# Blog: Mathematics

Formal Verification: History and Methods

In the first part of our series about formal verification, we discuss the roots of formal verification in fields such as mathematical logic and theoretical computer science.

Incomplete and Utter Introduction to Modal Logic, Pt. 2

In the first part, we introduced the reader to basic modal logic. In this part of the introduction to the modal logic, we observe use cases and take a look at connections of modal logic with topology, foundations of mathematics, and computer science.

Incomplete and Utter Introduction to Modal Logic, Pt. 1

Modal logic covers such areas of human knowledge as mathematics (especially topology and graph theory), computer science, linguistics, artificial intelligence, and philosophy. Explore this branch of logic together with Danya Rogozin.

Insane Properties of the Closed Long Ray and the Long Line

In topology, the long line, or Alexandroff line, is a space somewhat similar to the real line, but ‘longer’. To obtain the long line, one needs to put together a long ray in each direction. Closed long rays, as well as long lines, have remarkable properties.

Independent Functions or How to Create the Worst Random Number Generator

Random numbers are used in cryptography, and most of the cryptographic operations use computers. But a computer is a deterministic device, thus, it isn’t able to simply generate a truly random number. There are different approaches to solving the problem, and some of them are worse than others.

The Problem of Intermediate Recursively Enumerable Turing Degrees

The Turing degree of a set of natural numbers is a concept from computer science and mathematical logic that is a measure of the level of algorithmic unsolvability of the set. This post carries you deeper into the problem of the undecidable languages and the halting problem.

The Greatest Challenge of Modeling Large Quantum Computers

Perhaps you cannot quite picture how quantum computers work, but you definitely heard something about them. Nowadays, all rich, as well as not-so-rich states and corporations, are trying to build one. However, many face a problem of inability to emulate a large quantum computer. Let us figure out why.