A combinatory account of internal structure

Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. Some of this expressive power is captured by adding a factorisation combinator. It supports structural equality, and more generally, a large class of generic queries for updating of, and selecting from, arbitrary structures. The resulting combinatory logic is structure complete in the sense of being able to represent pattern-matching functions, as well as simple abstractions. §