Reduce vs. Fold in Common Lisp

[ruricolist]

The history here is that Common Lisp gets reduce from APL (https://aplwiki.com/wiki/Reduce). It's not an attempt an ML-style fold, but a different formalism from a different lineage.

[myco_logic]

While reading this, I was immediately reminded of the reduce operator, glad to see my intuition wasn't far off.

The nifty thing about this operator in the array-langs compared to the usual fold function is that they usually define identity elements for all primitive functions, which means that no initial value has to be provided: https://aplwiki.com/wiki/Identity_element

The downside of this approach though, is that using reduce with non-primitive functions can result in domain errors (at least in APL). I think BQN's version of the operator is a bit nicer, in that it allows you to specify an initial value in this situation: https://mlochbaum.github.io/BQN/doc/fold.html

[moonchild]

I never heard of apl's allowing for an explicit choice of initial value, nor direction control ('from-end'), which mls typically also provide (foldl vs foldr). On the other hand, the ad-hoc choice of result for an empty input is a flaw common to cl and apl.

[varjag]

…REDUCE functions can be called with zero, one or two list values.

No, it's either two or zero arguments.

[galdor]

Author here. I double checked the Hyperspec [1] and you're quite right. I'll fix the article tomorrow, thank you for noticing!

[1] http://www.lispworks.com/documentation/lw60/CLHS/Body/f_redu...

[somewhereoutth]

fold is a natural consequence of church lists, probably the most fundamental way of expressing the list abstraction:

\f \x (f a0 (f a1 (f a2 x)))

So fold is just applying a list (function) to 2 arguments. Or you can be helpful and make something like fold := \f \x \l l f x which is useful for binding the f and the x and applying to multiple lists (everything is Curried of course)

LISP is not quite based on lambda calculus, so it should be no surprise it doesn't quite get reduce(i.e. fold) right.

See also church numerals, which are like lists but without the elements, they also have a 'fold':

\f \x f (f (f x))) == 3

We can make trees! Which again also have a 'fold'

\f \x f a0 (x a1) (f a2 (x a3) (x a4))

And many other more exotic folding data structures.

[dreamcompiler]

> It then applies the function to successive pairs of sequence elements.

No. #'reduce may take the first pair as an optimization step, but from that point on it processes sequence elements one at a time. It passes an accumulated value and the next sequence value to the function.

[tangus]

>Fold is also simpler than REDUCE since it does not have any special case, making it easier to reason about its behaviour.

If returning the initial value when the list is empty is considered a special case (or "surprising aspect") of REDUCE, then it's the same for FOLD, no?

[thedufer]

The initial value is optional for `reduce`, but required for `fold`. If you don't pass the initial value to `reduce`, and the sequence argument is empty, then `reduce` calls the function with no arguments, which is unique - in all other cases, it is called with two arguments. `fold` always calls its function argument with two arguments.

This shows up more clearly in statically-typed functional languages, where variadic functions like this are far less common. In that case, you typically see that `reduce` returns an option type, whereas `fold` does not. The types would look something like `fold :: (a -> b -> a) -> a -> List b -> a` vs `reduce :: (a -> a -> a) -> List a -> Option a`.

[galdor]

The most surprising aspect of REDUCE is the way the callback can be called depending on both the length of the list and the presence of absence of an initial value.

[User23]

I don't have a running image handy, but

  (+) ; => 0
  (*) ; => 1

and

  (+ n) ; => n
  (* n) ; => n

which I expect has some bearing on the behavior of reduce in the examples given.

It's pretty obvious that any other function could either have or be advised to have whatever equivalent semantics are appropriate.

Of course

  (apply #'+ '(1 2 3 4 5)) ; => 15

So reduce can be obviated by just letting the function take variable args too.

[lispm]

> So reduce can be obviated by just letting the function take variable args too.

In Common Lisp the max number of arguments can be small.

  $ abcl
  Armed Bear Common Lisp 1.8.0
  Java 11.0.19 Ubuntu
  OpenJDK 64-Bit Server VM
  Low-level initialization completed in 0.304 seconds.
  Startup completed in 1.501 seconds.
  Type ":help" for a list of available commands.

  CL-USER(1): CALL-ARGUMENTS-LIMIT
  50

[lispm]

so foldl is basically

  (defun foldl (function value sequence)
    (reduce function sequence :initial-value value))

[Paul-Craft]

Yeah, I'm not seeing what's so special either. Maybe it's that you do have to specify that initial value, so your return types are never something you don't expect?

[mgaunard]

+ is not associative for many types.

[yxhuvud]

True, which is why Common Lisp accepts a parameter to specify which order the + should be applied - from left to right or vice versa. Left to right is default.