Prisms

So what are prisms?

Prisms are like lenses just for sum types … - the most memorable sentence, but I don’t think very helpful at first glance.

So let us start with defining a sum type.

> data Town = Prague | Vienna | Kingston | London | Hipsterhausen deriving (Show, Eq)

> data Pirate = Captain   { _attributes :: Human, _ship :: String}
>             | FirstMate { _attributes :: Human, _shanty :: String}
>             | Marauder  { _attributes :: Human, _hometown :: Town}

> makeLenses ''Human
> makeLenses ''Body
> makeLenses ''Pirate

Pirate is a sum type - with different records in each constructor.

A tangent on sums and products

Sums

Why is the above called a sum type, just let us calculate the number of options we have in a sum type:

data Sum = One | Two | Three

If I have a variable x :: Sum it has exactly three values it can be

• x = One
• x = Two
• x = Three

so the sum of the number of constructors is the solution of how many options x can be.

Product

So how come product types to their names - you might already guess it.

data Product = P Sum Sum

For a variable y :: Product we find y can be |Sum|×|Sum| many possible values:

• y = P One One
• y = P One Two
• y = P One Three
• y = P Two One
• y = P Two Two
• y = P Two Three
• y = P Three One
• y = P Three Two
• y = P Three Three

So can anybody find what exponential types are?

The answer is functions

data Two = One | Two
data THREE = ONE | TWO | THREE

So the functions from Two -> THREE have exactly |Two|^|THREE| many elements.

one,two,three,four,five,six,seven,eight :: Two -> THREE
one   One = ONE
one   Two = ONE
two   One = TWO
two   Two = TWO
three One = THREE
three Two = THREE
four  One = ONE
four  Two = TWO
five  One = ONE
five  Two = THREE
six   One = TWO
six   Two = THREE
seven One = THREE
seven Two = ONE
eight One = THREE
eight Two = TWO

Back to Prisms

The problem with lenses is that cpt some getters don’t make sense. For datatypes that have a special elements, so called monoids like String with "" we get special getters, but for Town for example …

Prelude > cpt^.shanty
""
Prelude > cpt^.hometown
...
error message complaining about Town not being a monoid
...

So an easy way to get special elements is combining it with Maybe. And that’s what prisms are - getters with Maybe-values instead of errors.

So how do we get them - again as with prisms we (have to) build them with template haskell magic using

> makePrisms ''Human
> makePrisms ''Body
> makePrisms ''Pirate

This magic words create “Constructors” _Captain, _FirstMate and _Marauder and the lens library provides functions preview, review and ^?.

Now what are those functions doing - they select one branch in a sum type.

Prelude > preview _Captain cpt
Just (Attributes {...},"SS Sea Serpent")
Prelude > preview _Captain mrd
Nothing

But there is also an infix shortcut for preview - (^?) so we could have written the above cpt^._Captain or use it with accessor-like functions as hometown,ship,shanty or attributes.

Prelude > cpt^?hometown
Nothing
Prelude > cpt^?ship
Just "SS Sea Serpent"
Prelude > mrd^?hometown
Just Hipsterhausen

So what about review?

The review function can create new things from the results of preview, well not exactly but almost and with use of (<$>) from Control.Applicative we can make them work together

Prelude> :t preview _Captain cpt
preview _Captain cpt :: Maybe (Human, String)
Prelude> let Just x = preview _Captain cpt
Prelude> :t review _Captain x
review _Captain x :: Pirate
Prelude> :t review _Captain <$> preview _Captain cpt
review _Captain <$> preview _Captain cpt :: Maybe Pirate
Prelude> review _Captain <$> preview _Captain cpt
Just Captain {Attributes {...},"SS Sea Serpent"}
Prelude> review _Captain <$> preview _Captain mrd
Nothing
Prelude> review _Captain <$> preview _Marauder mrd
... Error ... -- Captain needs String (ship name) where Marauder has a Town as
a second argument

So Prisms do not fix everything - but provide a safety layer for simple accessing stuff and sometimes for generating stuff as well.

So thats all I know about Lenses and Prisms - for understanding the type signatures - I still do not feel confident to present about.