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19 January 2023 — by Christopher Harrison
Functional Python, Part II: Dial M for Monoid
pythonprogramming-languagestutorial

Tweagers have an engineering mantra — Functional. Typed. Immutable. — that begets composable software which can be reasoned about and avails itself to static analysis. These are all “good things” for building robust software, which inevitably lead us to using languages such as Haskell, OCaml and Rust. However, it would be remiss of us to snub languages that don’t enforce the same disciplines, but are nonetheless popular choices in industry. Ivory towers are lonely places, after all.

Last time I wrote about how Python’s1 type system and syntax is now flexible enough to represent and utilise algebraic data types ergonomically. Here, I’ll develop that idea further by way of a motivating example, for which I shall make use of some functional programming “tricks” to arrive at an efficient Python implementation.

Typeclass ABCs

Typeclasses are famously a feature from Haskell, which have since spawned analogous features elsewhere (e.g., “traits” in Rust and “concepts” in C++20). They allow the definition of “interfaces for types”, where values of types which conform to that specification can be freely swapped out. This enables what’s known as “ad hoc polymorphism”… but perhaps an example would be more illuminating!

It is very common to want to encode and decode some value back and forth from a serialised representation (e.g., reading and writing to a byte stream). Say we have some arbitrary Python class, a first approximation may be to define encode and decode methods:

from typing import Self

class MyClass:
    ...

    def encode(self) -> bytes:
        # Implementation goes here
        ...

    @classmethod
    def decode(cls, data: bytes) -> Self:
        # Implementation goes here
        ...

We could then use this in, say, a function that writes an object’s representation to a file handler:

import io

def write_to_file(fd: io.BufferedWriter, obj: MyClass) -> int:
    return fd.write(obj.encode())

A reason for not implementing this as a method of MyClass could be because we may also want to write other types of objects to the file handler; let’s say the OtherClass and YetAnotherThing classes also conform to the interface. The naïve approach would be to use a typing.Union type annotation:

def write_to_file(
    fd: io.BufferedWriter,
    obj: MyClass | OtherClass | YetAnotherThing,
) -> int:

This will quickly get out of hand!

Instead, we can define an interface — what Python calls an “abstract base class” — that all our classes must provide implementations for, lest they fail at instantiation time. Then we can use that in the type annotation for our file-writing function:

from abc import ABC, abstractmethod
from typing import Self

class Codec(ABC):
    # NOTE Abstract base classes don't need implementations,
    # just method stubs to define their signatures

    @abstractmethod
    def encode(self) -> bytes: ...

    @classmethod
    @abstractmethod
    def decode(cls, data: bytes) -> Self: ...

# NOTE You may inherit from *many* abstract base classes
class MyClass(Codec):
    def encode(self) -> bytes:
        # Implementation goes here
        ...

    @classmethod
    def encode(cls, data: bytes) -> bytes:
        # Implementation goes here
        ...

def write_to_file(fd: io.BufferedWriter, obj: Codec) -> int:
    return fd.write(obj.encode())

We can thus make our code much easier to annotate by defining abstract base classes that outline related groups of capabilities. Indeed, as we’ll see next, some capabilities are so pervasive, it can be useful to consider them in their own right.

Monoidial Soup

In mathematics, a monoid, (M,)(M, \star), is a set MM, with a binary operation :M×MM\star : M\times M \to M that is associative2 and has an identity element.3 For non-mathematicians this probably sounds completely opaque, bordering on absurd. What possible application does this have in software engineering?

Well, it turns out that this occurs all the time! To offer a few examples:

  • Addition over arbitrary-precision integers, with 00 as the identity element. (Likewise for products, with 11 as the identity.)

  • String concatenation, with the empty string as the identity element.

  • Null coalescing over any type augmented with a null-value, which also serves as the identity element.

  • “Any” and “all” predication, over Boolean values, with False and True being the respective identities.

Combining values in such a way is very common and, provided the definition is satisfied, you have a monoid. Given the frequency at which monoids naturally occur, it makes sense to define an interface for them.

In Haskell, the Monoid typeclass4 defines three functions:

  • mempty
    A parameter-less function that returns the monoid’s identity element.

