## Functional and Reactive Domain Modeling

When we introduced Scala into Hello Heart, one of our greatest challenges was not having anyone on the team with any kind of functional programming experience. We grappled with functional concepts and how they fit together and ended up with an object oriented/semi functional style code that can’t be considered even remotely good in any of these paradigms.

One of my realisations at the time was that as developers, we do most of our learning by reading and imitating other people’s code. We don’t always realise it when it happens because reading and imitating is what most of us do for a living at the beginning if our career, but when you are required to introduce a new paradigm you have no experience with as lead developer, the lack of that kind of experience becomes apparent.

Another thing I was lacking at the time was some understanding of common functional programming idioms and design patterns. Despite the community’s distaste for GoF, I still appreciate the way it shaped my object oriented thinking and the understanding it gave me of how the different object oriented concepts fit together, and how design patterns can be composed to build  software. I spent quite some time looking for a resource that will give me a similar experience with functional concepts.

I am now reading the book Functional and Reactive Domain Modeling, and it gives me exactly that. The book presents one functional design approach from the ground up and explains the different functional concepts that come into play on the way. While I’m not sure this is an approach I would like to adopt as is, it gives me great insight into the thought process of one functional system designer.

In other words, sh!t functional programmers say is starting to make sense to me.

## Understanding the Y Combinator

I am now reading Types and Programming Languages, and I’ve been trying to wrap my head around the Y combinator and why it works. This is how I explained it to myself:

We want to define a recursive function, which means the function should be able to refer to itself. Unfortunately, Lambda calculus does not give us this option, so we define a function of the form:

$g = \lambda func. \lambda param.\space\text{return something or} func(param')$

Where $param'$ is some reduced version of $param$. In order for this thing to be a recursion, we need to find a parameter $func$ for which:

$func = g(func)$

We notice that $func$ is a fixed point of $g$ by definition, hence the name fixed point combinators. We also notice that it must be a function of $g$ (otherwise it’s a constant and we can show trivially that it doesn’t work, duh), so we can write:

$Y(g) = func = g(func) = g(Y(g))$

Cool. Now we are looking for $term_0$ such that:

$Y = \lambda f.\underbrace{\text{ } f(term_0) \text{ }}_{term_0}$

Hmmm… Ok. Let’s try to name $term_0$ and pass it as a parameter to $f$, will that work?

$Y = \lambda f.\underbrace{(\lambda x. f(x))}_{term_0} term_0 =\lambda f.\underbrace{(\lambda x. f(x))}_{term_0}\underbrace{(\lambda x. f(x))}_{term_0}$

This sucks. After our change we no longer need to pass $term_0$ to $f$, but $term_1$ :

$Y = \lambda f.\underbrace{(\lambda x. f(term_1))(\lambda x. f(term_1))}_{term_1}$

Fortunately, can easily express $term_1$ using our bound variable $x$ as $(x x)$! So if we write:

$Y = \lambda f.(\lambda x. f(x x))(\lambda x. f(x x))$

We win!

We can see that the Y combinator as we defined it is a fixed point of the function $g$, just like we wanted.
Great success :)

I ignored reduction rules for the sake of simplicity, but it’s interesting to note that the book talks about another type of fixed point combinator, which should be used under different reduction rules:

$fix = \lambda f. (\lambda x. f (\lambda y. x x y))(\lambda x. f (\lambda y. x x y))$

I wonder how many fixed points $g$ has under each set of reduction rules.