Which recursive functions cannot be rewritten using loops? [duplicate]

Answer 1

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 2

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 3

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

Answer 4

Every recursive function can be implemented with a single loop.

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

Answer 5

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 6

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.