Law abiding citizens

In the table below each row compares the operation and result on shape similar strings and ints for addition.

String operation String result Int operation Int result 1 "ab" + "c" "abc" 2 + 4 6 2 ("a" + "b") + "c" "abc" (1 + 2) + 3 6 3 "a" + ("b" + "c)" "abc" 1 + (2 + 3) 6 4 "" + "abc" "abc" 0 + 6 6

Notice the similarity between the operations on ints and strings. Rows 2 and 3 show associativity, which means the order of evaluation doesn't change the result as long as the order of the values are the same.

In mathematical terms we have the following law for associativity:

for any 𝑚1, 𝑚2, 𝑚3 ∈ 𝑀 the following equality holds: (𝑚1 + 𝑚2) + 𝑚3 = 𝑚1 + (𝑚2 + 𝑚3)

Row 4 reveals the second and last law - the law of identity. There exists a value such that combining with any other value, will return the other value unchanged. For numbers under addition this is zero.

In mathematical terms we have the following for identity:

There exists such 𝑒 ∈ 𝑀 that for any 𝑚 ∈ 𝑀 the following equalities hold: 𝑒 + 𝑚 = 𝑚 + 𝑒 = 𝑚

That's it! If some types have something that respects these two laws, then we have what the mathematicians have named a monoid 🎉

Why is this useful? When building software, I often think about it as taking building blocks and putting them together to create bigger blocks solving bigger problems. Naming such abstract behaviour is useful. If, for example, you discover that what you are doing is described as a monoid then you can test to verify that you respect the laws. More advance tools can help you prove the properties, but the simpler take is to use property-based testing where random data is generated to verify the properties. Look it up if it sounds interesting.

An example

Let us take a look at a text corpus analyser. The task is to find all unique words, total length of all words and how many occurrences there is of each word.

To solve this, we will write code in Scala, an object-oriented and functional programming language for the JVM and Javascript. If this is new to you then you can squint hard and see Python, Kotlin or Java. The brackets are the same as generics in Java. But the details in the code are not important, only the possibilities. Therefore, I will explain the reasoning behind each line. If you already know Scala and the Cats library then you can probably gloss right over it.

To start things of we have our text corpus to analyse in a list that is split on whitespace. Each word is an element that we need to process.

val message = List ( "Hello" , "world!" , "Have" , "you" , "learned" , "anything" , "new?" , "Hello" , "again" )

For each of these words we need to have a function that take one word and splits it into the tuple that contains all the information we can get from that word. Each element in that tuple will be:

1 word

word Length of the word

A hash map with the word as a key pointing to the count of occurrences for that word

In Scala such a function will look like the following code snippet.

def wordDetails ( word : String ) = ( 1 , word . length , Map ( word -> 1 ) )

Now, if we combine the data from using this function on all the words we will get:

Total count of words

Total length of all the words

A hash map with the words as keys pointing to the count of occurrences for that word

Does this sound familiar? Each of these types of data should have some default monoid implementation that defines this behaviour, and thanks to the Cats library in Scala we have just that.

import cats . _ , cats . implicits . _

With this we have imported the cats library for functional programming. The next step is to see if the compiler can give us an Monoid for the result type of the wordDetails function.

type WordDetailsResult = ( Int , Int , Map [ String , Int ] ) val monoidImpl = Monoid [ WordDetailsResult ]

An implementation of a monoid that can be used on the result from the wordDetails function is now stored in the value monoidImpl . Note that if our WordDetailsResult type contained some type that with no defined implementation for monoid then we would have a compiler error.

Using the monoid implementation we can now transform our input data and then aggregate it to one result. From the previous article on iterations we learned about folds. To reiterate they need an empty state value and a function that modifies the state for each element. Such a function signature in scala for a List[String] is shown below:

def foldLeft [ A ] ( z : A ) ( op : ( A , String ) => A ) : A

Putting these pieces together by first using wordDetails function and then folding over the result with it's corresponding monoid implementation:

message . map ( wordDetails ) . foldLeft ( monoidImpl . empty ) ( monoidImpl . combine )

Changing the laws

For our text corpus analyser we have the laws of associativity and identity. What would happen if we did not have the law of identity? If our text corpus was empty then we would still need to be able to produce a result that we got from the wordDetails function. But with no values we cannot run the function and therefore have no value to produce! Our function would have needed to be partial, that is to not be able to handle all cases of our input - the empty list. The result would have been an exception 😢 Thanks to having the law of identity we knew that we would always be able to produce some value that defines emptiness and so our problem is non existing.

But this does not mean that leaving out the identity laws is useless. In fact, having only associativity is named semigroup and is also a useful construct! Say you are creating some imaging software that uses bounding boxes. Naturally you can define a combine operator that represents the union between the boxes. But what should the representation of the empty element be? Being forced to make this construct will properly lead to messy code, so just avoid the issue and define a semigroup instead.

Summary

That concludes our introduction to the Monoid laws. We have seen some simple, yet practical application, that I hope to have helped to give some intuition on how it can be used. Furthermore, we have taken a peek into what the laws actually mean for the application of Monoids and how that relates to one other algebraic constructs. Note that there are more useful laws and algebraic constructs that one can look into. For a more in-depth explanation of these I can recommend the talk Monoids monoids monoids.

In real-world code there are many other hidden implementations, including for example web routes in web servers. Now go out and try to find some more! If these concepts were new to you then I hope this have piqued some interest.