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The algebra and calculus of algebraic data types (recurse.com)
170 points by alex_hirner 6 months ago | hide | past | web | favorite | 48 comments



That's fun and a bit surprising, but maybe it shouldn't be. I'm reminded that Dana Scott with Christopher Strachey showed that by using lattices or complete partial orders with a bit of topology to model graphs of possible computations of a program you could, just as in analysis, define least upper and lower bounds and a notion of continuity to derive a construction of limit for a computation which is analogous to a limit in analysis. They called this model a domain. That bit of of math is the basis of denotational semantics of programming languages and is necessary because sets are largely sufficient as the basis for algebra and analysis but not for programs which have evils like partial functions, side effects and variable assignment. I believe that Christopher Strachey with Scott also introduced the formal notions of sum, product and recusive types. They also showed how definitions or models of recursive types and functions could be well founded through their construction of limits on domains. An early tech report on it can be found here:

https://www.cs.ox.ac.uk/files/3228/PRG06.pdf

and here's a more recent free book from David Schmidt on the topic:

http://people.cs.ksu.edu/~schmidt/text/DenSem-full-book.pdf


Another interesting area of program semantics that hasn’t seen much attention afaik is in the design of languages whose programs form some algebraic structure.

My area of research is concatenative programming, where concatenation of two programs denotes the composition of those programs, and the empty program is the identity function (on the “program state”, which is usually a stack). That means that the syntax and semantics of the language both form monoids, and there’s a homomorphism from the syntax onto the semantics.

Several years ago, I was trying to come up with languages with other more interesting algebraic structures, so you could apply theorems from those structures to programs. A particular challenge is a language whose structure forms a nontrivial ring¹, in particular a Euclidean ring—then you could find the greatest common divisor of two programs, and because a Euclidean ring is a unique factorisation domain, you could also factor a program into prime subprograms. I was fascinated by what that might mean and whether it could be useful.

Unfortunately, it was that “nontrivial” aspect that I could never get past. The language always ended up insufficiently powerful to express anything of interest, its algebraic structure was trivial (e.g. there’s only one program), or it ended up having a weaker structure (e.g. an idempotent semiring). Chris Pressey also worked on this concept a bunch around 2007, and produced some results like Cabra² and Burro³, but ran into similar dead ends like Potro⁴.

¹ https://en.wikipedia.org/wiki/Ring_(mathematics)

² http://catseye.tc/article/Languages.md#cabra

³ http://catseye.tc/article/Languages.md#burro

https://github.com/catseye/Chrysoberyl/blob/master/article/L...


This algebraic approach to program semantics is basically how categorical semantics works. The syntactic equational theory of the simply typed lambda calculus with pairs, for example, is that of the "free" cartesian closed category. It can be mapped (in a way that respects the cartesian closed structure) into any cartesian closed category, giving you a semantics.

A monoid can be seen as a one-object category (the monoid elements are the morphisms on that object, and the monoid operator is morphism composition), so perhaps concatenative languages have categorical semantics too. (Although it seems like the semantics of "quote", or whatever [ foo bar baz ] is in forth, might make things a little interesting - ie. require more structure than a monoid homomorphism. I expect you really want cartesian closed categories, and quote is probably curry or apply.)


I don’t know how concatenative languages fit into category theory—I would guess Cartesian closed categories are the place to start, but that’s not really my area of expertise.

Maybe you can offer some insight if I tell you that the quotation syntax “[ … ]” in a concatenative language corresponds to lambda abstraction:

    e : a
    ---------------------
    [ e ] : ∀s. s → s × a
And the quotation operator “quote”, which takes a value on the stack and returns it quoted, corresponds to eta-expansion (or lifting lambda abstraction into a term, if you like) and has the type:

    quote : ∀sa. s × a → s × (∀t. t → t × a)
Generally the standard Turing-complete basis in concatenative calculus is the following set of combinators:

    compose : ∀rstu. r × (s → t) × (t → u) → r × (s → u)
    swap : ∀sab. s × a × b → s × b × a
    drop : ∀sa. s × a → s
    dup : ∀sa. s × a → s × a × a
    quote : ∀sa. s × a → s × (∀t. t → t × a)
    apply : ∀st. s × (s → t) → t
(Although you can get away with fewer if they’re equivalent in power.)

