**10 Part 1½Fundamentals Section
1 ½ Abstraction and Notation**

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 a graphical description of an inverter
circuit, as well as an algebraic and a 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 Q1, the resistor *R, *and
the stray capacitances *C*_{8}* *are given only token
values. The structure can be described symbolically by first writing the
relationship describing each of the components (Ohm's law, Faraday's law,
etc.) and then the equation which describes the interconnection of the
components (i.e., Kirchoff'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 that 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 disks are most likely to come to mind first, but card readers, card punches, line printers, and terminals are other examples. These devices obey laws of motion and are analyzed in units of mass, length, and time.

The next level is the *logic level. *It is unique to digital technology,
whereas the circuit level (and below) is what digital technology shares
with the rest of electrical engineering. The behavior of a system is now
described by discrete variables which take on only two values, called 0
and 1 (or + and -' true and false, high and low). The components perform
logic functions: AND, OR, NOT, NAND, etc. Systems are constructed in the
same way as at the circuit level, by connecting the terminals of components,
which thereby identify their behavioral values. The laws of boolean algebra
are used to compute the behavior of a system from the behavior and properties
of its components.

The previous paragraph described *combinational circuits *whose
outputs are directly related to the inputs at any instant of time. If the
circuit has the ability to hold values over time (store information), we
get *sequential circuits. *The problem that the combinational-level
analysis solves is the production of a set of outputs at time *t *as
a function of a number of inputs at the same time *t. *As described
in textbooks, the analysis abstracts from any transport delays between
input and output; however, in engineering practice the analysis of delays
is usually considered to be still part of the combinational level. In Fig.
3 we show a combinational network formed from combinational elements which
realize three boolean output expressions, 0_{1}, 0_{2}*,
*and 0_{3}*, *as a function of the input boolean variables
*A *and *B. *Note that in the symbolic representation of the
structure we can write an expression that reflects the structure of the
combinational network, but, on reduction, the boolean equations no longer
reflect the actual structure of the combinational circuit but become a
model to predict its behavior.

The representation of a sequential switching circuit is basically the
same as that of a combinational switching circuit, although one needs to
add memory components, such as a delay element (which produces as output
at time *t *the input at time *t *- t**).
**Thus the equations that specify structure must be difference equations
involving time. Again, there is a distinction (even in representation)
between *synchronous *circuits and *asynchronous *circuits, namely,
whether behavior can be represented by a sequence of values at integral
time points *(t *= 1, 2, 3, . . . ) or must deal in continuous time.
But this is a minor variation. Figure 4 gives a sequential logic circuit
in both an algebraic and a graphical form and shows also the representation
of the behavior of the system.

Now it is clear that logic circuits are simply a subspecies of general
circuits. Indeed, to design the logic components one