On Termination of Integer Linear Loops

We consider the problem of determining termination of single-path loops with integer variables and affine updates. The problem asks whether such a loop terminates on all integer initial values. This problem is known to be decidable for the subclass of loops whose update matrices are diagonalisable. In this paper we show decidability of determining termination for arbitrary update matrices, but with a single inequality as the loop guard. Our decision procedure relies on number-theoretic results concerning Diophantine approximation. For the class of loops considered in this paper, the question of deciding termination on a specific initial value is a longstanding open problem.