Why is function composition in APL cool?

A short article to highlight a cool feature of APL which doesn't exist in most other programming languages. It assumes absolutely no knowledge of APL.

Key features of APL

There are three features of APL that stand out to me:

1. It's an array language
2. It's very terse: its built-in functions are all one character each
3. It allows for unusual forms of function composition

If you've worked with R or Python's numpy library then the arrays will feel very familiar (and indeed APL inspired both). I don't get much benefit from the one-character functions yet - I'm still new enough to APL that I often have to look them up. And if you're willing to define one-character aliases for your functions, you can write some terse APL-like Python too.

Point 3 is to me the most interesting, because it's more unusual than point 1 and takes less experience to appreciate than point 2. In this article I'm going to try and explain point 3 in a way that makes sense even if you haven't seen any APL before.

Example: Palindromes

Let's use the example of checking if a string is a palindrome to show some interesting APL function composition. In APL we can check if a string is a palindrome with ⌽≡⊢. A result of 1 means 'true' and 0 means 'false'.

      (⌽≡⊢) 'aba'
1
      (⌽≡⊢) 'abc'
0

Each of ⌽, ≡ and ⊢ is a function. Most of the characters APL uses for its built-in functions are unusual characters like those. To concentrate on the function composition instead of the function names let's call them reverse, match and identity instead. (The function identity just returns its argument unchanged.)

      reverse ← ⌽
      match ← ≡
      identity ← ⊢
      (reverse match identity) 'abc'
0

This is saying, 'Does the REVERSE of the argument MATCH the IDENTITY of the argument?'. The argument 'abc' has been passed to identity (which it's next to) but also to reverse (three functions away). This is the feature of APL that I want to highlight as being cool. When an argument x is passed to a sequence of three functions (f g h)x, it's evaluated like g(f(x), h(x)).

Why is this good?

When I saw this for the first time I thought it was pretty bizarre and couldn't see the motivation, especially since it's common to see identity used in these cases. What other language makes common use of an identity function? But it turns out to allow for some very concise programming. Let's demonstrate this by making the palindrome task a bit more complicated: now we want to check whether a string is a palindrome ignoring whitespace.

First we'll define a function remove_spaces. (It's a few APL characters but we'll assign it a single name to make things clearer.)

      remove_spaces ← ~∘' '
      remove_spaces 'a ba'
aba

Now we can do this:

      (reverse match identity) remove_spaces 'a ba'
1

If we re-wrote this into Python-esque pseudocode we'd get a lot more repetition:

match(reverse(remove_spaces('a ba')), identity(remove_spaces('a ba')))

We could remove some of that repetition by defining an intermediate variable spaces_removed = remove_spaces('a ba'). But in APL we don't have to: we can pass around input arguments and intermediate results in a concise implicit way.

This three-function composition is called a 'fork' in APL. Because the middle function is called last, we can write something like (f+g)x and it will be evaluated as f(x)+g(x), which is what we might expect based on standard mathematical notation.

APL also supports several other function composition patterns using explicit 'operator' characters. As a side note, nothing special happens in APL when there are only two functions: (f g)x is evaluated as f(g(x)).

In conclusion

At work I write a lot of Python and no APL. Does knowing about this feature of APL help me when I write Python? Not really. As I write Python code I sometimes think, 'If Python had function composition like APL does, this code could be shorter', and that's about it. But learning about it made me aware of a feature a programming language could have that I hadn't thought of before.

How can I get started with APL?

If reading this has made you interested in APL and you'd like to give it a try, I recommend https://tryapl.org/. It runs APL in the browser, and you can click on the symbols to enter them - no need for a special keyboard or anything.