ON THE USE OF OPERATOR METHODS IN BOOLEAN ALGEBRA IN THE SYNTHESIS AND ANALYSIS OF POTENTIAL-PULSE RELAY DEVICES

E. L. Sobakin

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

The use of logical methods for formalizing the processes of synthesis and analysis of relay devices is one of the promising directions in the development of automation, telemechanics, and computer engineering. The formalization of scientific thought in design and analysis proceeds mainly by way of using the formal methods of mathematical logic and especially its apparatus—Boolean algebra.

At present, Boolean algebra is widely used to describe relay devices whose operation does not depend on time; such a description is carried out by means of combinational Boolean functions. However, such devices constitute only part of the general class of relay devices. The remaining part belongs to relay devices whose operation depends on time, to the so-called sequential automata. Sequential automata are described by temporal, or recurrent, Boolean functions, or by both together [3]. But even combinational, temporal, and recurrent Boolean functions cannot reflect (or reflect only in a very complex, implicit form) all the features and all the variety of connections that exist and arise between the signals and elements of real relay devices. This is why new formal methods arise that make it possible to take these features into account and to eliminate gaps in this area of knowledge. A certain gap now exists in the development of formal methods for the synthesis and analysis of potential-pulse relay devices [1]. Namely, this gap is manifested in the fact that the available methods for describing potential-pulse relay devices [1, 5] do not make it possible, from algebraic formulas, to judge the durations of pulse signals or to use recurrent Boolean functions for their description. Yet in the engineering application of these methods it is important not only to take into account the type of signals (pulse or static), but also to know and take into account the durations of pulse signals and, furthermore, to know as the result of changes in which signals the pulse signals appear.

The introduction into Boolean algebra of a special logical transition operator ([5]) made it possible, from the algebraic form of logical functions, to judge the presence or absence of pulse signals in synthe-

device, and in a number of cases to obtain the simplest fundamental solutions of practical problems.

In the cited work and in a number of other works [1, 2, 3, and 4], the unit value of a pulse signal is understood as the presence, and the zero value as the absence, of a pulse, whereas the unit value of a static signal is understood as one of its “permitted” levels, and the zero value as another “permitted” level. Such a specification of a pulse signal is valid only in the case when the duration of the pulse signal is many times smaller than the duration of the corresponding static signal, i.e., the duration of the pulse signal must be neglected. This is still done in the analysis of potential-pulse relay devices. Moreover, with such a specification of pulse signals the concept of an “inverse pulse signal” has no meaning, whereas in practice this concept corresponds to a quite definite meaning.

If the values of a pulse signal are defined in the same way as those of a static signal, the indicated shortcomings can be avoided. To do this, one should use special logical operators, by means of which it would be possible not only to indicate by what changes the pulse signal is generated, but also the durations of the pulse signals (pulses).

Since in Boolean algebra any logical variables can take only two values, 0 and 1, each variable can undergo two kinds of changes: from 0 to 1 (transition 0-1) and from 1 to 0 (transition 1-0). Therefore pulse signals can arise only at the instants when the initial signals change from zero value to unit value and from unit value to zero. It should be noted that, to denote static (potential) signals in Boolean algebra, Boolean variables are also used. In other words, all pulse signals can be divided into two types: pulse signals of the first type, depending on the transition 0-1, and pulse signals of the second type, depending on the transition 1-0.

Let us introduce into Boolean algebra, for denoting the operations of obtaining pulse signals, the corresponding operators: the first-type operator \(\tau\), to denote the operation of obtaining pulse signals of the first type, and the second-type operator \(\tau'\), to denote the operation of obtaining pulse signals of the second type. In addition, let us introduce the concept of the “modulus” of an operator. By the modulus of an operator \(|\tau|\) or \(|\tau'|\) we shall understand the duration of that pulse signal (pulse) whose operation of obtaining is denoted by the given operator. If it is necessary to indicate that operators of different moduli are being applied, then different decimal numbers should be attached to the operators as subscripts.

For example, the notation \(\tau_1\) and \(\tau_2\) indicates that these are first-type operators of different moduli, while the notation \(\tau_1\) and \(\tau'_1\) represents operators of different types but equal moduli.

By applying the introduced operators to Boolean variables, one can obtain pulse variables serving to denote the corresponding pulse signals. To obtain the notation of a pulse signal, one must use the following rule: to the notation of the initial signal from which the given pulse signal is formed, the corresponding operator is appended at the lower right as a subscript. In this case the initial expression is taken in parentheses.

