Much of the expressive power of array-oriented languages such as Iverson’s APL and J comes from their implicit lifting of scalar operations to act on higher-ranked data, for example to add a value to each element of a vector, or to add two compatible matrices pointwise. It is considered a shape error to attempt to combine arguments of incompatible shape, such as a 3-vector with a 4-vector. APL and J are dynamically typed, so such shape errors are caught only at run-time. Recent work by Slepak et al. (ESOP 2014) develops a custom type system for an array-oriented language, statically ruling out such errors. We show here that such a custom language design is unnecessary: the requisite compatibility checks can already be captured in modern expressive type systems, as found for example in Haskell; moreover, generative type-driven programming can exploit that static type information constructively to automatically induce the appropriate liftings. We show also that the structure of multi-dimensional data is inherently a matter of Naperian applicative functors - lax monoidal functors, with strength, commutative up to isomorphism under composition - that also support traversal.

I am Professor of Computing in the Department of Computer Science at the University of Oxford. I am currently Director of the Software Engineering Programme, which offers part-time professional Masters’ degrees in Software Engineering and in Software and Systems Security. I also lead the Algebra of Programming research group. I am Editor-in-Chief of the Journal of Functional Programming, Past Vice Chair of ACM SIGPLAN, Past Chair of IFIP WG2.1. Before taking up this post in 1999, I held lectureships at Oxford Brookes University and the University of Auckland, New Zealand.