Type algebra: the semantic ambiguity of nested lists

025 July 12, 2014 -- (math tech)

A while ago I had a short debate on the semantics and structure of the so-called "nested lists". For example Scheme naturally allows us to define structures such as:

'(2 (2 3) 4)

Of course, this is an inherent quality of Lisp dialects, where code and data blend and nested lists are required to express arbitrarily complex expressions. Moreover, we should note that Lisp languages usually don't impose any type strictness on lists (as lists are a particular case of pairs), allowing for truly arbitrary code to be embedded inside a list, e.g.:

'(1 #t 'a "yadda")

This, as you may know, is the worst nightmare of Haskellers, or, more generally, of the ML camp. We like our structures to be defined in terms of a single well-formed type, so if we had an arbitrary type \(A\) and a type of lists \(L\) defined in terms of \(A\), i.e. \(L\;A\), it would be inherently impossible to construct nested lists, due to it leading to the (impossible) equality \(A = L\;A\). Mathematics aside, if we were to define the following in Haskell:

> let f x = [x, [x]] in f 2

we would get quite a nasty error, along the lines of:

Occurs check: cannot construct the infinite type: t0 = [t0]
In the expression: x
In the expression: [x]
In the expression: [x, [x]]

given that it's quite impossible to unify an arbitrary type t0 with [t0].

It is however possible, albeit unnatural, as we will see, to define nested lists in Haskell. There exist at least two possible definitions, which I will explore in the remaining paragraphs.

Definining the type of nested lists

Before defining a nested-list type, which I take the liberty to name NList, I'll start with observing that Haskell's built-in list type can be redefined without all the sugary bells and whistles as follows:

data List a = Empty | Cons a (List a)

Its so-called "interface" consists of the two constructors Empty and Cons and of the functions head and tail, having the signatures:

head :: List a -> a
tail :: List a -> List a

where head returns the first element of the ordered value-list pair, and tail the second. We note that both are partial functions, i.e. applying them on Empty will result in an error.

Having said all this, there is an easy way to construct a nested-list type starting from the definition of List, and you might have already seen it: add a second constructor, one which would allow inserting a list inside another. The NList type would therefore look like something along the lines of:

data NList a = Empty | ConsElem a (NList a) | ConsList (NList a) (NList a)

This doesn't look particularly bad, except we have one problem: we can no longer define a single head function for this type. We would need to have two heads, such as:

headNElem :: NList a -> a
headNList :: NList a -> NList a

having the definitions:

headNElem (ConsElem x _) = x

and

headNList (ConsList x _) = x

Of course, we can also define a tail function:

tailNList :: NList a -> NList a
tailNList (ConsElem _ y) = y
tailNList (ConsList _ y) = y

The problem with headNElem and headNList is that, not only they're partial functions, but they're also unusable by themselves in practice. For the sake of testing our implementation, let us define a test_list [2, [2, 3], 4], or, without sugar:

test_list :: NList Integer
test_list = ConsElem 2 $ ConsList (ConsElem 2 $ ConsElem 3 $ Empty)
    $ ConsElem 4 Empty

If test_list were some arbitrary list, we'd have no way of knowing whether the first element was built using ConsElem or ConsList, leading to the following problem:

> headNList test_list 
*** Exception: drafts/NList1.hs:33:1-28:
    Non-exhaustive patterns in function headNList

This is indeed problematic. One way around the issue would involve defining a function isList that checks whether the head is a list or an element. Another, very similar method would involve giving up headNElem and making headNList have the return type Either a (NList a). Both approaches would result in code looking painfully similar to spaghetti, due to the necessity of case handling. This gets especially nasty when we don't immediately care about the nature of our elements, so we'd have to essentially duplicate code to do one thing for two different cases. In other words, welcome to type hell. Or limbo.

The third way consists in giving up headNElem and making headNList look somewhat similar to tailNList, by providing the former with an additional definition:

headNList (ConsElem x _) = ConsElem x Empty

loosely translated as "take the head and insert it into an empty list", ensuring that we always return a value of the type NList a.

To make our implementation more uniform, we can define some constructor functions:

emptyNList :: NList a
emptyNList = Empty

consElem :: a -> NList a -> NList a
consElem = ConsElem

consList :: NList a -> NList a -> NList a
consList = ConsList

and make our module export all the definitions useful for the outside world:

module NList
  ( NList
  , consElem
  , consList
  , emptyNList
  , headNList
  , tailNList
  ) where

Note that this more convenient implementation is also incorrect. Calling headNList on [2, [2, 3], 4] and returning a list defies the semantics of lists: what we're doing is taking the element and wrapping it in a list; what we should be doing is take the element and return it as it is, which we've established we can't do in Haskell without cluttering the implementation. To exemplify the ambiguity, we ponder what happens when we call headNList on a singleton:

*NList> headNList (ConsElem 2 Empty) == ConsElem 2 Empty 
True

meaning that we never know whether our head function returns a singleton or an element, i.e. if the original list contained 2 or [2]. Either way, this implementation of NList lacks consistency and would get our average programmer into trouble quite quickly, no matter how we put it.

There is at least one more way to define nested-list types in Haskell. There are probably more, but I will present only one more, and I will name it the "alternate".

