A Complete Method for the Synthesis of Linear Ranking Functions

We present an automated method for proving the termination of an unnested program loop by synthesizing linear ranking functions. The method is complete. Namely, if a linear ranking function exists then it will be discovered by our method. The method relies on the fact that we can obtain the linear ranking functions of the program loop as the solutions of a system of linear inequalities that we derive from the program loop. The method is used as a subroutine in a method for proving termination and other liveness properties of more general programs via transition invariants; see [PR03].