### begriffs

Day 5. I’m doing exercises from H-99, and challenging myself to make answers that don’t use named function arguments. Although arguments can help describe a function’s purpose they are often just clutter. In fact uniformly renaming a variable in a function has no effect on the function’s output, a phenomenon known as α-congruence in lambda calculus.

Because this is Haskell we’re talking about, my idle preoccupation (finding a point-free representation of functions) is of course already someone’s PhD thesis. I want to show you a cool tool that will help you strip out extraneous variables and examine the interesting results.

To see our code melt down our monitors and have the variables fall out like loose teeth we need to use GHCi on Acid. If you’re using a mac then turn on, tune in, and drop out.

The installation procedure is slightly different than what I read in the docs. Assuming your machine has homebrew and cabal installed, run this:

brew install pcre
cabal install lambdabot
cabal isntall goa

That will take a while. Now create a ~/.ghci file with the following

:m - Prelude
:m + GOA
:def bs        lambdabot "botsnack"
:def pl        lambdabot "pl"
:def unpl      lambdabot "unpl"
:def redo      lambdabot "redo"
:def undo      lambdabot "undo"
:def index     lambdabot "index"
:def docs      lambdabot "docs"
:def instances lambdabot "instances"
:def hoogle    lambdabot "hoogle"
:def source    lambdabot "fptools"
:def where     lambdabot "where"
:def version   lambdabot "version"
:def src       lambdabot "src"

Your setLambdabotHome will differ from mine. You may only need to change your home directory and the ghc version number to make it work. Worst case just search for the bin folder of lambdabot.

Now, if everything was successful, your GHCi prompt should say Prelude, GOA> and we can begin using the trippy pl command. Let’s start with a low dose and eliminate a variable that we all know shouldn’t be there

Prelude GOA> :pl \x -> succ x
succ

It detected that our anonymous function which takes a value only to pass it to succ behaves no differently from succ itself. The Greek alphabet lovers among you will be happy to know this is called η-conversion. Now a harder example. Let’s make a function similar to (!!) but which uses 1-based indexing. I can think of one that takes two arguments, but let’s let :pl attack it.

Prelude GOA> :pl \xs i -> xs !! (pred i)
(. pred) . (!!)

What an interesting result. Do you think it’s an improvement? I kind of like it, built as it is purely from simple functions. However its meaning doesn’t jump out at me. Let’s think it through.

Fist, what kind of a function is (. pred) and how is it suitable to compose with (!!)?

Prelude GOA> :t (. pred)
(. pred) :: Enum b => (b -> c) -> b -> c
Prelude GOA> :t (!!)
(!!) :: [a] -> Int -> a
Prelude GOA> :t (.pred) . (!!)
(.pred) . (!!) :: [c] -> Int -> c

Starting with the latter, it’s a function which takes a list and returns a new function – which is expecting an Integer argument and will give back a list item. Now (. pred) takes a function, and given this one it will return a function with the same domain. In this case a list. And what is the range? Functions from Int to list items. Hey, that’s the what we wanted originally with our 1-based element getter.

Honestly I can’t say it’s easy for me to think through this strange composition, but many of the point-free results do look pretty and symmetric. For instance, check out how to modify a function’s second or first argument

-- modify the second argument
Prelude GOA> :pl \a b -> f(a (g b))
(f .) . (. g)

-- modify the first argument
Prelude GOA> :pl \a b -> f((g a) b)
(f .) . g

It looks natural this way. Now the :pl command goes beyond stringing together composition and can summon strange functions I don’t know about yet. For instance, look how it rewrites a function that checks if a list is a palindrome.

Prelude GOA> :pl \xs -> xs == reverse xs
ap (==) reverse
Prelude GOA> :pl \xs -> reverse xs == xs
(==) =<< reverse

In either direction we get some crazy moon language! Do any of you, my dear readers, have ways to make sense of the point-free idioms? Perhaps I shouldn’t use the exotic ones and should stick to old fashioned named arguments.