Translation validation of loop and arithmetic transformations in the presence of recurrences

Compiler optimization of array-intensive programs involves extensive application of loop transformations and arithmetic transformations. Hence, translation validation of array-intensive programs requires manipulation of intervals of integers (representing domains of array indices) and relations over such intervals to account for loop transformations and simplification of arithmetic expressions to handle arithmetic transformations. A major obstacle for verification of such programs is posed by the presence of recurrences, whereby an element of an array gets defined in a statement S inside a loop in terms of some other element(s) of the same array which have been previously defined through the same statement S. Recurrences lead to cycles in the data-dependence graph of a program which make dependence analyses and simplifications (through closed-form representations) of the data transformations difficult. Another technique which works better for recurrences does not handle arithmetic transformations. In this work, array data-dependence graphs (ADDGs) are used to represent both the original and the optimized versions of the program and a validation scheme is proposed where the cycles due to recurrences in the ADDGs are suitably abstracted as acyclic subgraphs. Thus, this work provides a unified equivalence checking framework to handle loop and arithmetic transformations along with most of the recurrences -- this combination of features had not been achieved by a single verification technique earlier.