**4 Part 1 |
The structure of computers
**

Elementary circuit theory is an almost prototypic example. The components are R's, L's, C's, and voltage sources. The mode of combination is to run wires between the terminals of components, which corresponds to an identification of current and voltage at these terminals. The algebraic and differential equations of circuit theory provide the means whereby the behavior of a circuit can be computed from the properties of its components and the way the circuit is constructed.

There is a recursive feature to most system descriptions. A system, composed of components structured in a given way, may be considered a component in the construction of yet other systems. There are, of course, some primitive components whose properties are not explicable as the resultant of a system of the same type. For example, a resistor is not to be explained by a subcircuit but is taken as a primitive. Sometimes there are no absolute primitives, it being a matter of convention what basis is taken. For example, one can build logical design systems from many different primitive sets of logical operations (AND and NOT, NAND, OR and NOT, etc.).

A system level, as we have used the term in Fig. 1, is characterized by a distinct language for representing the system (that is, the components, modes of combination, and laws of behavior). These distinct languages reflect special properties of the types of components and of the way they combine. Otherwise, there would be no point in adopting a special representation. Nevertheless, these levels exist in the system analyst's way of describing the same

Fig. 2. Electronic-circuit level: inverter circuit.

physically existing system. The fact that the languages are highly distinct makes it possible to be confident about the existence of different system levels. Where we are fuzzy, as in the existence of an additional intermediate level, it is because new representations have not yet congealed into distinct formal languages. As we noted, within each level there exists a whole hierarchy of systems and subsystems. However, as long as these are all described in the same language, e.g., a subroutine hierarchy, all given in machine-assembly language, they do not constitute separate system levels.

With this general view, let us work through the levels of computer systems, starting at the bottom. Each level in Fig. 1 actually has two languages or representations associated with it: an algebraic one and a graphical one. These are isomorphic to each other, the same entities, properties, and relations being given in both.

The lowest level in Fig. 1 is the *circuit level. *Here the components are R's, L's, C's, voltage sources, and nonlinear devices. The behavior of the system is measured in terms of voltage, current, and magnetic flux. These are continuously varying quantities associated with various components, and so there is continuous behavior through time. The components have a discrete number of terminals, whereby they can be connected to other components. Figure 2 shows both an algebraic and graphical description of an inverter circuit, as well as an algebraic and graphical description of its behavior. We note that its structure is specified first as a circuit (a directed graph), with symbols for the arcs and nodes. The particular circuit still is an abstraction because the transistor Q 1, the resistor R, and the stray capacitors C_{s} are given only token values. The structure can be described symbolically by first writing the relationship describing each of the components (i.e., Ohm's law, Faraday's law, etc.) and then the equation which describes the interconnection of the components (i.e., Kirchhoff's laws). We observe the behavior of the circuit (probably using an oscilloscope) by applying an input e_{i}(t) and observing an output e_{0}(t). Alternatively, if we solve the equations which specify the structure, we obtain expressions which describe the behavior explicitly.

The circuit level is not in fact the lowest level that might be used in describing a computer system. The devices themselves require a different language, either that of electromagnetic theory or of quantum mechanics (for the solid-state devices). It is usually an exercise in a course on Maxwell's equations to show that circuit theory can be derived as a specialization under appropriately restricted boundary conditions. Actually, even at its level of abstraction, circuit theory is not quite adequate to describe computer technology since there are a number of mechanical devices which must be represented. Magnetic tapes and drums are most likely