11.6 Conclusion

12.1 Scheme Concepts

The fundamental units of Scheme programs are numbers (written like Java numbers ), symbols (written much like Java variables , except that dashes ( - ) and some other characters are permitted), and procedures . A procedure is a combination of a piece of code and an environment in which the code is to be evaluated.

A Scheme program is called a form . Here is an example of a form:

(+ 2 3)

This form denotes the list containing three entities: the symbol + , the number 2 , and the number 3 . To run a Scheme program, the Scheme evaluator takes the form and a binding environment . The binding environment maps symbols to values. Then it applies these rules to the form:

• A number evaluates to itself.

• A symbol is evaluated by looking it up in the binding environment. The results of the lookup are the value.

• A list is evaluated by evaluating each element of the list. The first element is treated as a procedure. A new binding environment is created, in which the parameters of the procedure are bound to the rest of the list. Then the body of the procedure is evaluated.

For example, in the list (+ 2 3) , the symbol + evaluates to a procedure that adds numbers together, and 2 and 3 evaluate to themselves . Then the evaluator calls the procedure, giving it the arguments 2 and 3 . The procedure adds 2 and 3 to get 5 , which is the final result.

There are a few exceptions to the third rule. Some lists have keywords in the first position; these are evaluated differently from other lists. These lists are called special forms .

To create a new procedure in Scheme, a special form called a lambda expression is used. Here is an example of a lambda expression:

(lambda (x) (* x x))

This expression evaluates to a new procedure, in this case, one that squares its parameter by multiplying it by itself. A lambda expression is written with the keyword lambda as the first element. The second element is a list of formal parameters. The third element is called the body of the lambda expression.

When the procedure is applied to a list of actual parameters, a new binding environment is created in which the symbols in the list of formal parameters are bound to the elements of the list of actual parameters. The new binding environment includes the one in which the lambda expression was first evaluated.

This example demonstrates the application of a lambda expression: ^[1]

^[1] As in Java, the carriage returns are not relevant. All examples are written with lots of carriage returns to make the syntax as clear as possible.

(
   (lambda (x) (* x x))
   5
)

This is a list with two elements. The first element is a lambda expression, which evaluates to a procedure, as described. The second element of the list is the number 5 , which evaluates to itself.

After evaluating the list elements, the evaluator evaluates the whole form by using the first element as a procedure and the second element as the argument to that procedure. To apply the procedure, a new binding environment is created, binding the symbol x to number 5 . The new binding environment also contains the environment in which the lambda expression was created, so any symbol not explicitly defined in the new environment is inherited.

Next, the body of the lambda expression is evaluated with respect to the new binding environment. The symbol * is not defined in the new binding environment, but it is defined in the environment in which the lambda was first evaluated. It evaluates to a procedure that multiplies its arguments together. The symbol x is evaluated, yielding the value 5 , since that's the result of looking up the symbol x in the binding environment.

Finally, the evaluator applies the multiplication procedure to the numbers 5 and 5 , returning 25 . The value of the entire form is 25 .

Another special form is the if special form:

(if
   (< a b)
   b
   a
)

This is an expression that returns the maximum of a and b . The if special form starts by evaluating the second element of the list, called the test . In this case, the test is the expression (< a b) . The symbol < evaluates to a procedure that compares its arguments. It returns the symbol #t if the first is less than the second, #f otherwise . The symbol #t is interpreted as "true," the symbol #f as "false."

The third and fourth elements of the if special form are called the true-expression and the false-expression . If the test returns #f , then the value of the if expression is the result of evaluating the false-expression. Otherwise, the value of the if expression is the result of evaluating the true-expression.

Only the true-expression or the false-expression is evaluated, not both. That's why if is a special form rather than a procedure. If it were a procedure, then both the true-expression and the false-expression would be evaluated. If they have side effects, like printing a value, then both sets of side effects would happen, which would be incorrect.

Suppose a is bound to 5 and b is bound to 26 . Consider the expression

(if (< a b) b a)

To evaluate this expression, the evaluator first evaluates the test: a and b evaluate to 5 and 26 , respectively, and < evaluates to the less-than comparison procedure. Then the comparison procedure is called with 5 and 26 , which returns #t , since 5 < 26. Because the test returned #t , the evaluator evaluates the true-expression, which is the form b . The form b evaluates to 26 , making 26 the result of the program.

The define special form creates new entries in the binding environment. Here's an example:

(define max
   (lambda (a b)
       (if (a < b)
           b
           a
       )
   )
)

When evaluated, the define special form defines a new symbol called max in the binding environment. The value of this symbol is the result of evaluating the form that is the third element of the list. That form is

(lambda (a b)
   (if (a < b) b a)
)

This code evaluates to a procedure that takes two arguments, returning the value of the first if it is the greater one, the value of the second otherwise. The effect is to create a new function called max that returns the greater of its two arguments. After max has been defined, the expression

(max 5 26)

evaluates to 26 .

Unlike Java, all the parentheses are important; you can't just add new ones wherever you want. For example, suppose you tried to evaluate

((max 5 26))

The evaluator would try to evaluate the first element of the list, (max 5 26) , which evaluates to 26 . Then it would try to apply 26 as a procedure. Since 26 is a number, not a procedure, an error would result.

A symbol may be rebound with the set! special form:

(set! x 10)

This special form rebinds x to 10 in the local binding environment. It should be used only on symbols that already have a binding, either from a lambda expression or from a define . For example,

(define sqr               ; Define sqr to square its
   (lambda (x)            ; argument
       (* x x)))
(sqr 10)                  ; This results in 100
(set! sqr                 ; Define sqr to double its
   (lambda (y)            ; argument
       (+ x x)))
(sqr 10)                  ; This results in 20

The first time (sqr 10) is evaluated, the evaluator uses the original definition found in the define , which multiplies the argument times itself. The second time (sqr 10) is evaluated, the symbol sqr has been redefined to be the sum of the argument and itself, so it returns a different value.