Reduction and Mediation of morphoCAs
                     Functional Analysis of morphoCAs
          

Dr. phil Rudolf Kaehr
copyright
© ThinkArt Lab Glasgow
ISSN 2041-4358
( work in progress, v. 0.3, October 2015 )

    Motivation

Despite the decision for a morphogrammatic foundation of morphic cellular automata, a more function-oriented analysis of morphoCAs is applied to give some new insights into the techniques of reduction and mediation of the complication/complexity of morphic CAs as they have been proposed in previous papers.

Well known features of polycontextural and morphogrammatic systems (structurations), like super-addivity of mediation and super-subtractivity of decomposition of morphoCAs will be sketched.

Stirling reductions

A very first reduction of the quantitative complexity of functions is given by the fact that morphoCAs are not based on exponential functions but on Stirling partitions. Hence, instead of having Reduction and Mediation_1.png ternary functions of a three-valued CA, morphoCAs are based on StirlingS2, Reduction and Mediation_2.png, hence Sum[StirlingS2[27,m],{m,3}]

The reduction is impressive:
From  
Reduction and Mediation_3.png= 7 625 597 484 987 three-valued ternary functions to Sum[StirlingS2[27,m],{m,3}] = 1 270 932 914 165 three-valued ternary Stirling functions.

There are Reduction and Mediation_4.png = 19 683 three-valued binary functions and just 3281 = Sum[StirlingS2[9,m],{m,3}] three-valued binary Stirling patterns. For the unary three-valued case there are Reduction and Mediation_5.png= 27 functions. And Sum[StirlingS2[4,m],{m,4}] = 15 for the number of morphograms.

Redundancy-Reductions

Reduction of redundancy in the symbolic (numeric) interpretation of morphogrammatic CA rules. This kind of reduction is not changing the morphogrammatic structure of the morphoCAs but is minimizing the redundant elements of its interpretation. Only the relevant elements (symbols, numbers) of the interpretation of the morphogrammatic compounds necessary for the realization of morphoCAs are considered. Until now, this reduction didn’t play a significant role in the definition of the newly introduced morphoCAs and their claviatures.

To make morphograms visible, they have to interpreted. But the complexity of the interpretations is contextually depending on the whole configuration of the morphogrammatic compounds. There is not just one reduced interpretation available for all situations. For non-reducible morphoCAs the full range of the possible interpretations is necessary to fulfill the definition and realization of the morphoCA.

Single morphograms get different complex interpretations depending on their morphoCA environment.

The morphogram [1] in ruleDM[{1,11,8,4,15}] or ruleDM[{1,11,8,9,15}] demands for an interpretation with the set of all possible mappings in the system of complexity 4: {0,0,0}→0, {1,1,1}→1, {2,2,2}→2, {3,3,3}→3.

In contrast, a similar constellation like ruleDM[{1, 11, 3, 9, 15}] demands in the same system for the interpretion of morphogram [1] just one mapping with {0,0,0}→0.

Depending on the context, there are other interesting cases too.
Just 3 interpretetions needed:
ruleDM[{1,2,12,13,14}] with the mapping {0,0,0}→0, {1,1,1}→1, {2,2,2}→2 for morphogram [1]. With a litle change by replacing the morphogram [14] by the morphogram [15], all interpretations accessible in the system are needed.

Because morphoCAs are dynamical systems, it might take a non-trivial amount of steps to observe the necessity of specific interpretations. The morphoCA may work without problems for some steps without a complementation of the interpretations. But it will show lacks of realizations some steps later.

It also turns out that the reductions are not necessarily fully working for arbitrary seeds, like Random seeds.

Nevertheless, not all morphoCAs need a full set of interpretations.

That shows nicely the context-dependence of the Interpretation of morphograms in morphoCAs.