For example, suppose that the Boolean variable \(a\) denotes a static signal (Fig. 1, \(a\)), which up to the instant \(t_0\) had value 0, and from the instant \(t_0\) to the instant \(t_1 = t_0 + T\), where \(T\) is the duration of the static sig-

the signal, assumes the value 1 and then, for all instants \(t>t_1\), again assumes the zero value. It is required to obtain the notation for a pulse signal of the first kind having duration \(|\tau_1|\). Applying the rule given above, we obtain \((a)_{\tau_1}\) and \((\bar a)_{\tau_1}\). Both expressions obtained are pulse variables of the first kind. They denote pulse signals of the first kind, one of which is the inversion of the other (Fig. 1, \(a\)).

Analogously, one can obtain pulse variables of the second kind, serving to denote pulse signals of the second kind. If it is assumed that the durations of these signals are equal to \(|\tau_2|\), then we obtain \((a)'_{\tau_2}\) and \((\bar a)'_{\tau_2}\) (see the same figure).

Since the Boolean variables \(a\) and \(\bar a\) can each have only two types of transitions, \(0—1\) and \(1—0\), and since the transition \(0—1\) of the variable \(a\) coincides in time with the transition \(1—0\) of the variable \(\bar a\), while the transition \(1—0\) of the variable \(a\) coincides with the transition \(0—1\) of the variable \(\bar a\), it is not difficult to establish the interrelation between pulse variables of the first kind and pulse variables of the second kind:

Fig. 1.

\[ a_{\tau_1}=(a)_{\tau_1}=(\bar a)'_{\tau_1}, \tag{1} \]

\[ \bar a_{\tau_1}=(\bar a)_{\tau_1}=(a)'_{\tau_1}, \tag{2} \]

\[ (a)'_{\tau_2}=a'_{\tau_2}=(\bar a)_{\tau_2}, \tag{3} \]

\[ \bar a'_{\tau_2}=(\bar a)'_{\tau_2}=(a)_{\tau_2}. \tag{4} \]

The validity of these identities is easily verified by considering the time diagrams of the corresponding signals (Fig. 1, \(a\), and Fig. 1, \(b\)). In Fig. 1, \(a\), diagrams are given of pulse signals of the first and second kind for the initial static signal \(a\), while in Fig. 1, \(b\), the same is shown, but for the initial static signal \(\bar a\). In other words, one may say that equalities (1) through (4) illustrate the rules of equivalent replacement of pulse variables, or the rules of equivalent replacement of an operator of one kind by an operator of the other kind.

Thus, the symbolism introduced makes it possible to take into account in algebraic form not only the type of signals, but also the durations of pulse signals.

Owing to the fact that pulse signals are specified by the same values as static ones, all the laws of Boolean algebra remain valid for pulse and static variables, and, moreover,

there arises a series of new consequences. Of these consequences, the following are the most interesting:

\[ a\cdot \bar a_{\tau}=(a)^{\tau}, \tag{5} \]

\[ \bar a+a_{\tau}=(\bar a)^{\tau}, \tag{6} \]

\[ \bar a\cdot a_{\tau'}=\overline{(a)}^{\tau'}, \tag{7} \]

\[ a+\bar a_{\tau'}=(a)^{\tau'}, \tag{8} \]

\[ a^{\tau}+\bar a_{\tau'}=(a)^{\tau\tau'}, \tag{9} \]

\[ \bar a^{\tau}\cdot a_{\tau'}=\overline{(a)}^{\tau\tau'}. \tag{10} \]

In these expressions the operators \(\tau\) and \(\tau'\) may have identical or different moduli. Moreover, in the right-hand sides of the equalities the operators are used to denote other operations, namely the operations written in the left-hand sides of the equalities.

The semantic meaning of these operations is not difficult to establish if one considers the operation of circuits implementing the dependences specified by the left-hand sides of the equalities. As a result, one may arrive at the following conclusions: the expression \((a)^{\tau}\) denotes a delay in switching on the signal \(a\), the expression \((\bar a)^{\tau}\) denotes a delay in switching on the signal \(\bar a\), the expression \((a)^{\tau'}\) denotes a delay in switching off the signal \(a\), the expression \(\overline{(a)}^{\tau'}\) denotes a delay in switching off the signal \(\bar a\), and the expressions \((a)^{\tau\tau'}\) and \(\overline{(a)}^{\tau'}\) denote delays in switching on and switching off, respectively, the signals \(a\) and \(\bar a\). In each case the magnitude of the delay is given by the modulus of the corresponding operator.

