Towards Abstract Memristic Machines

Outline of morphogrammatics as a formal model for tensed machines

Rudolf Kaehr Dr.phil.@

Copyright © ThinkArt Lab ISSN 2041-4358



Memristic machines are time-tensed machines of the nanosphere. Their definition and their rules are not covered by ordinary logic, arithmetics and semiotics, basic for a theory of abstract automata. The difference to classical concepts of machines to tensed, i.e. memristive machines is elaborated. As an attempt to develop memristive machines, basic constructs from morphogrammatics are applied.  
Properties of retro-gradeness (antidromicity), self-referentiality, simultaneity and locality (positionality) of operations as they occur in kenogramamtic and morphogrammatic basic operations, like the successor operations, ‘addition’ and ‘multiplication’ have to be realized on all levels of operativity in memristive systems.  
Hence, the tiny memristive properties of time- and history-dependence for kenomic successors are prsented for all further operations, like “addition" (coalition), "multiplication”, “reflection”, etc. Morphogrammatics will be further developed in Part II of the paper.
A new framework for design and analysis for memristive systems, i.e. memristics, shall be sketched as a complex methodology of Morphogrammatics, Diamond Category Theory, Diagrammatics and Nanotechnology.

Time- and history-dependence in history

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

„Soll die hierdurch bedingte Vieldeutigkeit begreiflich werden, so dürfen wir nicht
den jeweiligen Zustand allein ins Auge fassen. Wir müssen auch die Zwischenzustän-
de beachten, die der Körper bis zur Rückkehr in den anfänglichen Zustand durchläuft,
und sie für das veränderte Verhalten verantwortlichmachen. Dies führt dazu, die Ände-
rungsweise eines lebendigen Körpers ganz allgemein durch seine früheren Zustände
bedingt zu denken.“
G. F. Lipps, Mythenbildung und Erkenntnis, Leipzig u. Berlin
1907, S. 263 (Zitiert nach: Karl Faigl, Ganzheit und Zahl, Jena 1926, Herdflamme Bd. 2)

„Wenn ich aber jetzt ein kybernetisches System bauen will, das mindestens Spuren
oder Grade der Selbstreferenz zeigt, so setzt eine solche Selbstreferenz voraus, dass
das betreffende System eine innere Zeit hat, d.h., dass es auf einen früheren Zustand
seinerselbst zurückblicken kann. Auf das Früher kommt es an, also auf das Zeitmoment.
In diesem Fall genügt die einfache Alternative nicht mehr, dass etwas so oder nicht so
  Gotthard Gunther

1.  Memristors and non-trivial machines

1.1.  Trivial vs. non-trivial machines

"Devices whose resistance depends on the internal state of the system.” (Kim)

The cybernetician Heinz von Foerster introduced the distinctions of trivial, non-trivial and recursively operating non-trivial machines. Those distinctions got a wide use in cybernetics and in the theory of learning systems. Connection to the common theory of abstract machines had been studied too.
Instead to go primarily into the classic theory of automata to study the possibility of memristic machines, I will focus on this handy distinction of trivial, non-trivial and autopoietic machines.
Nevertheless it should be noticed that von Foerster models of machines are formal models in the sense of mathematical and programming systems and not in any way connected with the possibilities of emulations by material systems, like it is possible now with memristive systems. History-dependence is use in computing sense of recursive functions and not in the sense that the “hardware” (XX) in itself is history-dependent in its behaviour.

Furthermore, see Gordon Pask’s approach to cybernetics:
"Life and intelligence lie somewhere in the conflict between closed, unique, construction and open, shared, interaction. Between a specific material fabric, and a general conceptual/functional organization.”

Some criteria
Leading criteria of comparison shall be:
history-dependent vs. history-independent
determined vs indetermined
predictable vs. unpredictable
programmed vs. self-organized
simulation vs. emulation

Trivial Machines:
A trivial machine is modeled by the set-theoretic function f with the properties:
(i) Synthetically determined;
(ii) History independent;
(iii) Analytically determined;
(iv) Predictable.

[Graphics:HTMLFiles/Abstract Memristic Machines_1.gif]

Non-Trivial Machines:
In contrast to trvial machines, non-trivial machines have an internal state computed by z’:
(i)   Synthetically determined;
(ii)  History dependent;
(iii)  Analytically indeterminable;
(iv)  Unpredictable.

Driving function: y = F(x, y)
State function: z’  = Z (x, z)
Z = internal state.

N4= 4,294,967,296

[Graphics:HTMLFiles/Abstract Memristic Machines_2.gif]

(i)   Read the input symbol x.
(ii)  Compare x with z, the internal state of the machine.
(iii) Write the appropriate output symbol y.
(iv) Change the internal state z to the new state z’.
(v)  Repeat the above sequence with a new input state x'.

Further explanations of non-trivial machines: example 1

Recursively Operating Non-Trivial Machine:
  "Computing EigenValues, Eigen-Behaviours, Eigen-Operators, Eigen-Organizartions,  etc..."

