We begin by considering the factorial function, defined by \[ n!=n\cdot(n-1)\cdot(n-2)\cdots3\cdot2\cdot1 \] There are many ways to compute factorials. One way is to make use of the observation that $n!$ is equal to $n$ times $(n-1)!$ for any positive integer $n$: \[ n!=n\cdot\left[(n-1)\cdot(n-2)\cdots3\cdot2\cdot1\right]=n\cdot(n-1)! \] Thus, we can compute $n!$ by computing $(n-1)!$ and multiplying the result by $n$. If we add the stipulation that 1! is equal to 1, this observation translates directly into a function:

function factorial(n) { return n === 1 ? 1 : n * factorial(n - 1); }

Now let's take a different perspective on computing factorials. We could describe a rule for computing $n!$ by specifying that we first multiply 1 by 2, then multiply the result by 3, then by 4, and so on until we reach $n$. More formally, we maintain a running product, together with a counter that counts from 1 up to $n$. We can describe the computation by saying that the counter and the product simultaneously change from one step to the next according to the rule

product $\leftarrow$ counter $\cdot$ product

counter $\leftarrow$ counter $+$ 1

and stipulating that $n!$ is the value of the product when the counter exceeds $n$.Once again, we can recast our description as a function for computing factorials:[1]

function factorial(n) { return fact_iter(1, 1, n); } function fact_iter(product, counter, max_count) { return counter > max_count ? product : fact_iter(counter * product, counter + 1, max_count); }

Compare the two processes. From one point of view, they seem hardly
different at all. Both compute the same mathematical function on the
same domain, and each requires a number of steps proportional to $n$
to compute $n!$. Indeed, both processes even carry out the same
sequence of multiplications, obtaining the same sequence of partial
products. On the other hand, when we consider the
shapes

of the
two processes, we find that they evolve quite differently.

Consider the first process. The substitution model reveals a shape of
expansion followed by contraction, indicated by the arrow in
figure 1.3. The expansion occurs as the
process builds up a chain of
*deferred operations* (in this case,
a chain of multiplications). The contraction occurs as
the operations are
actually performed. This type of process, characterized by a chain of
deferred operations, is called a
*recursive process*. Carrying
out this process requires that the interpreter keep track of the
operations to be performed later on. In the computation of $n!$,
the length of the chain of deferred multiplications, and hence the amount
of information needed to keep track of it,
grows linearly with $n$
(is proportional to $n$), just like the number of steps.
Such a process is called a *linear recursive process*.

By contrast, the second process does not grow and shrink. At each
step, all we need to keep track of, for any $n$, are the current
values of the
names
`product`, `counter`, and
`max_count`. We call this an
*iterative process*. In general, an
iterative process is one whose state can be summarized by
the values of a fixed
number of
*state names*,
together with a fixed rule that
describes how
the values of the state names
should be updated as the process
moves from state to state and an (optional) end test that specifies
conditions under which the process should terminate. In computing
$n!$, the number of steps required grows linearly with $n$. Such a process is
called a
*linear iterative process*.

The contrast between the two processes can be seen in another way. In
the iterative case, the
values of the state names
provide a complete
description of the state of the process at any point. If we stopped
the computation between steps, all we would need to do to resume the
computation is to supply the interpreter with the values of the three
state names.
Not so with the recursive process. In this case
there is some additional hidden

information, maintained by the
interpreter and not contained in the
state names,
which
indicates where the process is

in negotiating the chain of
deferred operations. The longer the chain, the more information must
be maintained.[2]

In contrasting iteration and recursion, we must be careful not to
confuse the notion of a
recursive *process* with the notion of a
recursive
*function*.
When we describe a function as recursive,
we are referring to the syntactic fact that the
function declaration
refers (either directly or indirectly) to the function itself. But
when we describe a process as following a pattern that is, say,
linearly recursive, we are speaking about how the process evolves, not
about the syntax of how a function is written. It may seem
disturbing that we refer to a recursive function such as
`fact_iter`
as generating an iterative process. However, the process
really is iterative: Its state is captured completely by its three
state names,
and an interpreter need keep track of only three
names
in order to execute the process.

One reason that the distinction between process and procedure may be
confusing is that most implementations of common languages (including
Ada, Pascal, and C) are designed in such a way that the
interpretation of any recursive function consumes an amount of memory
that grows with the number of function calls, even when the process
described is, in principle, iterative. As a consequence, these
languages can describe iterative processes only by resorting to
special-purpose
looping constructs

such as `do`, `repeat`,
`until`, `for`, and `while`. The implementation of
JavaScript
we shall consider in chapter 5 does not share this defect. It will
execute an iterative process in constant space, even if the iterative
process is described by a recursive function. An implementation with
this property is called
*tail-recursive*. With a tail-recursive
implementation,
iteration can be expressed using the ordinary
function call mechanism, so that special iteration constructs are
useful only as
syntactic sugar.[3]

function plus(a, b) { return a === 0 ? b : inc(plus(dec(a), b)); }

function plus(a, b) { return a === 0 ? b : plus(dec(a), inc(b)); }

plus(4, 5) 4 === 0 ? 5 : inc(plus(dec(4), 5)) inc(plus(dec(4), 5)) ... inc(plus(3, 5)) ... inc(inc(plus(2, 5))) ... inc(inc(inc(plus(1, 5)))) ... inc(inc(inc(inc(plus(0, 5))))) inc(inc(inc(inc( 0 === 0 ? 5 : inc(plus(dec(0), 5)))))) inc(inc(inc(inc( 5 )))) inc(inc(inc( 6 ))) inc(inc( 7 )) inc( 8 ) 9

plus(4, 5) 4 === 0 ? 5 : plus(dec(4), inc(5)) plus(dec(4), inc(5)) ... plus(3, 6) ... plus(2, 7) ... plus(1, 8) ... plus(0, 9) 0 === 0 ? 9 : plus(dec(0), inc(9)) 9

function A(x,y) { return y === 0 ? 0 : x === 0 ? 2 * y : y === 1 ? 2 : A(x - 1, A(x, y - 1)); }

A(1, 10);

A(2, 4);

A(3, 3);

function f(n) { return A(0, n); } function g(n) { return A(1, n); } function h(n) { return A(2, n); } function k(n) { return 5 * n * n; }

[1] In a real program we would probably use the
block structure introduced in the last section to hide the
declaration of `fact_iter`:
We avoided doing this here so as to minimize the number of things to
think about at once.

function factorial(n) { function iter(product, counter) { return counter > n ? product : iter(counter * product, counter + 1); } return iter(1, 1); }

[2] When we discuss the implementation of
functions
on register machines in chapter 5, we will see that any
iterative process can be realized *stack*.

in hardwareas a machine that has a fixed set of registers and no auxiliary memory. In contrast, realizing a recursive process requires a machine that uses an auxiliary data structure known as a

[3] Tail recursion has long been
known as a compiler optimization trick. A coherent semantic basis for
tail recursion was provided by Carl Hewitt (1977), who explained it in
terms of the Steele 1975 )
constructed a tail-recursive
interpreter for Scheme. Steele later showed how tail recursion is a
consequence of the natural way to compile
function
calls (Steele 1977 ).
The IEEE standard for Scheme requires that Scheme implementations
be tail-recursive. The ECMA standard for JavaScript eventually followed
suit with ECMAScript 2015 (ECMA 2015 ). Note however,
that as of this writing (2019), most implementations of JavaScript do
not comply with this standard.

message-passingmodel of computation that we shall discuss in chapter 3. Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see

1.2.1 Linear Recursion and Iteration