determines whether a function definition is a logical act

Two ``defun-modes'' are supported, :program and :logic. Roughly speaking, :program mode allows you to prototype a function for execution without any proof burdens, while :logic mode allows you to add a new definitional axiom to the logic. The system comes up in :logic mode. Execution of functions whose defun-mode is :program may render ACL2 unsound! See defun-mode-caveat.

When you define a function in the ACL2 logic, that function can be run on concrete data. But it is also possible to reason deductively about the function because each definition extends the underlying logic with a definitional axiom. To ensure that the logic is sound after the addition of this axiom, certain restrictions have to be met, namely that the recursion terminates. This can be quite challenging.

Because ACL2 is a programming language, you often may wish simply to program in ACL2. For example, you may wish to define your system and test it, without any logical burden. Or, you may wish to define ``utility'' functions -- functions that are executed to help manage the task of building your system but functions whose logical properties are of no immediate concern. Such functions might be used to generate test data or help interpret the results of tests. They might create files or explore the ACL2 data base. The termination arguments for such functions are an unnecessary burden provided no axioms about the functions are ever used in deductions.

Thus, ACL2 introduces the idea of the ``defun-mode'' of a function. The :mode keyword of defun's declare xarg allows you to specify the defun-mode of a given definition. If no :mode keyword is supplied, the default defun-mode is used; see default-defun-mode.

There are two defun-modes, each of which is written as a keyword:

:program -- logically undefined but executable outside deductive contexts.

:logic -- axiomatically defined as per the ACL2 definitional principle.

It is possible to change the defun-mode of a function from :program to :logic. We discuss this below.

We think of functions having :program mode as ``dangerous'' functions, while functions having :logic mode are ``safe.'' The only requirement enforced on :program mode functions is the syntactic one: each definition must be well-formed ACL2. Naively speaking, if a :program mode function fails to terminate then no harm is done because no axiom is added (so inconsistency is avoided) and some invocations of the function may simply never return. This simplistic justification of :program mode execution is faulty because it ignores the damage that might be caused by ``mis-guarded'' functions. See defun-mode-caveat.

We therefore implicitly describe an imagined implementation of defun-modes that is safe and, we think, effective. But please see defun-mode-caveat.

The default defun-mode is :logic. This means that when you defun a function the system will try to prove termination. If you wish to introduce a function of a different defun-mode use the :mode xargs keyword. Below we show fact introduced as a function in :program mode.

(defun fact (n)
  (declare (xargs :mode :program))
  (if (or (not (integerp n)) (= n 0))
    (* n (fact (1- n)))))
No axiom is added to the logic as a result of this definition. By introducing fact in :program mode we avoid the burden of a termination proof, while still having the option of executing the function. For example, you can type
ACL2 !>(fact 3)
and get the answer 6. If you type (fact -1) you will get a hard lisp error due to ``infinite recursion.''

However, the ACL2 theorem prover knows no axioms about fact. In particular, if the term (fact 3) arises in a proof, the theorem prover is unable to deduce that it is 6. From the perspective of the theorem prover it is as though fact were an undefined function symbol of arity 1. Thus, modulo certain important issues (see defun-mode-caveat), the introduction of this function in :program mode does not imperil the soundness of the system -- despite the fact that the termination argument for fact was omitted -- because nothing of interest can be proved about fact. Indeed, we do not allow fact to be used in logical contexts such as conjectures submitted for proof.

It is possible to convert a function from :program mode to :logic mode at the cost of proving that it is admissible. This can be done by invoking

(verify-termination fact)
which is equivalent to submitting the defun of fact, again, but in :logic mode.
(defun fact (n)
  (declare (xargs :mode :logic))
  (if (or (not (integerp n)) (= n 0))
    (* n (fact (1- n)))))
This particular event will fail because the termination argument requires that n be nonnegative. A repaired defun, for example with = replaced by <=, will succeed, and an axiom about fact will henceforth be available.

Technically, verify-termination submits a redefinition of the :program mode function. This is permitted, even when ld-redefinition-action is nil, because the new definition is identical to the old (except for its :mode and, possibly, other non-logical properties).

See guard for a discussion of how to restrict the execution of functions. Guards may be ``verified'' for functions in :logic mode; see verify-guards.