Karnaugh reductions

A further functional reduction might be achieved with the help of Karnaugh maps. Pattern reductions are not the same as Karnough reductions. Pattern reductions are reducing the redundancy of the interpretations of the formulas without changing the formal structure of their definitions (literals, terms). In contrast, Karnaugh mappings are changing the definitions of the functions by reducing the numbers of variables of the original formula albeit without disturbing their functional purpose.

Equivalent to the Karnaugh maps, simplification of Boolean expressions are delivering the same result. For morphoCAs, Boolean expressions have to be adjusted to polycontextural logical formulas and their reduction techniques.

The sketched specifications of the structure of morphoCAs could be understood as a ‘blue print’ for a physical realization as a kind of new multi-layered chips for morphoCA machines (postToffoli).

UML process modelling

A first step to a UML modelling of morphoCAs could encourage to get a glimpse into the complex mechanism of morphoCAs.

Mediation Question:
Given two elementary CA (Boolean one - dimensional bi - infinite lattices of cells, the evolution of each cell being influenced by its direct neighbors) with known behaviors, what can be said on the behavior of the CA obtained by composing them (e.g., using a logic disjunction)?”

https://scs.carleton.ca/sites/default/files/tr/TR-96-31.pdf

Again, Reduction and Mediation

Functional interpretation of ruleDM[{1, 2, 12, 4, 5}] in Reduction and Mediation_6.png.

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transjunctional part of funct(ruleDM[{1, 2, 12, 4, 5}])

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mediating part of funct(ruleDM[{1, 2, 12, 4, 5}])

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Decomposition of ruleDM[{1, 2, 12, 4, 5}] into polylogical sub-systems

Junctional part with transjunctions

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mediating part

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Cellular automaton of ruleDM[{1, 2, 12, 4, 5}] in Reduction and Mediation_12.png

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Functional reductions

Example : ruleDM[{1, 2, 12, 4, 5}]

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Random seeds

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Without reduction, the random seeds are completely defining the morphoCA.

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ruleDM[{1, 2, 8, 4, 5}] : parallel

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Functional reduction of the morphic rules

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Mediation of Reduction and Mediation_27.png= (junctional) and Reduction and Mediation_28.png with functional reduction: final result is Reduction and Mediation_29.png= (Reduction and Mediation_30.png, Reduction and Mediation_31.png, mediating part). Additionally, the elimination of {0,1,0} of Reduction and Mediation_32.pngand {0,2,0}, {2,0,0} of Reduction and Mediation_33.png happens.

Is Reduction and Mediation_34.png a genuine CA like Reduction and Mediation_35.png?

(NKS page : 886 R)

Super-additivity of the mediation of Reduction and Mediation_36.pngand Reduction and Mediation_37.png

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Reduction and mediation

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Union of Reduction and Mediation_45.pngand Reduction and Mediation_46.png

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Yellow represents the background system of both, the blue and the red sub-system.
In the terms ‘blending theory’ (Goguen), the yellow system is the ‘base ontology’, while the red and the blue systems are the ‘input ontologies’  of the process of blending, resulting in the mediation, i.e. the ‘blendoid’, as the interaction of the red and the blue system on the background of yellow  system.

It shouldn’t be difficult to understand that the sketched method of separation and mediation works for more complex cases too.

Super-additivity and concept blending

One reason while blending is not an immanent feature of logic, semiotics and computation might be the fact that it happens only on the level of semantics. Like the term “houseboat” and “boathouse”, they are syntactically not produced by blending but by concatenation. The meaning of the composits is achieved by blending, and there is no simple concatenation of the meanings “boat” and “house” to “houseboat” and “boathouse” and super-additively to “amphibious vehicle". Both meaning of “boat” and “house” are per se autonomous.

