What are Prisms?

Question

I'm trying to achieve a deeper understanding of lens library, so I play around with the types it offers. I have already had some experience with lenses, and know how powerful and convenient they are. So I moved on to Prisms, and I'm a bit lost. It seems that prisms allow two things:

1. Determining if an entity belongs to a particular branch of a sum type, and if it does, capturing the underlying data in a tuple or a singleton.
2. Destructuring and reconstructing an entity, possibly modifying it in process.

The first point seems useful, but usually one doesn't need all the data from an entity, and ^? with plain lenses allows getting Nothing if the field in question doesn't belong to the branch the entity represents, just like it does with prisms.

The second point... I don't know, might have uses?

So the question is: what can I do with a Prism that I can't with other optics?

Edit: thank you everyone for excellent answers and links for further reading! I wish I could accept them all.

Answer 1

Lenses characterise the has-a relationship; Prisms characterise the is-a relationship.

A Lens s a says "s has an a"; it has methods to get exactly one a from an s and to overwrite exactly one a in an s. A Prism s a says "a is an s"; it has methods to upcast an a to an s and to (attempt to) downcast an s to an a.

Putting that intuition into code gives you the familiar "get-set" (or "costate comonad coalgebra") formulation of lenses,

data Lens s a = Lens {     get :: s -> a,     set :: a -> s -> s } 

and an "upcast-downcast" representation of prisms,

data Prism s a = Prism {     up :: a -> s,     down :: s -> Maybe a } 

up injects an a into s (without adding any information), and down tests whether the s is an a.

In lens, up is spelled review and down is preview. There’s no Prism constructor; you use the prism' smart constructor.

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What can you do with a Prism? Inject and project sum types!

_Left :: Prism (Either a b) a _Left = Prism {     up = Left,     down = either Just (const Nothing) } _Right :: Prism (Either a b) b _Right = Prism {     up = Right,     down = either (const Nothing) Just } 

Lenses don't support this - you can't write a Lens (Either a b) a because you can't implement get :: Either a b -> a. As a practical matter, you can write a Traversal (Either a b) a, but that doesn't allow you to create an Either a b from an a - it'll only let you overwrite an a which is already there.

Aside: I think this subtle point about Traversals is the source of your confusion about partial record fields.

^? with plain lenses allows getting Nothing if the field in question doesn't belong to the branch the entity represents

Using ^? with a real Lens will never return Nothing, because a Lens s a identifies exactly one a inside an s. When confronted with a partial record field,

data Wibble = Wobble { _wobble :: Int } | Wubble { _wubble :: Bool } 

makeLenses will generate a Traversal, not a Lens.

wobble :: Traversal' Wibble Int wubble :: Traversal' Wibble Bool 

For an example of this how Prisms can be applied in practice, look to Control.Exception.Lens, which provides a collection of Prisms into Haskell's extensible Exception hierarchy. This lets you perform runtime type tests on SomeExceptions and inject specific exceptions into SomeException.

_ArithException :: Prism' SomeException ArithException _AsyncException :: Prism' SomeException AsyncException -- etc. 

(These are slightly simplified versions of the actual types. In reality these prisms are overloaded class methods.)

Thinking at a higher level, certain whole programs can be thought of as being "basically a Prism". Encoding and decoding data is one example: you can always convert structured data to a String, but not every String can be parsed back:

showRead :: (Show a, Read a) => Prism String a showRead = Prism {     up = show,     down = listToMaybe . fmap fst . reads } 

To summarise, Lenses and Prisms together encode the two core design tools of object-oriented programming, composition and subtyping. Lenses are a first-class version of Java's . and = operators, and Prisms are a first-class version of Java's instanceof and implicit upcasting.

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One fruitful way of thinking about Lenses is that they give you a way of splitting up a composite s into a focused value a and some context c. Pseudocode:

type Lens s a = exists c. s <-> (a, c) 

In this framework, a Prism gives you a way to look at an s as being either an a or some context c.

type Prism s a = exists c. s <-> Either a c 

(I'll leave it to you to convince yourself that these are isomorphic to the simple representations I demonstrated above. Try implementing get/set/up/down for these types!)

In this sense a Prism is a co-Lens. Either is the categorical dual of (,); Prism is the categorical dual of Lens.

You can also observe this duality in the "profunctor optics" formulation - Strong and Choice are dual.

type Lens  s t a b = forall p. Strong p => p a b -> p s t type Prism s t a b = forall p. Choice p => p a b -> p s t 

This is more or less the representation which lens uses, because these Lenses and Prisms are very composable. You can compose Prisms to get bigger Prisms ("a is an s, which is a p") using (.); composing a Prism with a Lens gives you a Traversal.