Termination of Linear Loops over the Integers

Abstract: We consider the problem of deciding termination of single-path while loops with integer variables, affine updates, and affine guard conditions. The question is 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, but the general case has remained open since being conjectured decidable by Tiwari in 2004. In this paper we show decidability of determining termination for arbitrary update matrices, confirming Tiwari's conjecture. For the class of loops considered in this paper, the question of deciding termination on a specific initial value is a longstanding open problem in number theory. The key to our decision procedure is in showing how to circumvent the difficulties inherent in deciding termination on a fixed initial value.