"A classic example for this is the blending of the concepts house and boat, yielding as most straightforward blends the concepts of a houseboat and a boathouse, but also an amphibious vehicle.” (Kutz, Hois)

Reduction and Mediation_49.png

http : // memristors.memristics.com/Dissemination %20 as %20 Blending/Dissemination %20 as %20 Blending.html

http : // osl.cs.illinois.edu/media/papers/tosic - 2005 - parallel_vs _sequential _threshold _cellular _automata.pdf

“We shall illustrate the concept of sequential interleaving semantics of concurrency with a simple exercise.
Let' s consider the following question from a sophomore parallel programming class :
Find an example of two instructions such that, when executed in parallel, they give a result not obtainable from any corresponding sequential execution sequence.

A possible answer : Assume x = 0 initially and consider the following two programs

x ← x + 1; x ← x + 1
vs.
x ← x + 1 || x ← x + 1

where " || " stands for the parallel, and ";" for the sequential composition of instructions or programs, respectively.
Sequentially, one always gets the same answer : x = 2.
In parallel (when the two assignment operations are executed synchronously), however, one gets x = 1. It appears, therefore, that no sequential ordering of operations can reproduce parallel computation - at least not at the granularity level of high - level instructions as above.

Indeed, if we informally define Φ(P) to be the set of possible behaviors of program P, then the example above only shows that, for S1 = S2 = (x←x+1),

              Φ(S1||S2) !⊆ Φ(S1;S2) ∪ Φ(S2;S1)   (1)

However, it turns out that, in this particular example, it indeed is the case that
             
             Φ(S1||S2) ⊆ Φ(S1;S2) ∪ Φ(S2;S1) ∪ Φ(S1) ∪ Φ(S2)   (2)

and no finer granularity is necessary to model Φ(S1||S2), assuming that, in some of the sequential interleavings, we allow certain instructions not to be executed at all.” (Tosić)

Φ(S1||S2) ::
Φ(S1;S2) ∪ Φ(S2;S1) ∪ Φ(S1) ∪ Φ(S2) :
Φ(S1;S2) ∪ Φ(S2;S1) : double serial composition,
Φ(S1) ∪ Φ(S2)            : super-additivity

Reduction and Mediation_50.gif

Minimization of morphoCA functions

Reduction and Mediation_51.gif

http : // www.toves.org/books/logic/

MG - composition -> value-interpretation -> functional reduction -> MG-interpretation, with such formal reductions of morphoCAs a context of technical realizations seems to be sketched.  

Reduction and Mediation_52.png

The proposed functional reduction of morphoCA is not changing its morphogrammatic base, i.e. its morphograms, but the symbolic interpretation of the morphograms only. Therefore, functionally reduced morphoCAs are still morphoCAs, and nothing else.

ruleDM[{1, 2, 12, 4, 5}]

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The decomposition of ruleDM[{1, 2, 12, 4, 5}] into polylogical sub-systems might be generalized towards the following scheme.

Reduction and Mediation_56.png

Full rule scheme

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      Frame scheme

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      Transjunctional part                             mediative part

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As an example ruleDM[{1, 2, 12, 4, 5}] might be decomposed:

Reduction and Mediation_60.png

      Transjunctional part                                    mediative part

Reduction and Mediation_61.png

The decomposed scheme for ruleDM[{1, 2, 12, 4, 5}] resumed

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Reduced example of ruleDM[{1, 2, 12, 4, 5}]

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Application of the reduced ruleDM[{1, 2, 12, 4, 5}]

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Table notation of the reduced ruleDM[{1, 2, 12, 4, 5}]

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Elimination of the reduced values (in blue)

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A different notation

A/B 0 1 2 0 1 2 0 1 2
0 0 2 1 1 1 0 2 0 0
1 1 1 0 2 1 0 1 1 1
2 1 0 2 2 1 2 2 0 2
C 0 - - 1 - - 2 - -

val (ruleDM[{1, 2, 12, 4, 5}]) = 1 :

