Polyranking for Polynomial Loops

Although every terminating loop has a ranking function, not every loop has a ranking function of a restricted form, such as a lexicographic tuple of polynomials over program variables. We propose polyranking functions as a generalization of ranking functions for analyzing termination of loops. We define lexicographic polyranking functions in a general context and then specialize their synthesis and application to loops with polynomial guards and assignments of polynomial expressions. Finally, we demonstrate that our prototype implementation for C source code scales to large software projects, proving termination for a high percentage of targeted loops.