Fold: at the limit of comprehension

Fold” is a programming concept, a common name for a particular higher-order function that is widely used in functional programming languages. It’s a fairly simple thing, but in practice I think of it as representing the outer limit of concepts a normal programmer can reasonably be expected to grasp in day-to-day work.

What is fold? Fold is an elementary function for situations where you need to keep a tally of things. If you have a list of numbers and you want to tally them up in some way, for example to add them together, fold will do that.

Fold is also good at transforming sequences of things, and it can be used to reverse a list or modify each element of a sequence.

Fold is a useful fundamental function, and it’s widely used. I like using it. I just scanned about 440,000 lines of code (my own and other people’s) in ML-family languages and found about 14,000 that either called or defined a fold function.

Let me try to describe fold more precisely in English: It acts upon some sort of iterable object or container. It takes another function as an argument, one that the caller provides, and it calls that function repeatedly, providing it with one of the elements of the container each time, in order, as well as some sort of accumulator value. That function is expected to return an updated version of the accumulator each time it’s called, and that updated version gets passed in to the next call. Having called that function for every element, fold then returns the final value of the accumulator.

I tried, but I think that’s quite hard to follow. Examples are easier. Let’s add a list of numbers in Standard ML, by folding with the “+” function and an accumulator that starts at zero.

> val numbers = [1,2,3,4,5];
val numbers = [1, 2, 3, 4, 5]: int list
> foldl (op+) 0 numbers;
val it = 15: int

What’s difficult about fold?

  1. Fold is conceptually tricky because it’s such a general higher-order function. It captures a simple procedure that is common to a lot of actions that we are used to thinking of as distinct. For example, it can be used to add up a list of numbers, reverse a list of strings, increase all of the numbers in a sequence, calculate a ranking score for the set of webpages containing a search term, etc. These aren’t things that we habitually think of as similar actions, other than that they happen to involve a list or set of something. Especially, we aren’t used to giving a name to the general procedure involved and then treating individual activities of that type as specialisations of it. This is often a problem with higher-order functions (and let’s not go into monads).
  2. Fold is syntactically tricky, and its function type is confusing because there is no obvious logic determining either the order of arguments given to fold or the order of arguments accepted by the function you pass to it. I must have written hundreds of calls to fold, but I still hesitate each time to recall which order the arguments go in. Not surprising, since the argument order for the callback function differs between different languages’ libraries: some take the accumulator first and value second, others the other way around.
  3. Fold has several different names (some languages and libraries call it reduce, or inject) and none of them suggests any common English word for any of the actions it is actually used for. I suppose that’s because of point 1: we don’t name the general procedure. Fold is perhaps a marginally worse name than reduce or inject, but it’s still probably the most common.
  4. There’s more than one version of fold. Verity Stob cheekily asks “Do you fold to left or to the right? Do not provide too much information.” Left and right fold differ in the order in which they iterate through the container, so they usually produce different results, but there can also be profound differences between them in terms of performance and computability, especially when using lazy evaluation. This means you probably do have to know which is which. (See footnote below.)

A post about fold by James Hague a few years ago asked, “Is the difficulty many programmers have in grasping functional programming inherent in the basic concept of non-destructively operating on values, or is it in the popular abstractions that have been built-up to describe functional programming?” In this case I think it’s both. Fold is a good example of syntax failing us, and I think it’s also inherently a difficult abstraction to recognise (i.e. to spot the function application common to each activity). Fold is a fundamental operation in much of functional programming, but it doesn’t really feel like one because the abstraction is not comfortable. But besides that, many of the things fold is useful for are things that we would usually visualise in destructive terms: update the tally, push something onto the front of the list.

In Python the fold function (which Python calls reduce) was dropped from the built-in functions and moved into a separate module for Python 3. Guido van Rossum wrote, “apart from a few examples involving + or *, almost every time I see a reduce() call with a non-trivial function argument, I need to grab pen and paper to diagram what’s actually being fed into that function before I understand what the reduce() is supposed to do.” Instead the Python style for these activities usually involves destructively updating the accumulator.

Functional programming will surely never be really mainstream so long as fold appears in basic tutorials for it. Though in practice at least, because it’s such a general function, it can often be usefully hidden behind a more discoverable domain-specific API.


(Footnote. You can tell whether an implementation of fold is a left or right fold by applying it to the list “cons” function, which is often called “::”. If this reverses a list passed to it, you have a left fold. For example, the language Yeti has a function simply called fold; which is it? —

> fold (flip (::)) [] [1,2,3,4];
[4,3,2,1] is list<number>

So it’s a left fold.)