A/B 0 1 2 0 1 2 0 1 2
0 1 1 1
1 1 1 1 1 1 1
2 1 1
C 0 - - 1 - - 2 - -

val(ruleDM[{1,2,12,4,5}]) = 1 iff
A0B2C0 + A0B0C1 + A0B1C1 +
A1B0C0 + A1B1C0 + A1B1C1+ A1B0C2 + A1B1C2 + A1B2C2 +
A2B0C0 + A2B1C1

ruleDCl[{1, 2, 8, 4}]

Reduction and Mediation_70.png

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Reducible and non-reducible functional representations

Example : ruleDM[{1, 7, 3, 13, 10}]

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Functional reduction of ruleDM[{1,7,3,13,10}]

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

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Random seed

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Functional reduction of ruleDM[{1,11,3,13,15}]

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

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Random seed

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Reduction : ruleDM[{1, 11, 3, 4, 15}]

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ruleDM[{1, 11, 3, 4, 15}] is reducible to table:

Reduction and Mediation_101.png

Example ruleDCl[{1, 2, 3, 4}]

Reduction and Mediation_102.png

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Reductions by Karnaugh maps

“Karnaugh maps reduce logic functions more quickly and easily compared to Boolean algebra. By reduce we mean simplify, reducing the number of gates and inputs. We like to simplify logic to a lowest cost form to save costs by elimination of components. We define lowest cost as being the lowest number of gates with the lowest number of inputs per gate.”

http : // www.allaboutcircuits.com/textbook/digital/chpt - 8/karnaugh - maps - truth - tables - boolean - expressions/

x' y' z + y z' + x y

Reduction and Mediation_108.gif

http : // www.electronics - tutorials.ws/combination/comb_ 1. html

Further example

Reduction and Mediation_109.gif

http : // www.allaboutcircuits.com/textbook/digital/chpt - 8/karnaugh - maps - truth - tables - boolean - expressions/

http : // www.ijser.org/researchpaper %5 CFormulation - and - Design - of - Useful - Logic - Gates - Using - Quaternary - Algebra.pdf

Towards Karnaugh maps

x/yz 00 01 11 10
0 0 1 0 1
1 0 0 1 1

Reduction and Mediation_110.gif

http : // www.toves.org/books/logic/

Examples for Reduction and Mediation_111.png

Example1: ruleDM[{1, 11, 3, 13, 15}]

Reduction and Mediation_112.png

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ruleDM[{1, 11, 3, 13, 15}] is minimized manually to the table :

Reduction and Mediation_114.png

Example2: ruleDM[{1, 11, 3, 4, 15}]

Equality test: ruleDM[{1,11,3,4,15}] equal reduct(ruleDM[{1,11,3,4,15}])

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ruleDM[[{1, 11, 3, 4, 15}] is minimized manually to the table :

Reduction and Mediation_121.png

Not reduced table for ruleDM[{1, 11, 3, 4, 15}].

Reduction and Mediation_122.png

Reduction and Mediation_123.png

Irreducible rules

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.

The question for morphic patterns arises: How many irreducible patterns exist for Reduction and Mediation_124.png?

Do they have the same property to define all other functions?

With ruleDM[{1, 11, 8, 4, 15}] non-reducible, its dual function might be irreducible too. (?)

Reduction and Mediation_125.gif

Reduction and Mediation_126.png

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Example3 : ruleDM[{1, 11, 8, 4, 15}]

Reduction and Mediation_129.png

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ruleDM[{1, 11, 8, 4, 15}] can not be minimized without damaging the original result.

