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Digital analysis of a form

André Oksas

Research Fellow, Next Society Institute,

Kazimieras Simonavičius University, Vilnius, Lithuania

Abstract

Purpose –This paper aims to show how a sociological description –a swarm analysis of the Nazi

dictatorship –initially made with the means borrowed from George Spencer-Brown’s Calculus of Indications,

can be transformed into a digital circuit and with which methods and tools of digital mathematics this digital

circuit can be analyzed and described in its behavior. Thus, the paper also aims to contribute to a better

understanding of Chapter 11 of “Laws of Form.”

Design/methodology/approach –The analysis uses methods of automata theory for ﬁnite,

deterministic automata. Basic set operations of digital mathematics and special set operations of the Boolean

Differential Calculus are used to calculate digital circuits. The software used is based on ternary logic, in which

the binary Boolean logic ofthe elements {0, 1} is extended by the thirdelement “Don’tcare”to {0, 1, }.

Findings –The paper conﬁrms the method of transforming a form into a digital circuit derived from the

comparative functional and structural analysis of the Modulator from Chapter 11 of “Laws of Form”and

deﬁnes general rules for this transformation. It is shown how the indeterminacy of re-entrant forms can be

resolved in the medium of time using the methods of automata theory. On this basis, a reﬁned deﬁnition of the

degree of a form is presented.

Originality/value –The paper shows the potential of interdisciplinary approaches between sociologyand

information technology and provides methods and tools of digital mathematics such as ternary logic, Boolean

Differential Calculus and automata theoryfor application in sociology.

Keywords Systems theory, Automata theory, Niklas Luhmann, Calculus of Indications,

Digital mathematics, George Spencer-Brown

Paper type Research paper

Introduction

With his major work, “Laws of Form”(Spencer-Brown, 1972), George Spencer-Brown

presented a mathematical Calculus of Indications that has found widespread application in

the processing of sociological observations. In the basic distinction-and-indication operation,

the cross operator is used to indicate the presence of a marked state and its implied

contradictory, the unmarked state. The marked state and the unmarked state are

disjunctive, mutually exclusive and jointly exhaustive. The relation of the two

distinguishable states {marked, unmarked}, which are produced by one or more distinction-

and-indication operations, is called “form.”Furthermore, variables are introduced, but only

as expressions, the values of which indicate, again, the marked or the unmarked state:

“Let mstand for any number, greater than zero, of such expressions indicating the

marked state.

Let nstand for any number of such expressions indicating the unmarked state.”

(Spencer-Brown, 1972, p. 14).

The author is grateful to Franz Hoegl (Next Society Institute, Kazimieras Simonavičius University,

Vilnius) and Leon Conrad (Next Society Institute, Kazimieras Simonavičius University, Vilnius) for

active support and inspiring conversations about these subjects. Any errors in this article, however,

remain the author’s exclusive responsibility.

Digital

analysis

of a form

Received 21 October2020

Revised 30 December2020

Accepted 10 January2021

Kybernetes

© Emerald Publishing Limited

0368-492X

DOI 10.1108/K-10-2020-0682

The current issue and full text archive of this journal is available on Emerald Insight at:

https://www.emerald.com/insight/0368-492X.htm

Niklas Luhmann based his systems theory (Luhmann, 1995), which was formulated in

terms of difference theory and self-referentiality, on this Calculus of Indications. The

variables used in the differentiations of systems theory extend the Calculus of Indications,

as they no longer stand for mathematical expressions that themselves denote the marked or

unmarked state. As shown in Figure 1, the deﬁnition of a system can be formulated with the

Calculus of Indications as the difference between system and environment. System and

environment are disjunctive as variables, mutually exclusive and jointly exhaustive.

Beyond the difference-theoretical and self-referential aspects of the Calculus of Indications,

the visualization of the observer’s perspective implicit in the form has been widely accepted in

sociology. In the ﬁgure which symbolizes the formal deﬁnition of a system (Figure 1), the inner

distinction between system and environment implies an observer of the ﬁrst order. By contrast,

the outer distinction between the distinction of system/environment and the unmarked space

represents a second-order observer whose observation re-enters the form itself.

In Roth (2019), such distinctions between mutually exclusive and jointly exhaustive

variables are called “true distinctions”and the thesis is formulated that these “true

distinctions”are particularly suitable for digital processing.This thesis explores how digital

methods could be particularly suitable for observing the digitization of society. Before we

come back to this in the section “Swarm analysis of the Nazi dictatorship as a form”, the

method of transforming a form into a digital circuit of NOR gates will be established.

From universal NOR gate to the cross

Against the background of the assumption from Roth (2017) that George Spencer-Brown was

inspired by the Boolean NOR operation when developing the Calculus of Indications, it is worth

taking a more detailed look at the NOR operation and its electronic realization as a logic gate.

From the set of 2

4

or 16 possible Boolean logic gates with two inputs, the basic gates and the

universal gates stand out especially. Every Boolean function can be realized with a

combination of only the three basic gates: conjunction (AND), disjunction (OR) and negation

(NOT). Furthermore, according to the laws from De Morgan (1838) and using the negation

(NOT), any conjunction (AND) can be transformed into a disjunction (OR) and vice versa.

A complete algebraic logic can be represented by the combination of at least two of the

basic gates, where, in addition to the indispensable negation (NOT), either the conjunction

(AND) or the disjunction (OR) is needed. As the two universal gates, NAND (NOT AND) and

NOR (NOT OR) already contain the negation (NOT), one of these two universal gates is

sufﬁcient to represent a complete algebraic logic.

This universality and the negation included in the operation have particularly interested

mathematicians and philosophers. In Kauffman (2001), this history of the reception is traced

from the Peirce arrow of Charles Sanders Peirce, the Sheffer stroke of Henry Maurice

Sheffer, the Nicod arrow of Jean George Pierre Nicod to the cross of George Spencer-Brown.

This series could be extended to include Edward Stamm, as can be seen from a quotation

from George Spencer-Brown published in Roth (2017) from the typewriter manuscript

“Design with the NOR”(Roth, 2021), which was written in 1961. Furthermore, the use of the

Sheffer stroke in Ludwig Wittgenstein’s“Tractatus Logico-Philosophicus”(Wittgenstein,

1922) would also be worth mentioning in this context according to Hoegl (2021).

Figure 1.

System deﬁnition as

form, according to

Luhmann (1995)

K

How George Spencer-Brown’s understanding of the NOR operation and the NOR gate has

affected the development of his Calculus of Indications is of particular interest for our

investigations. The following quote by George Spencer-Brown from “Design with the NOR”

(Roth, 2021) deserves special attention:

“So whatever, in logic, the industrial NOR unit represents, it is certainly not the Sheffer stroke.

[...] To say that the industrial NOR unit represents an operation hitherto unknown to logic

would perhaps be too strong, although if we interpret ‘unknown’as ‘unfamiliar’the claim is

justiﬁed. For such an operation, even if known, has never been taken as standard. If we signify it

by a mark [...]called‘cross’we can compare it with the Sheffer stroke, thus.”(Roth, 2017 p. 1476)

At the time of his work on “Design with the NOR”(Roth, 2021), George Spencer-Brown

was employed as Chief Logic Designer by the manufacturer of electronic components

Mullard Equipment Limited (Baecker, 2015). According to the quote above, the industrial

NOR gate, thus, made special demands on the logic and led directly to the notation of the

cross. In principle, although the Sheffer stroke is also suitable as a notation to represent a

universal gate such as NAND or NOR, its limitation to two variables is an obstacle to the

requirements of circuit design. Furthermore, the negation of a variable in the sense of aby

means of the Sheffer stroke would be cumbersome to represent with aja.

With the cross, George Spencer-Brown created a topologically invariant notation with any

number of disordered, but bracketed variables to represent the logical “neither ... nor”,which

met his requirements from the circuit design with industrial NOR gates. How, eight years later,

this special notation for the circuit design developed into the philosophically more important

Calculus of Indications remains an exciting object of research and cannot be pursued further here.

A discussion of logic gates would be incomplete without the technological perspective

afforded by the semiconductor industry. As criteria such as integration density, power

consumption and delay times play decisive roles here, a different picture than from the

philosophical-logical one results. In the realization of logic gates based on semiconductors,

the transistor is the crucial element. This leads to the fact that NAND and NOR gates can

each be realized with one transistor less than their non-negated AND and OR gates. AND

and OR gates would have to be realized as negations of NAND or NOR, respectively, which

is unacceptable, based on the above-mentioned criteria.

