1.1.5 The Substitution Model for Procedure Application
To evaluate a combination whose operator names a compound procedure, the interpreter follows much
the same process as for combinations whose operators name primitive procedures, which we described
in section 1.1.3. That is, the interpreter evaluates the elements of the combination and applies the
procedure (which is the value of the operator of the combination) to the arguments (which are the
values of the operands of the combination).
We can assume that the mechanism for applying primitive procedures to arguments is built into the
interpreter. For compound procedures, the application process is as follows:
To apply a compound procedure to arguments, evaluate the body of the procedure with each
formal parameter replaced by the corresponding argument.
To illustrate this process, let’s evaluate the combination
(f 5)
where
f
is the procedure defined in section 1.1.4. We begin by retrieving the body of
f
:
(sum-of-squares (+ a 1) (* a 2))
Then we replace the formal parameter
a
by the argument 5:
(sum-of-squares (+ 5 1) (* 5 2))
Thus the problem reduces to the evaluation of a combination with two operands and an operator
sum-of-squares
. Evaluating this combination involves three subproblems. We must evaluate the
operator to get the procedure to be applied, and we must evaluate the operands to get the arguments.
Now
(+ 5 1)
produces 6 and
(* 5 2)
produces 10, so we must apply the
sum-of-squares
procedure to 6 and 10. These values are substituted for the formal parameters
x
and
y
in the body of
sum-of-squares
, reducing the expression to
(+ (square 6) (square 10))
If we use the definition of
square
, this reduces to
(+ (* 6 6) (* 10 10))
which reduces by multiplication to
(+ 36 100)
and finally to
136
The process we have just described is called the substitution model for procedure application. It can be
taken as a model that determines the ‘‘meaning’’ of procedure application, insofar as the procedures in
this chapter are concerned. However, there are two points that should be stressed:
The purpose of the substitution is to help us think about procedure application, not to provide a
description of how the interpreter really works. Typical interpreters do not evaluate procedure
applications by manipulating the text of a procedure to substitute values for the formal
parameters. In practice, the ‘‘substitution’’ is accomplished by using a local environment for the
formal parameters. We will discuss this more fully in chapters 3 and 4 when we examine the
implementation of an interpreter in detail.
Over the course of this book, we will present a sequence of increasingly elaborate models of how
interpreters work, culminating with a complete implementation of an interpreter and compiler in
chapter 5. The substitution model is only the first of these models -- a way to get started thinking
formally about the evaluation process. In general, when modeling phenomena in science and
engineering, we begin with simplified, incomplete models. As we examine things in greater
detail, these simple models become inadequate and must be replaced by more refined models.
The substitution model is no exception. In particular, when we address in chapter 3 the use of
procedures with ‘‘mutable data,’’ we will see that the substitution model breaks down and must
be replaced by a more complicated model of procedure application.
15
Applicative order versus normal order
According to the description of evaluation given in section 1.1.3, the interpreter first evaluates the
operator and operands and then applies the resulting procedure to the resulting arguments. This is not
the only way to perform evaluation. An alternative evaluation model would not evaluate the operands
until their values were needed. Instead it would first substitute operand expressions for parameters
until it obtained an expression involving only primitive operators, and would then perform the
evaluation. If we used this method, the evaluation of
(f 5)
would proceed according to the sequence of expansions
(sum-of-squares (+ 5 1) (* 5 2))
(+ (square (+ 5 1)) (square (* 5 2)) )
(+ (* (+ 5 1) (+ 5 1)) (* (* 5 2) (* 5 2)))
followed by the reductions
(+ (* 6 6) (* 10 10))
(+ 36 100)
136
This gives the same answer as our previous evaluation model, but the process is different. In
particular, the evaluations of
(+ 5 1)
and
(* 5 2)
are each performed twice here, corresponding
to the reduction of the expression
(* x x)
with
x
replaced respectively by
(+ 5 1)
and
(* 5 2)
.
This alternative ‘‘fully expand and then reduce’’ evaluation method is known as normal-order
evaluation, in contrast to the ‘‘evaluate the arguments and then apply’’ method that the interpreter
actually uses, which is called applicative-order evaluation. It can be shown that, for procedure
applications that can be modeled using substitution (including all the procedures in the first two
chapters of this book) and that yield legitimate values, normal-order and applicative-order evaluation
produce the same value. (See exercise 1.5 for an instance of an ‘‘illegitimate’’ value where
normal-order and applicative-order evaluation do not give the same result.)
Lisp uses applicative-order evaluation, partly because of the additional efficiency obtained from
avoiding multiple evaluations of expressions such as those illustrated with
(+ 5 1)
and
(* 5 2)
above and, more significantly, because normal-order evaluation becomes much more complicated to
deal with when we leave the realm of procedures that can be modeled by substitution. On the other
hand, normal-order evaluation can be an extremely valuable tool, and we will investigate some of its
implications in chapters 3 and 4.
16
1.1.6 Conditional Expressions and Predicates
The expressive power of the class of procedures that we can define at this point is very limited,
because we have no way to make tests and to perform different operations depending on the result of a
test. For instance, we cannot define a procedure that computes the absolute value of a number by
testing whether the number is positive, negative, or zero and taking different actions in the different
cases according to the rule
This construct is called a case analysis, and there is a special form in Lisp for notating such a case
analysis. It is called
cond
(which stands for ‘‘conditional’’), and it is used as follows:
(define (abs x)
(cond ((> x 0) x)
((= x 0) 0)
((< x 0) (- x))))
The general form of a conditional expression is
(cond (<p
1
> <e
1
>)
(<p
2
> <e
2
>)
(< p
n
> <e
n
>))
consisting of the symbol
cond
followed by parenthesized pairs of expressions
(<p> <e>)
called
clauses. The first expression in each pair is a predicate -- that is, an expression whose value is
interpreted as either true or false.
17
Conditional expressions are evaluated as follows. The predicate <p
1
> is evaluated first. If its value is
false, then <p
2
> is evaluated. If <p
2
>’s value is also false, then <p
3
> is evaluated. This process
continues until a predicate is found whose value is true, in which case the interpreter returns the value
of the corresponding consequent expression <e> of the clause as the value of the conditional
expression. If none of the <p>’s is found to be true, the value of the
cond
is undefined.
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