Section: New Results

Scaling Termination Proofs

Participants : Laure Gonnord, Gabriel Radanne [ENS Rennes] , David Monniaux [CNRS/VERIMAG] , Fernando Pereira [Univ. Mineas Gerais, Brasil] , Raphael Rodrigues [Univ. Mineas Gerais, Brasil] .

In  [15] , 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-intractable 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.

In 2014, we consolidated this approach with a work on complexity issues (inspired by  [19] ) and a new implementation: Termite (see Section  5.13 ). A corresponding paper is currently under submission for PLDI.

With Fernando Pereira's group in Brazil, we also studied the relevance of fast and simple solutions to compute approximations of the number of iterations of loops (loop trip count) of imperative real-world programs. The context of this work is the use of these approximations in compiler optimizations: most of the time, the optimizations yield greater benefits for large trip counts, and are either innocuous or detrimental for small ones. In our paper published at WST'14 [10] , we have shown that, most of the time, there is no need to use computationally-expensive state-of-the-art methods to compute (an approximation of) it. We support our position with an actual case study. We show that a fast predictor can be used to speedup the JavaScript JIT compiler of Firefox - one of the most well-engineered runtime environments in use today.