Eric at 12:49, 27 June 2016

2016-06-27T12:49:45Z

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			Jerermy Gibbons: APLicative Programming with Naperian Functors

	A lot 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. But it is 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. develops a custom static type system for an array-oriented language. 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, and moreover, generative type-driven programming can use 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.		A lot 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. But it is 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. develops a custom static type system for an array-oriented language. 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, and moreover, generative type-driven programming can use 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.

Eric: Created page with "A lot 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..."

2016-06-27T12:49:02Z

Created page with "A lot 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..."

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A lot 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. But it is 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. develops a custom static type system for an array-oriented language. 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, and moreover, generative type-driven programming can use 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.

WG211/M16Gibbons - Revision history

Eric at 12:49, 27 June 2016

Eric: Created page with "A lot 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..."