Thus, there a need for a direct comparison of NAND and NOR gates arises. In the 1960s,

the NOR gate was the central component of the NORBIT series YL6000 (from which it

derived its name) manufactured by Mullard Equipment Limited (Norbit, 1962). Nevertheless,

it lost almost all its importance compared to the NAND gate with the technological change

from the then common Resistor Transistor Logic (RTL) ﬁrst to Transistor Transistor Logic

(TTL) and subsequently to the CMOS technology widely used today.

In Sutherland (1999), Logical Effort was introduced as a benchmark for delay times

experienced at the different logic gates. As shown in Table 1, the delay time of a NAND gate

Table 1.

Logical effort of

different logic gates

No. of inputs

Gate type 1 2 3 4 5 n

Inverter 1

NAND 4/3 5/3 6/3 7/3 (n+ 2)/3

NOR 5/3 7/3 9/3 11/3 (2n+ 1)/3

Multiplexer 2 2 2 2 2

XOR (parity) 4 12 32

Source: Sutherland (1999,p.4)

Digital

analysis

of a form

with ninputs with a Logical Effort of (nþ2)/3 is signiﬁcantly lower compared to the

equivalent NOR gate, which has a value of (2nþ1)/3. This is the reason why the NAND gate

is the standard element in TTL and CMOS technology. However, due to its transferability,

according to De Morgan’s laws, this is not important for circuit technology.

For our further considerations, it remains to be noted that the cross was born from the NOR gate,

which supports the thesis of fundamental traceability of a form to a digital circuit of NOR gates.

The form as a digital circuit

Although the cross had its origin in the industrial NOR gate, it does not yet show the basic

transformability of a form into a digital circuit of NOR gates. To clarify this question and to determine

the basic rules of such a transformation it is worthwhile to have a look at Chapter 11 (Spencer-Brown,

1972, pp. 54–68), where the form E4 is introduced to realize a modulator function (Figure 2).

The modulator function to be realized is deﬁned as the conversion of a sequence of wave-

shaped signal sequences of the values 0 and 1, which are received at input a, into a signal

sequence with half the frequency at output f,i.e.a= 10101010 becomes f= 11001100. Tasks

of this kind are described in the circuit technology as modulo counters because the circuit

has to count the cycles of the input wave, to switch the output signal fafter the second cycle

in this example. A frequency halving would, therefore, correspond to a modulo-2 counter, a

frequency quartering would then correspond to a modulo-4 counter, etc.

The realization of the Modulator by means of so-called markers of the form E4 is

illustrated in Figure 3 and shows the complete function of the Modulator by means of the

wave patterns shown. According to the following quotation from George Spencer-Brown,

the Modulator only appears in a different shape than the form E4 itself:

“Transﬁgured in this way, E4 appears in a form in which it is easier to follow how the wave

structure of ais taken apart and recombined to give that of f.”(Spencer-Brown, 1972,p.67).

The markers introduced in Chapter 11 (Spencer-Brown, 1972, pp. 54–68) are described in

the work as follows:

“Let a marker be represented by a vertical stroke, thus,

Let what is under the marker be seen to be so by lines of connexion, called leads, thus,

(Spencer-Brown, 1972,p.66).

Figure 2.

Form E4 from

Chapter 11

K

However, as the states of the connecting lines have been deﬁned with the wave patterns

shown, a detailed description of the markers and the method used could be presented using

the methodology put forward by Kauffman (2006).

Beyond the description of the Modulator in Spencer-Brown (1972) and Kauffman (2006),

however, another perspective can be chosen. In the following, it shall be shown how, while

keeping all connections and their states described by the wave patterns, exactly the same

function can be realized only by replacing the markers with NOR gates. This interpretation

is also indicated by Spencer-Brown’s symbol for the marker, which shows a high level of

similarity to the transistor circuit diagram of a logic gate of the then common Resistor

Transistor Logic (RTL).

The digital circuit of NOR gates shown in Figure 4 was created by replacing all eight

markers of Figure 3 with eight NOR gates while maintaining their respective topological

positions and the connections between them. Furthermore, the wave patterns of each signal

Figure 3.

Modulator from

Chapter 11

Figure 4.

Modulator as digital

circuit of NOR gates

Digital

analysis

of a form

connection were represented as time sequences of the Boolean values 0 and 1. However, it

remains to be shown whether the structurally identical circuit of NOR gates has the same

behavior as the Modulator, i.e. whether it is functionally identical.

With the help of this chronological sequence of signal values, which was speciﬁed by

Spencer-Brown (1972), the behavior of each of the eight NOR gates can now be veriﬁed. The

ﬁrst NOR gate converts the signal sequence 1010 of the input variable atogether with the

signal sequence 1100 of the output variable finto the signal sequence NOR(a,f) of 0001. This

can be veriﬁed for all eight NOR gates and all given signal sequences. The circuit of NOR

gates, thus, realizes the same behavior as the modulator and is functionally identical to it.

If according to the above citation (Spencer-Brown, 1972,p.67),theModulatoronly

represents a different shape of form E4 and the digital circuit of NOR gates obtained

from the Modulator is structurally and functionally identical to the Modulator, the

principle of the transformability of a form into a digital circuit of NOR gates can be

derived from this.

From the transformation process from form E4 to a circuit design of NOR gates shown

above, we can now understand the basic conditions for a transformation from a form to a

digital circuit. Through a purely structural comparison of form E4 and the Modulator, the

following rules of transformation result, independent of the implementation of the switching

elements as markers or NOR gates:

Each cross corresponds to exactly one switching element.

The direct nesting of two crosses into each other is represented as a connection

between the output of the switching element belonging to the inner cross and an

input of the switching element belonging to the outer cross.

Each re-entry corresponds to a connection from the output of the respective cross to

the input of the switching element of the re-entry point.

If we now apply these rules to E4 –starting with the ﬁrst, innermost cross to the eighth,

outermost cross –and consider the symmetrical grouping of the form E4 into two groups of

four, we get the digital circuit shown in Figure 5.

For better assignability, the NOR gates were numbered from g1 to g8 according to the

position of their respective crosses and the signal sequences were transmitted according to

the wave patterns. With these signal sequences, the function for each individual NOR gate

Figure 5.

Digital circuit derived

from form E4 by

transformation

K

can now be traced. The circuit shown in Figure 5 is, thus, identical to the circuit shown in

Figure 4.

With this method of transforming a form into a digital circuit, derived from Spencer-

Brown (1972), we can now transform the selected sociological description of the swarm

analysis of the Nazi dictatorship from Baecker (2013) into a digital circuit.

Swarm analysis of the Nazi dictatorship as a form

In Baecker (2013), a sociological description of the Nazi dictatorship was presented by

means of swarm analysis, in two symbolic forms: ﬁrst, through extensive verbal description

and second, as a visual diagram, using Spencer-Brown’s Calculus of Indications (Spencer-

Brown, 1972). Figure 6 shows the second symbolic interpretation of the Nazi dictatorship, in

which the conceptual words chosen for historical-sociological observation were arranged as

variables according to rules taken from the Calculus of Indications in such a way that for an

observer familiar with the Calculus of Indications, references become visible that in classical

notation could only becommunicated with great effort and in an unclear manner.

As the digital analysis of the form presented here does not depend on the extensive

explanations of the swarm analysis method and the historical observations themselves, we refer

to Baecker (2013). However, for a better understanding of the intended semantics, the conceptual

words of the form used as variables will be described with quotations from Baecker (2013).

Hitler swarm:

“A swarm is a dense and self-organizing packaging of events and values feeding on each

other. Its structure deﬁnes the differentiation by which it relates to its social and natural

environment, its culture deﬁnes how it shapes its identity in strategy and reﬂection. The

structure is symmetrical, placing equal emphasis on all sides of differentiation (Simmel,

1989;Luhmann, 1997, pp. 595–609). Culture is asymmetrical, thus biasing the interpretation

of the world (Douglas, 1982;Swidler, 1986). A swarm is considered a social system

differentiating itself from its environment and reproducing in time, using its social

dimension to produce a kind of indeterminacy, which lends it ﬂexibility in differentiation

and robustness in reproduction (Parsons, 1951;Luhmann, 1995).”(Baecker, 2013,p.70).

Terror:

“Terror here means the threat of violence combined with the execution of physical violence,

a form of political communication constantly calculating how much physical violence was

necessary to maintain the threat of increasing it. With this procedure, the totalitarian state

documented that there were domains just outside its sphere of inﬂuence that were aware of

having to pay for this remaining outside in various forms.”(Baecker, 2013,p.79).

