The array programming paradigm adopts multidimensional arrays as the fundamental data structures of computation. Array operations process entire arrays instead of just single elements. This makes array programs highly expressive and introduces data parallelism in a natural way. Array programming imposes non-trivial structural constraints on ranks, shapes, and element values of arrays. A prominent example where such constraints are violated are out-of-bound array accesses. Usually, such constraints are enforced by means of run time checks. Both the run time overhead inflicted by dynamic constraint checking and the uncertainty of proper program evaluation are undesirable. We propose a novel type system for array programs based on dependent types. Our type system makes dynamic constraint checks obsolete and guarantees orderly evaluation of well-typed programs. We employ integer vectors of statically unknown length to index array types. We also show how constraints on these vectors are resolved using a suitable reduction to integer scalars. Our presentation is based on a functional array calculus that captures the essence of the paradigm without the legacy and obfuscation of a fully-fledged array programming language.
Abstract. Besides element type and values, a multidimensional array is characterized by the number of axes (rank) and their respective lengths (shape vector). Both properties are essential to do bound checking and to compute linear offsets into heap memory at run time. In order to have an array's rank and shape available at any time during program execution both are typically kept in an array descriptor that is maintained at run time in addition to the array itself. In this paper, we propose a different approach: we treat array rank and shape as first-class citizens themselves. Firstly, we use dependent types to reflect structural properties of arrays in the type system. Secondly, we annotate a program with the array explicit array properties wherever necessary. This choice not only renders implicit run time array descriptors obsolete, but exposing all rank and shape computations explicitly in intermediate code also allows us to perform extensive compile time optimisation on them. We have implemented the proposed approach in our experimental array language Qube; preliminary experimental results indicate the suitability of the proposed approach.
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