Combining Data of Different Types

We have seen how to define a unified arithmetic system that encompasses ordinary numbers, complex numbers, rational numbers, and any other type of number we might decide to invent, but we have ignored an important issue. The operations we have defined so far treat the different data types as being completely independent. Thus, there are separate packages for adding, say, two ordinary numbers, or two complex numbers. What we have not yet considered is the fact that it is meaningful to define operations that cross the type boundaries, such as the addition of a complex number to an ordinary number. We have gone to great pains to introduce barriers between parts of our programs so that they can be developed and understood separately. We would like to introduce the cross-type operations in some carefully controlled way, so that we can support them without seriously violating our module boundaries.

One way to handle cross-type operations is to design a different
procedure for each possible combination of types for which the
operation is valid. For example, we could extend the complex-number
package so that it provides a procedure for adding complex numbers to
ordinary numbers and installs this in the table using the tag `
(complex scheme-number)`:^{}

;; to be included in the complex package(define (add-complex-to-schemenum z x) (make-from-real-imag (+ (real-part z) x) (imag-part z))) (put 'add '(complex scheme-number) (lambda (z x) (tag (add-complex-to-schemenum z x))))

This technique works, but it is cumbersome. With such a system, the cost of introducing a new type is not just the construction of the package of procedures for that type but also the construction and installation of the procedures that implement the cross-type operations. This can easily be much more code than is needed to define the operations on the type itself. The method also undermines our ability to combine separate packages additively, or least to limit the extent to which the implementors of the individual packages need to take account of other packages. For instance, in the example above, it seems reasonable that handling mixed operations on complex numbers and ordinary numbers should be the responsibility of the complex-number package. Combining rational numbers and complex numbers, however, might be done by the complex package, by the rational package, or by some third package that uses operations extracted from these two packages. Formulating coherent policies on the division of responsibility among packages can be an overwhelming task in designing systems with many packages and many cross-type operations.

Coercion

In the general situation of completely unrelated operations acting on
completely unrelated types, implementing explicit cross-type
operations, cumbersome though it may be, is the best that one can hope
for. Fortunately, we can usually do better by taking advantage of
additional structure that may be latent in our type system. Often the
different data types are not completely independent, and there may be
ways by which objects of one type may be viewed as being of another
type. This process is called *coercion*. For example, if we are
asked to arithmetically combine an ordinary number with a complex
number, we can view the ordinary number as a complex number whose
imaginary part is zero. This transforms the problem to that of
combining two complex numbers, which can be handled in the ordinary
way by the complex-arithmetic package.

In general, we can implement this idea by designing coercion procedures that transform an object of one type into an equivalent object of another type. Here is a typical coercion procedure, which transforms a given ordinary number to a complex number with that real part and zero imaginary part:

(define (scheme-number->complex n) (make-complex-from-real-imag (contents n) 0))We install these coercion procedures in a special coercion table, indexed under the names of the two types:

(put-coercion 'scheme-number 'complex scheme-number->complex)(We assume that there are

Once the coercion table has been set up, we can handle coercion in a
uniform manner by modifying the `apply-generic` procedure of
section . When asked to apply an operation, we
first check whether the operation is defined for the arguments' types,
just as before. If so, we dispatch to the procedure found in the
operation-and-type table.
Otherwise, we try coercion. For simplicity, we consider only the case
where there are two arguments.^{} We
check the coercion table to see if objects of the first type can
be coerced to the second type. If so, we coerce the first argument and try the
operation again. If objects of the first type cannot in general be coerced to
the second type, we try the coercion the other way around to see if there is a
way to coerce the second argument to the type of the first argument.
Finally, if there
is no known way to coerce either type to the other type, we give up.
Here is the procedure:

(define (apply-generic op . args) (let ((type-tags (map type-tag args))) (let ((proc (get op type-tags))) (if proc (apply proc (map contents args)) (if (= (length args) 2) (let ((type1 (car type-tags)) (type2 (cadr type-tags)) (a1 (car args)) (a2 (cadr args))) (let ((t1->t2 (get-coercion type1 type2)) (t2->t1 (get-coercion type2 type1))) (cond (t1->t2 (apply-generic op (t1->t2 a1) a2)) (t2->t1 (apply-generic op a1 (t2->t1 a2))) (else (error "No method for these types" (list op type-tags)))))) (error "No method for these types" (list op type-tags)))))))

This coercion scheme has many advantages over the method of defining
explicit cross-type operations, as outlined above. Although we still
need to write coercion procedures to relate the types (possibly *n*^{2}
procedures for a system with *n* types), we need to write only one
procedure for each pair of types rather than a different procedure for
each collection of types and each generic operation.^{} What we are counting on here is the fact that the
appropriate transformation between types depends only on the types
themselves, not on the operation to be applied.

