" There are many bits and pieces of folklore in mathematics that are passed down from
advisor to student, or from collaborator to collaborator, but which are too fuzzy and non …

We consider two decision problems for linear recurrence sequences (LRS) over the
integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the …

Linear recurrence sequences permeate a vast number of areas of mathematics and
computer science. In this paper, we survey the state of the art concerning certain …

We consider the decidability and complexity of the Ultimate Positivity Problem, which asks
whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are …

Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem asks
whether all terms of the sequence are positive. We show that, for simple LRS (those whose …

A sequence of real numbers is said to be positive if all terms are positive, and ultimately
positive if all but finitely many terms are positive. In this article we survey recent progress on …

This paper investigates the problem of reasoning about non-linear behavior of simple
numerical loops. Our approach builds on classical techniques for analyzing the behavior of …

We consider the following decision problem: given a finite Markov chain with distinguished
source and target states, and given a rational number r, does there exist an integer n such …

The goal of this chapter is to demonstrate that undecidable decision problems arise naturally
in many branches of mathematics. The criterion for selection of a problem in this survey is …

A fundamental problem in program verification concerns the termination of simple linear
loops of the form: where x is a vector of variables, u, a, and c are integer vectors, and A and …