Metaphors of Dissemination and Interaction
                        of morphoCAs
              Functional Analysis of the Graphematics of morphoCAs
          

Dr. phil Rudolf Kaehr
copyright
© ThinkArt Lab Glasgow
ISSN 2041-4358
( work in progress, v. 0.5, Nov., Oct. 2015 )

Diagrams and Dissemination

Initialization

Initialization

morphoCA requisites

Keywords

ECA, morphoCA, diagrams, reduction, minimization, flow charts, interaction, transaction, mediation, heterogeneous structures, poly-, dis-, intra-, trans-contexturality, contextuality, morphogrammatics, recursion, memristivity  

Motivation

“Physical computing media are asymmetric.
Their symmetry is broken  by  irregularities,  physical  boundaries,  external connections  and  so  on.    
Such  peculiarities,  however,  are highly variable and extraneous to the fundamental nature of the media.  
Thus, they are
pruned from theoretical models, such as cellular automata, and reliance on them is frowned upon in programming practice.
However, computation, like many other highly organized activities, is incompatible with perfect  symmetry.    
Some standard  mechanisms  must  assure  breaking  the  symmetry inherent in idealized computing models.”
  Leonid A. Levin,
The Computer Journal Vol. 48, No. 6, 2005

Minimization and flow charts

It is not easy to explain how to understand the results of morphoCAs. It seems that there is a strong conflict between the millions of visualizations, sonifications and structurations managed by the approach of claviatures and the paradigmatic statement developed in the paper "Asymmetric Palindromes"  for morphoCAs that “What you see is not what it is”.

Instead of studying the multitude of the products of morphoCAs, another approach that is more focused on the mechanism of the production process of morphoCAs might help to uncover the deep-structural significance of morphogrammatic based cellular automata.

This paper offers some insights into the mechanism of production by the application of reductions (minimizations) of the functional interpretations of the morphoCA rules and by designing the network of the actions of the morphic automata by some flow charts.

There is not yet an algorthmic approach to reduce morphic CA functions accessible. But the distinction between reducible and non-reducible morphoCAs is well defined.

Hence, instead of considering the multi-millions of morphoCA productions, some specific flow charts of the mechanism of production is presented to continue the studies of morphoCAs. With that a kind of reflection a kind of a meta-theory of morphoCAs is introduced.

From this meta-theoretial point of view, morphoCAs might be involved into an introspection between Kaluzhnine-Graph-Schemata of recursivity and poly-contextural memristivity.

A further approach to study the deep-structure of the meaning of morphoCAs will be sketched in a further paper by an analysis of their underlying poly-contextural logics.

Contexturality vs. contextuality

The term “polycontexturality” occurs frequently in sociological studies. Often as a synonyme or replacement of ‘polycentricity’ and linguistically, modal-logically or semiotically identified with ‘contextuality’.

Polycontexturality refers to a trans-classical paradigm of thinkind and writing that is not compatible with established concepts of science, while ‘polycentricity’ and ‘contextuality’ are parts of classical logic (say, modal logic), ontology and semiotics.

“Polycentricity is similar to the concept of polycontexturality in logic. Polycontexturality represents a many-system logic, in which the classical logic systems (called contextures) interplay with each other, resulting in a complexity that is structurally different from the sum of its components (Kaehr and Mahler 1996).”
(Rajendra Singh, Towards Information Polycentricity Theory: Investigation of a Hospital Revenue Cycle, 2011)

A similar approach, chosen out of the ‘polycentricity or centextualist movement’, is proposed e.g. by Lars Qvortrup:

“The implicit idea behind the first three theses is that we are on our way into a society, which is radically different from the so-called modern society. It has been described as “functionally differentiated” (Luhmann 1997), as “polycontextural” (Günther 1979) or as “hypercomplex” (Qvortrup 1998), emphasising that it does not offer one single point of observation, but a number of mutually competing observation points with each their own social context.”
Lars Qvortrup, THE AESTHETICS OF INTERFERENCE: From anthropocentrism to polycentrism and the reflections of digital art

http://www.hotelproforma.dk/Userfiles/File/artikler/lq.pdf

It might provoke some progress if the distinctions proposed in this paper would be applied to systems theory of intra-, inter- and trans-contexturally mediated complex dis-contextural constellations and dynamics.

Diagram_1.gif

Diagram_2.gif

More entertainment with intra-, inter- and trans-disciplinarity of inter-, poly-, trans- and dis-contexturality at: “Modular Bolognese, Paradoxes of postmodern education”  in: Short Studies 2008. Adventures in Diamond Strategies of Change(s)

Diagram_3.gif

Three kinds of morphoCA diagrams

Three kinds of morphoCA diagrams have to be distinguished:

1. Mono-contextural diagrams (intra)

    Diagram_4.gif

The first kind is covered by the classical diagrams. These diagrams hold for classical ECAs as much as for mono-contextural morphoCAs of different topological complexity. Morphogrammatically, they are supported by the ‘classical’ morphograms of complexity 2.

2. Poly - contextural diagrams as interaction (inter, trans)

Diagram_5.gif

Diagram_6.gif

The second kind is based on a distribution of the diagram of at least 3 loci. This distribution is basic for the interactions between otherwise autonomous automata. The internal structure of the memory/logic unit of the single automata is intrinsically changed toward a chiastic, i.e. memristive behavior of internal and external events.

