Rudolf Kaehr Dr.phil^{@}

Copyright ThinkArt Lab ISSN 2041-4358

Abstract

Memristive cellular automata are introduced as a new
interpretation and model of general morphogrammatics. On the other
hand, memristive cellular automata based on morphogrammatics are sheding
some new light on the still little explored paradigm of morphogrammatic
thinking as it was invented by the cybernetician Gotthard Gunther
(1962) and elaborated by Kaehr/Mahler (1993).

Understanding
morphogrammatics as a system of kenomic cellular automata. Applying
retrograde recursivity to kenomic cellular automata by the definition of
the rules and by their applications. Hence, additional to the
properties of CAs, i.e. locality, uniformity, syncronicity, the property
of *memristivity* shall be implemented. This is possible only by a transformation from a symbolic to a kenomic concept of CA.

BETTER RESULTS AT:

http://memristors.memristics.com/Memristive Cellular Automata/Memristive Cellular Automata.pdf

"Rechnender Raum in denkender Leere” (SKIZZE-0.9.5)

„D´une certaine manière, ´la pensée‘ ne veut rien dire.” Derrida

„Seine
These, es gäbe weder die ´eine Wahrheit´ noch die ´eine Wirklichkeit‘,
sondern das Universum sei vielmehr als ein ´bewegliches Gewebe´
aufeinander nicht zurückführbarer Einzelwelten zu denken, formulierte
die entscheidende Aufgabe der Philosophie der Zukunft: eine Theorie
bereitzustellen, die es gestattet, die Strukturgesetze des organischen
Zusammenwirkens der je für sich organisierten Teilwelten aufzudecken.“
Gotthard Günther, 15. Juni 1980

Cellular Structured Space (Rechnender Raum)

Tom wrote:

> 1) Plankalkul

>

> :)-

Rechnender Raum. (Okay, that was cheap).

From: Eugene.Leitl@lrz.uni-muenchen.de

Date: Wed May 02 2001 - 15:24:32 PDT

http://www.thinkartlab.com/pkl/media/SKIZZE-0.9.5-Prop-book.pdf

"A cellular automaton is a
collection of "colored" cells on a grid of specified shape that evolves
through a number of discrete time steps according to a set of rules
based on the states of neighboring cells. The rules are then applied
iteratively for as many time steps as desired.”

http://mathworld.wolfram.com/CellularAutomaton.html

"The simplest class of
one-dimensional cellular automata. Elementary cellular automata have two
possible values for each cell (0 or 1), and rules that depend only on
nearest neighbor values."

Weisstein, Eric W. "Elementary Cellular Automaton." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ElementaryCellularAutomaton.html

Instead of the pre-defined {0,
1} values of each cell in a elementary cellular automaton, the value of
the neighbor cells gets defined by the kenomic successor rules.

Therefore,
the cellular automata rules for kenomic and morphogrammatic cellular
automata are defined by the memristivity of retrograde recursivity. In a
further step it will become obvious that the applications of the rules
will change memristively according to their retrofgradeness. Hence, the
‘set’ of ‘beginning’ rules is defined memristively as well as
the applications (iterability) of the rules. What is thus produced by
kenomic automata are memristive domains, worlds, universes of kenomic
computation.

Ruls of functorial interhangeability of guiding the
interactions of different cellular worlds enabled by kenomic cellular
automata.

(Recall: SKISZZE-0.9.5)

**Configurations: Transition in time***"The only reason for time is so that everything doesn’t happen at once.” *

-- Albert Einstein

The exact reason for morphogrammatics and its memristivity is the fact that things happens at once, all the time."One further ingredient that is needed for our cellular lattice to evolve with discrete time steps is a local rule or local

**Homogene **

One set of unambiguos rules are applied in time. Hence, a unambigous transition is defined.

c(t + 1) = φ [c(t), c(t), c (t)] .

c(t + 1) =

φ: [c(t), c(t), c (t)] --> [c(t), c(t), c (t)]

Retrogradness of memristic transitions

The transition ‘trans’ for a
CA at the time t to a new configuration at t+1 is depending on the
constellation of the CA of t-1. Hence, CA= trans(CA, CA^{t}).**Heterogene**

Several rules are applied and defining multiple transitions.

