Designing Register Machines

To design a register machine, we must design its *data paths*
(registers and operations) and the *controller* that sequences
these operations. To illustrate the design of a simple register
machine, let us examine Euclid's Algorithm, which is used to compute
the greatest common divisor (GCD) of two integers. As we saw in
section , Euclid's Algorithm can be carried out by an iterative
process, as specified by the following procedure:

(define (gcd a b) (if (= b 0) a (gcd b (remainder a b))))

A machine to carry out this algorithm must keep track of two numbers,
*a* and *b*, so let us assume that these numbers are stored in two
registers with those names. The basic operations required are testing
whether the contents of register `b` is zero and computing the
remainder of the contents of register `a` divided by the contents
of register `b`. The remainder operation is a complex process,
but assume for the moment that we have a primitive device that
computes remainders. On each cycle of the GCD algorithm, the contents
of register `a` must be replaced by the contents of register `
b`, and the contents of `b` must be replaced by the remainder of
the old contents of `a` divided by the old contents of `b`.
It would be convenient if these replacements could be done
simultaneously, but in our model of register machines we will assume
that only one register can be assigned a new value at each step. To
accomplish the replacements, our machine will use a third
``temporary'' register, which we call `t`. (First the remainder
will be placed in `t`, then the contents of `b` will be placed
in `a`, and finally the remainder stored in `t` will be placed
in `b`.)

We can illustrate the registers and operations required for this
machine by using the data-path diagram shown in
figure . In this
diagram, the registers (`a`, `b`, and `t`) are represented
by rectangles. Each way to assign a value to a register is
indicated by an arrow with an `X` behind the head, pointing from
the source of data to the register. We can think of the `X` as a
button that, when pushed, allows the value at the source to ``flow''
into the designated register. The label next to each button is the
name we will use to refer to the button. The names are arbitrary, and
can be chosen to have mnemonic value (for example, `a<-b` denotes
pushing the button that assigns the contents of register `b` to
register `a`). The source of data for a register can be another
register (as in the `a<-b` assignment), an operation result (as in
the `t<-r` assignment), or a constant (a built-in value that
cannot be changed, represented in a data-path diagram by a triangle
containing the constant).

An operation that computes a value from constants and the contents
of registers is represented in a data-path diagram by a trapezoid
containing a name for the operation. For example, the box marked `
rem` in figure represents an
operation that computes the remainder of the contents of the
registers `a` and `b` to which it is attached. Arrows
(without buttons) point from the input registers and constants to the
box, and arrows connect the operation's output value to registers.
A test is represented by a circle containing a name for the test. For
example, our GCD machine has an operation that
tests whether the contents of register
`b` is zero. A test also has arrows from its input
registers and constants, but it has no output
arrows; its value is used by the controller rather than by the data
paths. Overall, the data-path diagram shows the registers and
operations that are required for the machine and how they must be
connected. If we view the arrows as wires and the `X` buttons as
switches, the data-path diagram is very like the wiring diagram for a
machine that could be constructed from electrical components.

In order for the data paths to actually compute GCDs, the buttons must
be pushed in the correct sequence. We will describe this sequence in
terms of a controller diagram, as illustrated in
figure . The elements of the controller
diagram indicate how the
data-path components should be operated. The rectangular boxes in the
controller diagram identify data-path buttons to be pushed, and the
arrows describe the sequencing from one step to the next. The diamond
in the diagram represents a decision. One of the two sequencing
arrows will be followed, depending on the value of the data-path test
identified in the diamond. We can interpret the controller in terms
of a physical analogy: Think of the diagram as a maze in which a
marble is rolling. When the marble rolls into a box, it pushes the
data-path button that is named by the box. When the marble rolls into
a decision node (such as the test for `b`), it leaves the
node on the path determined by the result of the indicated test.
Taken together, the data paths and the controller completely describe
a machine for computing GCDs. We start the controller (the rolling
marble) at the place marked `start`, after placing numbers in
registers `a` and `b`. When the controller reaches `
done`, we will find the value of the GCD in register `a`.

**Exercise.** Design a register machine to compute factorials using the iterative
algorithm specified by the following procedure. Draw data-path and
controller diagrams for this machine.

(define (factorial n) (define (iter product counter) (if (> counter n) product (iter (* counter product) (+ counter 1)))) (iter 1 1))

- A Language for Describing Register Machines
- Abstraction in Machine Design
- Subroutines
- Using a Stack to Implement Recursion
- Instruction Summary