The Universe of Discourse
Tue, 03 Jan 2012

Eta-reduction in Haskell and English
The other day Katara and I were putting together a model, and she asked what a certain small green part was for. I said "It's a thing for connecting a thing to another thing."

Katara objected that this was a completely unhelpful explanation, but I disagreed. I would have agreed that it was an excessively verbose explanation, but she didn't argue that point.

Later, it occurred to me that Haskell has a syntax for eliding unnecessary variables in cases like this. In Haskell, one can abbreviate the expression

        λx → λy → x + y
to just (+). (Perl users may find it helpful to know that the Perl equivalent of the expression above is sub { my ($x) = @_; return sub { my ($y) = @_; return $x + $y } }.) This is an example of a general transformation called η-reduction. In general, for any function f, λxf x is a function that takes an argument x and returns f x. But that's exactly what f does. So we can replace the longer version with the shorter version, and that's η-reduction, or we can go the other way, which is η-expansion.

Anyway, once I thought of this it occurred to me that, just like the longer expression could be reduced to (+), my original explanation that the small green part was "a thing for connecting a thing to another thing" could be η-reduced to "a connector".

Perhaps if I had said that in the first place Katara would not have complained.

Happy new year, all readers.

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