## Function composition through operator overload

Let me show you how to do function composition through operator overloading, and other cool tricks!

Here is some Python 🐍 code using

• decorators
• callable classes
• custom operators
• “functional” function composition
• arbitrary args and kwargs
>>> class F:
...     def __init__(self, f):
...         self.f = f
...     def __call__(self, *args, **kwargs):
...         return self.f(*args, **kwargs)
...     def __or__(self, other):
...         return F(lambda *args, **kwargs: other(self(*args, **kwargs)))
...
>>> @F
...     return n + 2
...
>>> @F
... def divide_3_floor(n):
...     return n // 3
...
>>> s = F(str)
>>> f = add_two | divide_3_floor | s
>>> f(7)
'3'

Here is a breakdown of everything that is going on 👇

Let us start... from the end!

>>> f = add_two | divide_3_floor | s
>>> f(7)
'3'

The assignment to f takes 3 functions:

• add_two
• divide_3_floor
• s

and then combines them.

The | is in charge of doing this combination, but what combination is this?

The | is doing some “magic” I asked it to do, this is not default behaviour in Python.

f = g | h means that the function f corresponds to calling h after calling g.

So, f(x) actually means h(g(x)).

In my code, I wrote f = add_two | divide_3_floor | s.

So, when I write f(7), it actually means something else:

>>> f = add_two | divide_3_floor | s
>>> f(7)
'3'
'3'

Formally speaking, the operator pipe | is being used to do function composition.

In other words, by using | I have another way of composing functions in a convenient way.

But I lied 😅 add_two, divide_3_floor, and s, aren't “functions” per se...

Did you notice how I defined s?

s = F(str)

I created s by taking the built-in str and putting it through F.

But what IS F?

It's just a class I created, right?

But it's a very special class 😉

The class F only expects an argument f.

class F:
def __init__(self, f):
self.f = f
# ...

I used a lowercase f, not because I wanted to be mysterious, but because f is mathematician-speak for a function.

So, to create an instance of F, I just feed it a function.

So, s = F(str) creates an F out of str.

Then, the instance of F just stores that function in self.f.

Why?

For later use 🙃

The idea is that instances of F behave like the functions passed in originally.

So, if an instance of F must behave like a function, it must be callable.

Call-huh?

In other words, after I write s = F(str), I need to be able to write s(...).

I need to be able to call s (hence, callable).

That is why I needed to implement the dunder method __call__.

class F:
# ...
def __call__(self, *args, **kwargs):
return self.f(*args, **kwargs)
# ...

The dunder method __call__ is responsible for making instances of F callable.

So, what's the behaviour of s when we call it with s(...)?

To understand that, we need to look at __call__.

First, let us understand the signature:

def __call__(self, *args, **kwargs):
# ...

What do the *args* and **kwargs do?

Think with me:

Instances of F must act like the original functions, right?

But the original functions might be arbitrary functions, right?

The original function can be a function with 0 or 3 arguments.

It can be a function with an argument called this_is_just_an_argument.

We can't guess how many/what arguments we'll have...

So, we have to plan for any and all cases!

How do we do that?

With *args and **kwargs.

By using *args, a function can receive however many positional arguments:

>>> def func_with_args(*args):
...     print(args)
...
>>> func_with_args()
()
>>> func_with_args(15, True)
(15, True)

Similarly, by using **kwargs, a function can receive however many keyword arguments:

>>> def func_with_kwargs(**kwargs):
...     print(kwargs)
...
>>> func_with_kwargs(a=15, b=True)
{'a': 15, 'b': True}
>>> func_with_kwargs()
{}

Oh, by the way, here is an important remark:

This cool behaviour comes from using * and ** on the left of the parameters.

The words args and kwargs are just the common choice in Python!

We can pick other names:

>>> def func_with_args_kwargs(*banana, **split):
...     print(banana)
...     print(split)
...
>>> func_with_args_kwargs()
()
{}
>>> func_with_args_kwargs(15, True, stuff=None, info="hey")
(15, True)
{'stuff': None, 'info': 'hey'}

Now we can understand the signature of __call__:

def __call__(self, *args, **kwargs)

But there is more!

What about the body of the definition?

Why do we use *args and **kwargs again..?

class F:
# ...
def __call__(self, *args, **kwargs):
return self.f(*args, **kwargs)
# ...

By using *args and **kwargs again, we are trying to unpack the arguments we got so far.

