Systems of Claviatures for morphoCAs

    Applications of the sub-rules approach to cellular automata
          

Dr. phil Rudolf Kaehr
copyright
© ThinkArt Lab Glasgow
ISSN 2041-4358
( work in progress, v. 0.2, July 2015 )


   Systems of morphoCAs

Conceptual background

Claviatures gives a glimpse into the usefulness of the sub-rule approach for all kind of cellular automata. The merits of the sub-rule approach becomes evident for highly complex automata where it is practically not achievable to manipulate all single rules of the automaton explicitly.

In contrast to classical concepts of CAs, morphoCAs don’t come as a singular simple concept and apparatus. Also classical concepts of CAs have many different ways of extending the elementary framework of ECAs they are not changing their basic architectonics.

The morphogrammatic approach to CAs is based on morphograms. Morphogrammatics is a dynamic structuration (system) with at least two dimension: Evolution and emanation.

Hence, CAs based on morphogrammatics are inheriting such a dynamics of emanation and evolution.

Evolution is understood as a retro-grade recursive development towards higher complexity, reflected as a transition from Claviatures-(5,x,5+0)_1.gif to Claviatures-(5,x,5+0)_2.gif, while emanation is a differentiation of the structure of a Claviatures-(5,x,5+0)_3.gif to Claviatures-(5,x,5+0)_4.gif.

Embodiments of morphoCAs into morphoCAs of higher complexity is possible too.  

Therefore, a morphogrammatic system of CA is always embedded in the general system. This happens in 3 ways:
1. Over-balanced:  Claviatures-(5,x,5+0)_5.gif,  m>n,
2. Balanced: Claviatures-(5,x,5+0)_6.gif,            m=n,
3. Under-balanced:  Claviatures-(5,x,5+0)_7.gif, m<n.

What follows is an introduction of the balanced system Claviatures-(5,x,5+0)_8.gif that entails systems Claviatures-(5,x,5+0)_9.gifand Claviatures-(5,x,5+0)_10.gif. The present implementation is not yet working generally, i.e. for any number of steps.

The recently published system Claviatures-(5,x,5+0)_11.gif (DCKV) is morphogrammatically complete and correct, in the sense that all accessible tested functions are positively realized for any of the tested steps of application. This is not yet fully achieved for the case of Claviatures-(5,x,5+0)_12.gif.

Steps towards the under-balanced system Claviatures-(5,x,5+0)_13.gif are sketched for ‘experimental’ reasons.

There is certainly a possibility to formalize and program the development of the architectures of morphoCAs from one level to the next as the development of the morphogrammatic base of morphoCAs is elaborated correspondingly.

Therefore there is a new field to discover: the programming of new morphoCA rule sets out of the previous morphoCA rule sets.

This introduces a two-dimensional development of morphoCAs: the production of the rule sets of a specific complexity and complication of the dynamics and the study of the application of the morphoCA rules on such levels of the dynamics of the architectonics of the morphoCA systems.

From the point of view of morphogrammatic system analysis it is easy shown that the base of the ECA approach is incomplete.

morph(ECA) = Claviatures-(5,x,5+0)_14.gif,  Claviatures-(5,x,5+0)_15.gif⊂  Claviatures-(5,x,5+0)_16.gif.

   Claviatures-(5,x,5+0)_17.gif = [[1,1,1,1],[1,1,2,2],[1,2,1,2],[1,2,2,1],[1,1,1,2],[1,1,2,1],[1,2,1,1], [1,2,2,2]].
   Claviatures-(5,x,5+0)_18.gif = [[1,1,1,1],[1,1,2,2],[1,2,1,2],[1,2,2,1],[1,1,1,2],[1,1,2,1],[1,2,1,1],[1,2,2,2],
                    [1,1,2,3],[1,2,1,3],[1,2,3,1],[1,2,2,3],[1,2,3,2],[1,2,3,3],[1,2,3,4]].

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System of Claviatures

Initialization

Morphograms Claviatures-(5,x,5+0)_25.gif

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[1, 1, 1, 1, 1]  [1, 2, 3, 4, 5]

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Numeric representation of Claviatures-(5,x,5+0)_40.gif :

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Second refinement: D(TM[5,5]): 
1, (4+1, 6+4), (6+3+1, 12+2+1), 4+3+3, 1.

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Claviature Claviatures-(5,x,5+0)_44.gif

Examples and comparisons for Claviatures-(5,x,5+0)_46.gif

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Claviature Claviatures-(5,x,5+0)_78.gif, seed

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Examples Claviatures-(5,x,5+0)_92.gif

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Claviature for Claviatures-(5,x,5+0)_345.gif

Spikey Created with Wolfram Mathematica 9.0