Figure 6.

Hitler swarm as a

form

Digital

analysis

of a form

“The doubling of terror in the calculus shows that both imagining terror and experiencing it

were important to maintain the swarm. At the same time it shows that the swarm worked

under pressure which absorbed so many resources that, of its own accord, exhaustion

became a more and more attractive option when crossing the form back into its unmarked

state.”(Baecker, 2013,p.82).

n61:

“The number n61means that the swarm always counted its members by adding one

member, nþ1, namely, Hitler, who at the same time was lacking in any concrete

instantiation of the swarm, n1. He was, thus, added as lacking, n61, giving the swarm

its ﬂexible and situational coherence.”(Baecker, 2013 p. 79).

Loyalty:

“Then, loyalty means that the trust in present fellowship was more important than

programme, principle or static morality, let alone law or economic rationality (Weber, 1978,

pp. 241–246). Moreover, loyalty had the advantage that it could be tested by enforcing

immoral or illegal behavior, which was then even more binding.”(Baecker, 2013,p.79).

Mass media:

“This is also why the mass media became so important. They deal with actualities,

that is, with information distinguishing the new from the old. Here again, Gleichschaltung

was the rule, yet ironically it helped ﬁnd out that if the functional spheres were to be

aligned they had to develop their own professional understanding and practices in

dealing with tasks involved. The lasting effect of the suspicion of ‘manipulation’and

‘Kulturindustrie’only shows that, in developing their standards of information,

entertainment and advertisement, the mass media did in fact do a successful job of

differentiation (Horkheimer and Adorno, 2002;Enzensberger, 1974;Luhmann, 2000b).”

(Baecker, 2013, p. 81).

Social work:

“This is true for another public as well, that of social work (Volkswohlfahrt,

Arbeiterwohlfahrt,Winterhilfswerk, see Nolzen, forthcoming;Götz, 2001), which was so

important in guaranteeing support for the Nazi regime that the idea of calling the Nazi

regime an instantiation of an attempt by the functional system of social work to absorb the

whole of society seems not too farfetched. The functional system of social work is actually

the only system on the level of society that decides on questions of inclusion and exclusion

(Baecker, 1994), as otherwise only organizations can do by their usual practice of hiring and

ﬁring. It is, thus, the social system that ensures that the promises of the French Revolution

to include the whole reading population in society are fulﬁlled at least by proxy.”(Baecker,

2013, p. 81).

War economy:

“For industry, for men at the front and for men and women maintaining supplies, the war

economy literally meant counting the living and the dead, conveying a clear message to all

on how to secure their survival. Then, the terror everybody heard about fed back into the

terror everybody experienced so that the motives for selecting action appropriate to the

values and distinctions captured by the form of the swarm could be both evident and self-

fulﬁlling.”(Baecker, 2013,p.82).

Baecker’s symbolic form, shown in Figure 6, is analyzed below using methods of digital

mathematics: the Boolean Differential Calculus, automata theory and the software

XBOOLE.

K

The Boolean Diﬀerential Calculus

The origins of the Boolean Differential Calculus go back to applications of the work of Akers

(1959),Huffman (1958) and Reed (1954) in the synthesis and testing of integrated circuits in

the 1970s. Against the background of the constantly increasing integration density of

integrated circuits, it is easy to imagine the challenges for the generation of test vectors. For

example, if a function f(x1,...,x64) to be tested depends on 64 input variables, then all 2

64

test vectors would be needed for a complete test. To make these exponentially increasing

demands on computing time manageable, the question must be answered what the change

of a single input variable xi would mean for the change of the function result. With this kind

of dynamic consideration, the necessary amount of test vectors could be reduced

signiﬁcantly.

With Davio (1978) and Thayse (1981), comprehensive papers on the Boolean Differential

Calculus were then presented and a working group at the Chemnitz University of

Technology extended the Boolean Differential Calculus once again (Bochmann and Posthoff,

1981). At the same time, the software framework XBOOLE was developed and described in

Steinbach (1992), with which the operations of the Boolean Differential Calculus can be

calculated, even for large numbers of variables.

Analogous to the differential calculus, the Boolean Differential Calculus can be used to

analyze given functions according to their change behavior with the known operations of

Boolean algebra. The results of such derivatives of given Boolean functions are again

Boolean functions, where the results of the derivative are always simpler than the initial

functions. Boolean variables can take the values of the binary Boolean space {0, 1}.

However, these values are purely static and do not contain any information about their

change. There is no possibility of describing these values dynamically. To a certain extent,

the operations of the Boolean Differential Calculus closethis gap.

If xis a Boolean variable, then dx is the differential of this Boolean variable xwith its

values dx = 1 if the value of xchanges and dx = 0 if the value of xremains constant. The

differential dx is, thus, itself an independent variable, which –as it describes the change of

x–remains connected to the variable x. As the differential dx itself is also a Boolean

variable, all operations of Boolean algebra remain valid for dx.

The simple derivative operations are distinguished by the three basic operations

derivative, minimum and maximum. The derivative of a given function f(x,y) with respect to

xresults in the value 1 exactly if the function result changes due to the change of the value of

the input variable x–independent of the direction of change. As a Boolean expression, this

derivative corresponds to the antivalence (XOR) of the function result for x= 1 with the

function result for x=0:

df(x,y)/dx =f(1, y) XOR f(0, y)

For the minimum, the antivalence (XOR) is replaced by the conjunction (AND):

min

x

f(x,y)=f(1, y) AND f(0, y)

The minimum of f(x,y) with respect to xyields the value 1 exactly for the values of the input

variable x, where the function result f(x,y) is equal to 1 (and not equal to 0).

Finally, the maximumis formed with the disjunction (OR):

max

x

f(x,y)=f(1, y)ORf(0, y)

The maximum of f(x,y) with respect to xyields the value 0 exactly for the values of the input

variable x, where the function result f(x,y) is equal to 0. Or the other way round: the

Digital

analysis

of a form

maximum of f(x,y) with respect to xyields the value 1 if at least one value of xexists, where

the function result f(x,y) is equal to 1.

Using the NOR operation as an example, and thus to a certain extent also for the cross

operation itself, the following simple derivative outputs result for f(x,y) = NOR(x,y).

Derivative of NOR(x,y) with respect to x:

df(x,y)/dx = NOR(1, y) XOR NOR(0, y).

df(x,y)/dx = 0 XOR NOT y.

df(x,y)/dx = NOT y.

Minimum of NOR(x,y) with respect to x:

min

x

f(x,y) = NOR(1, y) AND NOR(0, y).

min

x

f(x,y) = 0 AND NOT y.

min

x

f(x,y)=0.

Maximum of NOR(x,y) with respect to x:

max

x

f(x,y) = NOR(1, y) OR NOR(0, y).

max

x

f(x,y) = 0 OR NOT y.

max

x

f(x,y) = NOT y.

These derivation results also correspond to the interpretation of the NOR operation with two

variables as an enableable negation operation, where an input with the value 0 enables the

negation of the remaining input to a certain extent; with only one value 1 at the inputs, the

function result keeps the value 0 constantly. This is described by the function result of

the derivation of NOR(x,y) with respect to xwith NOT y.

With a simple partial derivative, the effect of changing the value of exactly one input

variable xon the function result f(x,y) is examined. In a way, the question is answered, at

which initial assignment and a subsequent change of xthe function result f(x, y) changes as

well. If the result function is derived again after a further variable y, we speak of a double

partial derivative. The question is answered whether the function value f(x,y) changes if the

value of the input variable xis ﬁrst changed with the ﬁrst partial derivative and then

the derivative result is changed with another partial derivative of the value of the input

variable y. A further derivative of the second derivative result after another variable results

in a triple derivative and generally these multiple derivative operations are called k-fold

derivative, k-fold minimum or k-fold maximum.

While the simple derivative operations examine the dependence on an input variable x,

vectorial derivative operations extend this consideration in dependence on a set of input

variables. In contrast to the k-fold partial derivative operations, the term vectorial describes

the assumption that all input variables of the examined variable vector change

simultaneously. Together, the three k-fold derivative operations extended by three vectorial

derivative operations form the basic framework of the Boolean Differential Calculus, the use

of which is discussed in the following examples which draw on the software XBOOLE.