  • mappend
    A function that defines the monoid’s binary operation.

  • mconcat
    A convenience for applying the binary operation over a list of values. (A default implementation exists, but it can be overridden if there is opportunity for optimisation.)

If you recall our previous discussion — in which we define a List type5 and a right fold over it — we can follow Haskell’s lead and implement a monoidial abstract base class in Python as follows:

# This postpones type annotation evaluation until after the source is
# parsed, allowing structural type definitions that can return types of
# themself and references to types that are defined later in the code.
# As of Python 3.11, this is not yet the default.
from __future__ import annotations

from abc import ABC, abstractmethod
from typing import Generic, TypeVar

M = TypeVar("M")

class Monoid(Generic[M], ABC):
    @staticmethod
    @abstractmethod
    def mempty() -> Monoid[M]: ...

    @abstractmethod
    def mappend(self, rhs: Monoid[M]) -> Monoid[M]: ...

    @classmethod
    def mconcat(cls, values: List[Monoid[M]]) -> Monoid[M]:
        folder = lambda x, y: x.mappend(y)
        return foldr(folder, cls.mempty(), values)

Guess what? Lists also form a monoid over concatenation, with the empty list as the identity element. As such, we can update the List class to inherit from our Monoid abstract base class and plug in the implementations:6

from typing import TypeVar

T = TypeVar("T")

class List(Monoid[T]):
    @staticmethod
    def mempty() -> List[T]:
        return Nil()

    # WARNING mypy has trouble type checking this signature
    def mappend(self, rhs: List[T]) -> List[T]:
        return foldr(Cons, rhs, self)

Let’s test this out:7

>>> a = Cons(1, Cons(2, Cons(3, Nil())))
>>> b = Cons(4, Cons(5, Cons(6, Nil())))
>>> c = Cons(7, Cons(8, Cons(9, Nil())))

>>> a.mappend(b)
Cons(1, Cons(2, Cons(3, Cons(4, Cons(5, Cons(6, Nil()))))))

>>> List.mconcat(Cons(a, Cons(b, Cons(c, Nil()))))
Cons(1, Cons(2, Cons(3, Cons(4, Cons(5, Cons(6, Cons(7, Cons(8, Cons(9, Nil())))))))))

Perfect!

What the Thunk?

Now allow me to take you on a little digression. It’ll be worth it, I promise.

Consider a textbook recursive function. I’ll ignore type hints for now, so you can focus on what’s going on:

def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n - 1)

This executes as follows:

factorial(3)
→ 3 * factorial(2)
→ 3 * (2 * factorial(1))
→ 3 * (2 * (1 * factorial(0)))
→ 3 * (2 * (1 * 1))

The problem with this is that each recursion increases the depth of the call stack. In Python, by default, the call stack is limited to a depth of 1,000. We can increase that, but only insofar as the machine’s memory allows.

>>> factorial(4)
24

>>> factorial(1000)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 5, in factorial
  File "<stdin>", line 5, in factorial
  File "<stdin>", line 5, in factorial
  [Previous line repeated 995 more times]
  File "<stdin>", line 2, in factorial
RecursionError: maximum recursion depth exceeded in comparison

Let’s rewrite our function in so-called continuation-passing style, wherein the function takes an additional argument k,8 which represents the “continuation of the computation”. This is represented as a single-valued function and we return it embedded into the next step of the sequence:

def factorial(n, k):
    if n == 0:
        return k(1)
    else:
        return factorial(n - 1, lambda x: k(n * x))

Now the execution is like this:

factorial(3, k)
→ factorial(2, lambda x: k(3 * x))
→ factorial(1, lambda x: k(3 * (2 * x)))
→ factorial(0, lambda x: k(3 * (2 * (1 * x)))
→ k(3 * (2 * (1 * 1)))

We can retrieve the result from the continuation by setting k to be the identity function (lambda x: x). However, we see that this still fills up the stack. We can even make this explicit by adding some debugging output to our identity function:

import traceback

def identity_with_callstack_depth(x):
    print(f"Stack Depth: {len(traceback.extract_stack())}")
    return x

Now:

>>> factorial(3, identity_with_callstack_depth)
Stack Depth: 9
6

>>> factorial(123, identity_with_callstack_depth)
Stack Depth: 249
<< Number with 206 digits >>

>>> factorial(1000, identity_with_callstack_depth)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "<stdin>", line 5, in factorial
  File "<stdin>", line 5, in factorial
  File "<stdin>", line 5, in factorial
  [Previous line repeated 995 more times]
  File "<stdin>", line 2, in factorial
RecursionError: maximum recursion depth exceeded in comparison

The problem is that Python is strictly evaluated and so will still perform function evaluation on every recursion. So what if we didn’t give it functions to evaluate? What if instead of evaluating functions whenever we see them, we wrap them up in a thunk; that is, a “to-be-evaluated function”? That sounds complicated, but all it involves in Python is an argument-less lambda function, or functools.partial:

def factorial(n, k):
    if n == 0:
        return k(1)
    else:
        # Equivalently: partial(factorial, n - 1, lambda x: partial(k, n * x))
        return lambda: factorial(n - 1, lambda x: lambda: k(n * x))

So we return a thunk and the continuation returns a thunk. By doing so, the execution now looks like this:

factorial(3, k)
→ lambda: factorial(2, lambda x: lambda: k(3 * x))

That’s where it stops. There’s no recursion. What we get in return is a thunk that represents the next sequence of the iteration, the length of which will be twice the number of original recursive steps as we ping-pong between the thunk and its evaluation. Following from the above, you can convince yourself that the following execution holds:

factorial(3, k)()
→ factorial(2, lambda x: lambda: k(3 * x))

factorial(3, k)()()
→ lambda: factorial(1, lambda x: lambda: k(3 * (2 * x)))

factorial(3, k)()()()
→ factorial(1, lambda x: lambda: k(3 * (2 * x)))

factorial(3, k)()()()()
→ lambda: factorial(0, lambda x: lambda: k(3 * (2 * (1 * x))))

factorial(3, k)()()()()()
→ factorial(0, lambda x: lambda: k(3 * (2 * (1 * x))))

factorial(3, k)()()()()()()
→ k(3 * (2 * (1 * 1)))

This can be unrolled deterministically with a trampoline,9 which is just a simple loop:

def trampoline(thunk):
    while callable(thunk): thunk = thunk()
    return thunk

A loop will not blow up the stack:

>>> trampoline(factorial(6, identity_with_callstack_depth))
Stack Depth: 3
720

>>> trampoline(factorial(1000, identity_with_callstack_depth))
Stack Depth: 3
<< Number with 2,568 digits >>

What we’ve achieved by doing this transformation is to implement tail-call optimisation in Python; a language whose reference implementation does not support TCO. This allows us to implement recursive functions without O(n)O(n) space usage as we fill up the stack, while maintaining the same asymptotic time complexity as the original recursive version.

Origami

Now let’s jump back (pun intended) to folds. We can see that our recursive right fold will grow the stack linearly with respect to its depth, as any recursive function in Python will. As I’ve shown, this can be avoided by rewriting the function in continuation-passing style and using thunks to delay evaluation.

The conversion is quite a mechanical process:

def foldr(fn, acc, lst, k):
    match lst:
        case Nil():
            return k(acc)

        case Cons(x, xs):
            return lambda: foldr(fn, acc, xs, lambda v: lambda: k(fn(x, v)))

There we have it: an efficient right fold, which can be unrolled with a trampoline! That said, at this point it behooves me to point out that this technique is not always necessary and should be used judiciously. For example, a strictly evaluated left fold can easily be rewritten from a recursive version into a simple loop with an accumulator; no continuation-passing required.

Anyway, let’s see our right fold in action:

>>> trampoline(foldr(lambda x, y: x * y, 1, [1, 2, 3], identity_with_callstack_depth))
Stack Depth: 4
6

>>> trampoline(foldr(lambda x, y: x * y, 1, range(1, 1001), identity_with_callstack_depth))
Stack Depth: 4
<< Number with 2,568 digits >>

It remains an exercise for the reader to supplant this efficient fold implementation into our monoid abstract base class and List.

What About Types?