The expressions presented illustrate the second way of using operators, the second rule for their application. This rule says that operators may be assigned to initial expressions in the form of an exponent. In this case an operator of the first kind denotes the operation of delaying the transition \(0\text{-}1\), and an operator of the second kind denotes the operation of delaying the transition \(1\text{-}0\) of the initial signal. Both operators, standing simultaneously in the form of an exponent, denote, respectively, the operation of delaying the transitions \(1\text{-}0\) and \(0\text{-}1\) by intervals specified by the moduli of the corresponding operators. For this type of application of operators, identities can also be given that illustrate the interrelation of signal delays.

\[ a^{\tau_1}=(a)^{\bar \tau_1}=\overline{(\bar a)}^{\tau_1'}, \tag{11} \]

\[ \bar a^{\tau_1}=\overline{(a)}^{\tau_1}=(\bar a)^{\tau_1'}, \tag{12} \]

\[ a^{\tau_2'}=(a)^{\tau_2'}=(\bar a)^{\tau_2}, \tag{13} \]

\[ \bar a^{\tau_2'}=\overline{(a)}^{\tau_2'}=(\bar a)^{\tau_2}. \tag{14} \]

The validity of these equalities can be verified by comparing the corresponding signals (Fig. 2, \(a\), and Fig. 2, \(b\)).

In Fig. 2, \(a\), signal diagrams are given for the initial signal \(a\); in Fig. 2, \(b\), for the initial signal \(\bar a\).

Since the transition \(0\text{-}1\) may be the leading edge of the signal \(a\) and the trailing edge of the signal \(\bar a\), the element implementing the operation of delaying this transition will carry out a delay in switching on the signal \(a\) and a delay in switching off the signal \(\bar a\). However, these two types of delays cannot be implemented simultaneously by a single element, since the \(0\text{-}1\) transitions of the signals \(a\) and \(\bar a\) can never coincide in time.

time. Similarly, an element realizing the operation of delaying the transition 1–0 will carry out a delay when signal $a$ is switched off and a delay when signal $\bar a$ is switched on.

Thus we arrive at the conclusion that the same potential-pulse relay elements can perform different functions: the first element—the function of delay when $a$ is switched on, or the function of delay when $\bar a$ is switched off; the second—the function of delay when $a$ is switched off, or the function of delay when $\bar a$ is switched on.

Fig. 2.

The same conclusion can be reached by considering equalities (1)—(4). Namely: a relay element realizing the operation of obtaining a pulse signal of the first kind can, under certain conditions, realize the operation of obtaining a pulse signal of the second kind, and conversely. These conditions, in both the first and the second case, are the type of the input signal ($a$ or $\bar a$). By the type of a signal we shall understand its belonging to the group of signals which, up to the time instant under consideration (for example, the instant $t_0$ in Figs. 1 and 2), have the same value. In this connection, signals having zero value will be called signals of the first type, and signals having unit value—signals of the second type. For example, the signals $a$, $a_{\tau}$, $\bar a_{\tau'}$, $\bar a^{\tau}$, $a^{\tau'}$, etc., will belong to signals of the first type, while the signals $\bar a$, $\bar a_{\tau}$, $a_{\tau'}$, $a^{\tau}$, etc., will belong to signals of the second type. Such a division of signals reflects a very important point in the description of relay devices—the initial (original) state of the elements from which the relay device is constructed.

Thus, depending on the type of the initial signal, the kind of operators, and the rules for their application, one can establish the exact semantic content of a particular operator expression. Thus, for example, the expression $a_{\tau_1}^{\tau_2}$ denotes a delay, upon switching off, of the pulse signal $a_{\tau_1}'$, which arises at the instant when signal $a$ ends (at the instant of the transition 1–0) and has duration $|\tau_1'|$, over the interval $|\tau_2|$, since the signal $a_{\tau_1}$ belongs to the second type.