Circuit analysis with XBOOLE

Steinbach and Posthoff (2009) show how digital circuits can be analyzed and synthesized

with the methods of digital mathematics. Circuit analysis is used to calculate the behavior of

a given digital circuit, while circuit synthesis is used to calculate the best possible digital

K

circuit from a given behavior description. For tasks of circuit analysis and circuit synthesis,

the software XBOOLE-Monitor is a freely available tool, which can be downloaded here:

http://www.informatik.tu-freiberg.de/xboole/

The classical bivalent Boolean logic, as well as the digital circuit technology, are based

on the distinction of mutually exclusive and jointly exhaustive elements {0, 1}. Due to the

exponential dependence of the required memory capacity and computing time on the

number of Boolean variables used, complexity limits quickly arise and many problem

solutions belong to the NP-hard complexity class. This complexity can be reduced with the

methods of digital mathematics shown in Steinbach and Posthoff (2009), so that certain

Boolean problems can be solved at all or with signiﬁcantly reduced effort. The core of this

method is the extension of the bivalent Boolean logic by a third element “Don’t care”, which

can stand for the value 0 and for the value 1. This ternary logic {0, 1, –} results in a

considerable reduction of required storage capacity and computing time.

The behavior of Boolean functions can be represented as both a table and a list of all

those combinations of the values of the variables for which the Boolean function yields the

value 1 as result. All other combinations not contained in the list then inevitably result in

the value 0 as the result of the function. Each individual combination of the values of the

variables forms a variable vector (x1,...,xn) and these lists of any number of

interchangeable variable vectors are called vector lists. While the classical binary logic of

the elements {0, 1} works with binary vector lists, the ternary vector list of the elements

{0, 1, –} forms the essential data type of XBOOLE. Figure 7 shows the function of the well-

known NOR gate y= NOR(x1,x2) as a ternary vector list, which –unlike the classical four-

line binary vector list –only consists of three lines.

Alternately, any ternary vector list in XBOOLE can be visualized as a Karnaugh plan

using the representation commonly used in digital circuit technology, as shown in Figure 8

for the function y= NOR(x1,x2).

The Hitler swarm as a digital circuit

The swarm analysis of the Nazi dictatorship from Baecker (2013), there visualized as form,

can now be transformed into a digital circuit of NOR gates according to the rules found in

the section “The form as a digital circuit”. The conceptual words of the form terror,n61,

loyalty,mass media,social work and war economy used as variables are taken over as

Figure 8.

Karnaugh plan

y= NOR(x1, x2)

Figure 7.

Ternary vector list

y= NOR(x1, x2)

Digital

analysis

of a form

Boolean input variables of the digital circuit terror,n1,loyalty,massmedia,socialwork and

wareconomy in computer-readable notation and each NOR gate is assigned a Boolean output

variable from y1 to y7, where y7 corresponds to the variable Hitler swarm from the form

while maintaining the order of the cross from inside to outside. Figure 9 shows the digital

circuit of the Hitler swarm as the result of this transformation.

First statements about the behavior of the circuit can already be made with heuristic

means of traditional circuit analysis and without using digital mathematics or software. A

NOR gate with two inputs describes –as could be shown in the section “The Boolean

Differential Calculus”–an enableable negation operation. One input of the NOR gate

enables the negation operation of the other input with the value 0. Thus, for NOR gates with

ninputs, all n1 inputs must have the value 0 to negate the remaining input signal at the

output. Direct feedback of the output signal of a negation operator to its own input generates

a signal sequence 01010... at the output as an astable circuit. From this it follows that

astable oscillation behavior can be expected under the following conditions:

The output signal of a NOR gate (at least) is fed back to the input via an odd number

of preceding NOR gates, thus generating negative feedback and

All other inputs of these NOR gates not connected in cascade have the value 0.

We will describe this astable oscillation behavior in more detail in the section “Astable

oscillation behavior or imaginary state?”. Further statements about the behavior of the

digital circuit cannot be made by visual inspection alone, but require analysis with the

methods of digital mathematics.

Determination of the circuit as switching circuit

In principle, digital circuits can be divided into combinatorial circuits and switching circuits,

whereby switching circuits are created by feeding back output signals to preceding inputs.

While the output function of combinatorial circuits depends only on their input variables x

according to the formula y=f(x1,...,xn), the output function of switching circuits depends

not only on the input variables xbut also on the internal state variables zresulting from the

feedbacks: y=f(x1,...,xn,z1,...,zm). Switching circuits of this kind –i.e. without clock

signals, which control the synchronous switching –are called asynchronous automata and

can be analyzed in their behavior using the methods of automata theory. The essential step

is the correct deﬁnition of the internal state variables z1,...,zm because the number of

Figure 9.

Digital circuit

according to the form

Hitler swarm

K

states of the automaton depends exponentially on the number of internal state variables

according to 2

m

.

The digital circuit of the Hitler swarm can now be determined as a switching circuit

according to the above classiﬁcation because the output signals of the NOR gates y3,y4,y5,

y6 and y7 –due to the corresponding re-entries –are fed back to inputs of NOR gates which

precede these. In the following, these output functions are listed depending on their input

and output variables, with the relevant signal feedback marked in bold:

y1 =f(terror,y3,y7);

y2 =f(y1,n1,y4,y6);

y3 =f(y2,loyalty,y5);

y4 =f(y3,massmedia);

y5 =f(y4,socialwork);

y6 =f(y5,wareconomy); and

y7 =f(y6,terror).

The ﬁrst step in analyzing a switching circuit with the methods of automata theory is

the deﬁnition of the smallest set of state variables that completely describes the

behavior of the switching circuit. These state variables can be obtained by heuristic

analysis of the signal feedback that makes a purely combinatorial circuit a switching

circuit. In the present switching circuit, it can be seen that the output of the third NOR

gate y3 contains all ﬁve signal feedbacks, and thus can be deﬁned as state variable z.

All further output variables y4,y5,y6 and y7 are beyond y3, and thus are no longer

dependent on further state variables, but only on purely combinatorial connections

with the input variables.

With this one internal state variable z, the Hitler swarm switching circuit has exactly 2

1

or two internal states y3 = 0 and y3 = 1 and it can be concluded, based on the described

astable oscillation behavior of this kind of NOR gate cascades, that this switching circuit

will oscillate for certain input vectors (terror,n1,loyalty,massmedia,socialwork,

wareconomy) as long as these input vectors are present. Which input vectors excite this

astable oscillation behavior cannot be determined sufﬁciently by heuristic analysis.

Answering these and other questions about the behavior of the switching circuit requires

analysis with the methods of digital mathematics and can be done using speciﬁc software

such as the XBOOLE software.

Modeling of the switching circuit with XBOOLE

In the following, the main steps of analyzing the switching circuit with the XBOOLE

software are described. With the NEW command, XBOOLE is reset and all internal

rooms and objects such as ternary vector lists are deleted; with SPACE, a new room

number 1 with a maximum of 32 Boolean variables is created. In this room, the entire

switching circuit will be calculated.

new

space 32 1

Now the AVAR command is used to deﬁne and sort all input, output and state variables.

The question of why the variable pair zand zf are deﬁned, instead of an internal state

variable, shall be put aside for the time being. It will be dealt with in the section entitled

“Observability of re-entry signal feedbacks”, below.

Digital

analysis

of a form

avar 1

zzf

terror n1 loyalty massmedia socialwork wareconomy

y1 y2 y3 y4 y5 y6 y7.

The essential step for representing the switching circuit is now done by entering the Boolean

functions of the connected NOR gates. The SBE command stands for Solve Boolean

Equation and uses “/”as negation, “þ”as disjunction and “=”as equivalence operator. A

separate ternary vector list (TVL) TVL101 to TVL107 is calculated for each output variable.

sbe 1 101

y1=/(terrorþy7þz).

sbe 1 102

y2=/(y1þn1þy6þy4).

sbe 1 103

y3=/(y2þloyaltyþy5),

zf = y3.

sbe 1 104

y4=/(zþmassmedia).

sbe 1 105

y5=/(y4þsocialwork).

sbe 1 106

y6=/(y5þwareconomy).

sbe 1 107

y7=/(y6þterror).

With this step, all seven NOR gates are now completely described in terms of their ternary

vector lists and these can be merged into the global phase list TVL110 using the set operation

ISC. The command ISC stands for intersection and is a set operation that calculates the

average of two given ternary vector lists. This is essential for understanding the way in

which XBOOLE works because the identity of the variable names in space number 1 is used

to realize the connections between the individual NOR gates. The command OBBC reduces

the number of ternary vectors by means of the operations block formation and block

exchange while maintaining the orthogonality, which is only optimization ofcomputing time.

isc 101 102 110

isc 103 110 110

isc 104 110 110

isc 105 110 110

isc 106 110 110

isc 107 110 110

obbc 110

The result of the TVL110 shown in Figure 10 forms the global phase list of the switching circuit,

the 34 ternary vectors of which completely describe the behavior of the switching circuit.

