Morphic Cellular Automata Systems

Part III: Indicational CAs,

Dr. phil Rudolf Kaehr

copyright © ThinkArt Lab Glasgow

ISSN 2041-4358

( work in progress, vs. 0.2, July 2014 )

Part III: Indicational CA

Motivation

As it is reasonable to define “indicational” stack machines it is equally reasonable to define the concept of indicational cellular automata. In both cases, the definitions relies on the primary ‘semiotic’ properties and not on the specific features of the calculus of indication, like the double crossing. The same hold for all other definitions of CAs. The specific semiotic properties of digital CAs are not considered, what counts are the transition rules on semiotically identical signs. For kenomic CAs, it is the kenomic definition of the ‘data’ what is relevant. At least all this is correct at a first glance.

This study of “Indicational CAs” is focused on the proto-semiotic conditions of George Spencer Brown’s Calculus of Indication as developed in his work Laws of Form, and not on the ‘arithmetical’ and ‘algebraic’ rules of the calculus of indication (CI) itself.

Hence, the ‘arithmetical’ rules of the process of distinctions, the law of condensation and the law of cancellation are not thematized in this context but their deep-structure of inscription, the ‘topological invariance’ of the notation is chosen as the primary topic and studied in the framework of cellular automata.

This approach has not yet been studies properly in the literature about Spencer-Brown’s calculus of indication.

With this CA-oriented study of the proto-semiotic conditions of the CI, the field of research of the deep-structure of the CI is opened up.

The interesting results of this study are presented as such, and shall not yet be thematized or interpreted by graphematic and grammatological considerations.

Earlier graphematic characterizations of the Laws of Form are helpful for an understanding of the presented approach but shall not be involved into the present considerations.

Commutativity of terms for indCA

”A third deviation from classical semiotics is less obvious: the commutativity of the concatenation operation. For any two terms "a" and "b" the terms "ab" and "ba" are identical. That this is indeed a semiotic identity (and not just a logical equality) has been stressed by Varga."

"We call two elements x = (x, y = (y∈ Aequivalent, in short notation

(1) x ≈ y

if and only if there is a permutation f: {1, ..., n} → {1, ..., n} such that

(2) y= x i= 1,...,n

We could now say that a "string" in the Brownian sense is an equivalence class of A* with respect to ≈, i.e., an element of the quotient set A* / ≈.

(A) If the two given tokens of strings have different lengths, then they are different. If they have equal lengths, then go to (B'). (B') Check whether each atom appears equally often in both string-tokens. If this is the case, then they are equal, otherwise they are different.” (R. Matzka)

http://www.rudolf-matzka.de/dharma/semabs.pdf

http://www.thinkartlab.com/pkl/media/Diamond%20 Calculus/Diamond%20 Calculus.html

http://memristors.memristics.com/Complementary%20 Calculi/Complementary%20 Calculi.html

Toward indicational CAs

Topological invariance of the ‘heads’ of CA rules.

System of indicational CA rules

Alphabet = {a, b}

card(Ind(3, 2)) = {aaa, aab, abb, bbb} x {a,b}

Basic composition scheme

Rule set for

TabView for ind rules

MenuView for ind rules

Rule space for

{1,5}x{2,6}x{3,7}x{4,8} = 16

Program scheme for indicational CA

Indicational normal form (inf)

inf([ ■]) = inf([])= []) inf([]) = inf([]) = [)] inf([]) = [] inf([]) = ]

Indicational normal form (inf): inf([ ■]) = inf([])= []) inf([]) = inf([]) = [)] inf([]) = [] inf([]) = ]

Indicational normal forms are not identical with proto-normal forms, pnf, of kenogrammatics.

Calculus of indication: a=a, a≠b, ab=ba

For 1D : [aab] = [baa] = [aba] [abb] = [baa] = [bab] [aaa] = [aaa] [bbb] = [bbb]

pnf[([aaa]) = pnf([bbb]) but inf[([aaa]) ≠ inf([bbb]).

System of ind

Analysis of linear forms

Groups

Equalities

Examples

rule = 5.6.7.4

rule = 1274

Analysis of developping indCA forms

rule = 1634

seed = {0, 1, 0}

rule = 1638

CI - rule = 5.6 .3 .8

System of ind rules

Reflectional enaction for rules

Hence, the set of indCA rules might be extended by its interactional enaction rules which are relating to CA systems of neighbor contextures. This will be studied in another paper.

Combinatorics

{aaa, bbb, ccc, aab, aac, abb, acc, bcc, bbc, abc} x {a, b, c} =

Example

Rule scheme

Rule set for

Rule scheme ruleCIR, 10x3

CA Program scheme

Rule space for

{1,5}x{2,6}x{3,7}x{4,8} = 16

Indicational normal form (inf)

inf([]) = inf([]) = [)]

inf([]) = inf([]) = [)]

inf([]) = inf([]) = [)]

inf([ ■]) = inf([]) = [])

inf([ ■]) = inf([]) = [])

inf([ ■]) = inf([]) = [])

inf([]) = []
inf([]) = ]

inf([]) = ]

rule = 1.22.33 .4

ruleCIR = 1.4.x .22 .32.

corrected (no difference)

Double reflectional

Some examples

Rule table: perm{i,j,k}

Number of rules: 80 = (12+4+4) x 4

Rule set: ruleSetRCI

ruleSetRCI: MenuView (todo)

Rule scheme

Dynamic view of forms (todo)

Examples of complete forms:

First: Symmetric

Total trans-contextural patterns

alternative transcontextural

Deviant forms, overcomplete

Examples of complete forms:

Second: Asymmetric ??

Variations of the same asymmetric pattern: defect cases

Same head

Different head

and defectuos asymmetry

Non defect asymmetric pattern???

Overdetermination as defect

Asymmetric mutation by defect

Symmetric variations