Is my rec function tail recursive?

Question

Is this function tail-recursive ?

let rec rec_algo1 step J = 
    if step = dSs then J
    else
        let a = Array.init (Array2D.length1 M) (fun i ->         


        

Answer 1

You asked how we can formally check this so I'll have a stab. We first have to define what it means for a function to be tail-recursive. A recursive function definition of the form

let rec f x_1 ... x_n = e

is tail recursive if all calls of f inside e are tail calls - ie. occur in a tail context. A tail context C is defined inductively as a term with a hole []:

C ::= []
    | e
    | let p = e in C
    | e; C
    | match e with p_1 -> C | ... | p_n -> C
    | if e then C else C

where e is an F# expression, x is a variable and p is a pattern. We ought to expand this to mutually recursive function definitions but I'll leave that as an exercise.

Lets now apply this to your example. The only call to rec_algo1 in the body of the function is in this context:

if step = dSs then J
else
    let a = Array.init (Array2D.length1 M) (fun i -> minby1J i M J)
    let argmin = a|> Array.minBy snd |> fst
    []

And since this is a tail context, the function is tail-recursive. This is how functional programmers eyeball it - scan the body of the definition for recursive calls and then verify that each occurs in a tail context. A more intuitive definition of a tail call is when nothing else is done with the result of the call apart from returning it.

Answer 2

Yes, you can formally prove that a function is tail-recursive. Every expression reduction has a tail-position, and if all recursions are in tail-positions then the function is tail-recursive. It's possible for a function to be tail-recursive in one place, but not in another.

In the expression let pat = exprA in exprB only exprB is in tail-position. That is, while you can go evaluate exprA, you still have to come back to evaluate exprB with exprA in mind. For every expression in the language, there's a reduction rule that tells you where the tail position is. In ExprA; ExprB it's ExprB again. In if ExprA then ExprB else ExprC it's both ExprB and ExprC and so on.

The compiler of course knows this as it goes. However the many expressions available in F# and the many internal optimizations carried out by the compiler as it goes, e.g. during pattern match compiling, computation expressions like seq{} or async{} can make knowing which expressions are in tail-position non-obvious.

Practically speaking, with some practice it's easy for small functions to determine a tail call by just looking at your nested expressions and checking the slots which are NOT in tail positions for function calls. (Remember that a tail call may be to another function!)

Answer 3

This function is tail-recursive; I can tell by eyeballing it.

In general it is not always easy to tell. Perhaps the most reliable/pragmatic thing is just to check it on a large input (and make sure you are compiling in 'Release' mode, as 'Debug' mode turns off tail calls for better debugging).