## Sunday, October 19, 2014

### Tail-recursion

Stack overflow refers to a condition in the execution of a computer program whereby the stack pointer exceeds the address space allocated for the stack. The usual result of "blowing the stack" is swift and brutal abnormal termination of the program.

The amount of memory allocated by the operating system for a given program's stack is finite and generally little can be done by the programmer to influence the amount that will be made available. The best the programmer can really do is to use what's given wisely.

We can get a sense of the limits of the stack in practical terms with a program like the following.

let rec range s e =
if s >= e then []
else s :: range (s + 1) e

let rec loop i =
let n = 2.0 ** (i |> float_of_int) |> int_of_float in
try
let _ = range 0 n in
loop (i + 1)
with
| Stack_overflow ->
Printf.printf "Stack overflow at i = %d, n = %d\n" i n

let () = loop 0

range is a function that produces the half-open range $\left[s, e\right)$ - the ordered sequence $\left\{s, s + 1, s + 2, \dots, e - 2, e - 1\right\}$. Note that range is defined in terms of itself, that is, it is a recursive function. The idea is to use it in an unbounded loop to build sequences of increasing lengths of powers of $2$ : ${2^0, 2^1, 2^2, \dots}$. We set it off and when we encounter stack overflow, terminate the program gracefully reporting on the power of $2$ found to give rise to the condition. In my case I found that I was able to make $\approx 2^{19} = 524,288$ recursive calls to range before the stack limit was breached. That's very limiting. For realistic programs, one would hope to be able to produce sequences of lengths much greater than that!

What can be done? The answer lies in the definition of range and that thing called tail-recursion. Specifically, range is not tail-recursive. To be a tail-recursive function, the last thing the function needs do is to make a recursive call to itself. That's not the case for range as written as the recursive call to itself is the second to last thing it does before it returns (the last thing it does is to 'cons' a value onto the list returned by the recursive call).

Why being tail-recursive is helpful is that tail-calls can be implemented by the compiler without requiring the addition of a new "stack frame" to the stack. Instead, the current frame can be replaced in setting up the tail-call being modified as necessary and effectively the recursive call is made to be a simple jump. This is called tail-call elimination and its effect is to allow tail-recursive functions to circumvent stack overflow conditions.

Here's a new definition for range, this time implemented with tail-calls.
let range s e =
let rec aux acc s e =
if s >= e then acc
else aux (s :: acc) (s + 1) e
in List.rev (aux [] s e)

With this definition for range I find I can build sequences of length up to around $\approx 2^{26} = 67,108,864$ elements long without any sign of stack overflow which is a huge improvement! At around this point though, my sequence building capabilities start to be limited by the amount of physical memory present on my PC but that's a different story entirely.