The first three correspond to the standard B, C, K, and W combinators from combinatory logic, which give you all the substructural rules from logic—swapping is exchange, dropping is weakening, and copying is contraction. Without swap/drop/dup, it’s equivalent to ordered linear lambda calculus, which can be interpreted in any category (since it’s just B and I); what are the categorical equivalents of linear (BC), affine (BCK), ordered (BKW), and relevant (BCW) logics?


Oh neat! (I'm working on using Joy in a practical way, I had an implementation in Python but it turns out Prolog is more conducive: https://osdn.net/projects/joypy/scm/hg/Joypy/blobs/tip/thun/...)

> Unfortunately, it was that “nontrivial” aspect that I could never get past. The language always ended up insufficiently powerful to express anything of interest, its algebraic structure was trivial (e.g. there’s only one program), or it ended up having a weaker structure (e.g. an idempotent semiring). Chris Pressey also worked on this concept a bunch around 2007, and produced some results like Cabra² and Burro³, but ran into similar dead ends like Potro⁴.

Maybe that in itself is an important result?


> Maybe that in itself is an important result?

Well, I didn’t have the skills to actually prove the negative then, I was just never able to find a positive example. Still not convinced that it’s impossible, if I ever get back to it, or someone wants to pick it up as a problem to tinker with.


Also take a look at the Boom Hierarchy [1], which came from the same era (Bird-Meertens formalism, etc.). It describes relationships between data structures constructed from 4 algebraic properties: unit, commutative, associative and idempotent. The resulting 16 possibilities include the familiar set, bag, list and tree, but also other less useful combinations, such as non-empty mobile.

[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49....


This is almost identical to the study of combinatorial species (https://en.wikipedia.org/wiki/Combinatorial_species)


If you like this you'll want to look at "semiring programming" (just search on that phrase) and Categorical Programming, e.g.: "Compiling to categories" Conal Elliott http://conal.net/papers/compiling-to-categories/


I found the paper linked inside the article very interesting. It explores (one possible) meaning of negative and rational types. I'm still going through it:

https://www.cs.indiana.edu/~sabry/papers/rational.pdf


Nat = Nat + 1 does end up being meaningful: since ultimately the numbers that fall out (e.g. Bool = 2) end up describing the sizes of sets, an interpretation of Nat is some infinite cardinal (aleph-null), which is the size of the set of natural numbers.


That makes sense, but this case shows that apparently arbitrary algebraic simplifications aren't allowed because otherwise we could subtract Nat from each side and have trouble.


A cool aspect is how this algebra/calculus sheds light that maps(/dictionary/associative-arrays) have an equivalent cardinality to functions and that they are in many ways the same.


In Clojure, maps can be called as functions (equivalent to `get`). This isomorphism is also reflected in K.


I once tried to take the Taylor series idea further and define the "cosine" of a type as the Taylor expansion of cos(x), but didn't get anywhere with it.

Can you do anything else weird with ADTs?


Huh. Thinking about this example I'm confused, because the power series for exp(x) looks like the type for unordered lists of x: e.g. the term for lists of length three is x^3, and if you ignore order you get x^3/3!. The cosine is almost the same as the even terms of this series, but with a minus sign on every other term.

What confuses me (besides what to make of the minus sign) is that the type of sets of x ought to be 2^x -- that is, x -> Bool -- because it's isomorphic to the characteristic function of a set, and a function type is an exponential (https://bartoszmilewski.com/2015/03/13/function-types/). But 2^x differs from exp(x) by a constant factor. So I seem to be messing with surface-level analogies without real understanding.


You're comparing unordered lists of unlimited length with subsets.

To define a subset of x, you simply choose whether any element is present or not:

type Set x = x → Bool

Cardinality: 2^x

Defining an unordered list (of unlimited length, with possible repetitions) with algebraic data types is harder, if at all possible. Especially when you consider what "unordered" should mean in the face of repetitions. I wouldn't be surprised if it was not possible, or if the expansion turned out to have a strange cardinality such as e^x


Thanks, I forgot about repetitions of the same element. How could I miss that?


