The necessary metaphor 🌯🌯🌯

Let’s say that you work at a teddy bear factory. Your boss comes and tells you to remove all the faulty teddy bears from the line and package the good ones after putting a price tag on them.

Easy enough. You go to the conveyor belt where teddy bears are born and start looking at the teddy bears one by one. As you scan them, you throw away the faulty ones while labeling and boxing the good ones. If your quota for the day is 1000 bears, you would only have to scan 1000 bears.

Now, what would happen if we write a program for this? In Swift, we could do something like this

let packagedBox = bears . filter ( isValid ) . map ( putPriceTag )

Cool, but let’s see whats going on here. By doing bears.filter(isValid) , you are throwing away the faulty ones but also packaging the good ones into a box.

When the time comes for you to put a price tag on the good bears by doing map(putPriceTag) , you notice that you need to re-open the box and look at all the bears for the second time. At the end of the day, you feel twice as tired and you should because you just scanned twice as many teddy bears!

You end up getting home, feeling tired after looking at so many teddy bears and search for a solution on StackOverflow. And it turns out, there is an idea called ‘Transducers’ from Clojure that seems like a perfect solution to the problem. So you start Googling for answers…

What are Transducers?

Transducers modify a process by transforming their internal reducing functions.

The basic purpose is to look again at map and filter and see if there is some idea in them that can be made more reusable.

We can do this by recasting them as process transformations; or successions of steps that ingests an input and blots out an output.

If you think about it this way, map basically does what we said above and stores the collection of outputs into a collection. That’s a specialization of the idea. The generalized form of the idea is the “seeded left reduce”; taking something we are building up and a new thing and continuing to building up.

So, we want to get away from the idea that reduction is about creating a praticular thing. Instead, we should focus more on it being a process because some processes build things while others are infinite.

The concept may be hard to understand at first - it certainly took me awhile - so, here are some gifs because gifs are good…

Map Transducers

Implementation

Before going all crazy, let’s process lists in the naive/easy way.

Naive Way

I’m guessing that from all the FP buzz, you are familiar with map and filter . So I’m going to use them to combine multiple functions to process an array of integers.

func isEven ( _ x : Int ) -> Bool { return x % 2 == 0 } func incr ( _ x : Int ) -> Int { return x + 1 } let naive1 = ( 1 ... 10 ) . map ( incr ) . filter ( isEven ) >> [ 2 , 4 , 6 , 8 , 10 ]

¯_(ツ)_/¯

Analysis

The above function calls map and filter twice on a range of integers. And the performance-conscience you may say,

“Hey, that just looped over that range n * 2 times and created an intermediate array! Can we make it so that it iterates only n times?”

And I would say

“Yes, we can. We can use transducers for that”

But first, here’s some theoretical stuff we need to cover first.

Reduce everything

You need to understand that all list processing functions - such as map , filter - can be redefined in terms of reduce

But what does this have to do with transducers?

Recognizing this gives us regularity/uniformity because the things we can prove about reduce can also apply to the rest of the list processing functions as well.

Basically, if theory A applies to reduce and map can be expressed in terms of reduce , A must also apply to map .

Here are some examples to illustrate this.

func append < T > ( to accum : [ T ], with input : T ) -> [ T ] { return accum + [ input ] } extension Collection { typealias A = Iterator . Element func mmap < B > ( _ f : @escaping ( A ) -> ( B )) -> [ B ] { return reduce ([]) { accum , elem in append ( to : accum , with : f ( elem )) } } func mfilter ( _ f : @escaping ( A ) -> ( Bool )) -> [ A ] { return reduce ([]) { accum , elem in if f ( elem ) { return append ( to : accum , with : elem ) } else { return accum } } } }

Notice that mmap reduces with append and mfilter reduces with append as well. But it is important to recognize that we chose to append elem to accum to create a new collection in both cases.

let naive2 = ( 1 ... 10 ) . mmap ( incr ) . mfilter ( isEven ) naive1 == naive2 >> true

So, this works. But, if you haven’t noticed yet, both mmap and mfiler still use intermediate arrays to process elements. What this means is that everytime mmap prepares to process things, it starts with an empty array and so does mfilter .

This means that naive2 still had to iterate n * 2 times.

Transducers

We can do better than that. And this is where transducers come in.

Transducers allow us to use only one intermeidate array and one iteration through the array to apply many transformations while being in control of the way it reduces.

Thoughts before going in…

So, how should we go about this? Well right now, we know that both mmap and mfilter are implemented using reduce and that they both use a function called append . But why do we use append here? Do we even have to use it here? There’s nothing special about it. Afterall, it’s just a function 😃

If we think about it that way, I could use any function with the type (accum, elem) -> (accum) in place of append . Turns out, functions with the following type signitures are called reducing functions. Let’s go ahead and write a version of map/filter that can take a reducing function in its closure.

