analysis of boolean functions
Автор: CodeBeam
Загружено: 2025-06-17
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Analysis of Boolean Functions: A Comprehensive Tutorial
This tutorial dives deep into the analysis of Boolean functions, covering various aspects from their representation to their properties and applications. We'll explore different representations, learn how to manipulate and simplify Boolean functions, and finally, delve into essential concepts like sensitivity, influence, and Fourier analysis. We'll also provide Python code examples to illustrate these concepts and enable practical application.
*1. Introduction to Boolean Functions*
A Boolean function (also called a switching function or a logical function) is a function that takes n Boolean variables as input and produces a single Boolean value as output. A Boolean variable can take only two values: True (1) or False (0). Mathematically, a Boolean function f with n inputs can be represented as:
`f : {0, 1}^n - {0, 1}`
This means f maps an *n*-dimensional vector of 0s and 1s (representing the inputs) to either 0 or 1.
*Examples:*
*AND function (f(x, y) = x AND y):* `f(0, 0) = 0`, `f(0, 1) = 0`, `f(1, 0) = 0`, `f(1, 1) = 1`
*OR function (f(x, y) = x OR y):* `f(0, 0) = 0`, `f(0, 1) = 1`, `f(1, 0) = 1`, `f(1, 1) = 1`
*XOR function (f(x, y) = x XOR y):* `f(0, 0) = 0`, `f(0, 1) = 1`, `f(1, 0) = 1`, `f(1, 1) = 0`
*Majority function (f(x, y, z) = (x AND y) OR (x AND z) OR (y AND z)):* Returns 1 if at least two of its inputs are 1.
*2. Representations of Boolean Functions*
Boolean functions can be represented in various ways. Each representation has its advantages and disadvantages depending on the application. We'll cover the most common representations:
*Truth Table:* A truth table lists all possible input combinations and the corresponding output of the function. For n input variables, the truth table will have 2sup*n*/sup rows.
*Boolean Expression:* A Boolean expression uses logical operators (AND, OR, NOT) and variables to describe the func ...
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