The interactional activity of the second kind of diagrams is supported by the morphograms of complexity 3.
In this field of interactional activity of complexity 3, two different modi might be distinguished:
inter-actional with morphograms mg[5], mg[10] and mg[14], and head[{1,2,3}] -> i, i=1,2,3
• trans-contextural mg[11], mg[12] and mg[13] with head [{1,2}] -> 3.

3. Poly - contextural diagrams as mediation

Diagram_7.gif

The third kind is based on the second kind but is involving the whole structural complexion of the distributed morphoCAs. Only with this configuration the full graphematic character of morphoCAs enters the trans-classic game of computation, interaction, reflection and mediation.

Mediative actions are supported by morphograms of the minimal complexity 4, represented by the morphogram mg[15] with head[{1,2,3}]->4.

Poly - contextural basic component

    Diagram_8.gif

Examples

Functions

Diagram_9.gif

Action schemes

Diagram_10.gif

Morphograms

Diagram_11.gif

Diagram_12.gif

Diagram_13.gif

Diagram_14.gif

The compound morphogram ruleDM[{1, 3, 4, 11, 15}] inscribes the deep-structure of the mediation of intra- and inter-contextural actions.

Diagram_15.gif

Poly-contextural logic

Quite obviously, intra-contextural morphograms are representing the deep-structure of junctional mono- and poly-contextural operators.

As a first remark, inter- and trans-contextural morphograms are representing the deep-structures of transjunctional poly-contextural operators.

Morphogram mg[15] represents the full differentiations of the interplay of inter- and trans-contextural poly-contextural operators.

The proof-theoretical metaphor of polycontextural interplays is not anymore just a ‘tree’ but a ‘forrest of colored trees’.

Example: ternary 3-contextural transjunctions of ruleDCM[{1, 2, 12, 13, 5}]

Diagram_16.gif

1. Mono-contextural diagrams: ECA and Diagram_17.gif

Diagram of the ECA scheme

K. Salman’s paper “Elementary Cellular Automata (ECA) Research platform” gives an elaborated definition and explanation of the concept of ECAs.
For the purpose of an introduction of morphoCAs it suffice to connect to some of its terms and constructions.

“For ease of illustration we let the CA evolve according to one uniform neighborhood transition function and fixed radius which is a local function (rule) Diagram_18.gifwhere the CA evolves after a certain number of time steps T.
In this case we have a total of
Diagram_19.gifdistinct rules. It follows that a 1 - D CA is a linear lattice or register of KN memory cells. Each cell is represented by Diagram_20.gif , where k = [1: K], KN and t = [1:  T], TZ that describes the content of memory location at evolution time step t..”  (K. Salman)

Diagram_21.gif

                              Figure 5, Detailed Structure of a typical Cellular Automaton Cell for rule 30.

http : // www.cyberjournals.com/Papers/Jun2013/02. pdf

Diagram of the CA rule in respect of input and output cells in time t to t+1

Diagram_22.gif

Diagram_23.gif

Description of the mechanism of the CA calculation

The object Diagram_24.gif of Diagram_25.gif of Fig. 5 is a result of the calculation of the logical unit U, i.e. Transition Rule Logic, in relation to its inputs Diagram_26.gif and Diagram_27.gif but it is also at the same time the initial value, Diagram_28.gif, in the Memory Cell, of a new calculation of a next step of the CA.

This new calculation might happen intra-contexturally as a mapping from Diagram_29.gif as Diagram_30.gif to the logic unit Diagram_31.gif or trans-contexturally as a mapping from Diagram_32.gif as Diagram_33.gif to the new object Diagram_34.gif of Diagram_35.gif in Diagram_36.gifwhere it becomes the new value of Diagram_37.giffor a calculation in Diagram_38.gif.

The presumption of the classical model of ECAs is certainly that all components are from the same contexture, and having the same clock.

Mono-contextural CAs are homogeneous structures.

Classical Cellular Automata. Homogeneous Structures
 By V. Z. Aladjev

Diagram_39.gif

Diagram_40.gif

Diagram of the sub-rule definition of ECAs

A sub-rule implementation of the ECA rules might augment its computational efficiency and reduce numeric complexity for programmable hybrid ECA compositions.

As it is well known, CAs are understood as parallel computing concepts and devices.

There is no doubt that the sketched sub-rule appoach can be concretized and implemented as a ‘hybrid’ ECA on a hardware board like Spartan-6 FPGA Connectivity Kit or similar.
(http://www.xilinx.com/products/boards-and-kits.html)

A further step in augmenting the granularity of CAs might be achieved with the sub-rule approach for ECA rules. Each ECA rule is defined in a sub-rule oriented approach as a composition of sub-rules. Thus all compatible sub-rules can be applied in parallel to realize a single ECA rule.

Also the sub-rule approach is defining the ECA rules is not yet showing the flow chart of the interactions of the sub-rules to build the ECA rule.

ECA-rule = [eca1, eca2, ..., eca8]

Example: ECA rule 210

Diagram_41.gif

ruleECA[{6, 7, 3, 9, 10, 12, 13, 15}]

Diagram_42.gif

Diagram_43.gif

Diagram_44.gif

Hence the ECA rule 210 is represented by the tuple (6,7,3,9,10,12,13,15) of ECA sub-rules.

Diagram_45.gif

Flow chart of the parallel realization of the 8 sub - rules of an ECA.

