What is tail recursion?

Question

Whilst starting to learn lisp, I\'ve come across the term tail-recursive. What does it mean exactly?

Answer 1

This question has a lot of great answers... but I cannot help but chime in with an alternative take on how to define "tail recursion", or at least "proper tail recursion." Namely: should one look at it as a property of a particular expression in a program? Or should one look at it as a property of an implementation of a programming language?

For more on the latter view, there is a classic paper by Will Clinger, "Proper Tail Recursion and Space Efficiency" (PLDI 1998), that defined "proper tail recursion" as a property of a programming language implementation. The definition is constructed to allow one to ignore implementation details (such as whether the call stack is actually represented via the runtime stack or via a heap-allocated linked list of frames).

To accomplish this, it uses asymptotic analysis: not of program execution time as one usually sees, but rather of program space usage. This way, the space usage of a heap-allocated linked list vs a runtime call stack ends up being asymptotically equivalent; so one gets to ignore that programming language implementation detail (a detail which certainly matters quite a bit in practice, but can muddy the waters quite a bit when one attempts to determine whether a given implementation is satisfying the requirement to be "property tail recursive")

The paper is worth careful study for a number of reasons:

• It gives an inductive definition of the tail expressions and tail calls of a program. (Such a definition, and why such calls are important, seems to be the subject of most of the other answers given here.)

  Here are those definitions, just to provide a flavor of the text:

  Definition 1 The tail expressions of a program written in Core Scheme are defined inductively as follows.

  1. The body of a lambda expression is a tail expression
  2. If (if E0 E1 E2) is a tail expression, then both E1 and E2 are tail expressions.
  3. Nothing else is a tail expression.

  Definition 2 A tail call is a tail expression that is a procedure call.

(a tail recursive call, or as the paper says, "self-tail call" is a special case of a tail call where the procedure is invoked itself.)

• It provides formal definitions for six different "machines" for evaluating Core Scheme, where each machine has the same observable behavior except for the asymptotic space complexity class that each is in.

  For example, after giving definitions for machines with respectively, 1. stack-based memory management, 2. garbage collection but no tail calls, 3. garbage collection and tail calls, the paper continues onward with even more advanced storage management strategies, such as 4. "evlis tail recursion", where the environment does not need to be preserved across the evaluation of the last sub-expression argument in a tail call, 5. reducing the environment of a closure to just the free variables of that closure, and 6. so-called "safe-for-space" semantics as defined by Appel and Shao.

• In order to prove that the machines actually belong to six distinct space complexity classes, the paper, for each pair of machines under comparison, provides concrete examples of programs that will expose asymptotic space blowup on one machine but not the other.

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(Reading over my answer now, I'm not sure if I'm managed to actually capture the crucial points of the Clinger paper. But, alas, I cannot devote more time to developing this answer right now.)