This allows a choice of rules. Otherwise all possible transitions are holding.

c(t + 1) = φ ([c(t), c(t), c (t)] **|** [c_{j−1}(t), c_{j}(t), c (t)]).

"Therefore the three fundamental features of a cellular automaton are: **uniformity**: all cell states are updated by the same set of rules; **synchronicity**: all cell states are updated simultaneously; **locality**: the rules are local in nature."

The new property is **memristivity** of kenomic cellular automata.

**Locality versus retrogradness**

Locality of the rules is not meaning retrogradness of the applicability of the rules applied locally.

"A
fundamental precept of cellular automata is that the local transition
function determining the state of each individual cell at a particular
time step should be based upon the state of those cells in its immediate
neighborhood at the previous time step or even previous time steps.

Thus the rules are strictly *local*
in nature and each cell becomes an information processing unit
integrating the state of the cells around it and altering its own state
in unison with all the others at the next time step in accordance with
the stipulated rule.”

http://www.texnology.com/joel.pdf

“That is, complex global features can **emerge** from the strictly local interaction of individual cells each of which is only aware of its immediate environment.”**Emergent** features are not related to retrogradness.**First and second order automata"**These
elementary cellular automata are examples of ﬁrst order automata in the
sense that the state of a cell at time step t + 1 only depends on the
state of its neighbors at the previous time step t. Whereas in a second
order automaton a cell’s state at time step t + 1 is rather more
demanding and depends on the state of its neighbors at time steps t − 1
as well as t, analogous to the way the Fibonacci sequence was formed.”

Retrogradness is functionally of

Hence, for the cell (i, j) =
(a), the neighbor cells have the values (a) and (b). That is producing 6
kenomic patterns for {<a>, <b>} and not 8 distinct digital
patterns for {0, 1}. The next steps of the rules are depending on their
history which is not identical with an abstract continuation of the
application of rules. Hence, the “resulting value” of the rule, define
by the 2 “neighboring cells” is not abstractly defined by combinatorics
of possible valuations but by the possibilities opened up by the
predecessor states, i.e. by the history of the previous development.

Therefore, the combinatorics between digital and kenomic automata are not identical.

For a 3 cell grid with 2 states there are 2^{3}= 8 digital constellations.

For
a 3 cell grid (kenomic matrix)and 2 kenomic ‘states’ there are 4 =Sn(3,
2). Hence, for 3 states of a 3 cell grid there are Sn(3, 3) = 5 kenomic
constellations.

For 8 binary states there are a total of 2^{8}= 256 elementary cellular automata.

A
visualization of elementary cellular automata is quite straightforward.
There are no ambiguities and perplexities involved. Surprises are
appearing on an application level but not on the level of the basic
definitions of the rules and their graphic representations.

It is a principle of morphogrammatic thinking that ambiguity and perplexity comes first.

Morphograms
are interpretatively ambiguous. A solution of a graphic representation
of ambiguous cellular automata is not easily accessible.**Context rules of applicability**

In
contrast to the classical definitions of CAs, memristic CAs have to
realize their antidromic recursivity on all levels of the construction
and application, i.e. the context rules for antidromic repetitions have
to be explicitly defined to rule or guide the applications of the
elementary kenomic rules of kenomic CAs.

The combinatorics for cellular automata is exponential with the stable base 2 for two elements, i.e. 2^{m}.

Kenomic developments are combinatorially defined by the Stirling numbers of the second kind, Sn(m, n).

Therefore,
a kenomic definition of an “elementary cellular automaton” has to taken
all 4 cells into account to determine the “resulting value” of the next
generation. Otherwise, a value constellation for the 3 cells of (000)
and (111) would have to be considered as kenomically identical.

