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What is the name of λx.λf.fx in lambda calculus?

Does the corresponding function have a standard name in functional programming languages, like Haskell?

In object oriented programming, is there a usual name for a method foo which takes a function as an argument, such that x.foo(f) returns f(x)?

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You need to buy a dictionary. –  Robert Harvey May 17 '13 at 23:19
Which one? Where? How much? –  Alexey May 17 '13 at 23:21
I do not understand the downvotes. It seems a perfectly reasonable question to me. –  Giorgio May 18 '13 at 8:15
I have posted an answer below but I realised that the OP was asking what is the name of this lambda expression so I deleted =D –  TemporaryNickName May 18 '13 at 8:52
The concept you refer to is known as a combinator, have a read here if you're not familiar en.wikipedia.org/wiki/SKI_combinator_calculus though I'm not familiar with a known combinator of the nature you refer to though, C from BCKW is close (it's flip) but not the same en.wikipedia.org/wiki/B,C,K,W_system why not have some fun and come up with the combination of those combinators that results in your function :) –  Jimmy Hoffa May 18 '13 at 13:50
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3 Answers

In Haskell, \x.\f.f x is flip ($) which as $ is read as apply, I would read as reverse apply.

Roughly based on http://stackoverflow.com/questions/4090168/is-there-an-inverse-of-the-haskell-operator

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This wrong... flip is \f.\x.\y.f y x flip is C here en.wikipedia.org/wiki/B,C,K,W_system –  Jimmy Hoffa May 18 '13 at 13:46
also adding $ to flip like this is expecting a partially applied function that takes over 2 arguments where you're going to only flip the last 2 so you partially apply all but those 2 –  Jimmy Hoffa May 18 '13 at 13:52
So you doubt that (\f.\x.\y.f y x) (\f.\x.f x) reduces to (\x.\f.f x) modulo renaming? Which it does. –  Dan D. May 19 '13 at 0:02
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Using the standard combinators you can express this function as



C f x y = (f y) x

In Haskell it would be

flip id

Here id's type is specialized to (a -> b) -> a -> b and flip swaps a -> b and a.

You can also express it in the SKI calculus:

S (K (S I)) K
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+1 for deducing the combinators for this. I really don't suspect it's a much known thing, though if as he identified it's something related to church numerals, I'm inclined to wonder if CI is documented anywhere as relevant in anyones church numeral studies –  Jimmy Hoffa May 19 '13 at 7:17
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up vote 3 down vote accepted

In the original Alonzo Church's "The calculi of lambda-conversion", he denotes this combinator by T.

In the context of Church numerals, it is called exp. This matches the "shorthand" notation used by Church for the "application" of a term N to a term M: "[MN]" stands for "(NM)".

I haven't heard yet of a standard name for an analogous function in programming.

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