Interactions with Claviatures

Short ofClaviatures for morphic and indicational Sound and Graphic CAs’  

Dr. phil Rudolf Kaehr
© ThinkArt Lab Glasgow
ISSN 2041-4358
( work in progress, v. 0.5.5 July, May 2015 )

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.

With the sub-rule approach the single rule configuration that are defining an actual machine are constructed by the chosen keys of the claviature. Like for musical keyboards the melodies are composed by the chose of the keys and are not looked up from a list of stored melodies.

Even for a quite simple example of a CA based on the indicational rules indRCI, the complexity is not to handle by classical approaches. The sub-rule approach offers a claviature of the rule set so that all individual possibilities of the rule space of size 4^20=1’099’511’627’776 are manually accessible.

The complexity of claviatures remains in a finite range of small sets of rules measured by the sum of the Stirling Numbers of the Second Kind.

Hence the rule space of ruleDM of the first example of the Claviatures is defined by the 15 morphograms distributed over 15 places generating the combination of  2x3x3x3x4 = 216 single morphogrammatic compounds of ruleDM[{k,l,m,n,o}] with k={1,6}, l={2,7,11}, m={3,8,12}, n={4,9,13} and o={5,10,14,15}.

Therefore the claviature of ruleDM with its 15 keys defines all 216 different potential realizations of the automaton morphoDM. Because the number of functions for this morphoCA is small and manageable there is no barrier to define the functions explicitly.
But the rule space for ruleDCKV of Interaction with Claviatures_1.gif is 2x3^7x4^6 x 5 = 89’579’520. There is certainly no realistic chance to define this amount of rules and to handle it explicitly.

The case for the indicational automaton indRCI with its ruleRCI[{a,b,c,d,e, f,g,h,i, j,k,l,m,n,o,p,q,r,s,t}], where all components have 4 mutually excluding different indicational rules, the rule space is intriguingly less accessible without the sub-rule approach proposed in this paper.

The 20 positions of the automaton indRCI are defining 4^20=1’099’511’627’776 different potential realizations of the indicational rule space of indRCI. In contrast, the rule space of ECA is 2^8 = 256.

Rule space table

Interaction with Claviatures_2.gif

The current presentation of Claviatures for 1D automata is not restricting its application to 1D CAs, all higher order cellular automata of arbitrary dimensions are incuded to the application of claviatures.

Epistemologically, there is a paradigm change involved which turns the definition of classical CAs from an ‘algebraic’ to a ‘co-algebraic’ understanding of generalized CAs.

The co-algebraic approach emphasizes the ‘stream’ of computational events, and configurations are selected out of the stream by selectors like the proposed claviatures. Therefore there is no need to construct all the possible constellations step by step by CA rules.

Instead of developing reduction techniques to reduce the complexity of CAs, the claviatures approach plays on a meta-level with CAs that are accessible by selection. This leads to the well known automata theoretic concept of experiments with automata.

Algebraic structures have to be constructed, co-algebraic structurations have to be selected by interaction.

There is not just a simple duality between algebras and co-algebras in respect of constructors and destructors but also a not well recognized asymmetry between the pair “constructors/destructors” and the new deconstructors. A chiastic system change happens from the selectors of the destruction to the observators of co-algebras under the interaction of experiments.

Duality of algebras and co-algebras

Interaction with Claviatures_3.gif

Interaction with Claviatures_4.gif

Asymmetric shift from internal to external descriptions of selectors and observators

Interaction with Claviatures_5.gif

Interaction with Claviatures_6.gif

The organon of the claviatures

The claviature approach is exemplified with the morphogrammatic CAs for ruleDM, ruleDMN, ruleDMNP and ruleDCKV. Also for the indicational CAs for ruleCI, ruleCIR and ruleRCI. All are applied to the categories of graphics, sound, transition graphs and fixedpoint determination. The case for ECA is exemplified for all categories too.

The method of sub-rules for CAs is an abstraction and parametrization of the components of the rule schemes that allows a micro-analysis of the CAs. The CA sub-rule manipulator manages explicitly all CA rules of a 1D CA. The sub-rule manipulators enables a micro-analysis of the behavior of all CA rules. Comparisons of the behavior of rules, especially of groups, families and clusters of sub-rules, are part of a new kind of micro-analysis based comparatistics.

Further reading

Peter Wegner, Why Interaction is more Powerful than Algorithms, 1997

Dina Goldin, Peter Wegner, The Interactive Nature of Computing: Refuting the Strong Church-Turing Thesis

Skizze eines Gewebes rechnender Räume in denkender Leere (2004)

[[interactive view with .html and .cdf]]



Procedures morphoDCKV



Morphogram: ruleDM

Morphogram: Random ruleDM

Morphogram: ruleDMN

Morphogram:  ruleDMN, Random

Morphogram: ruleMNP

Morphogram: Random ruleDMNP

Morphogram: ruleDCKV

Morphogram: ruleDCKV, Random

Indication: ruleCI

Indication: ruleCI random

Indication: ruleCIR

Indication: ruleCIR Random

Indication: ruleRCI

Indication: Random ruleRCI


ECA Random



Morphogram: ruleDM

Morphogram: ruleDMN

Morphogram: ruleMNP

Morphogram: ruleDCKV

Indication: ruleCIR

Indication: ruleRCI

Structures: Transition Graphs


Morphograms: ruleDM

Morphograms: ruleDMN

Morphograms: ruleDCKV

Indication: ruleCIR

Indication: ruleRCI


FixedPoints: ECA

Number of events, rule number and sub - rules

FixedPoints: ruleM

Structure of self - modifications for the morphogrammatic calculus morphoCA DM

FixedPoints: ruleMN

Structure of self - modifications for the morphogrammatic calculus morphoCA DMN

Interaction with Claviatures_44.gif

FixedPoints: ruleCIR

FixedPoints: ruleDCKV

Spikey Created with Wolfram Mathematica 9.0