Induction Variable Analysis with Delayed Abstractions

This paper presents the design of an induction variable analyzer suitable for the analysis of typed, low-level, three address representations in SSA form. At the heart of our analyzer is a new algorithm recognizing scalar evolutions. We define a representation called trees of recurrences that is able to capture different levels of abstractions: from the finer level that is a subset of the SSA representation restricted to arithmetic operations on scalar variables, to the coarser levels such as the evolution envelopes that abstract sets of possible evolutions in loops. Unlike previous work, our algorithm tracks induction variables without prior classification of a few evolution patterns: different levels of abstraction can be obtained on demand. The low complexity of the algorithm fits the constraints of a production compiler as illustrated by the evaluation of our implementation on standard benchmark programs.