Introduction to Programming Languages/Type Definition

Data Types
The vast majority of the programming languages deal with typed values, i.e., integers, booleans, real numbers, people, vehicles, etc. There are however, programming languages that have no types at all. These programming languages tend to be very simple. Good examples in this category are the core lambda calculus, and Brain Fuc*. There exist programming languages that have some very primitive typing systems. For instance, the x86 assembly allows to store floating point numbers, integers and addresses into the same registers. In this case, the particular instruction used to process the register determines which data type is being taken into consideration. For instance, the x86 assembly has a subl instruction to perform integer subtraction, and another instruction, fsubl, to subtract floating point values. As another example, BCPL has only one data type, a word. Different operations treat each word as a different type. Nevertheless, most of the programming languages have more complex types, and we shall be talking about these typing systems in this chapter.

The most important question that we should answer now is "what is a data type". We can describe a data type by combining two notions:
 * Values: a type is, in essence, a set of values. For instance, the boolean data type, seen in many programming languages, is a set with two elements: true and false. Some of these sets have a finite number of elements. Others are infinite. In Java, the integer data type is a set with 232 elements; however, the string data type is a set with an infinite number of elements.
 * Operations: not every operation can be applied on every data type. For instance, we can sum up two numeric types; however, in most of the programming languages, it does not make sense to sum up two booleans. In the x86 assembly, and in BCPL, the operations distinguish the type of a memory location from the type of others.

Types exist so that developers can represent entities from the real world in their programs. However, types are not the entities that they represent. For instance, the integer type, in Java, represents numbers ranging from -231 to 231 - 1. Larger numbers cannot be represented. If we try to assign, say, 231 to an integer in Java, then we get back -231. This happens because Java only allows us to represent the 31 least bits of any binary integer.

Types are useful in many different ways. Testimony of this importance is the fact that today virtually every programming language uses types, be it statically, be it at runtime. Among the many facts that contribute to make types so important, we mention:
 * Efficiency: because different types can be represented in different ways, the runtime environment can choose the most efficient alternative for each representations.
 * Correctness: types prevent the program from entering into undefined states. For instance, if the result of adding an integer and a floating point number is undefined, then the runtime environment can trigger an exception whenever this operation might happen.
 * Documentation: types are a form of documentation. For instance, if a programmer knows that a given variable is an integer, then he or she knows a lot about it. The programmer knows, for example, that this variable can be the target of arithmetic operations. The programmer also knows much memory is necessary to allocate that variable. Furthermore, contrary to simple comments, that mean nothing to the compiler, types are a form of documentation that the compiler can check.

Types are a fascinating subject, because they classify programming languages along many different dimensions. Three of the most important dimensions are:
 * Statically vs Dynamically typed.
 * Strongly vs Weakly typed.
 * Structurally vs Nominally typed.

In any programming language there are two main categories of types: primitive and constructed. Primitive types are atomic, i.e., they are not formed by the combination of other types. Constructed, or composite types, as the name already says, are made of other types, either primitive or also composite. In the rest of this chapter we will be showing examples of each family of types.