How does lazy-evaluation allow for greater modularization?

Question

In his article "Why Functional Programming Matters," John Hughes argues that "Lazy evaluation is perhaps the most powerful tool for modularization in the functional programmer's repertoire." To do so, he provides an example like this:

Suppose you have two functions, "infiniteLoop" and "terminationCondition." You can do the following:

terminationCondition(infiniteLoop input)

Lazy evaluation, in Hughes' words "allows termination conditions to be separated from loop bodies." This is definitely true, since "terminationCondition" using lazy evaluation here means this condition can be defined outside the loop -- infiniteLoop will stop executing when terminationCondition stops asking for data.

But couldn't higher-order functions achieve the same thing as follows?

infiniteLoop(input, terminationCondition)

How does lazy evaluation provide modularization here that's not provided by higher-order functions?

Answer 1

Yes you could use a passed in termination check, but for that to work the author of infiniteLoop would have had to forsee the possibility of wanting to terminate the loop with that sort of condition, and hardwire a call to the termination condition into their function.

And even if the specific condition can be passed in as a function, the "shape" of it is predetermined by the author of infiniteLoop. What if they give me a termination condition "slot" that is called on each element, but I need access to the last several elements to check some sort of convergence condition? Maybe for a simple sequence generator you could come up with "the most general possible" termination condition type, but it's not obvious how to do so and remain efficient and easy to use. Do I repeatedly pass the entire sequence so far into the termination condition, in case that's what it's checking? Do I force my callers to wrap their simple termination conditions up in a more complicated package so they fit the most general condition type?

The callers certainly have to know exactly how the termination condition is called in order to supply a correct condition. That could be quite a bit of dependence on this specific implementation. If they switch to a different implementation of infiniteLoop written by another third party, how likely is it that exactly the same design for the termination condition would be used? With a lazy infiniteLoop, I can drop in any implementation that is supposed to produce the same sequence.

And what if infiniteLoop isn't a simple sequence generator, but actually generates a more complex infinite data structure, like a tree? If all the branches of the tree are independently recursively generated (think of a move tree for a game like chess) it could make sense to cut different branches at different depths, based on all sorts of conditions on the information generated thus far.

If the original author didn't prepare (either specifically for my use case or for a sufficiently general class of use cases), I'm out of luck. The author of the lazy infiniteLoop can just write it the natural way, and let each individual caller lazily explore what they want; neither has to know much about the other at all.

Furthermore, what if the decision to stop lazily exploring the infinite output is actually interleaved with (and dependent on) the computation the caller is doing with that output? Think of the chess move tree again; how far I want to explore one branch of the tree could easily depend on my evaluation of the best option I've found in other branches of the tree. So either I do my traversal and calculation twice (once in the termination condition to return a flag telling infinteLoop to stop, and then once again with the finite output so I can actually have my result), or the author of infiniteLoop had to prepare for not just a termination condition, but a complicated function that also gets to return output (so that I can push my entire computation inside the "termination condition").

Taken to extremes, I could explore the output and calculate some results, display them to a user and get input, and then continue exploring the data structure (without recalling infiniteLoop based on the user's input). The original author of the lazy infiniteLoop need have no idea that I would ever think of doing such a thing, and it will still work. If we've got purity enforced by the type system, then that would be impossible with the passed-in termination condition approach unless the whole infiniteLoop was allowed to have side effects if the termination condition needs to (say by giving the whole thing a monadic interface).

In short, to allow the same flexibility you'd get with lazy evaluation by using a strict infiniteLoop that takes higher order functions to control it can be a large amount of extra complexity for both the author of infiniteLoop and its caller (unless a variety of simpler wrappers are exposed, and one of them matches the caller's use case). Lazy evaluation can allow producers and consumers to be almost completely decoupled, while still giving the consumer the ability to control how much output the producer generates. Everything you can do that way you can do with extra function arguments as you say, but it requires to the producer and consumer to essentially agree on a protocol for how the control functions work; and that protocol is almost always either specialised to the use case at hand (tying the consumer and producer together) or so complicated in order to be fully-general that the producer and consumer are up tied to that protocol, which is unlikely to be recreated elsewhere, and so they're still tied together.