Dealing with arithmetic overflows in the polyhedral model

The polyhedral model provides techniques to optimize Static Control Programs (SCoP) using some complex transformations which improve data-locality and which can exhibit parallelism. These advanced transformations are now available in both GCC and LLVM. In this paper, we focus on the correctness of these transformations and in particular on the problem of integer overflows. Indeed, the strength of the polyhedral model is to produce an abstract mathematical representation of a loop nest which allows high-level transformations. But this abstract representation is valid only when we ignore the fact that our integers are only machine integers. In this paper, we present a method to deal with this problem of mismatch between the mathematical and concrete representations of loop nests. We assume the existence of polyhedral optimization transformations which are proved to be correct in a world without overflows and we provide a self-verifying compilation function. Rather than verifying the correctness of this function, we use an approach based on a validator, which is a tool that is run by the compiler after the transformation itself and which confirms that the code produced is equivalent to the original code. As we aim at the formal proof of the validator we implement this validator using the Coq proof assistant as a programming language [4].