Although this global phase list completely represents the behavior of the switching

circuit with all its output and transfer functions, this compact form of representation is only

conditionally suitable for interpretation. For example, the question of whether –and under

which assignment of the input variables –the switching circuit oscillates unstably can only

be answered by interpreting this global phase list, which requires considerable effort. To

answer these and other questions about the behavior of the switching circuit, suitable set

K

operations can be executed on this global phase list, the results of which allow the oscillation

behavior to be visualized more easily.

Observability of the re-entry signal feedbacks

At this point, before the oscillation behavior of the switching circuit can be analyzed,

however, it is useful to make a few comments on the observability of the signal feedback

resulting from the re-entries of the form. To observe the state transitions of a switching

circuit, the method of splitting the state variables zi into two-state variables zﬁand zi is

applied. As only one state variable zhas already been analyzed for the present switching

circuit, only zand zf will be used in the following instead of zi and zﬁ.

Figure 10.

Global phase list of

the switching circuit

Digital

analysis

of a form

The following rule applies to the method of splitting: the state variable zis read-only,

while the state variable zf is write-accessible only. The variable z, thus, represents the

current state of the switching circuit, while zf represents the subsequent state of the

switching circuit. With this method, the transition of the switching circuit from state zto

the subsequent state zf becomes observable because the global phase list contains all four

possible combinations between zand zf:

z=0,zf = 0: switching circuit remains in state 0.

z=0,zf = 1: transition of the switching circuit from state 0 and state 1.

z=1,zf = 1: switching circuit remains in state 1.

z=1,zf = 0: transition of the switching circuit from state 1 to state 0.

In general, asynchronous switching circuits are created by signal feedbacks resulting from

re-entries. In contrast to the synchronous switching circuits, the timing of the state

transitions is not determined by a uniform clock signal but directly by the timing of the

change of the input variables. These asynchronous switching circuits are preferred to slower

synchronous switching circuits in some applications, due to their higher switching speed.

As the actual runtime behavior is determined by the physical parameters of the

semiconductor material, interconnects, etc., of the microchip, such asynchronous switching

circuits can be fully simulated in the design process as stable, deterministic automatons and

can later be realized in a microchip.

Determination of the automaton graph of the switching circuit

The behavior of a switching circuit can be visualized as a ﬁnite, deterministic automaton by

means of its automaton graph.In a state transition diagram, the states of the automaton, the

possible state transitions and their conditions are visualized. The state transitions are also

called transitions or edges of the graph. The automaton graph of our switching circuit

shown in Figure 11 with its two states, z= 0 and z= 1, has exactly four edges, which

describe all possible state transitions visualized by arrows: (z,zf) = (0, 0), (z,zf) = (0, 1),

(z,zf) = (1, 0) and (z,zf)=(1,1).

The edges of the automaton graph can now be calculated from the global phase list.

Therefore, the respective intersection of the global phase list, with all four possible state

transitions (z,zf) is calculated. The four ternary vector lists, TVL121, TVL123, TVL125 and

TVL127, contain all transition conditions of the respective state transition:

Figure 11.

Automaton graph of

the switching circuit

K

state transition (z,zf) = (0, 0) from z=0tozf =0.

tin 1 120

z zf.

00.

isc 110 120 121

state transition (z,zf) = (1, 1) from z=1tozf =1.

tin 1 122

z zf.

11.

isc 110 122 123

state transition (z,zf) = (0, 1) from z=0tozf =1.

tin 1 124

z zf.

01.

isc 110 124 125

state transition (z,zf) = (1, 0) from z=1tozf =0.

tin 1 126

z zf.

10.

isc 110 126 127

These automaton edges are only dependent on the input variables and, therefore, all state

and output variables can be hidden in the ternary vector lists by using the variable tuple (z,

zf,y1,y2,y3,y4,y5,y6,y7). This hiding of variables is realized by a set operation from the

Boolean Differential Calculus, the so-called k-fold maximum.

vtin 1 130

zzfy1y2y3y4y5y6y7.

maxk 121 130 131

maxk 123 130 133

maxk 125 130 135

maxk 127 130 137

The following four possible subgraphs can now be calculated by intersecting the respective

edges already determined. With regard to the consideration of the astable oscillation

behavior, the subgraph which describes the state transitions (z,zf) = (0, 1) from z=0tozf =1

and (z,zf) = (1, 0) from z=1 to zf = 0 is of special interest. This subgraph, along with

TVL142, thus contains all input vectors, where the automat changes from z=0tozf = 1 and

from z=1tozf = 0, thus oscillating astably between the two states:

all input vectors where the stable state 0 is reached.

isc 137 131 140

Digital

analysis

of a form

all input vectors where the stable state 1 is reached.

isc 135 133 141

all input vectors where the state astable oscillates between 01010...

isc 137 135 142

all input vectors where the stable state does not change, i.e. 0 or 1 remain.

isc 131 133 143

At this point in the complete problem program, further set operations are used to check two

things: ﬁrst, whether the four automaton edges also cover all possible combinations of input

signals and second, whether none of the combinations belongs tomore than one of these four

subgraphs. As these checks are not essential for further analysis of the forms presented

here, we refer here to the XBOOLE problem program “Digital Analysis.PRP”in the

Appendix.

Calculation of astable oscillation behavior

With the calculation of all edges and subgraphs of the automaton, its behavior is completely

described. Thus, for example, the astable oscillation behavior of the switching circuit can be

analyzed and visualized. The subgraph shown in Figure 12 and calculated with TVL142

already contains all input vectors for which the switching circuit or its state variable z,

shows astable oscillation.

For each combination of the input variables terror,n1,loyalty,massmedia,socialwork and

wareconomy, where thevalue 1 can be seen in the Karnaugh plan of Figure 12, the switching

circuit shows astable oscillation between z= 0 and z= 1. As we have determined the output

of the third NOR gate y3 as state variable z, we can use the TVL142 to directly infer output

variable y3.

Figure 12.

Input vectors for

which the switching

circuit oscillates

K

To determine how the oscillation behavior of the state variable zaffects output variable

y7, for example, we have to consider the purely combinatorial dependence of this output

variable. From the functions discussed in the section “Determination of the circuit as

switching circuit”, we can derive the dependence of input variables and state variable

for y7:

y4 =f(y3,massmedia)

y5 =f(y4,socialwork)

y6 =f(y5,wareconomy)

y7 =f(y6,terror)

!

y7 =f(massmedia,socialwork,wareconomy,terror,z)

As these combinatorial dependencies are also already included in the global phase list of the

switching circuit, all that remains is for us to calculate the intersection of TVL142 with the

global phase list.

isc 142 110 162

To visualize the oscillation behavior of output variable y7, the intersection already

calculated with TVL162 is cut with the variable tuple (y7) = (1) and displayed as TVL17 per

k-fold maximum only depending on the input variables and theinternal state variable z.

tin 1 177

y7.

1.

isc 177 162 17

maxk 17 170 17

obbc 17

As shown with the Karnaugh plan in Figure 13 for the output variable y7, the astable

oscillation behavior of the internal state variable zonly results in an astable oscillation

between value 0 and 1: 01010...for the two orange marked input vectors (terror,n1,loyalty,

massmedia,socialwork,wareconomy) = (0, 0, 0, 0, 0, 0), as well as (0, 1, 0, 0, 0, 0) due to the

Figure 13.

Oscillation behavior

of output variable y7

Digital

analysis

of a form

further combinatorial dependencies of the output variable y7 described above. For the three

other ﬁve input vectors already visualized in Figure 12, the internal state variable zof the

switching circuit oscillates in the same way, but only between values 1 and 1: 11111....

In the complete problem program “Digital Analysis.PRP”outlined in the Appendix, the

astable oscillation behavior for all output variables y1 to y7 is calculated as TVL1 to TVL7

in the way shown here as an example for the output variables y7 and displayed with TVL11

to TVL17 depending on the input variables and the state variables.

Indeterminacy and degree of re-entrant forms

With the analytical results obtained through the application of automata theory, it is worth

taking a look at the deﬁnitions of the terms indeterminacy and degree in Chapter 11

(Spencer-Brown, 1972, pp. 54–68). Both the indeterminacy and the degree of an expression

are derived from a form called Echelon. First, in the ninth canon which outlines the rule that

“a demonstration rests in a ﬁnite number of steps”(rule of demonstration), it is shown how

the simple Echelon with a ﬁnite number of algebraic steps can be transformed into every n-

fold Echelon. Figure 14 shows the simple and the n-fold Echelon as form and the simple

Echelon as the transformed circuit of NOR gates.

