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Super combinator

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A super combinator is either a constant, or a Combinator which contains only super combinators as subexpressions.

To get a fuller idea of what a supercombinator is, it may help to use the following equivalent definition:

Any lambda expression is of the form \x1 x2 .. xn -> E, where E is not a lambda abstraction and n≥0. (Note that if the expression is not a lambda abstraction, n=0.) This is a supercombinator if and only if:

  • the only free variables in E are x1..xn, and
  • every lambda abstraction in E is a supercombinator.

So these are supercombinators:

  • 0
  • \x y -> x + y
  • \f -> f (\x -> x + x)

These are not combinators, let alone supercombinators, because in each case, the variable y occurs free:

  • \x -> y
  • \x -> y + x

This is a combinator, but not a supercombinator, because the inner lambda abstraction is not a combinator:

  • \f g -> f (\x -> g x 2)

Any lambda calculus expression (or, indeed, Haskell program) with no free variables can be converted into supercombinators using Lambda lifting. For example, the last example can be expressed as:

  • \f g -> f ((\h x -> h x 2) g)

A supercombinator which is not a lambda abstraction (i.e. for which n=0) is called a Constant applicative form.