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Arrows, Monads and Kleisli — part II

In part I I showed how Kleisli arrows could be used to implement domain modeling. Arrows serve as a foundation for a ‘DSL’ in which one can implement typical scenarios that arise in business-logic code: decoupling flow control from domain code, dealing with errors etc. Much to the spirit of Railway Oriented Programming but implemented in more generic terms.
In this part I’ll fill in the missing parts of this ‘framework’ — taking care of side-effects, conditional execution and mixing different monads in a single ‘pipeline’.

There is one more part planned where I’ll introduce kind-projector and arrow transformers.
Let’s pick up where we left:

Having glued together actions needed to carry out domain operations, we’d certainly like to be able to inspect what’s going on during execution. Typically that means plugging in some kind of logging framework. Let’s add another requirement: add logging. While logging itself is a no-brainer, it is problematic from functional point of view because of String => Unit signature. It returns no useful value, thus it is meant to be executed only for the side-effect occurring when it is being called. In the language of Railway Oriented Programming this type of function is called dead-end function. Dead-end means that it cannot be composed any further. Yet we certainly want to be able to use them in a computation, mainly because many tasks are naturally side-effecting actions (like it or not).

Deal with dead-end functions

Clearly we’re gonna need to reroute main flow around dead-end. Action does not produce any useful value (but probably needs access to the current input), so composition with dead-end will just return its input, capturing the side-effect within what looks like an identity process from the outside. Let’s make a small building block called teefor this purpose

If you read the first part, it probably rings a bell. Do you remember arrow split operator? Tee looks very much alike, doesn’t it?
That’s right — in arrow language: tee(f) = (f &&& action) >>> _._1.
Let’s add this little function to Arrow trait:


Using tee we can devise a compose-with-dead-end operator for ArrowOps (second diagram above). I thought that -| would be a pretty good symbol for this :-) (looks like turned dead-end sign, doesn’t it?)

This is sufficient to add logging to the ProductionLotService but looking at the flow we discover one quirk we’d yet like to address. How are we going to log the result?
Result is of Either[Error, Long] type. When computation signals error we might want to log with higher than usual level or even perform additional steps. This obviously could be achieved through pattern-matching but weren’t arrows supposed to free code from the unnecessary boilerplate?
It turns out that choice of execution path can be easily expressed using arrows. Enter ArrowChoice

Implement ArrowChoice

ArrowChoice is a specialization of arrow that can, as the name points out, choose which computation to perform next based on its input. It is the equivalent of if construct in for-comprehensions and is designed to work well with Either data type. The minimal definition of ArrowChoice calls for two primitive operations: leftand right from which two compound operations are built: multiplex(+++) and fan-in(|||). A picture is worth thousand words, so let’s start with diagrams.

Here we have two possible paths of execution (marked red and blue) based on data held in an Either instance passed as input.
left(f) feeds Left input through its argument arrow, leaving the other part unchanged. right(f) mirrors left. Thinking in terms of Scala library, these operators are maps on, respectively, Left and Right projections.
Multiplex(+++) simply combines both left and right into a single operation. It applies f if input is a Leftor g if it is a Right
Fanin(|||) is +++ which merges the output. Think Scala’s Either.fold.

Implementation is straightforward:


… and additional symbolic ops:

What’s left to be done is to implement Choice for arrow instances that support it. Both function and Kleisli do, but to keep it simple we’ll do that only for functions (after all, we’re using it for logging only):


Of course, it would have sufficed to implement just left and right but, as we observed, for functions it is perfectly possible to write simpler code, based on Scala standard library. Only multiplex is not equivalent to any method. It comes as no surprise because, as was said before, Either lacks bias, forcing us to choose explicitly which side we want to operate on. If it had been biased, +++ would have been map.

The stage is set to add logging to ProductionLotService. Let’s extend productionLotArrow to accept custom logging function of type (ProductionLot, Env) => Unit (after all, each caller wants a different message to be logged). Additionally, arrow implementation will log the outcome of flow (either success or error) by itself. Putting it together we get:


With a little help of -| operator, logging is painlessly stitched to where it belongs.
Curious reader might be asking, why is there a run called? (run extracts a function from Kleisli and there is an implicit conversion for that). Well, the short answer is that it’s because implicit conversions in Scala aren’t chained. Now, the long answer: to deduce that the correct call is actually -| coming from function arrow (not the one from Kleisli arrow), compiler would have had to construct chain of conversions leading from Kleislithrough Function1 (via run method) to ArrowOps(FunctionArrow). Instead it sees that -| can be obtained by one step conversion Kleisli -> ArrowOps(KleisliArrow), and sticks to that. But KleisliArrow requires Long => Unit action, while what we’ve got is a conditional Either[Error, Long] => Unit. So we have to manually unpack function from Kleisli.

After all this we can freely enhance different use cases with logging:

Mixing monads

I promised to show how to face the situation when functions that you want to build pipeline from can’t agree on what monad they use. Let’s imagine that save method wants to return Try instead of Either:

Obviously we also need to state that Try is actually a monad (there is some controversy about that, but I won’t delve into this matter):


There is no one-size-fits-all answer, I believe, but I’ll show you two possible ways of solving this awkwardness.


The crucial observation here is that if you have a function A => B and a functor F[A], you can easily transform the function into a function of F[A] => F[B] type. We’ve already seen that in action when we integrated non-monadic functions with Kleisli. It’s exactly what map does. If you recall, map‘s signature on Kleisli[M, A, B] (a function A => M[B]) is:

map composes B => C with A => M[B] to get A => M[C]. In other words it turns a function of type B => Cinto a M[B] => M[C] function.
This is possible with all functors, and, as you probably remember from part one, every monad is a functor.
Now, we can pretend that function A => Try[B] is kind of non-monadic from the perspective of Either monad and write this as follows:

Or, we can name this operation explicitly by adding liftM method to Kleisli class:


which turns arrow definition into:

This approach works fine, but it adds a lot of burden when you want to do anything with the result. The whole arrow definition after these changes looks like this:


The bad thing is that type of the constructed function changes from nice Either[Error, Long] into less-than-perfect Either[Error, Try[Long]]. You can see how it affected logging. logSuccess needs to accomodate the additional layer of types and be rewritten from Long => Unit into Try[Long] => Unit and that, in turn, forces us to pattern-match on input. It’s not the end of the world though. What are the other options?

Natural transformation

There are tons of material treating the subtleties of natural transformations, ranging from deep theoretical explanations to more practical ones.
The definition from Wikipedia states simply that:

a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms)

We’ll not be doing anything fancy though. For our purposes a natural transformation is just a generalization of map (A => B) that works on higher-order types (M[_] => N[_]).

The fancy ~> type used in transform behaves exactly like a generalized map that can transform one monad into another, irrespective of type of a value the monad carries inside. BTW, what a fine example of higher-order polymorphism.
And transform is very simple to use:


One can say that save‘s just got back on track :-)

The end

All right, that’s all folks. Stay tuned for the last part of series. Hope you enjoyed it so far.



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