One thing I’ve been wondering is how to interpret the function types `Void -> a` and `a -> Void`, where Void is the uninhabited type (unique up to isomorphism). The number of inhabitants of those types should be |a|^0=1 and 0^|a|=0, respectively, but what does that mean? In real world, function(s) of type a->Void certainly exist, such functions being those that diverge (loop forever or halt via some non-local means).


`Void` being the uninhabited type, in the light of the Curry-Howard isomorphism stands for a false proposition.

`a -> Void` get interpreted as "not a" or "from a follows contradiction" or equivalently "a is uninhabited". Combinatorially it's `0 ^ a` which for non-empty a is zero but is equal to 1 when a is empty (0^0=1). In other words there are no functions of type `a -> Void` for non-empty a and there's exactly one such function for uninhabited a (id :: Void -> Void).

`Void -> a` is interpreted "from falsehood, anything (follows)" https://en.wikipedia.org/wiki/Principle_of_explosion. Combinatorially a^0 = 1 for all a so there's exactly one such function. An easy way to define it is by induction on Void (which has no cases and you're done).


Yeah, Void makes for great for inductive definitions ;)


In mathematics, assuming 'void' is the empty set, the function void -> a exists, but is kind of trivial because it never returns anything, however functions a -> void can't exist because they'd have to return an element of void, which don't exist, the one exception being the function void -> void. Denoting A -> B as B^A this translates to:

    a^0 = 1  
    0^a = 0  
    0^0 = 1
The fact that there's precisely 1 function void -> a is considered rather special in category theory, and means that 'void' is the so called 'initial object' for the category (which is automatically unique up to isomorphism).


Yeah, it makes sense now. Thanks.


AFAIUI:

"Void -> a" means that the function cannot be called, because there no way to conjure a Void value. In Haskell, you can do it by (ab)using 'undefined', but that's kind of cheating. However, if you're using Idris with totality checking, I don't think you'll actually get any code calling such a function compile.

The "a -> Void" means that the function can never return.


`Void -> a` is inhabited by the slightly weird empty-case construct in (GHC) Haskell: `\x -> case x of {}` has type `Void -> a`


Also known as the `absurd` function IIRC?


Such functions can’t exist.

The “C” void is expressed by the unit singleton.

So it is |a|^1.


Yes, C void is a kinda-sorta unit type. But I’m talking about the real uninhabited type, ie. Void in Haskell, Nothing in Scala, ! (Never) in Rust. Real-world languages including Haskell are not total, so all functions a->b are in reality a->(b|Void). And for a good reason, as totality means loss of Turing completeness. In a total language, a function `a->Void` does not exist, but what about `Void->a`, then? What is the single inhabitant of that type?


Since one can prove anything if you're assuming nonsense, a function of the type 'Void -> a' should be able to just return its argument.

And assuming the theory is consistent(so no infinite loops etc.) this is the only instance of that function, so the cardinality is 1.


The "should be able to just return its argument" part is wrong; the types pretty clearly don't match, yeah? It is true that there is a unique function of type `Void -> a` for each `a`, but that isn't it.


When the Bottom type is included in a language it usually exhibits sub typing behavior. That can be expressed in Haskell, too:

  {-# LANGUAGE RankNTypes #-}

  type Bottom = forall a. a -- zero inhabitants

  f :: Bottom -> a
  f x = x

That passes the type checker just fine.


> That passes the type checker just fine.

I think you are running into some weird haskellisms with your Bottom-type.

The normal way of defining that in haskell is using the "EmptyDataDecls" pragma, like this:

    {-# LANGUAGE EmptyDataDecls #-}

    data Empty 

    g :: Empty -> a
    g x = x 
Which doesn't pass the type checker.

(From a theoretic standpoint, I would have thought your Bottom was a essentially a type-level identity function..)


Again, languages with a built in bottom usually have sub typing behavior that makes bottom a subtype of everything (so you can give a value of type bottom to everything).

