Which recursive functions cannot be rewritten using loops?

Question

As far as I know, most recursive functions can be rewritten using loops. Some maybe harder than others, but most of them can be rewritten. So the question is, under which conditions does it become impossible to rewrite a recursive function using a loop (if such conditions exist)?

---

Clarification: I know how to convert a recursive function to use a loop, but I am asking whether there are (corner) cases where this is impossible.

Answer 1

They all can be written as an iterative loop (but some might still need a stack to keep previous state for later iterations).

Answer 2

You can always use a loop, but you may have to create more data structure (e.g. simulate a stack).

Answer 3

It's not so much a matter of that they can't be implemented using loops, it's the fact that the way the recursive algorithm works, it's much clearer and more concise (and in many cases mathematically provable) that a function is correct.

Many recursive functions can be written to be tail loop recursive, which can be optimised to a loop, but this is dependent on both the algorithm and the language used.

Answer 4

See also this SO question

Answer 5

Indirect recursion comes to mind...

Edit: Mile's answer shows how to inline indirect recursion, so I was wrong.

Answer 6

I don't know about impossible but impractical->extremely inefficient examples are:

• tree traversal (a bitch in any loop)

• fast fourier

• quicksorts (and some others iirc)

Basically anything where you have to start keeping track of boundless potential states

Answer 7

Any recursive function can be made to iterate (into a loop) but you need to use a stack yourself to keep the state.

Normally, tail recursion is easy to convert into a loop:

A(x) {
  if x<0 return 0;
  return something(x) + A(x-1)
}

Can be translated into:

A(x) {
  temp = 0;
  for i in 0..x {
    temp = temp + something(i);
  }
  return temp;
}

Other kinds of recursion that can be translated into tail recursion are also easy to change. The other require more work.

The following:

treeSum(tree) {
  if tree=nil then 0
  else tree.value + treeSum(tree.left) + treeSum(tree.right);
}

Is not that easy to translate. You can remove one piece of the recursion, but the other one is not possible without a structure to hold the state.

treeSum(tree) {
  walk = tree;
  temp = 0;
  while walk != nil {
    temp = temp + walk.value + treeSum(walk.right);
    walk = walk.left;
  }
}

Answer 8

When you use a function recursively, the compiler takes care of stack management for you, which is what makes recursion possible. Anything you can do recursively, you can do by managing a stack yourself (for indirect recursion, you just have to make sure your different functions share that stack). So no, there is nothing that can be done with recursion and that cannot be done with a loop and a stack.

Answer 9

Indirect recursion is still possible to convert to a non-recursive loop; just start with one of the functions, and inline the calls to the others until you have a directly recursive function, which can then be translated to a loop that uses a stack structure.

Answer 10

Every recursive function can be implemented with a single loop.

Just think what a processor does, it executes instructions in a single loop.

Answer 11

In SICP, the authors challenge the reader to come up with a purely iterative method of implementing the 'counting change' problem (here's an example one from Project Euler).

But the strict answer to your question was already given - loops and stacks can implement any recursion.

Answer 12

Hi Hosam Aly,

Your stacking solution to the Ackermann problem is simply brilliant.

I am trying to solve the same for the Euler Project without any success.

Will now tinker with your idea/code.

Regards, nk