[rebeccaskinner]

> A function a->b is a container for bs

Anecdotally, this is one of those things that's trivially true to some people, but really hard for other people to internalize. I think it's why the "container" can lead people astray- if you haven't internalized the idea of functions as being indexed by their argument, it's a really mind twisting thing to try to make that leap.

[robto]

One of the fun things about Clojure that reinforces this "trivially true" perspective is that maps and sets are functions:

    ;; "maps" the keys to the values
    (map {1 "a" 2 "b"} (take 5 (cycle 1 2))) ;;=> '("a" "b" "a" "b" "a")
    ;; acts as a predicate that tests for membership
    (filter #{"a" "b" "c"} ["a" "b" "c" "d" "e" "f"]) ;;=> '("a" "b" "c")

Once you get used to this idiom you naturally start thing of other functions (or applicative functors) the same way. The syntax sugar makes for some very concise and expressive code too.

[magicalhippo]

If I give you a function "f(x) := 3 * x", is it really that useful to talk about it as a container of the natural numbers?

The reverse though is useful, a container looks like a function that takes one or more indices and returns a value or element.

[rebeccaskinner]

I think that understanding the (moral) equivalence is useful in both directions. In particular, I think helping people understand the "function-as-container" analogy is a useful way for people to understand pure functions- another thing that's conceptually simple but a lot of people struggle to really wrap their mind around it.

[magicalhippo]

Personally I would say it muddies the water for me, as "container" has strong connotations in other directions.

But then I've never thought the concept of a pure function to be particularly difficult, despite growing up on procedural languages.

It's other bits that I struggle with when it comes to functional programming.

[n_plus_1_acc]

I can recommend learnung some scala, where HashMap extends PartialFunction

[rebeccaskinner]

I've never looked at scala, but that's really interesting. Do you find that's useful in practice?

[mncharity]

> really hard [...] leap

Two stepping stones might be array getters (function that's array-ish), and arrays with an indexed default value function (array that's function-ish)?

[rebeccaskinner]

I've recently started writing a series of blog posts (https://rebeccaskinner.net/posts/2024-10-18-dictionaries-are...) trying to explain the idea and my approach has been to explain the idea using comprehensions. I haven't had a lot of people review the post yet, and I still have at least one if not two more follow-ups before it's done, so I'm not yet sure how well the idea will land.

[magicalhippo]

Nice introduction. Still not entirely sold that dictionaries are pure functions, though.

Will you be covering common dictionary operations like adding/removing elements and iterating over the dictionary keys?

I have some ideas on how one might frame it in a pure function setting but they all seem quite contorted in a similar way to your incrementDict, ie you'd never actually do that, so curious if there are better ways. Then maybe you'll sell me on the premise.

[rebeccaskinner]

I'm really focusing less on the idea that Dict the data type with it's associated methods is like a function, and more on the idea that a dictionary in the general sense is a mapping of input values to output values, and you can think of functions that way.

That said, there are some pretty reasonable analogies to be made between common dictionary operations and functions.

For example, adding and removing items can be done with function composition so long as you are okay with partial lookups. Here's a really short example I put together:

  module Example where
  import Control.Applicative

  type Dict a b = Eq a => a -> Maybe b

  emptyDict :: Dict a b
  emptyDict = const Nothing

  singleton :: a -> b -> Dict a b
  singleton k v target
    | k == target = Just v
    | otherwise = Nothing

  unionDict :: Dict a b -> Dict a b -> Dict a b
  unionDict dict1 dict2 k = dict1 k <|> dict2 k

  insertDict :: a -> b -> Dict a b -> Dict a b
  insertDict k v dict = singleton k v `unionDict` dict

  removeDict :: a -> Dict a b -> Dict a b
  removeDict k dict target
    | k == target = Nothing
    | otherwise = dict k

