04/01/2023

In Folha, Viana explains Boole's function.

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Our dear reader is applying for a great job. The company asks a series of questions which the reader must answer in a binary fashion: Yes or No. Based on these answers, the hiring manager decides whether to hire her, also in a binary fashion.

Unfortunately, the answer is no! And the reader wonders—hypothetically, of course!—if it wouldn't have been better to "trick" some of the answers… How many little lies would have been enough to make the result positive?

This type of procedure exemplifies what mathematicians and computer scientists call a "Boolean function," which is a rule for transforming a certain number of binary data points (the reader's answers) into a binary result (the decision about hiring her). The simplest and most interesting example is "negation," which uses only one piece of data: if its value is Yes, the result is No; and if the value of the data point is No, the result is Yes.

The name honors the English mathematician and philosopher George Boole (1815-1864), author of the book "The Laws of Thought," published in 1854. In it, Boole introduced what we now call "Boolean algebra," which is a mathematical formulation of logic. In this theory, binary values can be Yes and No, as in our example, or True and False, or even 1 and 0, which are my class's favorites: it makes no difference.

It's not a particularly difficult area of mathematics: I studied its fundamentals at the beginning of high school. But to say it's extremely important, useful, and current would be an understatement: everything electronic computers do is calculate Boolean functions, and therefore Boolean algebra is at the foundation of all computer science and technology.

The number of changes in the data (the reader's hypothetical little lies) sufficient to reverse the result is called the "sensitivity" of the Boolean function. It's a way to measure how complex the function is. There are others, but computer scientists have shown that all the others give similar results: the only one that resisted fitting into this theory was, precisely, sensitivity.

To read the full text, visit the newspaper's website.