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 = (xIndicational CA_1.gif, y = (yIndicational CA_2.gif∈ AIndicational CA_3.gifequivalent, in short notation
(1) x ≈Indicational CA_4.gif y
if and only if there is a permutation f: {1, ..., n} → {1, ..., n} such that
(2) yIndicational CA_5.gif= xIndicational CA_6.gif i= 1,...,n

We could now say that a "string" in the Brownian sense is an equivalence class of A* with respect to ≈Indicational CA_7.gif, i.e., an element of the quotient set A* / ≈Indicational CA_8.gif.

(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

Indicational CA_9.gif

Indicational CA_10.gif

Indicational CA_11.gif

Indicational CA_12.gif

Indicational CA_13.gif

Indicational CA_14.gif

Indicational CA_15.gif

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

Indicational CA_16.gif

Indicational CA_17.gif

Basic composition scheme

Indicational CA_18.gif

Indicational CA_19.gif

Rule set for Indicational CA_20.gif

Indicational CA_21.gif

Indicational CA_22.gif

TabView for indIndicational CA_23.gif rules

Indicational CA_24.gif

MenuView for indIndicational CA_25.gif rules

Indicational CA_26.gif

Rule space for Indicational CA_27.gif

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

Indicational CA_28.gif  Indicational CA_29.gif

Program scheme for indicational CA

Indicational CA_30.gif

Indicational normal form (inf)

inf([Indicational CA_31.gif ]) = inf([Indicational CA_32.gif])= [Indicational CA_33.gif])
inf([Indicational CA_34.gif]) = inf([Indicational CA_35.gif]) = [Indicational CA_36.gif)]
inf([Indicational CA_37.gif]) = [Indicational CA_38.gif]
inf([Indicational CA_39.gif]) = Indicational CA_40.gif]

Indicational normal form (inf):
inf([Indicational CA_41.gif ]) = inf([Indicational CA_42.gif])= [Indicational CA_43.gif])
inf([Indicational CA_44.gif]) = inf([Indicational CA_45.gif]) = [Indicational CA_46.gif)]
inf([Indicational CA_47.gif]) = [Indicational CA_48.gif]
inf([Indicational CA_49.gif]) = Indicational CA_50.gif]

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

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

For 1D Indicational CA_51.gif:
[aab] = [baa] = [aba]
[abb] = [baa] = [bab]
[aaa] = [aaa]
[bbb] = [bbb]

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

System of indIndicational CA_52.gif

Indicational CA_53.gif

Indicational CA_54.gif

Analysis of linear forms

Groups

Indicational CA_55.gif

Equalities

Indicational CA_56.gif

Indicational CA_57.gif

Indicational CA_58.gif

Examples

rule = 5.6.7.4

Indicational CA_59.gif

rule = 1274

Indicational CA_60.gif

Analysis of developping indCA forms

Indicational CA_61.gif   Indicational CA_62.gif


rule = 1634

Indicational CA_63.gif

Indicational CA_64.gif

Indicational CA_65.gif

Indicational CA_66.gif

seed = {0, 1, 0}

Indicational CA_67.gif

rule = 1638

Indicational CA_68.gif

Indicational CA_69.gif

Indicational CA_70.gif

CI - rule = 5.6 .3 .8

Indicational CA_71.gif

Indicational CA_72.gif

Indicational CA_73.gif

Indicational CA_74.gif

Indicational CA_75.gif

Indicational CA_76.gif

Indicational CA_77.gif

System of indIndicational CA_78.gif rules

Indicational CA_79.gif

Indicational CA_80.gif

Reflectional enaction for Indicational CA_81.gif rules

Indicational CA_82.gif

Indicational CA_83.gif

Indicational CA_84.gif

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

Indicational CA_85.gif

Indicational CA_86.gif

Indicational CA_87.gif

Indicational CA_88.gif

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

Example

Indicational CA_89.gif

Indicational CA_90.gif

Indicational CA_91.gif

Indicational CA_92.gif

Indicational CA_93.gif

Rule scheme

Indicational CA_94.gif

Rule set for Indicational CA_95.gif

Indicational CA_96.gif

Indicational CA_97.gif

Indicational CA_98.gif Rule scheme ruleCIR, 10x3

Indicational CA_99.gif

CA Program scheme

Indicational CA_100.gif

Rule space for Indicational CA_101.gif

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

Indicational CA_102.gif  Indicational CA_103.gif

Indicational normal form (inf)

inf([Indicational CA_104.gif]) = inf([Indicational CA_105.gif]) = [Indicational CA_106.gif)]
inf([Indicational CA_107.gif]) = inf([Indicational CA_108.gif]) = [Indicational CA_109.gif)]
inf([Indicational CA_110.gif]) = inf([Indicational CA_111.gif]) = [Indicational CA_112.gif)]

inf([Indicational CA_113.gif ]) = inf([Indicational CA_114.gif]) = [Indicational CA_115.gif])
inf([Indicational CA_116.gif ]) = inf([Indicational CA_117.gif]) = [Indicational CA_118.gif])
inf([Indicational CA_119.gif ]) = inf([Indicational CA_120.gif]) = [Indicational CA_121.gif])

inf([Indicational CA_122.gif]) = [Indicational CA_123.gif]
inf([Indicational CA_124.gif]) = Indicational CA_125.gif]
inf([Indicational CA_126.gif]) = Indicational CA_127.gif]

Indicational CA_128.gif

Indicational CA_129.gif

Indicational CA_130.gif

Indicational CA_131.gif

Indicational CA_132.gif

Indicational CA_133.gif

Indicational CA_134.gif

Indicational CA_135.gif

Indicational CA_136.gif

Indicational CA_137.gif

Indicational CA_138.gif

Indicational CA_139.gif

rule = 1.22.33 .4

Indicational CA_140.gif

Indicational CA_141.gif

Indicational CA_142.gif

Indicational CA_143.gif

ruleCIR = 1.4.x .22 .32.

