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Traditional optimizing compilers rely on rewrite rules to iteratively applyprogram transformations. This iterative approach hides optimizationopportunities behind intermediate transformation steps. For instance,vectorization can only be applied to the innermost loop in a nest: one mustfirst perform a loop interchange before even considering vectorization of anouter loop. In contrast, we propose an implementation framework representingprograms as sets of possible implementation decisions. Specifying one decisioncan have an impact on others in a bidirectional manner: specifying that a loopmust be vectorized prevents other loops from being nested inside it;conversely, specifying a loop as an outer loop will prevent it from beingvectorized. These optimization decisions commute, obviating the pass orderingproblem. We present a constraint programming system to formally define,represent and explore such implementation spaces. We also propose anexploration strategy combining tree search and branch-and-bound; the strengthand novelty of this strategy reside in an analytical model of the lower boundon the execution time of a set of possible implementations. We showcase ourapproach on the construction and exploration of an implementation space forlinear algebra kernels running on GPUs. We show this search space is expressiveenough to represent complex decisions that fundamentally change the structureof the generated code. We also present preliminary results competitive with theperformance of native GPU libraries.

On the Representation of Partially Specified Implementations and its Application to the Optimization of Linear Algebra Kernels on GPU