The Anatomy of Large Scale Systems

ions, Modules, Extensions and Layers The power of an abstraction is usually derived from creating a simplified domain where a set of details is hidden. A module is an abstraction in this sense if one can use the module without having to know its internal details. Yet the notion of a module does not fully encompass what is meant by an abstraction in abstract algebra. That is because many abstractions, in particular extensions, apply to sets of objects, rather than a single object, however complex. Consider the set of integers or even more simply just the binary numbers 0 and 1. One abstraction of the binary numbers is the set of polynomials, in x say, with binary coefficients. Let us call the set P(x). Consider an abstraction of P(x), the set of polynomials in y, say, whose coefficients are in P(x). Let us call it P(x,y). Note that in P(x) we can refer to either binary number as a coefficient. Likewise in P(x,y) we can refer to any polynomial in P(x) as a coefficient. This large number of references in P(x,y) to lower layers gives one an advantage that we relate to flexibility. For us flexibility often arises from the ability to switch from one path of interconnections to another. While mathematics does not clearly indicate connections, such as between the set of polynomials and the set of their coefficients, most of the physical world requires them. We show such connections in Figure 1. Whereas the set of polynomials even with just binary digit coefficients is infinite, the set of operations on such polynomials that are of interest is quite small. Thus another hierarchy related to that in Figure 1 is the hierarchy of operations, such as addition and multiplication on the integers, on polynomials in one variable with integer coefficients, and on polynomials in two variables. We show the interconnections between such operations in Figure 2. Taking a set of related objects or modules and creating a higher layer of abstractions is called, as we have previously noted, an extension. The ability to do so is called extensibility. Whereas an abstraction that forms a module is essentially a many-to-one operation, an extension usually is a many-to-many operation. The notions of base and extensions are relative, depending on the circumstances.