On the other hand, there may be applications for which our coercion scheme is not general enough. Even when neither of the objects to be combined can be converted to the type of the other it may still be possible to perform the operation by converting both objects to a third type. In order to deal with such complexity and still preserve modularity in our programs, it is usually necessary to build systems that take advantage of still further structure in the relations among types, as we discuss next.

Hierarchies of types

The coercion scheme presented above relied on the existence of natural
relations between pairs of types. Often there is more ``global''
structure in how the different types relate to each other. For
instance, suppose we are building a generic arithmetic system to
handle integers, rational numbers, real numbers, and complex numbers.
In such a system, it is quite natural to regard an integer as a
special kind of rational number, which is in turn a special kind of
real number, which is in turn a special kind of complex number. What
we actually have is a so-called *hierarchy of types*, in which,
for example, integers are a
*subtype* of rational numbers (i.e.,
any operation that can be applied to a rational number can
automatically be applied to an integer). Conversely, we say that
rational numbers form a
*supertype* of integers. The particular
hierarchy we have here is of a very simple kind, in which each type
has at most one supertype and at most one subtype. Such a structure,
called a *tower*, is illustrated in figure .

If we have a tower structure, then we can greatly simplify the problem
of adding a new type to the hierarchy, for we need only specify how
the new type is embedded in the next supertype above it and how it is
the supertype of the type below it. For example, if we want to add an
integer to a complex number, we need not explicitly define a special
coercion procedure `integer->complex`. Instead, we define how an
integer can be transformed into a rational number, how a rational
number is transformed into a real number, and how a real number is
transformed into a complex number. We then allow the system to
transform the integer into a complex number through these steps and
then add the two complex numbers.

We can redesign our `apply-generic` procedure in the following
way: For each type, we need to supply a `raise` procedure, which
``raises'' objects of that type one level in the tower. Then when the
system is required to operate on objects of different types it can
successively raise the lower types until all the objects are at
the same level in the tower. (Exercises
and
concern the details of implementing such a strategy.)

Another advantage of a tower is that we can easily implement the
notion that every type ``inherits'' all operations defined on a
supertype. For instance, if we do not supply a special procedure for
finding the real part of an integer, we should nevertheless expect
that `real-part` will be defined for integers by virtue of the
fact that integers are a subtype of complex numbers. In a tower, we
can arrange for this to happen in a uniform way by modifying `
apply-generic`. If the required operation is not directly defined for
the type of the object given, we raise the object to its supertype and
try again. We thus crawl up the tower, transforming our argument as we
go, until we either find a level at which the desired operation can be
performed or hit the top (in which case we give up).

Yet another advantage of a tower over a more general hierarchy is that
it gives us a simple way to ``lower'' a data object to the simplest
representation. For example, if we add 2+3*i* to 4-3*i*, it would be
nice to obtain the answer as the integer 6 rather than as the complex
number 6+0*i*. Exercise discusses a way to implement
such a lowering operation. (The trick is that we need a general way
to distinguish those objects that can be lowered, such as 6+0*i*, from
those that cannot, such as 6+2*i*.)

