Revised(5) Report on the Algorithmic Language Scheme - Tail recursion

## 3.5 Proper tail recursion

Implementations of Scheme are required to be properly tail-recursive. Procedure calls that occur in certain syntactic contexts defined below are `tail calls'. A Scheme implementation is properly tail-recursive if it supports an unbounded number of active tail calls. A call is active if the called procedure may still return. Note that this includes calls that may be returned from either by the current continuation or by continuations captured earlier by call-with-current-continuation that are later invoked. In the absence of captured continuations, calls could return at most once and the active calls would be those that had not yet returned. A formal definition of proper tail recursion can be found in [8].

Rationale:

> Intuitively, no space is needed for an active tail call because the continuation that is used in the tail call has the same semantics as the continuation passed to the procedure containing the call. Although an improper implementation might use a new continuation in the call, a return to this new continuation would be followed immediately by a return to the continuation passed to the procedure. A properly tail-recursive implementation returns to that continuation directly.

> Proper tail recursion was one of the central ideas in Steele and Sussman's original version of Scheme. Their first Scheme interpreter implemented both functions and actors. Control flow was expressed using actors, which differed from functions in that they passed their results on to another actor instead of returning to a caller. In the terminology of this section, each actor finished with a tail call to another actor.

> Steele and Sussman later observed that in their interpreter the code for dealing with actors was identical to that for functions and thus there was no need to include both in the language.

A tail call is a procedure call that occurs in a tail context. Tail contexts are defined inductively. Note that a tail context is always determined with respect to a particular lambda expression.

- The last expression within the body of a lambda expression, shown as <tail expression> below, occurs in a tail context.

          (lambda <formals>
            <definition>* <expression>* <tail expression>)

- If one of the following expressions is in a tail context, then the subexpressions shown as <tail expression> are in a tail context. These were derived from rules in the grammar given in chapter 7 by replacing some occurrences of <expression> with <tail expression>. Only those rules that contain tail contexts are shown here.

          (if <expression> <tail expression> <tail expression>)
          (if <expression> <tail expression>)

          (cond <cond clause>+)
          (cond <cond clause>* (else <tail sequence>))

          (case <expression>
            <case clause>+)
          (case <expression>
            <case clause>*
            (else <tail sequence>))

          (and <expression>* <tail expression>)
          (or <expression>* <tail expression>)

          (let (<binding spec>*) <tail body>)
          (let <variable> (<binding spec>*) <tail body>)
          (let* (<binding spec>*) <tail body>)
          (letrec (<binding spec>*) <tail body>)

          (let-syntax (<syntax spec>*) <tail body>)
          (letrec-syntax (<syntax spec>*) <tail body>)

          (begin <tail sequence>)

          (do (<iteration spec>*)
              (<test> <tail sequence>)
            <expression>*)

  where

          <cond clause> ---> (<test> <tail sequence>)
          <case clause> ---> ((<datum>*) <tail sequence>)

          <tail body> ---> <definition>* <tail sequence>
          <tail sequence> ---> <expression>* <tail expression>

- If a cond expression is in a tail context, and has a clause of the form

      (<expression1> => <expression2>)

  then the (implied) call to the procedure that results from the evaluation of <expression2> is in a tail context. <expression2> itself is not in a tail context.

Certain built-in procedures are also required to perform tail calls. The first argument passed to apply and to call-with-current-continuation, and the second argument passed to call-with-values, must be called via a tail call. Similarly, eval must evaluate its argument as if it were in tail position within the eval procedure.

In the following example the only tail call is the call to f. None of the calls to g or h are tail calls. The reference to x is in a tail context, but it is not a call and thus is not a tail call.

      (lambda ()
        (if (g)
            (let ((x (h)))
              x)
            (and (g) (f))))

Note: Implementations are allowed, but not required, to recognize that some non-tail calls, such as the call to h above, can be evaluated as though they were tail calls. In the example above, the let expression could be compiled as a tail call to h. (The possibility of h returning an unexpected number of values can be ignored, because in that case the effect of the let is explicitly unspecified and implementation-dependent.)