According to Spencer-Brown’sdeﬁnition (Spencer-Brown, 1972 p. 14), the variables m

and nare deﬁned as expressions, which themselves indicate marked and unmarked states,

respectively:

“Let mstand for any number, greater than zero, of such expressions indicating the

marked state.

Let nstand for any number of such expressions indicating the unmarked state.”

(Spencer-Brown, 1972, p. 14).

By using the 2

2

or four possible variable combinations for mand n, the solutions of the

simple Echelon, as well as of each ﬁnite n-fold Echelon can be determined with a ﬁnite

number of steps. The result of the simple and the n-fold Echelon is, thus:

f(n,n)=n;

f(n,m)=n;

f(m,n)=m; and

f(m,m)=n.

The calculation of the corresponding circuits from NOR gates gives the identical result for

both the simple and the n-fold Echelon, which is shownas Karnough plan in Figure 14.

This calculability of an expression in a ﬁnite number of steps, dependent only on the

(marked or unmarked) values of the variables, is classiﬁed according to George Spencer-

Figure 14.

Transformation from

simple to n-fold

Echelon

K

Brown as an expression of the ﬁrst degree. The indeterminacy and degree of an expression

are dealt with in Spencer-Brown (1972, p. 57) as follows:

“Equations of expressions with no re-entry, and thus with no unresolvable

indeterminacy, will be called equations of the ﬁrst degree, those of expressions with one re-

entry will be called of the second degree, and so on.”(Spencer-Brown, 1972,p.57).

Exactly what the “and so on”means for the relation of a count of re-entries and degree,

remains unclear in this deﬁnition. Probably a linear relation in the sense of degree =

count of re-entries þ1 is intended. In any case, both, the simple and every ﬁnite n-fold

Echelon are expressions of the ﬁrst degree and correspond to purely combinatorial digital

circuits.

The introduction of re-entry in the case of the re-entrant Echelon shown in Figure 15

results in an indeﬁnite regression and the result of the expression cannot be determined with

aﬁnite number of steps. This corresponds exactly to the indeterminacy of an expression

according to Spencer-Brown (1972, p. 57). The re-entrant Echelon, having exactly one re-

entry, is, thus, of the second degree and its transformation into a digital circuit results in a

switching circuit which can be analyzed as follows.

With the rules from the section “The form as a digital circuit”, the re-entrant Echelon of

the second degree can be transformed into a digital circuit of NOR gates, also shown in

Figure 15. Based on this, theautomaton graph of this re-entrant Echelon can be calculated as

shown in Figure 15, using the methodology described in the section “Determination of the

automaton graph of the switching circuit”. Thus, the re-entrant Echelon has exactly one

state variable z. From this, the 2

1

or two states of the automaton result. At this point, it is

important to note that this automaton graph not only describes the switching circuit of NOR

gates but also the form itself. The output variable yis not only dependent on the input

variables aand bbut also on the state variable z; the Karnaugh plan of y=f(a,b,z) is also

shown in Figure 15.

Figure 15.

Re-entrant Echelon,

Karnaugh plan of y

and automaton graph

Digital

analysis

of a form

The functionality of the re-entrant Echelon can now be described completely. The output

signals of the two NOR gates result with the input vector (a,b) = (0, 0) from the negation of

the respective other input. However, as these respective other inputs depend on the value of

the internal state variable z, we have to consider both possibilities z= 0 and z= 1 for the

input vector (a,b) = (0, 0).

In the case of the internal state variable z=0, the second NOR gate would set the output

variable yto the value 1 and return this value 1 to the other input of the ﬁrst NOR gate. By

negating this 1 again by the ﬁrst NOR gate, the output state z= 0 is stabilized.

In the case of the internal state variable z= 1, the input vector (a,b) = (0, 0) results in the

output signal y= 0 and this returned value 0 stabilizes the output state z= 1 via the ﬁrst

NOR gate. Thus, if “no”signal (a,b) = (0, 0) is present at the two inputs of the re-entrant

Echelon, it stores its present state with the value of its internal state variable z.

In addition to this memory function, the re-entrant Echelon can be selectively set to its

two states with the corresponding input vectors. The resetting to the state z= 0 is done with

a= 1 and this independent of the current state. With a= 1 the value 0 results for the ﬁrst

NOR gate, and thus for zand this also independently from the input variable band in

ternary logic the corresponding input vector (a,b) = (1, ) can be written. The output

variable ycorresponds in the state z= 0 to the negation of the input variable b, which is not

important for the memory function or resetting.

With the input vector (a,b) = (0, 1), the setting to the state z= 1 is carried out, likewise

independently of the current state. The value of the input variable bof 1 results in the value

0 for the output of the second NOR gate, and thus for the output variable yand this returned

value results in the value 1 for the output of the ﬁrst NOR gate, and thus for the internal

state variable z.

These observations show that George Spencer-Brown’s indeterminacy can be described

more precisely by means of automata theory and with the help of the medium of time. The

above-quoted analysis from Spencer-Brown (1972, p. 57) is correct in that the re-entrant

Echelon as a form of higher degree cannot be determined by a ﬁnite number of algebraic

steps from the input variables aand b. However, the re-entrant Echelon can be brought to a

deﬁned state with a calculable and ﬁnite sequence of signals at the inputs aand band then

be completely determined in this state. Thus, the determination of a form of higher degree is

no longer done purely algebraically in the Calculus of Indications, but by a ﬁnite number of

arithmetic steps in themedium of time, which will be showna bit later in this section also for

the negative self-reference. The indeterminacy of a form of higher degree is not a total

indeterminacy if the automaton can be set to a deﬁned state by a ﬁnite number of steps,

caused by calculable input vectors and then is completely calculable using the output

function and the transition function.

George Spencer-Brown deﬁnes the degree of an expression in direct dependence of the

number of re-entries according to the quotation above (Spencer-Brown, 1972, p. 57): degree =

number of re-entries þ1. However, although the Hitler swarm has altogether ﬁve re-entries,

the automaton can be completely described with only one internal state variable zand 2

1

or

two states. According to the deﬁnition above, the Hitler swarm would have to be an equation

of the sixth degree, which the analysis results do not conﬁrm. In the author’s opinion, the

deﬁnition of the degree of an expression should be coupled to the number of state variables,

which are created by re-entries instead of the number of re-entries: degree = number of state

variables þ1. With this improved deﬁnition, the Hitler swarm is a form of the second degree.

Although according to this proposed deﬁnition, both the Hitler swarm and the re-entrant

Echelon are expressions of the second degree, the ﬁrst shows an astable oscillation behavior;

the other, a self-stabilizing memory behavior. This difference corresponds to the difference

K

between negative and positive self-referentiality, i.e. whether the corresponding re-entry has

been returned by a sequence of odd or even negations. Whether a form in the medium of

time now realizes memory or oscillation behavior would be a further, determinable

parameter and could play a respective role for the comparability of forms. In any case, the

algebraic indeterminacy of all forms of higher than ﬁrst degree deﬁned in Spencer-Brown

(1972) is extended by their determinability in the medium of time.

At this point, a remark on the importance of the re-entrant Echelon for digital and

computer technology is in order. As shown in Figure 16, the re-entrant Echelon represents

the asynchronous, non-clocked RS ﬂip-ﬂop or RS latch, where the internal state variable zis

also available as output variable Qof the ﬁrst NOR gate. The input Rlike “Reset”resets the

ﬂip-ﬂop with the value 1 to the state zor Qequals 0, while the input Slike “Set”sets the ﬂip-

ﬂop with the value 1 to the state zor Qequals 1.

Together with their synchronous, clocked variants, the ﬂip-ﬂops with their one-bit memory

function form the basic elements of digital and computer technology and also constitute the

heart of every modulo counter. The typical ﬂip-ﬂop pattern of a crosswise interconnection of

input and output signals of a pair of NOR gates is, of course (color-coded) found in the

already discussed form E4 shown in Figure 17, as well as in the switching circuit of the

Modulator shown in Figure 18.