Diagram_46.gif

“If in a CA the same rule applies to all cells, then the CA is called a uniform CA; otherwise the CA is called a hybrid CA (Fig. 1).”

Theory and Applications of Cellular Automata in Cryptography
S.Nandi, B.K.Kar and P.Pal Chaudhuri

ECA sub-rule manipulators

The method of sub-rules for ECAs is an abstraction and parametrization of the components of the rule schemes that allows a micro-anlysis of the ECAs. The ECA sub-rule manipulator manages all ECA rules of a 1D ECA. The sub-rule manipulators enables a micro-analysis of the behavior of all Diagram_47.gif ECA rules.  

Each 1-D ECA rule number has a sub-rule number representation. There are just 8 disjunct pairs of sub-rules to define a 1D ECA rule.

The results are visualized below. The combination of the 8 sub-rules covers all the 256 well known ECA rules.

Mono-contextural Diagram_49.gif

Reduction (trivial)

Diagram_51.gif

Diagram_52.gif

Diagram_53.gif

Diagram_54.gif

Diagram_55.gif

Scheme: (r1, r2, r3, r4) ⇒ r5, Diagram_56.gif = {0, 1}

Diagram_57.gif

2. Poly-contextural diagrams with interactions

General approach

Internal structure of the memory unit of the second kind

Following for example K. Salman’s classical modelling approach in “Elementary Cellular Automata (ECA) Research platform“  a more explicit modelling of the mechanism of morphoCAs might be achieved.

A first crucial difference to the classical concept is the fact that the memory unit is not just passively receiving (D) and sending data (Q) but is also actively deciding to which system of its computational environment they belong and if the data remain in its domain or not. If not, the activity of the memory unit is deciding where that data belong and sends it to the evoked computational unit of the complexion.

In terms of actors, the memory unit is receiving, sending and deciding about the contextures of data. Classical memory actions are strictly intra-contextural. This holds in the same sense for multi-processor systems too. They are acting strictly intra-contexturally, keeping their distributed data together.

Hence, the logic devices in the modified diagram, Fig. 5b, have two function towards its memory units:
1. a decision operation over the logical operations, i.e. to decide if an operation stays inside the contexture or if it leaves trans-contexturally the contexture for another contexture on another layer of the complex poly-layered morphoCA system.
2. the intra-contextural function of producing the junctional values for the corresponding intra-contextural memory in the sense of ordinary logical functions, like NAND or NOR.

The object Diagram_58.gif of the CA diagram receives a value from the logic unit and it delivers it to Q as Diagram_59.giffor the new calculation with Diagram_60.gif.
Secondly, Q receives the value from D as a value, not for Diagram_61.gif in Diagram_62.gif but for Diagram_63.gifof the neighbor layer Diagram_64.gif. This new value is memorized in the neighbor Diagram_65.gif as the new positive value for calculation in Diagram_66.gif, hence it is placed in Diagram_67.gifand not as a genuine value of Diagram_68.gif as Diagram_69.gif.

The result of the application of the rule in all 3 sub-systems is delivered with the multi-layered system as a whole, i.e. with Diagram_70.gif and its rules Diagram_71.gif.

Obviously, the whole automaton with its different layers has to be designed in the epistemological mode of the ‘as-abstraction’, i.e. as “A as B is C” and not in the mode of identity with “A is A”.

The modified diagram is introducing an environment to the original mono-contextural CA diagram that implies the possibility of interactions. The environment of a CA system is the primary condition for a possible self-reflection of the complex system of different and interacting CAs.

The logic behind this construction was first introduced by Gotthard Gunther’s “Cognition and Volition” (1970) which gives a profound explanation of the new concept of the ‘proemial relation’.

Modified diagram Fig. 5b

Diagram_72.gif

Memristive properties of the memory/logic unit

Why and how is the behavior of the memory units of morphoCAs of second-order and memristive and not just defined as first-order actions of storage and transformations? The main strategy of the whole maneuver is to avoid ‘information processing’. Interaction is prior to information exchange.

It could be said: morphoCAs without memristivity are reducible without loss to classical CAs.

Diagram_73.gif     Diagram_74.gif

The diagram below, Fig. 1, shows again the chiastic interaction between operators (M) and operands ‘σ’ distributed over different loci of the kenomic matrix.

“M as σ” is obviously not the so called self-reference of “M is σ”.

Diagram_75.gif

Diagram_76.gif

Fig . 1 Chiasm (M, σ)

Diagram_77.gif

Explanation of Fig. 1

Diagram_78.gif

Thus, "a type as a term becomes a term and as a type it remains a type".
And the same round for terms.

Full wording for a chiasm between terms and types over two loci

Explicitly, first the green round,
"A type Diagram_79.gif as a term Diagram_80.gif becomes a term Diagram_81.gif
and as a type Diagram_82.gifit remains a type Diagram_83.gif  for a term Diagram_84.gif".
And,
"A type Diagram_85.gif  as a term Diagram_86.gif becomes a term Diagram_87.gif
  and as a typeDiagram_88.gif it remains a type Diagram_89.gif for a term Diagram_90.gif".

And simultaneously, mediated,
   the second round in red, the same for terms:
"A term Diagram_91.gif as a type Diagram_92.gif becomes a type Diagram_93.gif
   and as a term Diagram_94.gif it remains a term Diagram_95.gif for a type Diagram_96.gif".
   And,
   "A term Diagram_97.gifas a type Diagram_98.gif becomes a type Diagram_99.gif
  and as a term Diagram_100.gifit remains a term Diagram_101.gif for a type Diagram_102.gif".

