[jgg]

so I think they went a little over your head.

I understand lazy evaluation and tail recursion fine. I interpreted your comment as presenting corecursion as the only logical alternative to naive recursive algorithms with or without memoization.

You've tacked on the part where you say the latter algorithm is equivalent (due to Haskell's evaluation) - I get that. I'm still not understanding what you mean by only knowing inefficienct, naive recursion in contrast to corecursion. In practice, I have rarely seen corecursion or naive recursion used, but maybe we read different code.

In that toy example, it's easy to get between the three different forms, but in many cases the connection is far less obvious. We need principles that unite the different forms, and allow us to move between them. Co-recursion is one of those principles.

Uh, okay.

[Pitarou]

I once read a textbook introducing the theory of algorithms that started with the Fibonacci series. The author gave a formal definition, and translated this directly into a naive, recursive algorithm. Then he showed how to improve the algorithm with memoization. This was a real revelation to me: using this one simple, generally applicable transformation, all kinds of algorithms can be improved.

Then the author introduced the iterative Fibonacci algorithm, and formally proved that the iterative and recursive algorithms were equivalent. But the author failed to explain where that iterative algorithm came from: there was no general purpose tool like memoization that could be used to derive the iterative form from the recursive form. I still remember the intellectual disappointment.

Now, with co-recursion and a couple of other techniques, that gap is finally bridged. Using co-recursion, I can derive the stream algorithm directly from the recursive definition. And then I can apply deforestation techniques and tail-call optimization to derive the iterative algorithm from the stream algorithm. That's a pretty powerful intellectual toolset, don't you think?

[jgg]

I once read a textbook

I think I've read that book too.

For the benefit of anyone following along at home: The unproven-but-accepted Church-Turing thesis states that any algorithm that can be done on a Turing machine (i.e., a Turing-complete language) can be represented as a recursive function and a function in the lambda calculus, and vice-versa.

And then I can apply deforestation techniques and tail-call optimization to derive the iterative algorithm from the stream algorithm.

How do you use deforestation and tail-call optimization to derive an iterative function from a stream in the general case?

You're also pulling a false comparison: you've jumped from a 'general' algorithm to "what a corecursive function is evaluated as under Haskell's semantics".

Corecursion builds a data structure. A TCO function, for example, won't produce that kind of output. The corecursive function could only be directly equivalent to the linear time, constant memory TCO function in a lazily-evaluated runtime (if that's true - tail recursive functions in Haskell can actually blow up the stack due to lazy evaluation).

[Pitarou]

How do you apply deforestation ... in the general case?

How do you cut an iron bar in half with a pair of scissors?

Deforestation is a tool; not a universal truth.

You're also pulling a false comparison ... The corecursive function could only be directly equivalent to the linear time, constant memory TCO function in a lazily-evaluated runtime

But I'm using a lazily evaluated runtime. My code is Haskell. And in any case, I used deforestation in the last step to remove the lazily evaluated data structure.

If I understand you correctly, you're telling me that high-level mathematical formalisms work better in Haskell. Yes, they do.