Example4 : ruleDM[{1, 11, 12, 4, 15}]

Reduction and Mediation_132.gif

Reduction and Mediation_133.png

ruleDM[{1, 11, 12, 4, 15}] can not be minimized without damaging the original result.

http : // www.isaet.org/images/extraimages/S1213024.pdf

http : // www.ijser.org/researchpaper %5 COptimization - of - Ternary - Combinational - System.pdf

Reduction and Mediation_134.png

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Random seed

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

Reduction and Mediation_137.png

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Without {1, 2, 3} -> 0:

Reduction and Mediation_139.gif

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Reducible examples

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Mapping morphoCA rules onto computational diagrams

ruleDCl[{1, 2,3,4}]

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ruleDM[{1, 11, 3, 9, 15}]

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A0B0C0 + A1B0C0 + A2B0C0 + A0B0C1 + A0B0C2 + A0B1C0 + A0B2C0

A0B0C0 +  A0B0C2 + A0B1C0 + A0B2C0 + A1B0C0 + A2B0C0 + A0B0C1

A0 (B0C0 + B0C2 + B1C0 + B2C0) + A1B0C0 + A2B0C0 + A0B0C1

A0B0C0 + A1B0C0 + A2B0C0 + A0B0C1 + A0B0C2 + A0B1C0 + A0B2C0

A0B0C0 +  A0B0C2 + A0B1C0 + A0B2C0 + A1B0C0 + A2B0C0 + A0B0C1

A0 (B0C0 + B0C2 + B1C0 + B2C0) +(A1+A2+A0(C0 + C1))

Multi-layer structure of morphoCAs

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

start (init) : yellow, red

Reduction and Mediation_155.gif

construction: blue, yellow, red, yellow

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iteration of construction

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Polylogical interpretation of the reduction

S1 = {0, 1}, S2 = {1, 2}, S3 = {0, 2}

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Systems:   ruleDM[{1,11,3,9,15}]
Reduction and Mediation_162.png

Internal mappings:  ruleDM[{1,11,3,9,15}]
Reduction and Mediation_163.png

Reduction and Mediation_164.gif

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Again, another formal approach

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ruleDM[{1, 11, 3, 9, 15}]

Reduction and Mediation_167.gif

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The automaton at the level of a single layer of the multilayer morphoCAs is not necessarily running on its own without the inclusion of its neighbor components. Multi-layer moprhocAs are defined by their interaction with the partial automata of their neighbor layers and are therefore depending on the definitions of those partial CAs of the complementing layers of the whole morphoCA.

In this case it is supposed that the clocks of the different automata of different layers are synchronized.

Reduction and Mediation_169.png

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Visualization of classical morphograms

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Example

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The morphoCA of ruleDM[{1, 11, 12, 4, 15}] needs, to work as a CA, additionally to the morphic skeleton values of the morphograms [{1, 11, 12, 4, 15}], some embeddings as semiotic or symbolic interpretations of the rules. That doesn't come as a surprise. The skeleton ‘values’ of the morphograms are also just interpretations of the morphograms and not their direct representations.

Further simple examples for parallelism

Reduction and Mediation_177.png

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

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

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Gray code

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Reduction of ECA rules via sub-rule elimination

Example1

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Reducts = {{1, 0, 1} -> 1, {1, 1, 1} -> 1}: sub-rules 11 and 16.

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Example2

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Reduct = {{1, 0, 1} -> 1}: sub-rule 12.

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Reducts : {0, 1, 1} -> 1, {1, 1, 0} -> 1, {1, 1, 1} -> 0

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Reduction and Mediation_201.gif

Reduction and Mediation_202.gif

x/yz 00 01 11 10
0 0 1 - 0
1 1 0 - -

“Situations can arise where a circuit has N input signals, but not all Reduction and Mediation_203.png combinations of inputs are possible. Or, if all Reduction and Mediation_204.pngcombinations of inputs are possible, some combinations might be irrelevant.”

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Non - reducible ECA example

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x/yz 00 01 11 10
0 1 1 0 0
1 0 1 0 1

The ECA rule 110 belongs to the balanced non-reducible rules.

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More complex examples: Reduction and Mediation_214.png

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Manual reduction of ruleDCKV[{1111,1123,1211,1221,2121,2211,2221,2112,2132,1231,2231,2311,2321,2331,2301}]

Some sub - rules are redundant, the rest gets a significant reduction.

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Result of reduction

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Output - oriented notation

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Created with the Wolfram Language