And finally, between terms Diagram_103.gif and Diagram_104.gif and types Diagram_105.gif and Diagram_106.gif,
  a categorial coincidence is realized.  
  While between terms and types a morphism (order relation) exists.

Fig . 2  Complete interactional scheme

Diagram_107.gif

Diagram_108.gif

Hence, this kind of memory is a complexion of ‘memory’ and ‘logic’ as it is supposed for memristive behavior.

There are four basic components plus the clock in the interaction paradigm of morphoCAs.

Diagram_109.gif

In contrast to the classical CA with its send/receive properties, there are four basic components plus the clock in the paradigm of morphoCAs. The sens/receive or read/write mechanism is augmented in morphoCAs by a decision-making (trans-logical) component of accept/reject in regard of the sub-system property.

The contrast to Konrad Zuse's conception of calculation is obvious :

“Rechnen heisst : Aus gegebenen Angaben nach einer Vorschrift neue Angaben bilden.” (Konrad Zuse)

The discontexturaliy of morphoCAs is certainly also not in hamony with Karl Hewitt’s monolithic actor approach to computation.

A systematic deconstruction has obviously to deconstruct all 4+1 components of the diagram.

The very first deconstruction happens by parametrizing the inputs. Each input/output, i.e. send/receive action might belong to a different contexture. Hence, the very first task of the automaton is to handle such profound diversity. This job is obsolete for classical CAs because all data are from/in the same contexture.

This contextural embodiment of the fourth term, Diagram_110.gif, explains why the term is not just an extensional result of a mapping but is structurally depending on the conceptual ‘history’ of the 3 previous actions.

This understanding of the morphoCA rules relates back to the concept of the ε/ν-structure of morphic objects and actions within the concept of the proposed memristive automata.

Diagram_111.gif

Diagram_112.gif

Diagram_113.gif

From memristive flip-flop to memristive interactions

Finite state machines and morphoCAs

“A Cellular Automaton (CA) is an infinite, regular lattice of simple finite state machines that change their states synchronously, according to a local update rule that specifies the new state of each cell based on the old states of its neighbors.” (Kari)

http://users.utu.fi/jkari/ca/CAintro.pdf

“Furthermore, since the ECA is actually a finite state machine then the present state of the neighborhood Diagram_114.gif of cell Diagram_115.gif at time step t and the next state Diagram_116.gif at time step t + 1, can be analyzed by the state transition table and the state diagram depicted in figure 4.”  (K. Salman)

Diagram_117.gif

Figure 4, state machine analysis of Rule 30

Diagram_118.gif

ECA Rule 30

Diagram_119.gif

Diagram_120.gif

Diagram_121.gif

Diagram_122.gif

Diagram_123.gif

     ruleECA[{1, 2, 3, 9, 5, 11, 13, 15}] = rule30

Diagram_124.gif

Diagram_125.gif

System of elementary kenomic cellular automata rules in trito-difference form

Diagram_126.gif

Diagram_127.gif

Interpretation

Difference scheme

The difference scheme is a scheme of differences, and not just a relational mapping from Diagram_128.gifto C.

Also an evolution from Diagram_129.gif] to Diagram_130.gif is defined by all previous elements of time t of the specified CA rule there is no concrete differentiation between the new state of  Diagram_131.gifand the previous states defined.

Hence, the new state Diagram_132.gif of a classical CA might incorporate any arbitrary value from a pre-given set of values and is not retro-recursive characterized by the differences of the previous constellation it depends.