By unpacking those arguments, we pass them down to self.f, which is the original function:

>>> s = F(str)
>>> s(15)
'15'

Can you trace the journey that the 15 makes?

One good thing you can do is modify __call__ to print args and kwargs before the return.

It's also something I do very often:

I tweak the code I'm studying to see what is really going on.

Now we know what is going on with __call__...

But what's up with __or__..?

The dunder method __or__ is what is called when we use | next to instances of F.

But what gives?

By implementing __or__, we hijack the operator | and we get to use it for whatever we want!

In this case, we want | to be able to combine instances of F with each other.

How do we do that?

“Simple”:

Remember that Fs are supposed to look like functions.

When we combine two Fs, we want to create a new one that represents the two Fs in succession.

Let us use the built-ins int and bin to see what happens when we do F(int) | F(bin).

>>> int("15")  # String to integer.
15
>>> bin(15)    # Integer to binary representation.
'0b1111'

When we write F(int) | F(bin), we want to combine the F(int) with the F(bin).

Also, when we write F(int) | F(bin), the dunder __or__ is called.

When that happens, we go inside the __or__ that looks like this:

class F:
# ...
def __or__(self, other):
return F(lambda *args, **kwargs: other(self(*args, **kwargs)))

What's what?

Inside __or__, after writing F(int) | F(bin), these are the arguments:

• self is F(int); and
• other is F(bin).

We want the combination to be an F that calls bin after calling int:

>>> int_then_bin = F(int) | F(bin)
>>> int_then_bin("15")
'0b1111'

We want the function inside other after the function inside self.

And we want the whole thing to be an F.

Why?

So that we can later combine that again if we want to!

That explains part of what is going on:

class F:
# ...
def __or__(self, other):
return F(...other(self(...)))

What are we missing?

Not too much, now!

Remember how instances of F are created?

We need to feed F with a function.

So, in order to combine F(int) with F(bin), which are self and other, respectively, we need to wrap them in a function.

We could define a regular function with def and whatnot, but I opted for something leaner.

By using a lambda, I can write a function that wraps self and other:

>>> self = F(int)
>>> other = F(bin)
>>> func = lambda *args, **kwargs: other(self(*args, **kwargs))
>>> func("15")
'0b1111'

In the end, we just add the F around that so that the lambda can also be combined with other things.

The lambda, by itself, is a regular lambda.

If I put it inside an F, we get one of these cool things that can be combined with each other:

>>> self = F(int)
>>> other = F(bin)
>>> func = lambda *args, **kwargs: other(self(*args, **kwargs))
>>> func
<function <lambda> at 0x000001E60733E940>
>>> F(func)
<__main__.F object at 0x000001E6073169D0>

The final piece of the puzzle is the @F on top of the definitions of add_two and divide_3_floor:

>>> @F
...     return n + 2
...
>>> @F
... def divide_3_floor(n):
...     return n // 3
...

By using the keyword def I can create regular Python functions, right?

But I want these to be cool functions I can combine.

So, they need to go through F.

But how?

They could've gone through F like str did:

>>> def add_two(n):
...     return n + 2
...
>>> def divide_3_floor(n):
...     return n // 3
...
>>> divide_3_floor = F(divide_3_floor)

Because I only care about the cool version of these functions, I can reuse the same name.

(For s and str I used two names: s = F(str).)

Now, take a look at the pattern that emerges!

I took the functions add_two and divide_3_floor and tweaked them.

I decorated them with some bells and whistles.

I decorated them...

This is the pattern of decorators!

Instead of doing the final assignment by hand, after making the functions go through F, I can just write @F at the top.

By writing the @F, Python does that assignment for me.

This is a lot to digest!

Write the code and play with it.

This means you can bookmark the link or go there and copy & paste the code: https://mathspp.com/blog/twitter-threads

Here are some of the things we have seen:

• callable classes
• custom operators for classes (the |)
• decorators (and classes as decorators)
• *args and **kwargs
• function composition
• lambdas
• and more!

Follow me @mathsppblog for more 😉

But wait, there is even MORE!

My original snippet of code wasn't like this.

There was a subtle (but not so subtle!) difference.

Here is the original code: https://twitter.com/mathsppblog/status/1512495768300572674

Can you spot the differences?

Can you explain them?

Hint: look at the final result.

I hope you learned something new! If you did, consider following the footsteps of the readers who bought me a slice of pizza 🍕. Your small contribution helps me produce this content for free and without spamming you with annoying ads.