An alternate definition

To provide this "alternate" definition, I propose the clean approach of reusing the existing list type [a] to build our new type. Reiterating, our goal is to let the Haskell programmer build lists with nested structure, as illustrated by our earlier example, [2, [2, 3], 4]. Let's start by defining NList as a type synonym:

type NList a = [a]

This (quite obviously) doesn't work, because we want to be able to (possibly) store lists inside other lists, intuitively leading us to the initial problem of NList a = [NList a]. We can however express this type recurrence relation in Haskell in a plain manner, using data:

data NList a = List [NList a]

Notice that this is a legitimate Haskell type. It isn't however very useful to us, since it only allows us to construct arbitrarily nested empty lists, such as:

List [List [], List [List []]]

By taking another look at the example above, we notice that the NList type describes the proper structure, without however giving us the proper tools to populate the lists with elements. We'll solve this by "lifting" elements into lists, turning the definition into:

data NList a = Elem a | List [NList a]

Given this context, the example test_list will have the definition:

test_list :: NList Integer
test_list = List [Elem 2, List [Elem 2, Elem 3], Elem 4]

The rest of the interface stays the same: we can still build headNList and tailNList with the previously defined interface. Note the partialness, i.e. the impossibility of defining the functions on values constructed using Elem:

headNList (List (x : _)) = x
tailNList (List (_ : xs)) = List xs

This holds as well for emptyNList, consElem and consList, which now become a sort of "pseudo-constructors", as we can't define them purely as the two constructors provided above. We can however define the former in terms of the latter:

emptyNList = List []
consElem x xs = consList (Elem x) xs
consList x (List xs) = List $ x : xs

The implementation of consElem and consList looks as elegant as it can get, although we no longer have some of the nonsensical cases handled statically (i.e. at type level); for example consList (Elem 1) (Elem 2) needs to be kept undefined, either explicitly or implicitly.

To express the problem in semantical terms, we're rid of the ugly spaghetti implementation and/or ambiguities, only we're now smashing our heads against the dirty issue of having elements treated as lists! This might make sense in the formal contexts of singletons or applicative functors, but here we're just using this as a hack to obtain a practical advantage, which, in a way, beats the whole purpose of correctness and clean design that Haskell programs should stand for.

But before admitting defeat, let's make a short analysis of the problem at a purely algebraic level.

Some abstract nonsense

If you've got some insight of how Haskell works, you are probably aware of the fact that Haskell types are built upon a mechanism called "algebraic data types". Now, if you're a mathematician, you are very probably aware of the fact that the various type-theoretical disciplines use only a few fundamental operations to construct types, the most important being products, sums (also called disjoint unions) and functions, with the possibility of reducing them only to the former two if we view functions as a particular type of binary relations (themselves expressed as products).

Getting back to programming for a bit, all (or most?) languages provide some form of product and sum types: C has structs and unions respectively, and C++ also introduces classes and objects, which are in fact a whole different story. Java doesn't have a union type, but the canonical Either type can be build on top of classes; this is also true for Python, maybe due to the fact that disjoint unions can cause real trouble in dynamically-typed languages. Finally, Haskell user types are constructed exclusively based on products (e.g. data Pair a b = MkPair a b) and sums (e.g. data T = T1 | T2). Mathematical notation uses \(\times\) to denote products and \(+\) to denote sums.

Moreover, instead of using the set-theoretical \(x \in A\), we would use \(x : A\) for types, and we would define higher-order types by separating the type from its parameter using a space character, e.g. \(F\;X\). Also, the empty set would by replaced by a so-called "unit" type, i.e. the type with a single value, denoted as \(\mathbb{1}\).

Using this language, we can define the type of lists as:

\(L\;A = \mathbb{1} + A \times L\;A\)

(Note that we assume that the precedences of the binary operators \(\times\) and \(+\) are the same as those of arithmetic products and sums respectively.)

Looking back at List, we notice that the two definitions are isomorphic. We can thus define the two NList types in the same way, I will call them \(\text{NL}_1\) and \(\text{NL}_2\):

\(\text{NL}_1\;A = \mathbb{1} + A \times \text{NL}_1\;A + \text{NL}_1\;A \times \text{NL}_1\;A\).

\(\text{NL}_2\;A = A + L\;(\text{NL}_2\;A)\)

Now, the first thing that pops to my mind by looking at the definitions of \(\text{NL}_1\) and \(\text{NL}_2\) is that they look different; but can we prove that the two types are equivalent? Assuming that \(\times\) and \(+\) are distributive, could we juggle the two expressions in some way so that one leads to the other? I'm not sure that we can. The best I can think of is doing some factoring in \(\text{NL}_1\)'s equation, getting to:

\(\text{NL}_1\;A = \mathbb{1} + (A + \text{NL}_1\;A) \times \text{NL}_1\;A\)

The part \(A + \text{NL}_1\;A\) seems to suggest that we can "select" between values and lists, but other than that it doesn't seem that we can obtain the second type from the first; nor do I venture to try and obtain the first from the second, since \(\text{NL}_2\)'s definition depends on \(L\). I will leave this for the reader to explore further, I'm really curious if one can prove, or even better, disprove the equivalence between the two.

Conclusion

I've attached the Haskell source code for the two definitions as NList1.hs and NList2.hs respectively. Feel free to browse through them, modify them and get your own insight on the matter.

Ultimately, I believe that "nested lists" are an ill-posed problem. Ok, we have at least two ways to define them, but what are they useful for? If we want to model nested data, then maybe we're better off defining multi-way trees and using them as such. If, on the other hand, we want to model arbitrary expressions like Scheme does, then we'd better use the semantics of a "type of expressions", along with the algorithms appropriate for this application.

Jonathan Tang's Write Yourself a Scheme in 48 Hours defines Lisp values in a way looking strikingly similar to our second NList definition. This makes a lot more sense, since an Atom (the equivalent of our Elem) may be a Lisp value, although it's not necessarily a nested list. While this narrows down the scope to the parsing and evaluation of Scheme expressions, it also looks much clearer and cleaner, and it's unquestionably the manner in which programming must be done in Haskell, or in any other strongly-typed functional language for that matter.