'forall a. a' is just how you define the bottom type in System F, which Haskell is somewhat based on. As a type it describes that a term of that type can just conjure a value of any type out of thin air, which is obviously nonsense. That what makes it the bottom type.

The sub typing behavior associated with that is just the normal subsumption rule of polymorphic functions. The same reason why you can pass a function of type 'forall a. a -> a' to something expecting a function of type 'Int -> Int'. The types don't 'match' directly, but they do under the subsumption rule.


Yeah, I just wanted to show that (GHC) haskell has a way of getting a type with "zero" inhabitants that actually gets caught by the type checker. (I say "zero" since of course my Empty-type above does contain non-terminating programs and (undefined :: a).)

With "forall a. a" I was coming from the persepctive of intuisionistic type theory where the usual parametric polymorphism just get subsumed by Π-types and universal quantification is usually represented with Π-types, so you would have:

         forall a.a       (Universal Quantification)
    <==> Π(a : Type) a    (Π-type)
    <==> (a : Type) -> a  (Agda-notation)
     ==> a -> a           (Π-type as regular function type)
where Type represents any type. That's what I meant by "type-level identity function".


Interesting, thanks! I hadn't considered using "forall a. a" as the bottom type, but it makes sense.


And correspondingly `a->Void` must be empty because you can’t prove a contradiction given a true proposition in a consistent system? Makes sense from the Curry-Howard point of view.


Where does an instance of a come from in that case? A function of that type has no way to construct one.


Yes, there are no possible return values. But there is a unique vacuous total function because no return values are ever needed. A function can be considered as a set of ordered pairs, one for each value in the domain. Since there are no values in the domain, no pairs need exist.

Another way to look at it is as a constrained subset of the Cartesian product of domain and codomain. In this case the domain is the empty set, so the Cartesian product is empty too.


The key is that you can (in theory, I don't know if Haskell has a proper bottom that can act as a type) give a value of Void/Bottom to anything, analogous to logic in that a proof of nonsense can be used as a proof of anything.

Take this Haskell code that passes the type checker:

  bottom = bottom

  f :: Int -> a
  f _ = bottom

Here 'bottom' has the conceptual type Void/Bottom and checks successfully against the 'a'.

So assuming we have this Void type in Haskell (I don't know if Haskell has it or not), this should also type check:

  f :: Void -> a
  f x = x
Just like bottom (of type Void) checked against the 'a' earlier, an argument of that type should also type check.


That makes sense, thanks!


It is the so-called empty function, which diverges, e.g. throws an exception if called.


This one typechecks but loops forever:

  f :: Void -> a
  f _ = let x = x in x


Very neat insights into data type differentiation! Minor nitpick: the article uses

    data Unit = Unit
but the Haskell standard Prelude already has the empty tuple type (), with single element (), functioning as the standard unit type.


Does distributivity hold? a x (b + c) = a x b + a x c


Up to isomorphism, yes. That is, you can write a pair of functions that are mutual inverses that witness the fact that those two types are isomorphic:

    distl :: (a, Either b c) -> Either (a, b) (a, c)
    distl (a, b_c) = case b_c of
      Left b -> Left (a, b)
      Right c -> Right (a, c)

    factl :: Either (a, b) (a, c) -> (a, Either b c)
    factl ab_ac = case ab_ac of
      Left (a, b) -> (a, Left b)
      Right (a, c) -> (a, Right c)
With the “TypeOperators” feature enabled (and “UnicodeSyntax” for pretty arrows and such, because why not), you can write it more literally:

    type (×) a b = (a, b)
    infixl 7 ×

    type (+) a b = Either a b
    infixl 6 +

    distl ∷ a × (b + c) → a × b + a × c
    factl ∷ a × b + a × c → a × (b + c)


Not literally, but there is a 1-1 correspondence between terms of the former and terms of the latter. Modulo technicalities in the presence of non-termination.


Logically, it would seem to. If you can have an apple and either a ball or a cat, that's logically the same as having either an apple and a ball, or an apple and a cat. In terms of types, they are the same.


What happened to Code Words that are not new issues?




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