Graphics:                                t                    t                t                                                                                                 t + 1 PlotLabel /. Options[{RowBox[{ C     , &nbsp;&nbsp; C , &nbsp;&nbsp; C      &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; C      &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;, &nbsp;&nbsp;&nbsp;, &rArr;, &nbsp;&nbsp;&nbsp;, GraphicsBox[TagBox[RasterBox[RawArray[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, ConvertClosed -> False, ConvertReverseClosed -> False, ConvertLinkedNotebooks -> False, CharacterEncoding -> Automatic, ConversionStyleEnvironment -> None, ConversionRules -> Automatic, HeadAttributes -> {}, HeadElements -> {}, CSS -> Automatic, ConvertLinkedNotebooks -> False, MathOutput -> GIF, GraphicsOutput -> GIF, Graphics3DOutput -> Automatic, ManipulateOutput -> CDF, ConvertClosed -> True, ConvertReverseClosed -> False, FullDocument -> True, AltMathOutput -> FileName, TableOutput -> {TextForm, Automatic}, AnimationOutput -> Automatic, FilesDirectory -> HTMLFiles, LinksDirectory -> HTMLLinks, HTMLEntities -> {HTML}, AllowBlockMathML -> False, ShowStyles -> True, DataUri -> False, MathMLOptions -> {UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], ArrayObject[Byte, <989400>]], {{0, 425}, {582, 0}}, {0, 255}, ColorFunction -> RGBColor], BoxForm`ImageTag[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, ConvertClosed -> False, ConvertReverseClosed -> False, ConvertLinkedNotebooks -> False, CharacterEncoding -> Automatic, ConversionStyleEnvironment -> None, ConversionRules -> Automatic, HeadAttributes -> {}, HeadElements -> {}, CSS -> Automatic, ConvertLinkedNotebooks -> False, MathOutput -> GIF, GraphicsOutput -> GIF, Graphics3DOutput -> Automatic, ManipulateOutput -> CDF, ConvertClosed -> True, ConvertReverseClosed -> False, FullDocument -> True, AltMathOutput -> FileName, TableOutput -> {TextForm, Automatic}, AnimationOutput -> Automatic, FilesDirectory -> HTMLFiles, LinksDirectory -> HTMLLinks, HTMLEntities -> {HTML}, AllowBlockMathML -> False, ShowStyles -> True, DataUri -> False, MathMLOptions -> {UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], ColorSpace -> System`Convert`CommonDump`ConvertText[RGB, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, ConvertClosed -> False, ConvertReverseClosed -> False, ConvertLinkedNotebooks -> False, CharacterEncoding -> Automatic, ConversionStyleEnvironment -> None, ConversionRules -> Automatic, HeadAttributes -> {}, HeadElements -> {}, CSS -> Automatic, ConvertLinkedNotebooks -> False, MathOutput -> GIF, GraphicsOutput -> GIF, Graphics3DOutput -> Automatic, ManipulateOutput -> CDF, ConvertClosed -> True, ConvertReverseClosed -> False, FullDocument -> True, AltMathOutput -> FileName, TableOutput -> {TextForm, Automatic}, AnimationOutput -> Automatic, FilesDirectory -> HTMLFiles, LinksDirectory -> HTMLLinks, HTMLEntities -> {HTML}, AllowBlockMathML -> False, ShowStyles -> True, DataUri -> False, MathMLOptions -> {UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], Interleaving -> True], Selectable -> False], BaseStyle -> System`Convert`CommonDump`ConvertText[ImageGraphics, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, ConvertClosed -> False, ConvertReverseClosed -> False, ConvertLinkedNotebooks -> False, CharacterEncoding -> Automatic, ConversionStyleEnvironment -> None, ConversionRules -> Automatic, HeadAttributes -> {}, HeadElements -> {}, CSS -> Automatic, ConvertLinkedNotebooks -> False, MathOutput -> GIF, GraphicsOutput -> GIF, Graphics3DOutput -> Automatic, ManipulateOutput -> CDF, ConvertClosed -> True, ConvertReverseClosed -> False, FullDocument -> True, AltMathOutput -> FileName, TableOutput -> {TextForm, Automatic}, AnimationOutput -> Automatic, FilesDirectory -> HTMLFiles, LinksDirectory -> HTMLLinks, HTMLEntities -> {HTML}, AllowBlockMathML -> False, ShowStyles -> True, DataUri -> False, MathMLOptions -> {UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], ImageMargins -> 0., ImageSize -> {226.905, Automatic}, ImageSizeRaw -> {582, 425}, PlotRange -> {{0, 582}, {0, 425}}]}], }]                                 k - 1                k                k + 1                                                                                             k

Diagram_134.gif

Diagram_135.gif

FSA Example

FSA(aabc)                                        MorphoFSA[aabc]Diagram_136.gif

Diagram_137.gif

Diagram_138.gif

Monomorphic prolongation

First aspect: iteration
Given a morphogram MG, which is always a localized pattern in a kenomic matrix, a prolongation (successor, evolution) of the morphogram is achieved with the successor operator sDiagram_139.gif. To each prolongation a further prolongation is defined by the iterated application of the operator sDiagram_140.gif. 
The morphogrammatic succession Diagram_141.gif is founded by its model Diagram_142.gif and the morphism f, guaranteeing the commutativity of the construction.  
As a third rule, the iterability of the successor operation is arbitrary, which is characterised by the commutativity of the diagram. Hence, the conditions for a (retrograde) recursive formalisation are given.

Second aspect: anti-dromicity
Each prolongation is realized simultaneously by an iterative progression and an antidromic retro-gression. That is, the operation of prolongation of a morphogram is defined retro-grade by the possibilities given by the encountered morphogram. A concrete prolongation is selecting out of those possibilities its specific successions. All successions are to be considered as being realized at once.

Third aspect: simultaneity and interchangeability
This simultaneity of different successions defines the range of the prolongation. This definition of morphogrammatic prolongation is not requiring an alphabet and a selection of a sign out of the alphabet. Hence, the concept of morphogrammatic prolongation is defined by the two aspects of iteration and antidromic retro-gradeness of the successor operation. The simultaneity of the prolongations is modeled by the interchangeability of its actions.

Fourth aspect: diamond characterization of antidromicity
Both aspects together, repeatability and antidromicity with its simultaneous and interchangeable realizations, are covered by the diamond-theoretic concept of combination of operations and morphisms, i.e. composition and saltisition, between morphogramatic prolongations.

The philosophical status of morphoCAs has yet to be determined.“What’s after digitalism?” might give a hint.

Diagram_143.gif

Internal structure of the morphogrammatic transition rule

Recall definitions: classical transition rule

"Rigid computations have another node parameter: location or cell. Combined with time, it designates the event uniquely. Locations have structure or proximity edges between them. They (or their short chains) indicate all neighbors of a node to which pointers may be directed.

