Several of the Scheme procedures that we have written or studied in
preceding labs presuppose that their arguments will meet specific
preconditions -- constraints on the types or values of its
arguments. For instance, we saw in the lab on recursion with lists that the
greatest-of-list procedure requires its argument to be a
non-empty list of real numbers. If some careless programmer invokes
greatest-of-list and gives it, as an argument, the empty list,
or a list in which one of the elements is not a real number, or perhaps
even some Scheme value that is not a list at all, the computation that the
longest-in-list describes cannot be completed.
A procedure definition is like a contract between the author of the
definition and someone who invokes the procedure. The
postconditions of the procedure are what the author guarantees:
When the computation directed by the procedure is finished, the
postconditions shall be met. Usually the postconditions are constraints
on the value of the result returned by the procedure. For instance, the
postcondition of the
(define square (lambda (root) (* root root)))
is that the result is the square of the argument
The preconditions are the guarantees that the invoker of a procedure makes
to the author, the constraints that the arguments shall meet. For
instance, it is a precondition of the
square procedure that
root is a number.
If the invoker of a procedure violates its preconditions, then the contract
is broken and the author's guarantee of the postconditions is void. (If
root is, say, a list of symbols, then the author can't very
well guarantee to return its square.) To make it less likely that an
invoker violates a precondition by mistake, it is usual to document
preconditions carefully and to include occasional checks in one's programs,
ensuring that the preconditions are met before starting a complicated
Many of DrScheme's primitive procedures have such preconditions, which they enforce by aborting the computation and displaying a diagnostic message when the preconditions are not met:
> (/ 1 0) /: division by zero > (log 0) log: undefined for 0 > (length 116) length: expects argument of type <proper list> given 116
To enable us to enforce preconditions in the same way, DrScheme provides a
error, which takes a string as its argument.
error procedure aborts the entire computation of
which the call is a part and causes the string to be displayed as a
For instance, we could enforce
by rewriting its definition thus:
(define greatest-of-list (lambda (ls) (if (or (not (list? ls)) (null? ls) (not (all-real? ls))) (error "greatest-of-list: requires a non-empty list of reals") (if (null? (cdr ls)) (car ls) (max (car ls) (greatest-of-list (cdr ls)))))))
all-real? is a predicate that takes any list as its
argument and determines whether or not all of the elements of that list are
(define all-real? (lambda (ls) (or (null? ls) (and (real? (car ls)) (all-real? (cdr ls))))))
greatest-of-list procedure enforces its precondition:
> (greatest-of-list 139) greatest-of-list: requires a non-empty list of reals > (greatest-of-list null) greatest-of-list: requires a non-empty list of reals > (greatest-of-list (list 71/3 -17 23 'oops 16/15)) greatest-of-list: requires a non-empty list of reals
Including precondition testing in your procedures often makes them markedly easier to analyze and check, so I recommend the practice, especially during program development. There is a trade-off, however: It takes time to test the preconditions, and that time will be consumed on every invocation of the procedure. Since time is often a scarce resource, it makes sense to save it by skipping the test when you can prove that the precondition will be met. This often happens when you, as programmer, control the context in which the procedure is called as well as the body of the procedure itself.
For example, in the preceding definition of
although it is useful to test the precondition when the procedure is
invoked ``from outside'' by a potentially irresponsible caller, it is a
waste of time to repeat the test of the precondition for any of the
recursive calls to the procedure. At the point of the recursive call, you
already know that
ls is a list of strings (because you tested
that precondition on the way in) and that its cdr is not empty (because the
body of the procedure explicitly tests for that condition and does
something other than a recursive call if it is met), so the cdr must also
be a non-empty list of strings. So it's unnecessary to confirm this again
at the beginning of the recursive call.
One solution to this problem is to replace the definition of
greatest-of-list with two separate procedures, a ``husk'' and
a ``kernel.'' The husk interacts with the outside world, performs the
precondition test, and launches the recursion. The kernel is supposed to be
invoked only when the precondition can be proven true; its job is to
perform the main work of the original procedure, as efficiently as
(define greatest-of-list (lambda (ls) ;; Make sure that LS is a non-empty list of real numbers. (if (or (not (list? ls)) (null? ls) (not (all-real? ls))) (error "greatest-of-list: requires a non-empty list of reals") ;; Find the greatest number in the list. (greatest-of-list-kernel ls)))) (define greatest-of-list-kernel (lambda (ls) (if (null? (cdr ls)) (car ls) (max (car ls) (greatest-of-list-kernel (cdr ls))))))
The kernel has the same preconditions as the husk procedure, but does not need to enforce them, because we invoke it only in situations where we already know that the preconditions are satisfied.
The one weakness in this idea is that some potentially irresponsible caller might still call the kernel procedure directly, bypassing the husk procedure that he's supposed to invoke. In later labs, we'll see that there are a couple of ways to put the kernel back inside the husk without losing the efficiency gained by dividing the labor in this way.
February 4, 2000 (John Stone)
March 17, 2000 (John Stone)
Friday, 15 September 2000 (Sam Rebelsky)
Disclaimer Often, these pages were created "on the fly" with little, if any, proofreading. Any or all of the information on the pages may be incorrect. Please contact me if you notice errors.
This page may be found at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2000F/Readings/prepost.html
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