Array Types for a Graph Processing Language

Graphs are frequently represented and manipulated in the form of arrays encoding their adjacency matrices. Programs that operate on these arrays, for example, using linear algebra operations to encode graph algorithm primitives, can benefit from optimization techniques such as operation and loop fusion, data layout tuning, and other classical compilation methods. To support the construction of sound transformations and optimizations we present HAL (Hierarchical Array Language), a typed language for array processing. HAL provides first class index spaces, programming via type-directed transformations, and an expressive type-system that allows for encoding data structure properties. We show laws that can be derived for HAL to establish a basis for many semantics preserving transformations. This work also briefly introduces methods for expressing computations over these types via higher order functions.