Inadequacies of hierarchies

If the data types in our system can be naturally arranged in a tower,
this greatly simplifies the problems of dealing with generic operations
on different types, as we have seen. Unfortunately, this is usually
not the case. Figure illustrates a
more complex arrangement of mixed types, this one showing relations
among different types of geometric figures. We see that, in general,
a type may have more than one subtype. Triangles and quadrilaterals,
for instance, are both subtypes of polygons. In addition, a type may
have more than one supertype. For example, an isosceles right
triangle may be regarded either as an isosceles triangle or as a right
triangle. This multiple-supertypes issue is particularly thorny,
since it means that there is no unique way to ``raise'' a type in the
hierarchy. Finding the ``correct'' supertype in which to apply an
operation to an object may involve considerable searching through the
entire type network on the part of a procedure such as `
apply-generic`. Since there generally are multiple subtypes for a
type, there is a similar problem in coercing a value ``down'' the type
hierarchy. Dealing with large numbers of interrelated types while
still preserving modularity in the design of large systems is very
difficult, and is an area of much current research.
^{}

**Exercise.**
Louis Reasoner has noticed that `apply-generic` may try to
coerce the arguments to each other's type even if they already have
the same type. Therefore, he reasons, we need to put procedures
in the coercion table to "coerce" arguments of each type to their
own type. For example, in addition to the `scheme-number->complex`
coercion shown above, he would do:

(define (scheme-number->scheme-number n) n) (define (complex->complex z) z) (put-coercion 'scheme-number 'scheme-number scheme-number->scheme-number) (put-coercion 'complex 'complex complex->complex)

a. With Louis's coercion procedures installed, what happens if `apply-generic`
is called with two arguments of type `scheme-number` or two arguments of
type `complex` for an operation that is not found in the table for those
types? For example, assume that we've defined a generic exponentiation
operation:

(define (exp x y) (apply-generic 'exp x y))and have put a procedure for exponentiation in the Scheme-number package but not in any other package:

What happens if we call;; following added to Scheme-number package(put 'exp '(scheme-number scheme-number) (lambda (x y) (tag (expt x y)))); using primitiveexpt

b. Is Louis correct that something had to be done about
coercion with arguments of the same type, or does `apply-generic`
work correctly as is?

c. Modify `apply-generic` so that it doesn't try coercion if
the two arguments have the same type.

**Exercise.**
Show how to generalize `apply-generic` to handle
coercion in the general case of multiple arguments. One strategy is
to attempt to coerce all the arguments to the type of the first argument, then
to the type of the second argument, and so on. Give an example of a situation
where this strategy (and likewise the two-argument version given
above) is not sufficiently general. (Hint: Consider the case where
there are some suitable mixed-type operations present in the table
that will not be tried.)

**Exercise.**
Suppose you are designing a generic arithmetic system for dealing with
the tower of types shown in figure :
integer, rational, real, complex. For
each type (except complex), design a procedure that raises objects of
that type one level in the tower. Show how to install a generic `
raise` operation that will work for each type (except complex).

**Exercise.**
Using the `raise` operation of exercise , modify the `
apply-generic` procedure so that it coerces its arguments to have the
same type by the method of successive raising, as discussed in this
section. You will need to devise a way to test which of two types is
higher in the tower. Do this in a manner that is ``compatible'' with
the rest of the system and will not lead to problems in adding new
levels to the tower.

**Exercise.**
This section mentioned a method for ``simplifying'' a data object
by lowering it in the tower of types as far as possible. Design
a procedure `drop` that accomplishes this for the tower described
in exercise . The key is to decide, in some general way, whether
an object can be lowered. For example, the complex number 1.5+0*i*
can be lowered as far as `real`, the complex number 1+0*i* can be
lowered as far as `integer`, and the complex number 2+3*i* cannot
be lowered at all. Here is a plan for determining whether an object
can be lowered: Begin by defining a generic operation `project`
that ``pushes'' an object down in the tower. For example, projecting
a complex number would involve throwing away the imaginary part. Then
a number can be dropped if, when we `project` it and `raise`
the result back to the type we started with, we end up with something
equal to what we started with. Show how to implement this idea in
detail, by writing a `drop` procedure that drops an object as far
as possible. You will need to design the various projection
operations^{} and install `project` as a generic operation in
the system. You will also need to make use of a generic equality
predicate, such as described in exercise . Finally, use `drop`
to rewrite `apply-generic` from exercise so that it
``simplifies'' its answers.

**Exercise.**
Suppose we want to handle complex numbers whose real
parts, imaginary parts, magnitudes, and angles can be either ordinary
numbers, rational numbers, or other numbers we might wish to add to
the system. Describe and implement the changes to the system needed
to accommodate this. You will have to define operations such as `
sine` and `cosine` that are generic over ordinary numbers and
rational numbers.