A higher-order function is a function that takes another
function as an argument or
returns another function as a result. More specifically, a
first-order function takes
and returns base types, such as integers or lists. A kth-order
function takes or
returns a function of order k−1. Currying often artificially
order of a function, so we will ignore all inessential currying.
(Whether a particular instance of
currying is essential or inessential is open to debate, but we expect that
will be uncontroversial.) In addition, when calculating the order of a
function, we instantiate all type variables with base types. Under these
most common higher-order functions, such as map and foldr,
are second-order, so
beginning functional programmers often wonder: What good are functions
three or above? We illustrate functions of up to sixth-order with examples
from a combinator parsing library.
Combinator parsing is a classic application of functional programming,
back to at least Burge (1975). Most combinator parsers are based on Wadler's
list-of-successes technique (Wadler, 1985). Hutton (1992) popularized the
idea in his
excellent tutorial Higher-Order Functions for Parsing. In spite
of the title, however,
he considered only functions of up to order three.