Difference between revisions of "Super combinator"
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A super combinator is either a constant, or a [[Combinator]] which contains only super combinators as subexpressions. |
A super combinator is either a constant, or a [[Combinator]] which contains only super combinators as subexpressions. |
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| + | To get a fuller idea of what a supercombinator is, it may help to use the following equivalent definition: |
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| − | ''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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| + | Any lambda expression is of the form <code>\x1 x2 .. xn -> E</code>, 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: |
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| + | * the only [[free variable]]s in E are x1..xn, and |
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| − | '''Question:''' Is \x y -> x+y a supercombinator? You could get this by lambda lifting \x -> x+y in some context, but as in Haskell all functions are of single argument only it's really \x -> \y -> x+y, where x is free in the inner lambda. In addition to lambda lifting, do you have to uncurry it to \(x, y) -> x+y? |
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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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| + | A supercombinator which is not a lambda abstraction (i.e. n=0) is called a [[Constant applicative form]]. |
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| − | See also [[Constant applicative form]] |
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[[Category:Glossary]] |
[[Category:Glossary]] |
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[[Category:Combinators]] |
[[Category:Combinators]] |
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Revision as of 03:51, 1 February 2010
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)
A supercombinator which is not a lambda abstraction (i.e. n=0) is called a Constant applicative form.
Any Haskell program can be converted into supercombinators using Lambda lifting.