“CA are a parallel rigid model. Its sequential restriction is the Turing Machine (TM). The configuration of CA is a (possibly multi-dimensional) grid with a fixed (independent of the grid size) number of states to label the events. The states include, among other values, pointers to the grid neighbors. At each step of the computation, the state of each cell can change as prescribed by a transition function of the previous states of the cell and its pointed-to neighbors. The initial state of the cells is the input for the CA. All subsequent states are determined by the transition function (also called program)." Leonid A. Levin. Fundamentals of Computing.

http://www.cs.bu.edu/fac/lnd/toc/z/z.html

Morphogrammatic transition rule

Diagram_144.gif

Diagram_145.gif

NextGen is in this morphoCA context a retrograde recursive action and not to be confued by a classical recursion.

What makes the difference?

1. retro - grade recursivity
2. irreducible heterogeneity
3. interactivity and reflectionality

Diagram_146.gif

Diagram_147.gif

Diagram_148.gif

Flow charts for morphoCAs

Full mediation of input

Diagram_149.gif

Diagram_150.gif

Diagram_151.gif

Diagram_152.gif

Diagram_153.gif

Diagram_154.gif

Diagram_155.gif

Diagram_156.gif

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Diagram_158.gif

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Diagram_160.gif

Diagram_161.gif

Diagram_162.gif

Full interaction and mediation table for Diagram_163.gif

Diagram_164.gif

Diagram_165.gif

Diagram_166.gif

Discontexturality of distributed CAs

Diagram_167.gif

Poly-layered grid structure

Diagram_168.gif

An interpretation of the discontexturality diagram shows that the grid structure of distributed CAs of the morphoCAs are in fact not 1-D CAs but disseminated 1-D CAs. It also shows that disseminated CAs are not necessarily 2- or 3-dimensional or higher. What we see as a linear 1-D grid by the visualization of morphoCA actions is in fact a composition of different parallel 1-D grids projected onto an 1-D grid of an uninterpteted output.

Hence, in functional terms, there is no mapping from Diagram_169.gif -> {0,1,2,3} but a composition of partial sub-maps.

Nevertheless, poly-layered grids are not multi-layered, because the layers of a multi-layered system are unified under the umbrella of First-Order Logic with Modal logic and General Ontology (Upper Ontology). While dis-contexturality implies an interplay of a multitude of irreducibly different logics, each containing their inter- and trans-logical operators, additionally to the full set of intra-logicial operators too.

Multi-layerd systems are logically defined by the basic intra-logical operations only. Poly-layerd systems are involved in an interplay of dis-contextural operations of inter- and trans-contextural actions.

This discontextural approach obviously is in strict conflict with Proposition1 of category theory and its unique universe U:

                If x ∈ U and y ⊆ x, then y ∈ U.

As usual in such fundamental situation, the proposition is circular. It presumes the uniqueness of its logical universe to work for a definition of its unique category-theoretical universe which is taken as the base for the definition of First-Order Logic and its unique universe of terms.

“Polycontexturality alone is not enough to realize the interwoven dynamics a new world-view is desperate for. Gotthard Gunther introduced his proemial relationship to dynamize his contextures, albeit still restricted to a uni-directional movement.The concept of metamorphosis as part of the diamond strategies, based on polycontexturality and disseminated over the kenomic matrix, is a further step to realize a radical paradigm change in our way of thinking and designing futures.”

http : // memristors.memristics.com/Polyverses/Polyverses.html

The projection marks the difference of the deep-structure and the surface structure of the productions of morphoCAs.

It makes it clear, again, that “What you see is not what it is”. Hence, any ontologizing will fail.

Diagram_170.gif

Diagram_171.gif

Diagram_172.gif

The difference between multi-layered and poly-layered systems got a conceptual sketch with the paper:

Memristics: Dynamics of Crossbar Systems
Strategies for simplified polycontextural crossbar constructions for memristive computation

“Interchangeability is part of a new axiomatics of poly-categorical diamond systems still to be developed. Interchangeability is defined intra-contextural for composition and yuxtaposition, and trans-contextural for interactions, like mediation, replication, iteration and transposition.”

http://www.thinkartlab.com/Memristics/Poly-Crossbars/Poly-Crossbars.pdf

Claviatures for morphoCAs

Claviature: ruleDM

Claviature: Random ruleDM

Claviature: ruleDCKV

Analysis of ruleDM[{1,11,3,9,x}]

Diagram_176.gif

Diagram_177.gif

Diagram_178.gif

Diagram_179.gif

Distribution density

Diagram_180.gif

Flow chart for ruleDM[{1,11,3,9,x}]

Diagram_181.gif

Diagram_182.gif

Explicit transition system table for ruleDM[{1,11,3,9,x}]

Diagram_183.gif

Diagram_184.gif

Diagram_185.gif

Diagram_186.gif

Diagram_187.gif

Diagram_188.gif

Example: ruleDM[{1, 11, 3, 9, 15}] step-wise realization

start (init) : yellow, red

Diagram_190.gif

Diagram_191.gif

Diagram_192.gif

Start of the morphoCA with init {{1}, 0} producing the entry “red” with an environment “yellow” with the properties defined at step 37 by the rules:
{0,0,0}→0 of sys1||sys3 and  {0,1,0}→0, {1,0,0}→0  of sys1.

construction: blue, yellow, red, yellow

Diagram_193.gif

Diagram_194.gif

Diagram_195.gif

At step 44, the memory decides that the received value “2” doesn’t belong to its range, i.e. the system 1, defined by the values {0,1}.
The value “2” of system 3 defines a new start at the system 3 with the properties of  {0,0,2}→1, {0,2,0}→0, {2,0,0}→0.