Code

func mapping < A , B , C > ( f : @escaping ( A ) -> ( B )) -> ( @escaping (( C , B ) -> ( C ))) -> ( C , A ) -> ( C ) { return { reducer in return { accum , input in reducer ( accum , f ( input )) } } } func filtering < A , C > ( f : @escaping ( A ) -> ( Bool )) -> ( @escaping (( C , A ) -> ( C ))) -> ( C , A ) -> ( C ) { return { reducer in return { accum , input in if f ( input ) { return reducer ( accum , input ) } else { return accum } } } }

😵😵😵😵

I know it looks like a lot but stay with me. It’s not TOO complicated. Here’s what each parameter/generic type is trying to say.

A -> Input type

-> Input type B -> Output type

-> Output type C -> Accumulated data type (Array, Int, etc)

-> Accumulated data type (Array, Int, etc) f: (A) -> (B) Transformation function that takes an input and returns an ouput

((C, B) -> (C)) -> ((C, A) -> (C)) Returning closure parameters Input : Reducing function . Takes (accum, output) and returns a new accumulated output. Output : A closure fed into reduce . Takes a (accum, input) and applies f() to the input and calls reducer



Example

To use these, let’s start off with a simple reducing function; function that adds two numbers and returns a number.

func add ( l : Int , r : Int ) -> Int { return l + r } // or just (+)

In the first reduce , we choose to append the new element to the array and in the second reduce , we choose to numerically add the new element to the initial value.

let clever = ( 1 ... 10 ) . reduce ([], mapping ( f : incr )( append )) . reduce ( 0 , filtering ( f : isEven )( + )) let oldWay = ( 1 ... 10 ) . map ( incr ) . filter ( isEven ) . reduce ( 0 , + ) clever == oldWay >> true

👍👍👍👍

In our old naive way, we have to add one more reduce after the filter to do the adding because we have no control over how these functions reduce.

To see what mapping and filtering does in detail, let’s play with it. As a reminder, they both return (accum, input) -> (accum) so we can feed it to reduce .

assert ( mapping ( f : incr )( append )([], 1 ) == [ 2 ] ) assert ( mapping ( f : incr )( append )([ 0 , 0 ], 1 ) == [ 0 , 0 , 2 ] ) assert ( mapping ( f : incr )( add )( 0 , 1 ) == 2 ) assert ( filtering ( f : isEven )( append )([ 1 , 1 ], 2 ) == [ 1 , 1 , 2 ] ) assert ( filtering ( f : isEven )( append )([ 1 , 1 ], 3 ) == [ 1 , 1 ] )

The Big Moment.

Wait a sec… Did I just say mapping and filtering returns a (accum, input) -> (accum) ? Didn’t I say that functions with that type signiture are called reducing functions ?

So, mapping / filterting are the same things as append and add . This is the moment we have been waiting for. Get excited…

Let’s put a filtering where the reducing closure used to be.

//mapping(f: inr)(Any reducing functions here) let incrAndFilterEvens : ([ Int ], Int ) -> ([ Int ]) = mapping ( f : incr )( filtering ( f : isEven )( append )) let transducerRes = ( 1 ... 20 ) . reduce ([], incrAndFilterEvens ) let oldRes = ( 1 ... 20 ) . map ( incr ) . filter ( isEven ) transducerRes == oldRes >> true

As you see above, incrAndFilterEvens is also a reducing type. This means we can keep on composing functions until we drop.

Getting fancy 💃

This is cool but a bit messy. So let’s make the process of combining functions a little more “pretty” by making it functional.

Here are some function composition operators.

// (f --> g)(x) = f(g(x)) infix operator --> : AdditionPrecedence func --> < A , B > ( x : ( A ), f : ( A ) -> ( B )) -> ( B ) { return f ( x ) } // (f --> g)(x) = f(g(x)) func --> < A , B , C > ( f : @escaping ( A ) -> ( B ), g : @escaping ( B ) -> C ) -> ( A ) -> C { return { g ( f ( $0 )) } } let transduceFTW = ( append ) --> mapping ( f : incr ) --> filtering ( f : isEven ) ( 1 ... 10 ) . reduce ([], transduceFTW )

And if you print every time append is called, you get the following output.

[] [2] [2, 3] [2, 3, 4] [2, 3, 4, 5] [2, 3, 4, 5, 6] [2, 3, 4, 5, 6, 7] [2, 3, 4, 5, 6, 7, 8] [2, 3, 4, 5, 6, 7, 8, 9] [2, 3, 4, 5, 6, 7, 8, 9, 10]

The array was created after going through the range only n times!

🎉🎉🎉

We managed to apply multiple transformations to an array whilst in full control of the reduction process! In addition, it looks pretty as hell. Here’s a link to the git repo containing the .playground file if you want to play around with it.