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 function 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 function for adding complex numbers to ordinary numbers and installs this in the table using the tag list("complex","javascript_number"):[1]
// to be included in the complex package function add_complex_to_javascript_num(z, x) { return make_from_real_imag(real_part(z) + x, imag_part(z)); } put("add", list("complex", "javascript_number"), (z, x) => tag(add_complex_to_javascript_num(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 functions for that type but also the construction and installation of the functions 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.
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 functions that transform an object of one type into an equivalent object of another type. Here is a typical coercion function, which transforms a given ordinary number to a complex number with that real part and zero imaginary part:
function javascript_number_to_complex(n) { return make_complex_from_real_imag(contents(n), 0); }
put_coercion("javascipt_number", "complex", javascript_number_to_complex);
Once the coercion table has been set up, we can handle coercion in a uniform manner by modifying the apply_generic function of section 2.4.3. 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 function found in the operation-and-type table. Otherwise, we try coercion. For simplicity, we consider only the case where there are two arguments.[2] 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 function:
function apply_generic(op, args) { const type_tags = map(type_tag, args); const fun = get(op, type_tags); if (fun !== false) { return fun(map(contents, args)); } else { if (length(args) === 2) { const type1 = head(type_tags); const type2 = head(tail(type_tags)); const a1 = head(args); const a2 = head(tail(args)); const t1_to_t2 = get_coercion(type1, type2); const t2_to_t1 = get_coercion(type2, type1); if (t1_to_t2 !== false) { return apply_generic(op,list(t1_to_t2(a1), a2)); } else if (t2_to_t1 !== false) { return apply_generic(op, list(a1, t2_to_t1(a2))); } else { return Error("No method for these types", list(op, type_tags)); } } else { return 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 functions to relate the types (possibly $n^2$ functions for a system with $n$ types), we need to write only one function for each pair of types rather than a different function for each collection of types and each generic operation.[3] 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.
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 2.25.
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 function integer_to_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
function
in the following
way: For each type, we need to supply a raise
function, 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 2.83
and 2.84
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
function
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 $2+3i$ to $4-3i$, it would be
nice to obtain the answer as the integer 6 rather than as the complex
number $6+0i$. Exercise 2.85 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 $6+0i$, from
those that cannot, such as $6+2i$.)
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 2.26 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
function
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.[4]
coercearguments of each type to their own type. For example, in addition to the javascript_number_to_complex coercion shown above, he would do:
function javascript_number_to_javascript_number(n) { return n; } function complex_number_to_complex_number(n) { return n; } put_coercion("javascript_number", "javascript_number", javascript_number_to_javascript_number); put_coercion("complex_number", "complex_number", complex_number_to_complex_number);
function exp(x, y) { return apply_generic("exp", list(x, y)); }
// following added to JavaScript-number package put("exp", list("javascript_number", "javascript_number"), (x, y) => tag(math_exp(x, y))); // primitive math_exp
compatiblewith the rest of the system and will not lead to problems in adding new levels to the tower.
simplifyinga data object by lowering it in the tower of types as far as possible. Design a function drop that accomplishes this for the tower described in exercise 2.83. The key is to decide, in some general way, whether an object can be lowered. For example, the complex number $1.5+0i$ can be lowered as far as "real", the complex number $1+0i$ can be lowered as far as "integer", and the complex number $2+3i$ 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
pushesan 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 function that drops an object as far as possible. You will need to design the various projection operations[5] 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 2.79. Finally, use drop to rewrite apply_generic from exercise 2.84 so that it
simplifiesits answers.
graphof relations among types and automatically generate those coercion functions that can be inferred from the ones that are supplied explicitly.
ontology) seems intractably difficult. The main difference between the confusion that existed ten years ago and the confusion that exists now is that now a variety of inadequate ontological theories have been embodied in a plethora of correspondingly inadequate programming languages. For example, much of the complexity of object-oriented programming languages—and the subtle and confusing differences among contemporary object-oriented languages—centers on the treatment of generic operations on interrelated types. Our own discussion of computational objects in chapter 3 avoids these issues entirely. Readers familiar with object-oriented programming will notice that we have much to say in chapter 3 about local state, but we do not even mention
classesor
inheritance.In fact, we suspect that these problems cannot be adequately addressed in terms of computer-language design alone, without also drawing on work in knowledge representation and automated reasoning.