Diagram_196.gif

Diagram_197.gif

Diagram_198.gif

Diagram_199.gif

Again, at the step 66, the decider of the memory unit of system 3 decides that the value “1” doesn’t belong to its range, i.e. the system 3, defined by the values {0,2}.
The value “1” of memory 3 defines a continuation in the system 1 with the background properties of {0,2,0}→0, {2,0,0}→0.

The background is symbolized numerically by 0, i.e. yellow. But “0” belongs to 2 different sub-systems defined by {0, 1} and {0, 3}.
What counts is not just the value in a system but its contextual relation or difference to other values. Hence the presupposed rule: {0,0,0} -> 0, holds in general but its significance depend on its context.

iteration of construction

Diagram_200.gif

Diagram_201.gif

Diagram_202.gif

At step 88, the memory decides that the received value “2” doesn’t belong to its range, i.e. the system 1, defined by the values {0,1}.
The value “2” of system 3 defines a new start at the system 3 with the properties of  {0,0,2}→1, {0,2,0}→0, {2,0,0}→0.

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2}, {23, 2}}, {{0, 1}, {23, 1}}, {{0, 0}, {23, 0}}}], Antialiasing -> False]}, {GrayLevel[0.618034], StyleBox[LineBox[{{{0, 0}, {0, 23}}, {{1, 0}, {1, 23}}, {{2, 0}, {2, 23}}, {{3, 0}, {3, 23}}, {{4, 0}, {4, 23}}, {{5, 0}, {5, 23}}, {{6, 0}, {6, 23}}, {{7, 0}, {7, 23}}, {{8, 0}, {8, 23}}, {{9, 0}, {9, 23}}, {{10, 0}, {10, 23}}, {{11, 0}, {11, 23}}, {{12, 0}, {12, 23}}, {{13, 0}, {13, 23}}, {{14, 0}, {14, 23}}, {{15, 0}, {15, 23}}, {{16, 0}, {16, 23}}, {{17, 0}, {17, 23}}, {{18, 0}, {18, 23}}, {{19, 0}, {19, 23}}, {{20, 0}, {20, 23}}, {{21, 0}, {21, 23}}, {{22, 0}, {22, 23}}, {{23, 0}, {23, 23}}}], Antialiasing -> False]}}}, {RGBColor[1, 0, 0], Thickness[0.01]}}, ImageSize -> {74.3477, Automatic}],  }], , step 110 :  {0, 0, 2} &rarr;1, {0, 2, 0} &rarr;0, {2, 0, 0} &rarr;0,}]

Diagram_204.gif

Diagram_205.gif

Again, at the step 110, the decider of the memory unit of system 3 decides that the value “1” doesn’t belong to its range, i.e. the system 3, defined by the values {0,2}. The value “1” of memory 3 defines a continuation in the system 1 and in system 2 with the properties of  {0,1,0}→0, {1,0,0}→0.
And so on.

Unfortunately it is necessary to go through these tedious phenomenological interpretations of the mechanism of morphoCAs because without this kind of modelling it isn’t possible to understand the nature of their outcome. Just to enjoy interesting pictures and listening to unheard sounds is not yet enough to understand the novelty of the morphogrammatic approach towards cellular automata and automata in general.

The switch from one automaton to the net of automata is not just ruled by the clock but also by the logic of the unit. If there is a transjunctional result of the logical unit, the calculations have to switch to another automaton. Different types of polycontextural transjunctions are ruling such interactions. Otherwise, without a switch, it stays inside the domain of the automaton for further intra-contextural calculations.

http://www.thinkartlab.com/pkl/media/Dynamic%20 Semantic %20 Web.pdf

http : // memristors.memristics.com/Notes %20 on %20 Polycontextural %20 Logics/Notes %20 on %20 Polycontextural %20 Logics.pdf

PCA, programmable CAs

“As the matter of fact, PCA are essentially a modified CA structure. It employs some control signals on a CA structure. By specifying certain values of control signals at run time, a PCA can implement various functions dynamically in terms of different rules.”

http://infonomics-society.ie/wp-content/uploads/ijicr/published-papers/volume-3-2012/Security-of-Telemedical-Applications-over-the-Internet-using-Programmable-Cellular-Automata.pdf

For morphoCAs, the range of reconfiguring processors is not limited to the range of classical CAs but spans over a wide range of trans-classical paradigms of morphoCAs also including classical CAs.

The specification of morphoCAs shows clearly the paradigmatical difference between morphoCAs, ECAs and PCAs.

Concerning the sub-rule approach, morphoCAs might be seen as ‘hybrid’ CAs with transjunctional functions and mediation to be considered.

3. PCL diagrams for Diagram_206.gif with interaction and mediation

Analysis of minimized ruleDCM[{1, 2, 12, 13, 5}]

Diagram_207.gif

Diagram_208.gif

Analysis

Diagram_209.gif

Distribution density

The distribution density of a morphoCA constellation gives a simple measure for classification and comparison of morphoCAs.
It holds for reduced and non-reduced morphoCA constellations.

