Python exposes a number of built-in functions enriched with a plethora of modules composing The Python Standard Library. It is a pragmatic language that does not confine the developer in a specific programming paradigm. A Python developer can write imperative, procedural, object oriented or functional code. In Python, common functional constructs are available as built-in functions (e.g. `map`

, `filter`

, `all`

, `any`

, `sum`

…). Additional higher-order functions are regrouped in the functools module (e.g. `reduce`

, `partial`

…). Prior to crafting some Python code, let’s take a detour in some potentially more arcane areas of functional programming, in particular surrounding the `fold`

concepts.

Despite being primary a Python article, some of the initial code examples are in Haskell. Haskell is the perfect cradle for concepts like *Folding*. In addition, Haskell code, like Python code, reads like pseudo-code.

## Fold

Fold regroups a family of higher-order functions pertaining to the functional programming domain. At a high level, *folding* allows to deconstruct or reduce data. A typical signature for a generic *fold* function is the following:

$$ fold\ f\ z\ xs $$

Where:

`f`

is a higher-order function taking two arguments, an accumulator and an element of the list`xs`

. It is applied recursively to each element of`xs`

.`z`

is the initial value of the accumulator and an argument of the function`f`

.`xs`

is a collection.

Fold functions come in different kinds, the two main linear ones are `foldl`

and `foldr`

.

### foldl

`foldl`

, for “fold left”, is left associative. Think of foldl as “fold from the left” (image courtesy of Wikipedia):

Let `⊗`

be a variable bound to the function of two arguments `f`

in the diagram above. The `foldl`

function can be defined as follows:

$$ foldl\ (\otimes)\ z\ [1, 2, 3, 4, 5] = ((((z\ \otimes\ 1)\ \otimes\ 2)\ \otimes\ 3)\ \otimes\ 4)\ \otimes\ 5 $$

To cement the concept, here is an example in Haskell with the *subtraction* operator:

```
> foldl (-) 0 [1,2,3]
-6
> (((0 - 1) - 2) - 3)
-6
```

### foldr

`foldr`

, for “fold right”, is right associative. Think of `foldr`

as “fold from the right” (image courtesy of Wikipedia):

As for `foldl`

in the previous section, let `⊗`

be a variable bound to the function of two arguments `f`

. The `foldr`

operator can be defined as:

$$ foldr\ (\otimes)\ z\ [1, 2, 3, 4, 5] = 1\ \otimes\ (2\ \otimes\ (3\ \otimes\ (4\ \otimes\ (5\ \otimes\ z)))) $$

Following is an example with the *division* operator:

```
> foldr (/) 10 [1,2,3]
0.15
> 1 / (2 / (3 / 10))
0.15
```

## Laws of fold

In the book Introduction to Functional Programming using Haskell, the authors Richard Bird and Philip Wadler wrote a section on the *Laws of fold*. The first three laws are called *duality theorems* and concern the relationship between `foldl`

and `foldr`

. For simplification and in the context of this article, let’s focus on the first and third duality theorems.

### First duality theorem

For all finite lists `xs`

, if `f`

is associative and has identity element `e`

, then

$$ foldr\ f\ e\ xs\ =\ foldl\ f\ e\ xs $$

To concretely illustrate this principle in the Haskell REPL, see the following example using the *addition* as operation and *0* as the identity element for *addition*:

```
> foldl (+) 0 [1,2,3]
6
> foldr (+) 0 [1,2,3]
6
> foldl (+) 0 [1,2,3] == foldr (+) 0 [1,2,3]
True
```

### Third duality theorem

For all finite lists `xs`

,

$$ foldr\ f\ e\ xs\ =\ foldl\ (flip\ f)\ e\ (reverse\ xs) $$

where,

$$ flip\ f\ x\ y = f\ y\ x $$

To illustrate this principle, let’s take another simple example in Haskell, this time with *substraction* that is not associative:

```
> foldl (-) 0 [1,2,3]
-6
> foldr (-) 0 [1,2,3]
2
> foldl (flip(-)) 0 (reverse [1,2,3])
2
> foldr (-) 0 [1,2,3] == foldl (flip(-)) 0 (reverse [1,2,3])
True
```

## Fold in Python

Despite some *resistance* from Guido Van Rossum (see The fate of reduce() in Python 3000), Python has a *Fold* function. It is named reduce and is a built-in function in Python 2. In Python 3, it can be found in the functools module: functools.reduce.

**Note**: The snippets of code used as examples in this article target Python 3.

### foldl in Python

Python already has `foldl`

because functools.reduce is a `foldl`

construct. As an exercise and to mimic Haskell, a `foldl`

function can be written as follows with a *lambda*:

```
>>> import functools
>>> import operator
>>> foldl = lambda func, acc, xs: functools.reduce(func, xs, acc)
>>> foldl(operator.sub, 0, [1,2,3])
-6
>>> foldl(operator.add, 'L', ['1','2','3'])
'L123'
```

Or more formally as a function:

```
import functools
import operator
def foldl(func, acc, xs):
return functools.reduce(func, xs, acc)
# tests
print(foldl(operator.sub, 0, [1,2,3])) # -6
print(foldl(operator.add, 'L', ['1','2','3'])) # 'L123'
```

### foldr in Python

Relying on the *third duality theorem* evoked in the **Laws of fold** section above, `foldr`

can be crafted as a *lambda*:

```
>>> import functools
>>> import operator
>>> foldr = lambda func, acc, xs: functools.reduce(lambda x, y: func(y, x), xs[::-1], acc)
>>> foldr(operator.sub, 0, [1,2,3])
2
>>> foldr(operator.add, 'R', ['1', '2', '3'])
'123R'
```

**Note**: `xs[::-1]`

is the Python idiomatic way to return the reverse of a list (see this answer from Alex Martelli on stackoverlow). The other option, more readable, is to use the built-in reversed function.

