Ranking function synthesis is a key aspect to the success of modern termination provers for imperative programs. While it is well known how to generate linear ranking functions for relations over (mathematical) integers or rationals, effcient synthesis of ranking functions for machine-level integers (bit-vectors) is an open problem. This is particularly relevant for the veri cation of low-level code. We propose several novel algorithms to generate ranking functions for relations over machine integers: a complete method based on a reduction to Presburger arithmetic, and a template-matching approach for prede ned classes of ranking functions based on reduction to SAT- and QBF-solving. The utility of our algorithms is demonstrated on examples drawn from Windows device drivers.

}, URL = {http://www.kroening.com/publications/view-publications-ckrw2010.html}, author = {Byron Cook and Daniel Kroening and Philipp Ruemmer and Christoph Wintersteiger} }