AlphaZ and the Polyhedral Equational Model

With the emergence of multi-core processors, parallelism has gone main-stream. However, parallel programming is difficult for many reasons. Programmers now must think about which operations can legally be executed in parallel, when to insert synchronizations, and so on. In addition, parallelism and non-determinism nature of it makes debugging much harder. One approach to address this problem is automatic parallelization, where programmers still write sequential code, and it is left to the compiler to parallelize the program. However, automatic parallelization is extremely difficult, and even after decades of research it still remains largely unsolved. An alternative approach is to develop parallel programming languages, designed to write parallel programs from the beginning [1], [2], [3], [4], [5], [6]. One area where automatic parallelization has been successful is regular and dense computations that fit the polyhedral model. Although the applicable class of programs are restricted, fully automatic parallelization has been achieved for polyhedral programs. Approaches based on polyhedral analyses are now part of production compilers [7], [8], and many research tools [9], [10], [11], [12], [13], [14] that use the polyhedral model have been developed. The polyhedral model has its origins in reasoning of systems of equations [15]. When programs are defined as equations, there is no notion of memory or execution order. What needs to be computed are defined purely in terms of values produced by other equations. The connection between loop programs and equational representations was later made by Feautrier [16] through array dataflow analysis. Array dataflow analysis gives precise valuebased dependence information, which can be used to take loop programs into equational view. This equational view is synonimous to the polyhedral representation of programs that the polyhedral compilers manipulate. By giving concrete syntax to polyhedral representations of programs, we have an equational language to specify polyhedral programs. In this paper, we illustrate some of the benefits of equational programming, from perspectives of both users and compilers. We also present the AlphaZ [17] system that provides polyhedral analyses, transformations, and code generators for such equational language.