Analysis and synthesis of certain electric circuits by means of special logical operators

An algebraic theory of the logical operation of some electric circuits is developed. This theory takes into account the time behaviour of circuit processes. The logical variable $Z$ when represented in the form of voltage-state signal is called «potential variable» denoted as $X$ and is called «pulse variable» denoted as $Y$ when represented in the form of pulse signal. The special logical operators are defined as follows. If at the moment of time $t$ a true value of variable $Z$ goes to a false value it is expressed by $dZ(t)=1$ and $dZ(t)=0$ is assumed in all other cases. The symbol $d$ is called «transition operator». The operator $D$ is defined by the formula (1). These operators describe changes of Boolean functions when their arguments change. Some of their properties are investigated (2)–(6).
Applying operators $d$ and $D$ to elementary function $\vee$ and & we obtain relationships (7)–(10). Applying operator $d$ to an arbitrary function of $n$ variables we obtain the formula (11). These relationships are outside the province of Boolean algebra. The new logical operators may be employed to characterize the behaviour of certain electric circuits called «homogeneous potential-pulse circuits. Transformation procedure for such circuits is developed.
The application of new operators to the design of certain digital control circuits is illustrated by an actual example.