I’ve deliberately avoided adding type hints in the above discussion for clarity’s sake. (You’re welcome.) However, our code should be annotated. We already know the signature for recursive foldr, where S and T are type variables:

def foldr(fn: Callable[[S, T], T], acc: T, lst: List[S]) -> T: ...

The continuation-passing style version returns a thunk that will ultimately return a T; let’s call this type Thunk[T]. The continuation itself is a single-valued function that takes a T and returns a thunk that returns a T; that is, Callable[[T], Thunk[T]]. So how should we define the Thunk type?

The definition of a thunk that we’re using is a zero-valued function that returns either another thunk or a value of some type. Ideally, therefore, we’d like to write something like this:

Thunk[T] = Callable[[], Thunk[T] | T]

The problem is, we can’t; this is not valid Python syntax. Generics — to account for the type variable — are not used like that, but the bigger problem is that mypy, for example, does not currently support this kind of recursive type definition. To get around both of these problems, we have to write a recursive callback protocol type:

from __future__ import annotations

from typing import TypeVar, Protocol

# The type variable has to be marked as covariant to type check
T_co = T.TypeVar("T_co", covariant=True)

class Thunk(T.Protocol[T_co]):
    def __call__(self) -> Thunk[T_co] | T_co: ...

We have to be explicit about the covariance of the type variables, which are expected to take a _co suffix. So putting this altogether gives the following signatures for our continuation-passing style version of foldr and trampoline:

def foldr(
    fn: Callable[[S, T_co], T_co],
    acc: T_co,
    lst: List[S],
    k: Callable[[T_co], Thunk[T_co]]
) -> Thunk[T_co]: ...

def trampoline(thunk: Thunk[T_co]) -> T_co: ...

Phew!

Conclusion

Continuing our theme of building software from composable parts, the concept of typeclasses, from Haskell, can be simulated in Python using abstract base classes. This allows classes to be categorised by their capabilities, simply by virtue of inheritance, enabling you to build more generic functions that utilise your object models, without upsetting the type checker.

One such typeclass is the “monoid”, which is a pattern that is regularly seen in day-to-day software engineering. By abstracting this interface with a typeclass, we can ensure consistency amongst all our monoid implementations and arrive at familiar patterns throughout our code. This is a boon for reasoning about the software we write.

Finally, I have shown how using continuation-passing style can improve the performance of traditional functional programming patterns in a more imperative and strictly evaluated context. While the indirection breeds a certain complexity — and it is not something that needs to be done on the regular — this has its place in, for example, the depths of library code, where performance matters.

In the next and final episode, I’ll cover testing strategies that can be learnt from functional programming and applied to Python.

Thanks to Connor Baker, Guillaume Desforges, Johann Eicher, Johan Herland and Maria Knorps for their reviews of this article.


  1. We are not limited to Python; these techniques can be applied in any language with suitable support, libraries and tooling.
  2. For all a,b,cMa,b,c \in M, (ab)c=a(bc)(a\star b)\star c = a\star(b\star c).
  3. There exists an eMe\in M such that, for all aMa\in M, ea=ae=ae\star a = a\star e = a.
  4. In Haskell, a monoid is actually defined in terms of a semigroup. This is a further abstraction that we could also implement in Python as an abstract base class, leveraging object inheritance in our monoid ABC.
  5. We can do the same with regular Python lists, which would be more efficient, but we use our definition for consistency (and dog-fooding purposes!)
  6. The type annotations are circular, here: Monoid.mconcat is dependent on List, which is dependent on Monoid. Enabling postponed type annotation evaluation resolves this. However, I suspect this may be the reason why mypy cannot (as of writing) validate the annotation for List.mappend.
  7. I have not documented the custom __str__ implementations for Nil and Cons, which render this output, for sake of brevity.
  8. Conventionally, the continuation is denoted by the letter k. I cannot find from where this convention originated; even when the concept was introduced, by Sussman and Steele, they used the letter C.
  9. Care must be taken with a trampoline when you want to return a function. In our case, it won’t work — a function is callable and so the trampoline will try to evaluate it — but this can be resolved easily by wrapping it in some kind of container type, for example.
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