Beautiful Code

This bit of code for converting numbers into their Roman numeral equivalents was posted by geezusfreeek on reddit, and I found it so alarmingly pretty that I'm going to spend some time to break it down. This exercise is largely for my own edification, but perhaps somebody else will learn a bit along the way. Here's the code:

import List
import Control.Arrow

romanize = concat . unfoldr next
    where next 0 = Nothing
          next x = Just $ second (x-) $ head $ filter ((<=x) . snd) numerals
          numerals = [("M", 1000), ("CM", 900), ("D", 500), ("CD", 400),
                      ("C", 100),  ("XC", 90),  ("L", 50),  ("XL", 40),
                      ("X", 10),   ("IX", 9),   ("V", 5),   ("IV", 4),
                      ("I", 1)]

Starting from the inside out is usually the easiest method of understanding blocks like this, so we'll look first at the "numerals" list. It is, simply, a list of tuples matching Roman numerals to their value. Importantly, it's ordered from largest to smallest; we'll see why shortly.

The meat of this snippet comes in the "next" function:

next 0 = Nothing
next x = Just $ second (x-) $ head $ filter ((<=x) . snd) numerals

This function is going to return either Nothing (in the base case) or Just a tuple of type (String, Integer), where the string is the largest roman numeral that's smaller than x and the int is the difference between x and the value of that roman numeral.

The first thing we do is filter out the numerals larger than x; we can run some quick tests at the interactive console to see how this works:

Prelude> let numerals = [("M", 1000), ("CM", 900), ("D", 500)]
Prelude> filter ((<=900) . snd) numerals
[("CM",900),("D",500)]

We use head to grab the largest numeral we can squeeze into the current value:

Prelude> head $ filter ((<=900) . snd) numerals 
("CM",900)

Then we subtract the value of the roman numeral from the current value of x, in order to return the remainder to be operated on next:

Prelude> :m Control.Arrow
Prelude Control.Arrow> second (900-) $ head $ filter ((<=900) . snd) numerals
("CM",0)
Prelude Control.Arrow> second (915-) $ head $ filter ((<=915) . snd) numerals
("CM",15)

And, finally, we wrap it up in a Maybe monad, so that the main function can handle the base case transparently:

Prelude Control.Arrow> Just $ second (915-)  $ head $ 
    filter ((<=915) .  snd) numerals
Just ("CM",15)

And now we're getting pretty close! What we have is a function that returns either Nothing or a tuple containing a letter to add to our roman numeral and the next value to operate on. All we have left is the code to drive the operation:

romanize = concat . unfoldr next

The use of unfoldr helps explain why we've wrapped next up in Maybe; here's its type:

Prelude Control.Arrow> :m List
Prelude List> :t unfoldr
unfoldr :: (b -> Maybe (a, b)) -> b -> [a]

So we'll take a function from b -> Maybe (a, b), a value of type b, and return a list of [a]. In our case, a = String and b = Integer.

All unfoldr does is call the function it's given as a first argument, which returns a tuple or Nothing. If it's a tuple, it appends its first element to a list and calls the same function with the second element. If it's Nothing, it returns the list it's built up to this point.

As we can see, that matches up perfectly with the "next" function we've already defined, and the unfoldr will build up a list of roman numeral strings:

Prelude> :m Control.Arrow List
Prelude Control.Arrow List> let numerals = [("X", 10), ("IX", 9), ("V", 5), 
("IV", 4), ("I", 1)]
Prelude Control.Arrow List> let next x = if x==0 then Nothing               
    else Just $ second (x-) $ head $ filter ((<=x) . snd) numerals
Prelude Control.Arrow List> unfoldr next 9
["IX"]
Prelude Control.Arrow List> unfoldr next 8
["V","I","I","I"]

(We had to mangle the definition of next a bit to fit into the interactive interpreter, but we haven't changed anything fundamental about it.)

Finally, all we have to do is concatenate the strings into our final result:

Prelude Control.Arrow List> (concat . unfoldr next) 8
"VIII"

Conclusion

Phew - that's an awful lot of information stuffed into a few pretty lines! I still have a hell of a time geting the types to match up so that I can write stuff like this, but that doesn't mean that I can't appreciate the beauty of how it all fits together. The original function presented here is tight in the very best sense of the word.