# Which recursive functions cannot be rewritten using loops?

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.

Under which conditions does it become impossible to rewrite a recursive function using a loop (if such conditions exist)?

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.

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

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

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.

Indirect recursion comes to mind...

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

I don't know about examples where recursive functions cannot be converted to an iterative version, but impractical or extremely inefficient examples are:

• tree traversal

• fast Fourier

• quicksorts (and some others iirc)

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

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;
}
}
``````

One example which would be extremely difficult to convert from recursive to iterative would be the Ackermann function.