It should be noted that the conclusion obtained concerning the ambiguity of the functioning of a particular simplest potential-pulse relay device does not contradict the conclusions of Boolean algebra, but merely says that these devices are described by elementary sequential functions. Owing to the use of operators, these functions can be considered as combinational logical functions, and this should simplify the analysis and synthesis of sequential circuits, to which potential-pulse relay circuits must also be assigned.

Functions obtained as a result of applying the operators $\tau$ and $\tau'$ to Boolean variables may be called potential-pulse logical functions; however, these functions differ substantially from functions of the same name introduced by A. D. Talantsev [5].

Analyzing expressions (9) and (10), one can establish a connection between operator expressions and recurrent Boolean functions. Thus, the expression $a^{\tau_1\tau_2'}$ denotes a signal whose leading edge is delayed by the interval $|\tau_1|$, and whose trailing edge is delayed by the interval $|\tau_2'|$ relative to the corresponding edges of the original signal $a$ (Fig. 3, a). If these intervals are equal, i.e., $|\tau_1|=|\tau_2'|=|\tau|$, then the dependence between the signal $a$ and the signal $a^{\tau\tau'}$ can be expressed by the following recurrent relation:

Fig. 3.

\[ a^{\tau\tau'} \doteq y_{1t}=a_{(t-|\tau|)} . \tag{11} \]

Here, instead of the modulus of the operators $|\tau|$, one must substitute its concrete value, expressed in some conditional units; then expression (11) will not differ in any way from ordinary recurrent expressions (see [3]). For example, with operator moduli $|\tau|=|\tau'|=3$ units, we obtain

\[ y_{1t}=a_{(t-3)} . \]

This recurrent expression corresponds to the signal diagrams shown in Fig. 3, b, for the following assignment of the original signal:

\[ \{a_t\}=1,1,1,1,1,1,1,1,0,0,0,0\ldots,\quad \text{for } t=0,1,2,3\ldots \]

\[ \text{and for } t<0\quad a_t=0 . \]

In this figure, transitions of the static signal $a_t$ occur at the instants $t_0=0$ and $t_1=8$ conditional units, while the transitions of the signal $y_{1t}$ are delayed by three conditional units of time.

It can be proved that any operator expression obtained from one Boolean variable can be expressed through recurrent Boolean functions (RBFs). For example,

\[ a_{\tau}\doteq y_{2t}=a_t\cdot a_{(t-|\tau|)}, \tag{12} \]

\[ a_{\tau'}\doteq y_{3t}=a_t+\overline{a}_{(t-|\tau'|)}, \tag{13} \]

\[ a^{\tau}\doteq y_{4t}=a_t\cdot a_{(t-|\tau|)}, \tag{14} \]

\[ a^{\tau'}\doteq y_{5t}=a_t+a_{(t-|\tau'|)}, \tag{15} \]

\[ a^{\tau_1\tau_2'}\doteq y_{6t}=a_t\cdot a_{(t-|\tau_1|)}+\overline{a}_t\,a_{(t-|\tau_2'|)} . \tag{16} \]

From this one may conclude that the proposed method for describing potential-pulse relay devices is based chiefly on the use of RBFs in operator form and may be espe-

especially effective in describing potential-pulse circuits whose operation can in principle be conditioned by a single external (initial) action.

In conclusion it should be said that the operator relations obtained can be implemented not only with contact or contactless elements, but also with both of these simultaneously. In the latter case it should be taken into account that, in comparison with contactless elements, any electromagnetic relay realizes the function \(a^{\tau_1 \tau_2}\).

When implementing expressions with contact elements, the temporal relations will, of course, differ significantly from those assumed, since in no electromagnetic relay are the operating time and the release time equal to zero. However, the basic functional dependences between pulse and static signals can also be implemented with contact elements. As a result, one may conclude that the proposed method is more convenient for describing electronic and semiconductor circuits.

REFERENCES

1. O. N. Ivanova, V. G. Lazarev, E. I. Piil. Synthesis of Electronic Circuits of Discrete Action. Svyaz Publishing House, 1964.

2. S. Caldwell. Logical Synthesis of Relay Devices. Foreign Literature Publishing House, 1962.

3. D. A. Pospelov. Logical Methods for the Analysis and Synthesis of Circuits. Energia Publishing House, 1964.

4. V. N. Roginsky. Construction of Relay Control Circuits. Energia Publishing House, 1964.

5. A. D. Talandsev. On the analysis and synthesis of certain electrical circuits using special logical operators, A i T, No. 7, 1959.