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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.
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A supercombinator is either a constant, or a [[combinator]] which contains only supercombinators as [[subexpression]]s.
   
''This definition is bewildering until you realize that a [[Combinator]] can have non-Combinator internal subexpressions. A super combinator is "internally pure" (every internal lambda is a combinator) as well as externally.''
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To get a fuller idea of what a supercombinator is, it may help to use the following equivalent definition:
   
Any Haskell program can be converted into super combinators using [[Lambda lifting]].
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Any lambda expression is of the form <code>\x1 x2 .. xn -> E</code>, where E is not a lambda abstraction and n&ge;0. (Note that if the expression is ''not'' a lambda abstraction, n=0.) This is a supercombinator if and only if:
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* the only [[free variable]]s in E are x1..xn, and
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* every lambda abstraction in E is a supercombinator.
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So these are supercombinators:
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* <code>0</code>
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* <code>\x y -> x + y</code>
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* <code>\f -> f (\x -> x + x)</code>
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These are not combinators, let alone supercombinators, because in each case, the variable y occurs free:
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* <code>\x -> y</code>
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* <code>\x -> y + x</code>
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This is a combinator, but not a supercombinator, because the inner lambda abstraction is not a combinator:
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* <code>\f g -> f (\x -> g x 2)</code>
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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:
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* <code>\f g -> f ((\h x -> h x 2) g)</code>
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Because supercombinators have no free variables, they can always be given their own names and [[lifting pattern|let floated]] to the top level. The last example, for example, can be rewritten as:
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* <code>let { scF h x = h x 2; scE f g = f (scF g) } in scE</code>
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Some older compilers for Haskell-like languages, such as [[Gofer]], used this as part of the compilation process. Converting a whole program into supercombinators leaves no internal lambda abstractions left, and each supercombinator can then be compiled more or less directly into a lower-level language, such as the [[G machine]].
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A supercombinator which is not a lambda abstraction (i.e. for which n=0) is called a [[constant applicative form]].
   
See also [[Constant applicative form]]
 
 
[[Category:Glossary]]
 
[[Category:Glossary]]
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[[Category:Combinators]]

Latest revision as of 06:11, 20 July 2010

A supercombinator is either a constant, or a combinator which contains only supercombinators 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)

Because supercombinators have no free variables, they can always be given their own names and let floated to the top level. The last example, for example, can be rewritten as:

  • let { scF h x = h x 2; scE f g = f (scF g) } in scE

Some older compilers for Haskell-like languages, such as Gofer, used this as part of the compilation process. Converting a whole program into supercombinators leaves no internal lambda abstractions left, and each supercombinator can then be compiled more or less directly into a lower-level language, such as the G machine.

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