1 research outputs found
A Lambda Calculus for Transfinite Arrays: Unifying Arrays and Streams
Array programming languages allow for concise and generic formulations of
numerical algorithms, thereby providing a huge potential for program
optimisation such as fusion, parallelisation, etc. One of the restrictions that
these languages typically have is that the number of elements in every array
has to be finite. This means that implementing streaming algorithms in such
languages requires new types of data structures, with operations that are not
immediately compatible with existing array operations or compiler
optimisations.
In this paper, we propose a design for a functional language that natively
supports infinite arrays. We use ordinal numbers to introduce the notion of
infinity in shapes and indices. By doing so, we obtain a calculus that
naturally extends existing array calculi and, at the same time, allows for
recursive specifications as they are found in stream- and list-based settings.
Furthermore, the main language construct that can be thought of as an -fold
cons operator gives rise to expressing transfinite recursion in data, something
that lists or streams usually do not support. This makes it possible to treat
the proposed calculus as a unifying theory of arrays, lists and streams. We
give an operational semantics of the proposed language, discuss design choices
that we have made, and demonstrate its expressibility with several examples. We
also demonstrate that the proposed formalism preserves a number of well-known
universal equalities from array/list/stream theories, and discuss
implementation-related challenges