This particular representation of dictionaries isn't necessarily something you'd really want to do, but the general approach can be quite useful when you start working with something like GADTs and you end up with things like:

  data Smaller a where
    SmallerInt :: Smaller Int
    SmallerBool :: Smaller Bool
  
  data Larger a where
    LargerInt :: Larger Int
    LargerBool :: Larger Bool
    LargerString :: Larger String
  
  someLarger :: Larger x -> x
  someLarger l =
    case l of
      LargerInt -> 5
      LargerBool -> True
      LargerString -> "foo"
  
  embedLarger ::
    (forall x. Larger x -> Smaller x) ->
    (forall smallerI. Smaller smallerI -> r) ->
    (forall largerI. Larger largerI) -> r
  embedLarger mapping fromSmaller larger = fromSmaller (mapping larger)

(I'm actually co-authoring a talk for zurihac this year on this pattern, so I have quite a bit more to say on it, but probably not ideal to cram all of that into this comment).

[magicalhippo]

> and more on the idea that a dictionary in the general sense is a mapping of input values to output values, and you can think of functions that way.

So what's the difference between a map and a dictionary then?

> Here's a really short example I put together

Much appreciated. I don't really know Haskell (nor any other functional language), but I'm pretty sure I understood it.

> This particular representation of dictionaries isn't necessarily something you'd really want to do

Yeah that's pretty much what I had in mind, and yes it's possible but it feels forced. For one you're not actually removing an element, you just make it impossible to retrieve. A distinction that might seem moot until you try to use it, depending on the compiler magic available.

> I'm actually co-authoring a talk for zurihac this year on this pattern

Sounds interesting, will check it out when it's published.

[rebeccaskinner]

> So what's the difference between a map and a dictionary then?

You're asking good questions and catching me being imprecise with my language. Let me try to explain what I'm thinking about more precisely without (hopefully) getting too formal.

When I say "a function is a mapping of values" I'm really trying to convey the idea of mathematical functions in the "value goes in, value comes out" sense. In a pure function, the same input always returns the same output. If you have a finite number of inputs, you could simply replace your function with a lookup table and it would behave the same way.

When I talk about dictionaries, I'm speaking a little loosely and sometimes I'm taking about particular values (or instances of a python Dict), and other times I'm being more abstract. In any case though, I'm generally trying to get across the idea that you have a similar relationship where for any key (input) you get a particular output.

(aside: Literally right now as I'm typing this comment, I also realize I've been implicitly assuming that I'm talking about an _immutable_ value, and I've been remiss in not mentioning that. I just want to say that I really appreciate this thread because, if nothing else, I'm going to edit my blog post to make that more clear.)

The main point is that dictionaries are made up of discrete keys and have, in Python at least, a finite number of keys. Neither of those constraints necessarily apply to functions, so we end up in an "all dictionaries are functions, but not all functions are dictionaries" situation.

> Yeah that's pretty much what I had in mind, and yes it's possible but it feels forced. For one you're not actually removing an element, you just make it impossible to retrieve. A distinction that might seem moot until you try to use it, depending on the compiler magic available.

This is a great example of the kind of thinking I'm trying to address in the article. You're completely right in a very mechanical "this is what memory is doing in the computer" sort of sense, but from the standpoint of reasoning about the problem space deleting an element and being unable to access the element are the same thing.

Of course in the real world we can't completely handwave away how much memory our program uses, or the fact that a function encoding of a dictionary turns a constant time lookup into a linear time lookup. Those are real concerns that you have to deal with for non-trival applications, even in a pure functional language.

The benefit you get, and I apologize because this is hard to explain- let alone prove, is that you can often end up with a much _better_ solution to problems when you start by handwaving away those details. It opens up the solution space to you. Transformations to your architecture and the way you think about your program can be applied regardless of the specific representation, and it's a really powerful way to think about programming in general.

[mncharity]

Interesting. I liked how "Dictionaries are Pure Functions" set up currying as JSON nested dictionaries.

Curiously, I've a backburnered esolang idea of gathering up the rich variety of dict-associated tooling one never gets to have all in one place, and then making everything dict-like. Permitting say xpath sets across function compositions.