Section: New Results

Synthesis of Ranking Functions using Extremal Counter-Examples

Participants : David Monniaux [Verimag, Grenoble] , Lucas Séguinot [Student at ENS Cachan Bretagne] , Laure Gonnord.

In  [14] , we presented a new algorithm adapted from scheduling techniques to synthesize (multi-dimensional) affine functions from general flowcharts programs. But, as for other methods, our algorithm tried to solve linear constraints on each control point and each transition, which can lead to quasi-untractable linear programming instances.

In contrast to these approaches, we proposed a new algorithm based on the following observations:

• Searching for ranking functions for loop headers is sufficient to prove termination.

• Furthermore, there exist loops such that there is a linear lexicographic ranking function that decreases along each path inside the loop, from one loop iteration to the next, but such that there is no lexicographic linear ranking function that decreases at each step along these paths. For these reasons, it is tempting to treat each path inside a loop as a single transition.

Unfortunately the number of paths may be exponential in the size of the program, thus the constraint system may become very large, even though it features fewer variables. To face this theoretical complexity, even though the number of paths may be large, we argue that, in practice, few of them actually matter in the constraint system (we formalize this concept by giving a characterization as geometric extremal points). Our algorithm therefore builds the constraint system lazily, taking paths into account on demand.

We are currently testing our preliminary implementation and submitting a paper on these new results.