Evaluating Combinations

One of our goals in this chapter is to isolate issues about thinking procedurally. As a case in point, let us consider that, in evaluating combinations, the interpreter is itself following a procedure.

- To evaluate a combination, do the following:

- 1.Evaluate the subexpressions of the combination.
- 2.Apply the procedure that is the value of the leftmost subexpression (the operator) to the arguments that are the values of the other subexpressions (the operands).

Even this simple rule illustrates some important points about
processes in general. First, observe that the first step dictates
that in order to accomplish the evaluation process for a combination
we must first perform the evaluation process on each element of the
combination. Thus, the evaluation rule is
*recursive* in nature;
that is, it includes, as one of its steps, the need to invoke the rule
itself.^{}

Notice how succinctly the idea of recursion can be used to express what, in the case of a deeply nested combination, would otherwise be viewed as a rather complicated process. For example, evaluating

(* (+ 2 (* 4 6)) (+ 3 5 7))requires that the evaluation rule be applied to four different combinations. We can obtain a picture of this process by representing the combination in the form of a tree, as shown in figure . Each combination is represented by a node with branches corresponding to the operator and the operands of the combination stemming from it. The terminal nodes (that is, nodes with no branches stemming from them) represent either operators or numbers. Viewing evaluation in terms of the tree, we can imagine that the values of the operands percolate upward, starting from the terminal nodes and then combining at higher and higher levels. In general, we shall see that recursion is a very powerful technique for dealing with hierarchical, treelike objects. In fact, the ``percolate values upward'' form of the evaluation rule is an example of a general kind of process known as

Next, observe that the repeated application of the first step brings us to the point where we need to evaluate, not combinations, but primitive expressions such as numerals, built-in operators, or other names. We take care of the primitive cases by stipulating that

- the values of numerals are the numbers that they name,
- the values of built-in operators are the machine
instruction sequences that carry out the corresponding operations, and
- the values of other names are the objects associated with those names in the environment.

We may regard the second rule as a special case of the third one by
stipulating that symbols such as `+` and `*` are also included
in the global environment, and are associated with the sequences of
machine instructions that are their ``values.'' The key point to
notice is the role of the
environment in determining the meaning of
the symbols in expressions. In an interactive language such as
Lisp, it is meaningless to speak of the value of an expression such as
`(+ x 1)` without specifying any information about the environment
that would provide a meaning for the symbol `x` (or even for the
symbol `+`). As we shall see in chapter 3, the general notion of
the environment as providing a context in which evaluation takes place
will play an important role in our understanding of program execution.

Notice that the
evaluation rule given above does not handle definitions.
For instance, evaluating
`(define x 3)` does not apply `define` to two arguments, one
of which is the value of the symbol `x` and the other of which is
3, since the purpose of the `define` is precisely to associate
`x` with a value.
(That is, `(define x 3)` is not a combination.)

Such exceptions to the general evaluation rule are called *special
forms*. `Define` is the only example of a special form that we
have seen so far, but we will meet others shortly.
Each special form
has its own evaluation rule. The various kinds of expressions (each
with its associated evaluation rule) constitute the
syntax of the
programming language. In comparison with most other programming
languages, Lisp has a very simple syntax; that is, the evaluation rule
for expressions can be described by a simple general rule together
with specialized rules for a small number of special
forms.^{}