Symbolic Evaluation of Chains of Recurrences for Loop Optimization

This paper presents a novel method for loop optimization that exploits symbolic evaluation of chains of recurrences (CRs). The method generalizes loop induction expression recognition, loop parallelization by induction variable substitution, loop strength reduction , and loop invariant expression elimination (code motion). Symbolic diierencing of loops has been extensively studied by Haghighat for these type of transformations. The diierencing method detects generalized loop induction variables and approximates the corresponding closed-form functions by polynomials bounded to a predetermined maximum order. We will show that compared to diierencing methods, symbolic evaluation with CRs is safe, more powerful, and simpler to implement. We prove that CRs are unique normal forms, which is crucial for the use of CRs as intermediate program representations of recurrences. Based on the algorithms developed by Bachmann, Zima, and Wang, we develop a CR-based algorithm for loop analysis. We present a set of symbolic transformations that deene a partial mapping of CRs to closed-form functions. We also demonstrate the potential of the method for analyzing power series computations in loops. Finally, we demonstrate that the CR intermediate program representation provides a very eeective means for non-linear loop dependence testing.