it is not possible to construct the toffoli gate using only CNOT gates, given we are allowed to choose any number of ancilla bits. My Attempt The action of a toffoli gate can be defined as, ...
• 139
59 views

### Is there an algorithm to implement N-input-gates using smaller gates?

To borrow part of a description from a similar but distinct question: there exist 2^(2^N) different functions which accept N binary inputs and return a 1 bit output. For the purposes of my question I ...
62 views

### what is the smallest 3-CNF possible that enforces the boolean expression: a = b + c?

What is the smallest 3-CNF system of equations possible that enforces the boolean expression: a = b + c for boolean variables a, b, c? 'Smallest' can be defined as: (1) Number of 3 CNF clauses. (2) ...
• 789
1 vote
23 views

### CNF Horn-renamability to 3-CNF Horn-renamability reduction?

A CNF formula is Horn-renamable if you can invert variables in such a way that each clause has at most one positive literal. There is an algorithm based on a reduction to 2-SAT given in Renaming a Set ...
• 1,664
70 views

### Number of n-variable symmetric boolean functions that are linear

How many symmetric boolean functions exist that are linear? Let $f$ be a $n$-variable boolean function. $f$ is said to be symmetric if it is unchanged by any permutation of its variables, i.e. for 2-...
• 33
32 views

### Expressing Boolean Functions In Terms of Another Function

Given two boolean functions f1 and f2, are there any tools available that could be used to automate the process to represent f1 in terms of f2? I understand the process of this doing this by hand ...
712 views

### Representing binary functions with a finite gate set without exponential blow-up?

It is pretty well taught that any binary function can be represented using CNF. But conversion to CNF can take an exponential number of gates. The truth table is exponentially sized relative to the ...
• 327
80 views

### 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 ...
26 views

1 vote
96 views

### 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 ...
• 11
46 views

### 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:
187 views

### 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.
22 views

### 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 ...
1 vote
92 views

### 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 ...
131 views

### 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 ...
94 views

### 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 ...
1k views

### 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)....
112 views

### 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 ...
• 113
1 vote
297 views

### 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 ...
1 vote
82 views

### 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 ...
1 vote
57 views

### 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}?
• 11
1 vote
35 views

### 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 ...
• 123
1 vote
247 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 ...
• 13
1 vote
23 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 ...
• 11
1 vote
69 views

### 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 ...
• 123
67 views

### 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 ...
• 429
1 vote
56 views

### 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.
43 views

### 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.
• 113
1 vote
31 views

### 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 ...
• 111
29 views

### 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 ...
33 views

### 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 $x\in{0,1}$. ...
• 121
56 views

### 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 ...
• 123
78 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 ...
• 53
1 vote
50 views

### 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 ...
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\} .\$...