Indicational CA_144.gif

Indicational CA_145.gif

Indicational CA_146.gif

Indicational CA_147.gif

Indicational CA_148.gif

Indicational CA_149.gif

Indicational CA_150.gif

Indicational CA_151.gif

Indicational CA_152.gif

Indicational CA_153.gif

Indicational CA_154.gif

Indicational CA_155.gif

Indicational CA_156.gif

Indicational CA_157.gif

Indicational CA_158.gif

Indicational CA_159.gif

Indicational CA_160.gif

Indicational CA_161.gif

Indicational CA_162.gif

Indicational CA_163.gif

Indicational CA_164.gif

Indicational CA_165.gif

Indicational CA_166.gif

Indicational CA_167.gif

Indicational CA_168.gif

Indicational CA_169.gif

Indicational CA_170.gif

Indicational CA_171.gif

Indicational CA_172.gif

Indicational CA_173.gif

Indicational CA_174.gif

Indicational CA_175.gif

Indicational CA_176.gif

Indicational CA_177.gif

Indicational CA_178.gif

Indicational CA_179.gif

Indicational CA_180.gif

Indicational CA_181.gif

Indicational CA_182.gif

Indicational CA_183.gif

Indicational CA_184.gif

Indicational CA_185.gif

Indicational CA_186.gif

Indicational CA_187.gif

Indicational CA_188.gif

corrected  (no difference)

Indicational CA_189.gif

Double reflectional Indicational CA_190.gif

Indicational CA_191.gif

Some examples

Indicational CA_192.gif

Indicational CA_193.gif

Indicational CA_194.gif

Indicational CA_195.gif

Indicational CA_196.gif

Indicational CA_197.gif Rule table: perm{i,j,k}

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

Indicational CA_198.gif

Indicational CA_199.gif

Indicational CA_200.gif

Indicational CA_201.gif Rule set: ruleSetRCI

Indicational CA_202.gif

Indicational CA_203.gifruleSetRCI: MenuView (todo)

Indicational CA_204.gif

Indicational CA_205.gif Rule scheme

Indicational CA_206.gif

Dynamic view of Indicational CA_207.gif forms (todo)

Examples of complete forms:
First: Symmetric Indicational CA_208.gif

Indicational CA_209.gif

Indicational CA_210.gif

Indicational CA_211.gif

Indicational CA_212.gif

Indicational CA_213.gif

Indicational CA_214.gif

Indicational CA_215.gif

Indicational CA_216.gif

Indicational CA_217.gif

Indicational CA_218.gif

Indicational CA_219.gif

Indicational CA_220.gif

Indicational CA_221.gif

Indicational CA_222.gif

Indicational CA_223.gif

Indicational CA_224.gif

Indicational CA_225.gif

Indicational CA_226.gif

Indicational CA_227.gif

Indicational CA_228.gif

Indicational CA_229.gif

Indicational CA_230.gif

Indicational CA_231.gif

Indicational CA_232.gif

Indicational CA_233.gif

Indicational CA_234.gif

Indicational CA_235.gif

Indicational CA_236.gif

Indicational CA_237.gif

Indicational CA_238.gif

Indicational CA_239.gif

Total trans-contextural patterns

Indicational CA_240.gif

Indicational CA_241.gif

alternative transcontextural

Indicational CA_242.gif

Indicational CA_243.gif

Indicational CA_244.gif

Indicational CA_245.gif

Deviant forms, overcomplete

Indicational CA_246.gif

Indicational CA_247.gif

Indicational CA_248.gif

Indicational CA_249.gif

Indicational CA_250.gif

Indicational CA_251.gif

Examples of complete forms:
Second: Asymmetric Indicational CA_252.gif ??

Indicational CA_253.gif

Indicational CA_254.gif

Variations of the same asymmetric pattern: defect cases

Same head

Indicational CA_255.gif

Indicational CA_256.gif

Indicational CA_257.gif

Indicational CA_258.gif

Indicational CA_259.gif

Indicational CA_260.gif

Indicational CA_261.gif

Indicational CA_262.gif

Indicational CA_263.gif

Indicational CA_264.gif

Indicational CA_265.gif

Indicational CA_266.gif

Indicational CA_267.gif

Indicational CA_268.gif

Indicational CA_269.gif

Indicational CA_270.gif

Indicational CA_271.gif

Indicational CA_272.gif

Indicational CA_273.gif

Indicational CA_274.gif

Different head
and defectuos asymmetry

Indicational CA_275.gif

Indicational CA_276.gif

Indicational CA_277.gif

Indicational CA_278.gif

Indicational CA_279.gif

Indicational CA_280.gif

Indicational CA_281.gif

Indicational CA_282.gif

Non defect asymmetric pattern???
Overdetermination as defect

Indicational CA_283.gif

Indicational CA_284.gif

Asymmetric mutation by defect

Indicational CA_285.gif

Indicational CA_286.gif

Indicational CA_287.gif

Indicational CA_288.gif

Indicational CA_289.gif

Indicational CA_290.gif

Indicational CA_291.gif

Indicational CA_292.gif

Indicational CA_293.gif

Indicational CA_294.gif

Symmetric variations

Indicational CA_295.gif

Indicational CA_296.gif

Indicational CA_297.gif

Indicational CA_298.gif

Indicational CA_299.gif

Indicational CA_300.gif

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