Although ﬂip-ﬂops play such a fundamental role for the digital and computer industry as

basal switching and memory elements, George Spencer-Brown did not mention these ﬂip-

ﬂop patterns either in the E4 form or in the Modulator. However, as the ﬂip-ﬂop was

invented as early as 1918 by two British physicists and also patented with Eccles and Jordan

(1920), he should have been familiar with the concept of ﬂip-ﬂops. In addition, as Chief Logic

Designer for Mullard Equipment Limited, he would also probably have been responsible for

ﬂip-ﬂop products, as advertised in Norbit (1960) as “Prefabricated Electronic Bricks”(i.e.

early integrated circuits).

Figure 17.

Form E4 with color-

coded ﬂip-ﬂops

Figure 16.

Re-entrant Echelon as

RS ﬂip-ﬂop

Digital

analysis

of a form

The determination of the degree of the form E4 is now almost impossible with the above-

cited deﬁnition from Spencer-Brown (1972, p. 57). E4 has seven re-entries in total and would,

thus, be of eighth degree, but of these seven re-entries only the three backward-facing ones

contribute to the emergence of internal state variables. The four forward-facing re-entries

have an only a combinatorial inﬂuence on the values calculated with them. Based on the two

internal state variables g2z and g4z from the two RS ﬂip-ﬂops, automata theory provides the

automaton graph shown in Figure 19, which results in a total of 2

2

or four internal states.

Thus, the form E4 is according to our improved deﬁnition degree = number of internal state

variables þ1 of the third degree.

Astable oscillation behavior or imaginary state?

A consideration of Chapter 11 (Spencer-Brown, 1972, pp. 54–68) would probably be

incomplete without a discussion of the imaginary state. In Spencer-Brown (1972, p. 58), a

function of negative self-reference is examined with form E3 shown in Figure 20, in which

the negation operation is reintroduced into itself.

This equation, leading into an inﬁnite regress or into a paradox, results neither in the

marked state nor the unmarked state as a solution and leads to the introduction of the

imaginary state. As in his consideration of indeﬁnitely extended negative self-reference

(Spencer-Brown, 1972, p. 58), the Calculus of Indications is not abandoned, this paradox is

solved with the means of time:

Figure 18.

Switching circuit of

the Modulator with

color-coded ﬂip-ﬂops

Figure 19.

Automaton graph of

E4 and Modulator

K

“Time

As we do not wish, if we can avoid it, to leave the form, the stage we envisage is not in

space but in time.”(Spencer-Brown, 1972,p.58).

With the assumption that each state transition between the two states of the Calculus of

Indications {marked, unmarked} takes a certain amount of ﬁnite time, the solution of E3 is a

resulting oscillation in the form of a rectangular wave marked-unmarked-marked-

unmarked-marked-.... But at this point, George Spencer-Brown has no further reason for

the statement of an additional imaginary state and the extension of space to {marked,

unmarked, imaginary}. Neither the effects of this new state on the operations introduced so

far are shown, nor does he describe what exactly can be done with this new state within the

Calculus of Indications. By merely naming the imaginary state, the text, at most, opens a

door for all conceivable metaphors of the imaginary, to which the explicit, graphically

produced reference to an imagined tunnel in a higher dimension also contributes. It is simply

assumed, that only in this way –and without crossing the border –state transitions between

the two states {marked, unmarked} are conceivable. As summarized in Fuchs and Hoegl

(2011, p. 194), however, a graphical representation of a tunnel remains only a representation

and is not a tunnel itself.

The introduction of the new imaginary state is compared in Spencer-Brown (1972, p. ix)

with the introduction of complex numbers in mathematics. However, while there are

practical applications for complex numbers, e.g. with the conversion between Cartesian

coordinates and polar coordinates –which, in our estimation, have contributed signiﬁcantly

to the inclusion of complex numbers in the mathematical canon –, no application of the

imaginary state is known in circuit technology and digital circuits continue to operate in the

binary space of Boolean logic {0, 1}. The following quote by George Spencer-Brown from

EE Times (1994, pp. 31–35) must rather be attributed to a euphemistic self-description:

“Armed with these imaginary values, digital engineers can now analyze circuits with

equations, rather than antiquated state diagrams.”(EE Times, 1994, pp. 31–35).

In digital circuit technology, such an astable oscillation behavior resulting in a

rectangular wave 01010... is handled according to the exception/rule scheme. With the

methods of automata theory shown, all possible states and their corresponding transition

conditions can be determined for any switching circuit. If an automaton with the same

transition conditions can change from one state to another, as well as from that state back to

the ﬁrst one, then the automaton oscillates back and forth between these states as long as

these transition conditions are present at the inputs. As it is known from the analysis, with

which transition conditions the switching mechanism can be brought into one of these two

oscillation states at all, exactly these assignments of the input variables, and thus the

oscillation behavior, can be speciﬁcallyprevented with circuit-technical measures.

Beyond the avoidance of oscillation, however, there are applications in which oscillation

behavior is speciﬁcally used. In Sutherland (1999 p. 7), for example, a ring oscillator shown

in Figure 21 is described as a chain of an odd number of NOT gates, which can be used to

generate 01010 ... rectangular signals for test purposes. As already shown in the section

“From universal NOR gate to the cross”using the Logical Effort approach outlined by

Sutherland (1999), the delay times of the logic gates, and thus the frequency of the

Figure 20.

Form E3 of negative

self-reference

Digital

analysis

of a form

rectangular signals, together with the odd number of NOT gates connected in the ring

oscillator, can be calculated.

But independent of the treatment as an exception or as a rule, neither an imaginary state

nor access to higher dimensions is needed to describe astable oscillation behavior in Boolean

space {0, 1}.

Observation of the results with regard to the form

The above demonstration showed how methods of digital mathematics can be used to

analyze the digital circuit of NOR gates derived from the form of the Hitler swarm with

software support. Thereby, the so-called indeterminacy of this second-order form could be

determined more precisely using state variables and the automaton graph in the medium of

time. These structural properties of the form are exclusively determined by the topological

nesting of the seven crosses with their ﬁve re-entries and the positioning of the six input

variables within the form.

In no case are the structural properties of forms determined in a speciﬁc way by the

semantics of formalized conceptual relations. The following thought experiment will clarify

this. In the form shown in Figure 6, conceptual words chosen for the historical-sociological

swarm analysis of the Nazi dictatorship have been arranged in such a way that different

observation perspectives of relevant distinctions can be communicated. If, while retaining

the structure of the form, the conceptual words terror,n61,loyalty,mass media,social work

and war economy were replaced by infection risk,incidence,trust,mass media,intensive care

and vaccine development, the form could be interpreted as a description of an observation of

the COVID-19 pandemic. The analysis results that could be obtained with the methods

shown would be identical for (terror,n1,loyalty,massmedia,socialwork,wareconomy) and

(infectionrisk,incidence,trust,massmedia,intensivecare,vaccinedevelopment), but no

conclusions can be drawn from this about similarities or comparability on the sociological-

historical level.

To answer the question of how the results of the analysis relate to the original

sociological-historical observation of the Nazi dictatorship, we must take a closer look at the

two transformations that took place:

(1) transformation of swarm analysis from the medium of natural language into the

form.

(2) transformation of the form into the digital circuit of NOR gates.

As we have seen with the example of the Modulator from Chapter 11 (Spencer-Brown, 1972,

pp. 54–68), the second transformation from the form to the digital circuit is already

established in the Calculus of Indications itself. This is further conﬁrmed by the described

genesis of the cross from the circuit technology issues of the early industrial NOR gates.

However, one has to ask oneself what historically relevant meaning the statement can

have that the Hitler swarm gets into an astable oscillation state when terror and n61and

loyalty and mass media and social work and war economy equal 0. The same meaning would

remain for all other conceptual words in the same formal structural context of the form.

Thus, following the above thought experiment, with infection risk and incidence and trust

and mass media and intensive care and vaccine development equal to the value 0, the

pandemic would oscillate in an astable condition. These and the other statements made

about the functional result of the Hitler swarm depending on the internal state and the input

variables are mathematically correct but do not make sense in relation to the sociological

problem. As the second transformation from the form into the digital circuit is established in

K

the Calculus of Indications itself, there remains the view of the ﬁrst transformation from the

medium of natural language into the form at this point.

While the Calculus of Indications in Spencer-Brown (1972) allows the use of variables only

as expressions that themselves result in the marked or unmarked state, Baecker (2013) overcomes

this limitation and uses conceptual words designating abstract sociological categories as

variables that cannot be reduced to the marked or unmarked state. Already with this use as

variables in the form and the implied assignment of sociological categories such as terror,loyalty,

mass media,social work,war economy and abstract quantity concepts such as the number of

members n61to the marked and unmarked state or the Boolean values 0 and 1, the historical-

sociological meaning is lost. Conversely, this means that the results of the analysis can no longer

be projected back onto the initial problem formulated in the medium of natural language.