Diagram_210.gif

Analysis of the interaction patterns

Diagram_211.gif

Diagram_212.gif

Diagram_213.gif

Diagram scheme for ruleDCM[{1, 2, 12, 13, 5}]

Diagram_214.gif

Diagram_215.gif

Diagram_216.gif

Analysis of ruleDM[{1, 2, 12, 13, 5}]

ruleDM[{1, 2, 12, 13, 5}]

Diagram_217.gif

Analysis

intra + inter + trans

Diagram_218.gif

Distribution density

Diagram_219.gif

Analysis of the interaction patterns

Diagram_220.gif

Diagram_221.gif

Diagram_222.gif

Diagram_223.gif

4. PCL diagrams with interactions and mediations: Diagram_224.gif

Analysis of ruleDM[{1,11,3,4,15}]

Diagram_225.gif

Diagram_226.gif

Reduced

Diagram_227.gif

Diagram_228.gif

Diagram_229.gif

Random (restricted by reduction)

Diagram_230.gif

Analysis

Analysis of ruleDM[{1,11,3,4,15}]

Diagram_231.gif

Distribution density

Diagram_232.gif

Diagram_233.gif

Analysis of the interaction patterns

Calculation: intra-contextural action

Diagram_234.gif

Diagram_235.gif

Diagram_236.gif

Alternation: trans-contextural action from sys1 to sys2||sys3 and from sys3 to sys2||sys1

Diagram_237.gif

Mediation: poly-layered action

Diagram_238.gif

Interpretation of mediation

Diagram_239.gif

Diagram_240.gif

Transition system table for ruleDM[{1,11,3,4,15}]

Diagram_241.gif

Diagram scheme for ruleDM[{1, 11, 3, 4, 15}]

Diagram_242.gif

The rules placed in the first half are the rules of intra-contextural actions. They don’t refer to other contextures. The rules in the upper part represent the trans-contextural actions between different contextures depicted as directed arrows.

The compound morphogram of ruleDM[{1, 3, 4, 11, 15}] reflects the mediation of intra- and inter-contextural actions of the flow chart. It is the morphogram compound of the flow chart of the actions of the morphoCA ruleDM[{1, 3, 4, 11, 15}] .

Diagram_243.gif

Non-reducible examples

Non-reducible automata definitions might be used as complete irreducible building-blocs for complex morphoCAs.

For complete irreducible building-blocs, all entries of the transition table are occupied. In other terminology, all intra-, inter- and trans-contextural sections of the flow-chart scheme are occupied.

Irreducible rules are playing the same role for morphoCAs as the irreducible binary functions like NAND, XOR for binary reductions. With NAND or NOR, all other two-valued binary function are defined.  Because they are not reducible they are used as elementary devies in electronic circuit consturctions.

Unfortunately, there is not yet an algorithmic procedure to minimize (reduce) the functional representation of morphoCA rules.

The question for morphic patterns arises: How many irreducible patterns exist for Diagram_244.gif?

In analogy:
“No logic simplification is possible for the above diagram. This sometimes happens. Neither the methods of Karnaugh maps nor Boolean algebra can simplify this logic further. [..] Since it is not possible to simplify the Exclusive-OR logic and it is widely used, it is provided by manufacturers as a basic integrated circuit (7486).”

http://www.allaboutcircuits.com

http : // memristors.memristics.com // Reduction %20 and %20 Mediation/Reduction %20 and %20 Mediation.pdf

Example : ruleDM[{1, 2,12,4,15}]

reducible to steps 22

Diagram_245.gif

Diagram_246.gif

Reducts

Diagram_247.gif

Diagram_248.gif

Not reduced

Diagram_249.gif

Diagram_250.gif

Random

Diagram_251.gif

Analysis

Analysis of the interaction patterns

Computation: intra-contextural actions

Diagram_252.gif

Diagram_253.gif

Diagram_254.gif

Diagram_255.gif

Diagram_256.gif

Diagram_257.gif

Alternation: inter-contextural actions

Diagram_258.gif

Mediation: poly-layered trans-contextural action

Diagram_259.gif

Example: ruleDM[{1, 11, 12, 9, 15}]

ruleDM[{1, 11, 12, 9, 15}] : reducible with {3,3,3}-> 3 for steps <33

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Non - reducible for steps >22

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Random

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Example : ruleDM[{1, 11, 12, 4, 15}]

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Random

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Analysis

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Distribution density

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Analysis of the interaction patterns

Computation: intra-contextural actions

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Alternation: inter-contextural actions

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Mediation: poly-layered trans-contextural action

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Example : ruleDM[{1, 11, 8, 4, 15}]

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Analysis

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Analysis of the interaction patterns

Computation: intra-contextural action

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Alternation: inter-contextural actions

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Mediation: poly-layered trans-contextural action

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ruleDCKV reduction examples

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Analysis

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distribution - type (reduced)

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Analysis of the interaction patterns

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Further example: non-reducible

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Not reduced

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Random

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Layers of morphoCA sub-systems of ruleDM[{1,11,3,4,15}]

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All together

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Step - wise developments of ruleDM[{1,11,3,4,15}]

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Transition graphs of reduced ruleDM[{1,11,3,4,15}]

Additionally to the difference of reduced and non-reduced morphoCA rule in respect to their seed structure, there is also an interesting difference between ArrayPlot visualizations and transition graph representations by the GraphPlot of reduced morphoCA rules to observe. All reductions are conserving the full visualization of the original, while the transition structure is significantly reduced.

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Reduction steps

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Full pattern

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Without {1, 1, 1} -> 1, {2, 2, 2} -> 2, {3, 3, 3} -> 3,

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Fully reduced

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Created with the Wolfram Language