## DEV Community is a community of 617,782 amazing developers

We're a place where coders share, stay up-to-date and grow their careers.

# Getting recursively drunk with monoids

Disclaimer: I am not a mixologist. This is not professional cocktail advice!

Sam Horvath-Hunt blogged about modelling cocktails as monoids. This is a really cool example of FP modelling that I want to expand on. (Lennart Kolmodin once wrote that the dance steps of Tango form a monoid.)

First, I will demonstrate my ignorance by assuming that cocktail recipes are free commutative monoids over ingredients:

• The order in which you add ingredients does not matter,
• If you add two cocktails together, you get another cocktail,
• The identity cocktail is the empty cocktail with no ingredients in it.

Second, I want to up the ante with a recursive cocktail recipe.

It comes from a sketch in the computer science student revue at DIKU (University of Copenhagen): Superdrinks (2002); credits go to Uffe Christensen, Uffe Friis Lichtenberg, Jonas Ussing, Niels H. Christensen, Torben Æ. Mogensen, Jørgen Elgaard Larsen who either co-wrote or enacted the sketch.

A superdrinks consists of:

• 1 part gin,
• 2 parts lemon,
• 3 parts superdrinks.

Now, according to the sketch there are plenty of bad ways to materialize this drink; one such is through approximation: take 1 part gin, 2 parts lemon and 3 parts of whatever is your current best approximation of superdrinks. Iterate this process enough times and you will have a gradually finer superdrinks.

Recursively,

``````superdrinks(n) = 1 × gin
+ 2 × lemon
+ 3 × superdrinks (n-1)
``````

As for `superdrinks(0)`, it could be water. It could be gin!

Experimenting a little,

``````superdrinks(1) = 1 × gin + 2 × lemon + 3 × superdrinks(0)

superdrinks(2) = 1 × gin + 2 × lemon + 3 × superdrinks(1)
= 1 × gin
+ 2 × lemon
+ 3 × (1 × gin + 2 × lemon + 3 × superdrinks(0))
= 4 × gin + 8 × lemon + 9 × superdrinks(0)
``````

The relationship between the number of parts of each ingredient can be expressed in closed form eliminating recursion:

``````superdrinks(n) = (3ⁿ - 1)/2 × gin
+ (3ⁿ - 1) × lemon
+ 3ⁿ × superdrinks(0)
``````

(You can find the closed form either by recognizing that the series 3 × 3 × ... with n occurrences is 3ⁿ, that there's always one less part lemon than superdrinks(0), and that there's always half the amount of gin of that; or you can solve their recurrence relation; or you can expand the three number series using a function,

``````> let superdrinks (gin, lemon, super) = (1 + 3*gin, 2 + 3*lemon, 3*super)
> unzip3 \$ take 6 \$ iterate superdrinks (0,0,1)
([0,1,4,13,40,121],[0,2,8,26,80,242],[1,3,9,27,81,243])
``````

and look them up.)

It is time to get schwifty.

The following ingredients are enough to make gin-tonic and superdrinks:

``````data Ingredient = Gin | Tonic | Lemon
deriving (Eq, Ord, Show)
``````

A cocktail is any set of ingredients and their multiplicity:

``````newtype Cocktail = Cocktail (Map Ingredient Natural)
deriving (Eq, Ord, Show)

emptyCocktail :: Cocktail
emptyCocktail = Cocktail Map.empty
``````

For convenience,

``````parts :: Natural -> Ingredient -> Cocktail
parts n ingredient =
Cocktail (Map.singleton ingredient n)

combine :: Cocktail -> Cocktail -> Cocktail
combine (Cocktail c1) (Cocktail c2) =
Cocktail (Map.unionWith (+) c1 c2)
``````

One consequence of this modelling is:

``````> let gintonic = combine (1 `parts` Gin) (2 `parts` Tonic)
> gintonic == combine gintonic gintonic
False
``````

Since these are cocktail recipes, I'd like to normalize the quantities of each ingredient so that recipes don't eventually say "2 parts gin, 4 parts tonic" or "0 parts gin":

``````normalize :: Cocktail -> Cocktail
normalize (Cocktail ingredients) =
Cocktail . normalize' \$ ingredients
where
scale = foldr1 gcd (Map.elems ingredients)
normalize' = Map.map (`div` scale) . Map.filter (/= 0)
``````

(Note that while `foldr1` is partial, because of Haskell's non-strict semantics, it is never evaluated when `ingredients` is empty because it is used within `Map.map` zero times.)

Demonstrating `normalize`:

``````> normalize emptyCocktail
Cocktail (fromList [])

> normalize (0 `parts` Gin)
Cocktail (fromList [])

> normalize \$ combine (2 `parts` Gin) (4 `parts` Tonic)
Cocktail (fromList [(Gin,1),(Tonic,2)])
``````

It would be tempting to specialize the `Eq Cocktail` instance to use `normalize` so that `c == combine c c` for all `c`. But I don't like to do that because if you ever need to compare for structural equality, you can't, whereas equality under normalization can be achieved with:

``````> let (=~) = (==) `on` normalize
> (1 `parts` Gin) =~ (2 `parts` Gin)
True
``````

It would also be tempting to add normalization to `combine` so that the combination of two cocktails is a normalized cocktail. But since this blog post is about monoidal cocktails and `combine` is the best candidate for a composition operator, such choice actually breaks the law of associativity:

``````> let norm_combine c1 c2 = normalize (combine c1 c2)

> gin1 `norm_combine` (gin1 `norm_combine` tonic1)
Cocktail (fromList [(Gin,2),(Tonic,1)])

> (gin1 `norm_combine` gin1) `norm_combine` tonic1
Cocktail (fromList [(Gin,1),(Tonic,1)])
``````

So while I like the notion of normalizing cocktail recipes, making it a part of the `Semigroup Cocktail` instance would probably be a bad idea, leaving the much simpler instances:

``````instance Semigroup Cocktail where
(<>) = combine

instance Monoid Cocktail where
mempty = emptyCocktail
``````

As for superdrinks, the recipe can now be expressed as:

``````superdrinks :: Natural -> Cocktail -> Cocktail
superdrinks n base = mconcat
[ ((3^n - 1) `div` 2) `parts`  Gin
,  (3^n - 1)          `parts`  Lemon
,  (3^n)              `rounds` base
]

rounds :: Natural -> Cocktail -> Cocktail
rounds n = mconcat . genericReplicate n
``````

Whether the best approximation is using a base of `mempty` or, as the revue sketch suggests, `n `parts` Gin`, is highly subjective. For a 5th approximation of superdrinks using pure gin as 0th approximation,

``````> normalize \$ superdrinks 5 (1 `parts` Gin)
Cocktail (fromList [(Gin,182),(Lemon,121)])
``````

Cheers!

## Discussion (3)

Taylor Fausak

Great post! We liked it so much we recorded a podcast about it :) haskellweekly.news/episode/26.html

Janus Troelsen

You joke around that 2 gallons of gin is so much. But a "part" is not an actual physical unit, is it? I thought it only is a fraction, constructed using the other parts.

Also, the 21 year age limit does not apply to Simon since he is not US-American.

Ok, maybe I am taking the podcast too literally :P

Simon Shine

Ha, I just realized.

Thanks for making new podcast episodes again. :)