Section: New Results

Program Equivalence Modulo A/C (Associativity/Commutativity)

Participants : Guillaume Iooss [PhD student] , Christophe Alias, Sanjay Rajopadhye [Colorado State University] .

Program equivalence is a well-known problem with a wide range of applications, such as algorithm recognition, program verification, and program optimization. This problem is also known to be undecidable if the class of programs is rich enough, in which case semi-algorithms are commonly used. We focus on programs represented as a system of affine recurrence equations (SARE), defined over parametric polyhedral domains, a well-known formalism for the polyhedral model, which includes as a proper subset, the class of affine control loop programs. Several semi-algorithms for program equivalence have already been proposed for this class. A few of them take into account algebraic properties such as associativity and commutativity. However, to the best of our knowledge, none of them is able to manage reductions, i.e., accumulations of a parametric number of sub-expressions using an associative and commutative operator.

Our contributions are:

• An equivalence checking algorithm able to manage associativity and commutativity properties. Our method subsumes the previous approaches and is, to the best of our knowledge, the first one able to manage these properties over a parametric number of expressions.

• A semi-algorithm to construct a perfect matching problem on a parametric bipartite graph. We partially solve this problem through a heuristic based on the augmenting path algorithm. This heuristic is able to find a set of non-interfering augmenting paths to improve a proposed maximum matching, as long as these augmenting paths do not have a parametric length.

A preliminary implementation is under development. This work has been submitted to ESOP'14.