Lambdas implemented as above are not generally conducive of good code readibilty. The following code, although longer, may be arguably more maintainable:

```
import functools
import operator
def flip(func):
@functools.wraps(func)
def newfunc(x, y):
return func(y, x)
return newfunc
def foldr(func, acc, xs):
return functools.reduce(flip(func), reversed(xs), acc)
# test
print(foldr(operator.sub, 0, [1,2,3])) # 2
print(foldr(operator.add, 'R', ['1','2','3'])) # '123R'
```

**Note**: the `flip`

function above is courtesy of Raymond Hettinger in a stackoverflow answer

## Now what?

We now have new toy functions in Python, `foldl`

and `foldr`

, what can we do with those?

### Reimplementing existing functions with reduce (foldl)

The folding concept opens the doors to build many other functions. It allows to be done without having recourse to writing explicit recursive code or managing loops. For example, `max`

, `min`

, `sum`

, `prod`

, `any`

, `all`

, `map`

, `filter`

among others, can all be defined with folding/reduce functions.

Here are some simplistic examples, using lambdas for conciseness:

```
>>> import functools
>>> import operator
>>> lmax = lambda xs: functools.reduce(lambda x, y: x if x > y else y, xs)
>>> lmax([1,2,3,4,5])
5
>>> lmin = lambda xs: functools.reduce(lambda x, y: x if x < y else y, xs)
>>> lmin([1,2,3,4,5])
1
>>> lsum = lambda xs: functools.reduce(operator.add, xs)
>>> lsum([1,2,3,4,5])
15
>>> product = lambda xs: functools.reduce(operator.mul, xs)
>>> product([1,2,3,4,5])
120
>>> lany = lambda pred, xs: functools.reduce(lambda x, y: x or pred(y), xs, False)
>>> lany(lambda x: x > 3, [1,2])
False
>>> lany(lambda x: x > 3, [1,2,3,4,5,6])
True
>>> lall = lambda pred, xs: functools.reduce(lambda x, y: x and pred(y), xs, True)
>>> lall(lambda x: x > 3, [4,5,6,7])
True
>>> lall(lambda x: x > 3, [1,2])
False
>>> lmap = lambda func, xs: functools.reduce(lambda x, y: x + [func(y)], xs, [])
>>> lmap(lambda x: x + 2, [1,2,3,4,5])
[3, 4, 5, 6, 7]
>>> lfilter = lambda func, xs: functools.reduce(lambda x, y: x + [y] if func(y) else x, xs, [])
>>> lfilter(lambda x: x % 2 == 0, [1,2,3,4,5,6,7,8,9])
[2, 4, 6, 8]
```

All the examples above, except `product`

(in that regard see another stackoverflow response from Raymond Hettinger), have an existing implementation in Python. Also, all of the functions above are relying on `reduce`

(`foldl`

) and none are taking advantage of `foldr`

.

### Final Example

To avoid ending on a dried note and to justify the *functional* workout executed in the sections above, here is a simple scenario that may demonstrate a decent usage of `foldl`

and `foldr`

in Python. Peter Drake presents this construct in his Lambdas and folds Youtube video. Imagine that, given a list, we need to identify the last and/or the first element that satisfies a certain predicate. This could be written as follows:

```
import functools
def foldl(func, acc, xs):
return functools.reduce(func, xs, acc)
def flip(func):
@functools.wraps(func)
def newfunc(x, y):
return func(y, x)
return newfunc
def foldr(func, acc, xs):
return functools.reduce(flip(func), reversed(xs), acc)
def first(func, acc, xs):
return foldr(lambda x, y: x if func(x) else y, acc, xs)
def last(func, acc, xs):
return foldl(lambda x, y: y if func(y) else x, acc, xs)
print(last(lambda x: x<8, 99, [1,2,3,4,5,6,7,8,9])) # => 7
print(first(lambda x: x>3, 99, [1,2,3,4,5,6,7,8,9])) # => 4
print(first(lambda x: x>20, 99, [1,2,3,4,5,6,7,8,9])) # => 99
```

`first`

and `last`

don’t require any loop or explicit recursion. `first`

uses `foldr`

taking advantage of the *right folding* whereas `last`

relies on `foldl`

.

## Conclusion

In this era of rediscovery of functional programming, there is much more to explore and to apply to languages that are not inherently functional. Arguably, from a pragmatic perspective, there may be little we need that is not already provided in the current versions of Python and that would require some sophisticated folding mechanisms. Further more, other higher-order functions flagships along with `fold`

, like `map`

and `filter`

, can be expressed in Python with elegant *list comprehensions* and *generator expressions*, but this should be the subject of a different article.

## Resources

- Python
- Haskell
- Haskell REPL
- Introduction to Functional Programming using Haskell
- Thinking Functionally with Haskell
- Fold
- Richard Bird
- Philip Wadler
- Alex Martelli
- Raymond Hettinger

## Legal

The source code accompanying this article is released under the MIT License.

## Release Notes

- 08/28/2016: Initial publication of this article.