Normal Order and Applicative Order

Lazy Evaluator

To start, get our version of the lazy evaluator:

cp ~cs61as/lib/lazy.scm .

Now that we have an evaluator expressed as a Lisp program, we can experiment with alternative choices in language design simply by modifying the evaluator. Indeed, new languages are often invented by first writing an evaluator that embeds the new language within an existing high-level language.

For example, if we wish to discuss some aspect of a proposed modification to Lisp with another member of the Lisp community, we can supply an evaluator that embodies the change. The recipient can then experiment with the new evaluator and send back comments as further modifications. Not only does the high-level implementation base make it easier to test and debug the evaluator; in addition, the embedding enables the designer to snarf features from the underlying language, just as our embedded Lisp evaluator uses primitives and control structure from the underlying Lisp. Only later (if ever) need the designer go to the trouble of building a complete implementation in a low- level language or in hardware.

In this section and the next we explore some variations on Scheme that provide significant additional expressive power.

Review of Normal and Applicative Order

In Lesson 1, where we began our discussion of models of evaluation, we noted that Scheme is an applicative-order language, namely, that all the arguments to Scheme procedures are evaluated when the procedure is applied. In contrast, normal-order languages delay evaluation of procedure arguments until the actual argument values are needed. Delaying evaluation of procedure arguments until the last possible moment (e.g., until they are required by a primitive operation) is called lazy evaluation.

Consider the procedure

(define (try a b)
  (if (= a 0) 1 b))

Evaluating (try 0 (/ 1 0)) generates an error in Scheme. With lazy evaluation, there would be no error. Evaluating the expression would return 1, because the argument (/ 1 0) would never be evaluated.

An example that exploits lazy evaluation is the definition of a procedure unless

(define (unless condition usual-value exceptional-value)
    (if condition

that can be used in expressions such as

(unless (= b 0)
        (/ a b)
        (begin (display "exception: returning 0")

This won't work in an applicative-order language because both the usual value and the exceptional value will be evaluated before unless is called. An advantage of lazy evaluation is that some procedures, such as unless, can do useful computation even if evaluation of some of their arguments would produce errors or would not terminate.

Test Your Understanding

Consider the following:
> (define (double x) (+ x x))
> (double (+ 2 1))
In applicative order, how many times does + get called?

In normal order, how many times does + get called?

Strict vs. Non-Strict

If the body of a procedure is entered before an argument has been evaluated we say that the procedure is non-strict in that argument. If the argument is evaluated before the body of the procedure is entered we say that the procedure is strict in that argument. In a purely applicative-order language, all procedures are strict in each argument. In a purely normal-order language, all compound procedures are non-strict in each argument, and primitive procedures may be either strict or non-strict. There are also languages (see SICP Exercise 4.31) that give programmers detailed control over the strictness of the procedures they define.

A striking example of a procedure that can usefully be made non-strict is cons (or, in general, almost any constructor for data structures). One can do useful computation, combining elements to form data structures and operating on the resulting data structures, even if the values of the elements are not known. It makes perfect sense, for instance, to compute the length of a list without knowing the values of the individual elements in the list. We will exploit this idea later in the lesson to implement the streams of Lesson 11 as lists formed of non-strict cons pairs.