The comparison of the system deﬁnition according to Luhmann (1995) from Figure 1

with the structurally identical re-entrant Echelon gives a similar result. The memory

behavior of the re-entrant Echelon could be related to the fact that the form system/

environment quasi stores the last marked state system or environment, but this does not

seem to be very useful to us. Also worth mentioning for this interpretation is the fact, that

system and environment are disjunctive to each other, mutually exclusive and jointly

exhaustive. However, as Niklas Luhmann with his difference-theoretical and self-referential

theoretical approach only uses the unity of the operation of distinction and indication and

the re-entry and, as he does not refer to the mathematics of the Calculus of Indications, norto

indeterminacy, degree or imaginary state, questions of the traceability of the results of the

analysis to systems theory are not important.

But due to the principal algebraic indeterminacy of higher degree forms, the use of the

equals sign in the system deﬁnition from Figure 1 does not satisfy George Spencer-Brown’s

deﬁnition of value and equality (Spencer-Brown, 1972,p.5):

Value

Call a state indicated by an expression the value of the expression.

Equivalence

Call expressions of the same value equivalent.

Let a sign

=

of equivalence be written between equivalent expressions. (Spencer-Brown, 1972,p.5).

Figure 21.

Ring oscillator of

2nþ1 NOT gates

Figure 22.

System deﬁnition as

form according to

Luhmann (1995) with

assignment

Digital

analysis

of a form

With the use of the assignment sign “:=”shown in Figure 22, this algebraic contradiction to

the Calculus of Indications can be avoided. Furthermore, the assignment sign corresponds

more to the character of the deﬁnition than the equals sign. As the Calculus of Indications

itself gets along completely without deductive deﬁnitions and with “Draw a distinction.”

(Spencer-Brown, 1972, p. 3), there was no need in the Calculus of Indications itself to deﬁne

the assignment operation.

Perhaps, the assumption from Roth (2019) that “true distinctions”are of special importance

is a good starting point for assessing the general applicability of the presented digital methods

of analysis to sociological observations noted by means of Calculus of Indications.

Furthermore, the determinability of higher-degree forms in the medium of time, as shown

above, points to possible applications in which given sequences of input signals reproduce the

desired behavior. However, it should be noted that higher-degree forms represent ﬁnite and

deterministic automata, and thus describe non-contingent behavior, which seems rather

unsuitable for sociological applications. A further aspect of this consideration is due to the fact

that higher-degree forms describe the structure of the automaton and not its behavior, which

has to be derived from the structure by the methods shown.

Forms used in sociology such as the Hitler swarm from Figure 6 or the system deﬁnition

in Figure 1 or Figure 22, describe andcommunicate the different observation perspectives of

relevant distinctions of the respective sociological question. In doing so, they use rules

derived from the Calculus of Indications to make references visible that could be

communicated in the classical notation in the medium of natural language albeit more

obscurely and with greater effort. However, these applications of the Calculus of Indications

seem to elude purely mathematical processing by means of algebraic and arithmetic

operations. In this respect, the question of which sociological questions are at all suitable for

the mathematical processing shown by means of digital analysis opens up a wide ﬁeld for

further research.

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Digital

analysis

of a form

Further reading

Spencer Brown, G. (1961), Design with the NOR, Mullard Equipment Limited, Crawley, in Roth, S.,

Heidingsfelder, M., Clausen, L. and Laursen, K. (2021), George Spencer Brown’s“Design with the NOR:

With Related Essays”, Bingley: Emerald Publishing Limited (8 Mar. 2021), ISBN: 9781839826115.

Appendix. XBOOLE problem program “Digital Analysis.PRP”

; reset XBOOLE system

new

;deﬁne space with maximal 32 boolean variables

space 32 1

;deﬁne all input variables

;deﬁne all internal state variables zi as the ﬁrst yi that

; includes all feedback loops

; typically, GSB-forms should have only one internal state

; variable, but larger chains could have more than one

; zi should be used as read-onlyof the internal state variable

;zﬁshould be used as write-only of the internal state

; variable just at the exit of thegate

;deﬁne all output variables

avar 1

zzf

terror n1 loyalty massmedia socialwork wareconomy

y1 y2 y3 y4 y5 y6 y7.

;deﬁne all boolean functions

sbe 1 101

y1=/(terrorþy7þz).

sbe 1 102

y2=/(y1þn1þy6þy4).

sbe 1 103

y3=/(y2þloyaltyþy5),

zf=y3.

sbe 1 104

y4=/(zþmassmedia).

sbe 1 105

y5=/(y4þsocialwork).

sbe 1 106

y6=/(y5þwareconomy).

sbe 1 107

y7=/(y6þterror).

; calculate global phase list tvl110 as the intersection of all

; boolean functions

isc 101 102 110

isc 103 110 110

isc 104 110 110

isc 105 110 110

isc 106 110 110

isc 107 110 110

obbc 110

K

; analyze the automaton graph

;deﬁne all transitions between internal state variables

; zi/zﬁin tvl12i

tin 1 120

z zf.

00.

tin 1 122

z zf.

11.

tin 1 124

z zf.

01.

tin 1 126

z zf.

10.

; calculate all edges of the automata as transitions in

; tvl13i based on global phase list tvl110

; show all edges of the automat as dependent on input variables in

; tvl13i, only

vtin 1 130

zzfy1y2y3y4y5y6y7.

; z/zf 0–>0

isc 110 120 121

maxk 121 130 131

; z/zf 1–>1

isc 110 122 123

maxk 123 130 133

; z/zf 0–>1

isc 110 124 125

maxk 125 130 135

; z/zf 1–>0

isc 110 126 127

maxk 127 130 137

; calculate all input vector leading to stable internal state 0

; z/zf –>0

isc 137 131 140

obbc 140

; calculate all input vector leading to stable internal state 1

; z/zf –>1

isc 135 133 141

obbc 141

; calculate all input vector leading to instable state oscillation

; between 0 and 1

Digital

analysis

of a form

; z/zf 0–>0, 1–>1

isc 137 135 142

obbc 142

; calculate all input vector leading to stable states 0 and 1

; (empty)

; z/zf 1–>0, 0–>1

isc 131 133 143

obbc 143

; calculate all partial automaton graphs

; check if all input vectors are covered (tvl150 should be all -)

uni 140 141 150

uni 150 142 150

uni 150 143 150

obbc 150

; calculate phase list leading to stable internal state 0

; z/zf –>0

isc 140 110 160

obbc 160

; calculate phase list leading to stable internal state 1

; z/zf –>1

isc 141 110 161

obbc 161

; calculate phase list leading to instable state oscillation

; between 0 and 1

; z/zf 1–>0, 0–>1

isc 142 110 162

obbc 162

; calculate phase list leading to stable states 0 and 1 (empty)

; z/zf 0–>0, 1–>1

isc 143 110 163

obbc 163

; variable tuple to show all input variables and internal

; state variable z

vtin 1 170

zf y1 y2 y3 y4 y5 y6 y7.

; variable tuple to show all output variables = 1

tin 1 171

y1.

1.

tin 1 172

y2.

1.

tin 1 173

y3.

K

1.

tin 1 174

y4.

1.

tin 1 175

y5.

1.

tin 1 176

y6.

1.

tin 1 177

y7.

1.

; calculate all output variables yi on basis of global phase list

isc 171 110 1

maxk 1 170 1

obbc 1

isc 172 110 2

maxk 2 170 2

obbc 2

isc 173 110 3

maxk 3 170 3

obbc 3

isc 174 110 4

maxk 4 170 4

obbc 4

isc 175 110 5

maxk 5 170 5

obbc 5

isc 176 110 6

maxk 6 170 6

obbc 6

isc 177 110 7

maxk 7 170 7

obbc 7

; calculate all input vector leading to instable state

; oscillation between 0 and 1

; z/zf 1–>0, 0–>1

isc 171 162 11

maxk 11 170 11

obbc 11

isc 172 162 12

maxk 12 170 12

Digital

analysis

of a form

obbc 12

isc 173 162 13

maxk 13 170 13

obbc 13

isc 174 162 14

maxk 14 170 14

obbc 14

isc 175 162 15

maxk 15 170 15

obbc 15

isc 176 162 16

maxk 16 170 16

obbc 16

isc 177 162 17

maxk 17 170 17

obbc 17

Corresponding author

André Oksas can be contacted at: andre.oksas@gmx.de

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K