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A combinatory account of internal structure. (English) Zbl 1248.03025

Summary: Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure-complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure-complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

MSC:

03B40 Combinatory logic and lambda calculus
03B70 Logic in computer science
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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