But then the constellations and which are different couldn’t be produced.**Cyclic applications**

This is nicely depicted in the paper:

Pascal Bouvry et al, Cellular automata computations and secret key cryptography

http://pascal.bouvry.org/ftp/parco04.pdf

**Ambiguity versus computational reduction **

Mirrored rules, which are the same as their mirrored rule are called **amphichiral **(64).

Complementary rules are changing the roles of 0 and 1 in the rule definition.

Number sequences defined by elementay cellular automata: Jacobsthal, Pascal, etc.

Stirling numbers of the second kind are crucial for the architectonics of kenomic cllular automata.

"For example, the table giving the evolution of rule 30 (30 = 00011110_{2})
is illustrated above. In this diagram, the possible values of the three
neighboring cells are shown in the top row of each panel, and the
resulting value the central cell takes in the next generation is shown
below in the center.” (Weisstein, ibd.)

In contrast to the abstract combinatorial definition of the elementary cellular automata rules as a product of the the 28 states of neighboring cells and the binary next generation states 2^{8}=256
the kenomic cells and ther kenomic states are building a ‘holistic’
pattern. Therefore, the whole structure of the base of the elementary
rules is changed.

Because the next generation state of the fourth
cell is depending on the previous kenomic states, the number of the
whole pattern is maximally .

Following the standard definitions for dyadic CA presented in a systematic way by Jaime Rangel-Mondragón:

That is, a rule of the form
axb -> y describes the new value y of a given cell, given its present
value x and those of its two neighbors a and b. In the case of dyadic
CA, there are 2^2^3=256 possible rules. It is possible to prove that
linear CA do not need to have large neighborhoods; [...]."

http://library.wolfram.com/infocenter/MathSource/505/CAcatalog.nb

**Kenomic CA scheme**

This representation of the
kenomic scheme is conventional. Every other representation which fulfils
the epsilon/nu-structure of the patterns is accepted. **Epsilon/Nu-structure of morphograms**

How to define the elements of the rule schemata?

Obviously,
the elements are not considered as semiotic or syntactic entities and
are therefore not primarily determined by their atomic identity.

The
two sign sequences (aba) and (bab) are seen as structurally, i.e.
kenomically equivalent. Instead of using abstractions to form
equivalence classes of signs the simple method of relational equality
and non-equality of pairs of signs shall be used. This defines the
epsilon/nu-structure of morphograms,(epsilon=equal, nu=non-equal).

Hence both tuples are
structurally equivalent because their relations are equal. This allows
to give a precis definition of the transition rules for Cas.

For reasons of convenience and aesthetics the relational symbols shall be replaced by the usual set of marks {•, O, }.

This construction of the rule-schemata is using the *retrograd recursivity*
of the continuation operation for morphograms. Therefore, retrogradness
as a memristive property is implemented at the very beginning of
kenomic cellular automata. In other words, th rules of kenomic cellular
automata are memristive. This holds for all further extensions of the
memristive construction to higher order and more complex kenomic
cellular automata.

http://memristors.memristics.com/MorphoReflection/Morphogrammatics%20of%20Reflection.html

Kenomic rules are build in analogy to the CA rules. Thus the *"resulting value the central cell takes in the next generation is shown in the center".*

1. (a, a, a) (a, a, a), (a, b, a).

In fact, the rule might also be interpreted not as disjunctive but as a simultaneity of both: (a, a, a) | (a, b, a).

For a≡ •, b≡O, c≡ and d ≡ the following symbolic and graphical representation is depicted.

**Wolfram**

CellularAutomaton[150, {1, 0, 1, 1}, 3]

{{1, 0, 1, 1},

{0, 0, 0, 1},

{1, 0, 1, 1},

{0, 0, 0, 1}}**sub-rules for rule 150****(10010110)**

r1,-,8

**Kenomic model for rule 150**

kenom([150], {1,0,1,1}, 3]):

[[150] = r1, r7, r8, r9], [dual[150] = r6, r2, r3, r4]

