The above examples demonstrate how the ability to pass procedures as arguments significantly enhances the expressive power of our programming language. We can achieve even more expressive power by creating procedures whose returned values are themselves procedures.
We can illustrate this idea by looking again at the fixed-point example described at the end of section 1.3.3. We formulated a new version of the square-root procedure as a fixed-point search, starting with the observation that √x is a fixed-point of the function y → x/y. Then we used average damping to make the approximations converge. Average damping is a useful general technique in itself. Namely, given a function f, we consider the function whose value at x is equal to the average of x and f(x).
We can express the idea of average damping by means of the following procedure:
(define (average-damp f) (lambda (x) (average x (f x))))
Average-damp is a procedure that takes as its argument a procedure
f and returns as its value a procedure (produced by
lambda) that, when applied to a number
x, produces the average of
(f x). For example, applying
average-damp to the
square procedure produces a procedure whose
value at some number x is the average of x and x2. Applying this resulting procedure to 10 returns the average of 10 and 100, or
((average-damp square) 10)55
average-damp, we can reformulate the square-root procedure as follows:
(define (sqrt x) (fixed-point (average-damp (lambda (y) (/ x y))) 1.0))
Notice how this formulation makes explicit the three ideas in the method: fixed-point search, average damping, and the function y → x/y. It is instructive to compare this formulation of the square-root method with the original version given in section 1.1.7. Bear in mind that these procedures express the same process, and notice how much clearer the idea becomes when we express the process in terms of these abstractions. In general, there are many ways to formulate a process as a procedure. Experienced programmers know how to choose procedural formulations that are particularly perspicuous, and where useful elements of the process are exposed as separate entities that can be reused in other applications. As a simple example of reuse, notice that the cube root of x is a fixed point of the function y → x/y2, so we can immediately generalize our square-root procedure to one that extracts cube roots:
(define (cube-root x) (fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0))
When we first introduced the square-root procedure, in section 1.1.7, we mentioned that this was a special case of Newton’s method. If x → g(x) is a differentiable function, then a solution of the equation g(x) = 0 is a fixed point of the function x → f(x) where
and Dg(x) is the derivative of g evaluated at x. Newton’s method is the use of the fixed-point method we saw above to approximate a solution of the equation by finding a fixed point of the function f For many functions g and for sufficiently good initial guesses for x, Newton’s method converges very rapidly to a solution of g(x) = 0.
In order to implement Newton’s method as a procedure, we must first express the idea of derivative. Note that “derivative,” like average damping, is something that transforms a function into another function. For instance, the derivative of the function x → x3 is the function x → 3x2. In general, if g is a function and dx is a small number, then the derivative Dg of g is the function whose value at any number x is given (in the limit of small dx) by
Thus, we can express the idea of derivative (taking dx to be, say, 0.00001) as the procedure
(define (deriv g) (lambda (x) (/ (- (g (+ x dx)) (g x)) dx)))
along with the definition
(define dx 0.00001)
deriv is a procedure that takes a procedure as argument and returns a procedure as value. For example, to approximate the derivative of x → x3 at 5 (whose exact value is 75) we can evaluate
(define (cube x) (* x x x)) ((deriv cube) 5)75.00014999664018
With the aid of
deriv, we can express Newton’s method as a fixed-point process:
(define (newton-transform g) (lambda (x) (- x (/ (g x) ((deriv g) x))))) (define (newtons-method g guess) (fixed-point (newton-transform g) guess))
newton-transform procedure expresses the formula at the beginning of this section, and
newtons-method is readily defined in terms of this. It takes as arguments a procedure that computes the
function for which we want to find a zero, together with an initial
guess. For instance, to find the square root of x, we can use
Newton’s method to find a zero of the function y → y2 - x starting with an initial guess of 1. This provides yet another form of the square-root procedure:
(define (sqrt x) (newtons-method (lambda (y) (- (square y) x)) 1.0))
Abstractions and first-class procedures
We’ve seen two ways to express the square-root computation as an instance of a more general method, once as a fixed-point search and once using Newton’s method. Since Newton’s method was itself expressed as a fixed-point process, we actually saw two ways to compute square roots as fixed points. Each method begins with a function and finds a fixed point of some transformation of the function. We can express this general idea itself as a procedure:
(define (fixed-point-of-transform g transform guess) (fixed-point (transform g) guess))
This very general procedure takes as its arguments a procedure
that computes some function, a procedure that transforms
an initial guess. The returned result is a fixed point of the
Using this abstraction, we can recast the first square-root computation from this section (where we look for a fixed point of the average-damped version of y → x/y) as an instance of this general method:
(define (sqrt x) (fixed-point-of-transform (lambda (y) (/ x y)) average-damp 1.0))
Similarly, we can express the second square-root computation from this section (an instance of Newton’s method that finds a fixed point of the Newton transform of y → y2 - x) as
(define (sqrt x) (fixed-point-of-transform (lambda (y) (- (square y) x)) newton-transform 1.0))
We began section 1.3 with the observation that compound procedures are a crucial abstraction mechanism, because they permit us to express general methods of computing as explicit elements in our programming language. Now we’ve seen how higher-order procedures permit us to manipulate these general methods to create further abstractions.
As programmers, we should be alert to opportunities to identify the underlying abstractions in our programs and to build upon them and generalize them to create more powerful abstractions. This is not to say that one should always write programs in the most abstract way possible; expert programmers know how to choose the level of abstraction appropriate to their task. But it is important to be able to think in terms of these abstractions, so that we can be ready to apply them in new contexts. The significance of higher-order procedures is that they enable us to represent these abstractions explicitly as elements in our programming language, so that they can be handled just like other computational elements.
In general, programming languages impose restrictions on the ways in which computational elements can be manipulated. Elements with the fewest restrictions are said to have first-class status. Some of the “rights and privileges” of first-class elements are:
- They may be named by variables.
- They may be passed as arguments to procedures.
- They may be returned as the results of procedures.
- They may be included in data structures.
Lisp, unlike other common programming languages, awards procedures full first-class status. This poses challenges for efficient implementation, but the resulting gain in expressive power is enormous.