1001 r4.4.2.4 1001
r9.9.7.9 1011 r6.3.4.6 1011
r9.8.9.1

1111 r6.6.6.6 0000
r1.1.1.1 0110 r4.4.2.4 0001
r7.1.7.8

0000 r6.6.6.6 1111
r1.1.1.1 1111 r6.6.6.6 0100
r1.8.9.1

1111 1111 0000 1001

**Dual kenomic rules**

r1, r6,

r2, r7; r11

r3,r8, r12

r4, r9; r13

r5, r10; r14, r15**Strict** **Combinations **

type1
=
r1.2.3.4.5/10 dual-type1
= r6.7.8.9.10/5

type2 =
r1.7.8.9.5/10 dual-type2
= r6.2.3.4.10/5

type3 = r1.2.8.9. 5./10 dual-type3 = r6.7.3.4.10/5

type4
=
r1.2.3.9.5/10 dual-type4
= r6.7.8.4.10/5

11.12.13.14

11.12.13.5

11.12.13.10

11.12.13.15

1.7.3.9.5 6.2.8.4.10

1.11.12.13.14/5/10/15

1.11.12.13.15/5/10/14

1.2.12.13.14/

1.2.12.13.15/

**Combinations with ambiguity or decision space of application**type3 = r1.2.

**Logical representation of symbolic rules**

CellularAutomaton[30,init,t] : 00011110

Mod[p + q + r + q r, 2]

(p

O••

128: and(p q r)

252: or(pq) -- non(or(pq)): 3

60: non(and(pq))**Morphogrammatic representation of kenomic rules**

LOG([252, 3]) = MG[8] .

http://atlas.wolfram.com/01/01/30/

Morphogrammatics was well
formalized and implemented as a theory of form and its transformations.
One specific transformation is produced by the so called “reflector”. A
reflector is reversing the order of a basic morphogram. The results of
such reflections are obviously very simple but elementary. But
morphogrammatics is studying the behavior of complex compounds of basic
morphograms and their transitions.

It might be of interest to
transform the results of reflectional morphogrammatics into the
framework of memristive cellular automata.

Hence, there shall be a
transition from a patter [10, 2,10] to a pattern [2, 2, 11] of
morphogramatics on the base of kenomic cellular automata
transformations.

=>

"The simplest neighborhood is
an elementary system consisting of a one-dimensional row of cells, each
of which can contain the value 0 or 1 (depicted as two colors), with a
local neighborhood of size 3 (range or radius of 1).

More complex CA can be deﬁned on two- or higher-dimensional arrays with multicolored cells and larger ranges. Each rule

is represented as an array of cells.

For
the case of a local neighborhood of size 3, each triplet determines a
single output cell in an array. A triplet with binary values can have
eight possible patterns from 111 to 000. A local neighborhood of size 3
thus can generate 256 possible rules.

The formula for calculating the rule size space in a one-dimensional system is k,
where k represents the color possibilities for each state and r is the
range or radius of the neighborhood. It is interesting to note that
merely increasing r from 1 to 2 and maintaining the colors at two
increases the rule space from 256 to 4.3 billion.”

http://www.wolframscience.com/conference/2006/presentations/materials/speller-complex_systems-17-1-2.pdf

Symbolic CA are representable by keno CA.

The morphogrammatic base of symbolic CAs are the 8 basic morphograms with 2 kenograms.

It is shown that a pattern of 4 places gives space for morphograms with up to 4 kenograms.

Hence, the morphogrammatic base of kenoCA are 15 morphic patterns and not just 8 like for symbolic CAs.

In this sense, symbolic CA are incomplete in respect of their kenomic deep-structure.

Interaction between basic kenoCAs is established with a mediation of the 15 basic kenoCAs to compound kenoCAs.

The
crucial question where are the new ‘values’ coming from that appeared
for kenomic cellular automata with complexity m=3 and n=2 has an answer
in the theory of compound kenomic automata.

**Composition**

**Homogene composition****keno[150]**** =[****, ****, **]; ((, ), ,)

**Interactional constellations**

The question is what kind of
physical realization of kenomic cellular automata could be imagined to
transform kenomic CAs into kenomic automata machines?

Interacting grids of memristors is the answer.

Is it time for a new *“Cellular Automata Machine" *(Toffoli/Margolus)?