[Graphics:HTMLFiles/Abstract Memristic Machines_3.gif]

What kind of machines are memristive systems representing?
Amazingly, it will turn out that within the possibilities of memristive systems very different types of machines are constructible.
Common to all three types of von Foerster’s machine is that they are dealing with data, information, objects, etc., and not with their own conceptual domains and definitions. Hence, even if they are classified as second-order machines, they are not in any sense interactional, reflectional or interventional.

Because memristive behaviors occur in different physical media, a dissemination over disjunct media to realize polycontextural emulations of memristors appears quite natural. This fact might contribute, together with the mem-properties of memristive devices to realize polycontextural machines.
Both, properties, the mem-property, allowing the realization of chiastic self-referentiality and the localization-property doesn’t exist in the framework of von Foerster’s second-order machines.

The ‘property’ of localization (or positionality) obviously has nothing in common with localization theories of Sir John Eccles and his engrams or Pribram’s hologram theory of memory. Localization in the context of polycontextural studies of computation and memory refers to different contexture. It is well known, that the notion “contexture” and the concept of a multitude of distributed and mediated, i.e. disseminated contextures has no equivalent in classical theories.

Both kinds of machine, trivial and non-trivial, are depending on a pre-given data set of a single general domain.Their languages (Chomsky-Hierarchy) are presuming a pre-given alphabet (sign repertoire), which is, in general, stable and  is not changing during computation. In contrast, autopoietic co-creative machines are not dealing with signs therefore they don’t presume an alphabet. Their beginnings are changing depending of their usage. Unfortunately, this point never got a clarification in the second-order cybernetic literature.

1.2.  Memristors as trivial machines

1.2.1.  Memristors, semiotics and categories

Also memristors are not genuinely representing trivial machines because their “resistance depends on the internal state of the system”, while trivial machines are “history independent”, memristors might be used to construct, nevertheless, trivial machines.
Because the behavior of memristors might be interpreted as a material implication, and material implication plus the value F are building a logically complete set for propositional connectives, and propositional logic is additionally to the completeness of the junctional set and its axiomatization, decidable, memristors are able to perform decidable, i.e. trivial machines, too.

Also the point that memristors are passive elements in contrast to the active time-clocked transistors has to be considered in more detail it will not necessarily change the base of the argument.

Semiotically, trivial machines are based on the operation of concatenation, which is the basic construct underlying state transitions. Concatenation is an abstract ‘addition’ without any possibility to refer to past semiotic events. The mechanisms of feedback loops are not escaping this restriction of concatenation but accelerating it as a function in time.

The concatenation mechanism for sign-sequences consists in a two-level action: selection of a sign out of the pre-given sign repertorire (alphabet) and a ‘linear addition’ of the selected sign to the existing chain of signs.

          Semiotic concatenation <br /> <br />    ...                ] <br />

      <br />      Alphabeth     &n ... atenation       <br />         chain

This concept of semiotics might be simplified without loss to a semiotic system with one element only, typeset structure. And the empty sign to mark the blank between signs. Then, all different alphabetic elements are produced by the one-element system as abbreviations. Say, | = a, || = b, etc.
The first rule typeset structure of this stroke calculus is introducing a stroke, |, the second rule typeset structure says, if there are n strokes produced a further stroke might be added, n|, the third rule typeset structure states the iterativity of the second rule. Hence, there is no need to refere to a “history” of the stroke production. There are obviously some crucial implications involved which are not mentioned in the context of a stroke claculus. Funny enough, all that is repeated, nevertheless, with George Spencer’s Calculus of Indication.

typeset structure                           

Category theory
Trivial machines are perfectly modeled by category-theoretical approaches. Classical category theory is studying the constructions based on composition. Composition is interpreted generally as a serial connectivity (Abramsky, Coecke), while an additional operator, like yuxtaposition, is understood as parallel connectivity. Both, composition and yuxtaposition, nevertheless, are defined strictly within the paradigm of a-temporal and “history independent” structures and operativity.

This “history-independency” is axiomatized in category theory by the axioms of associativity and identity for composition of morphisms. Both axioms are not referring in their definition retro-grade to previous events.

<br />    Identity : I  _ A o f = f = f o I  _ A <br />    Associativity : ( f o g) o h = f o (g o h)    <br />

Category Theory and Computer Programming (Eds., David Pitt, Samson Abramsky et al, 1985, Springer LNCS 240) gives a definitive introduction, overview and programatics for future studies.

The axiom of associativity guarantees the “history-independent” charactor of the operation of composition. But this is necessarily secured by the first axiom, the identity of typeset structureand typeset structureas f . There is no such property like retro-grade reference for categorical composition, and for yuxtaposition too.

In a interesting remark, John Baez points to the fact that non-commutative systems got studied in category theory, and elsewhere, but there seems to be not much interest for non-associative formalisms for category theory. Non-associativity might be well studied for general algebras but not as categories.

An up-to-date approach of modern category theory is presented by Peter Selinger:22

In contrast to the memristor-papers, this important research report is free accessible at:

Finite State Machine
    • A deterministic finite state machine or acceptor deterministic finite state machine is a quintuple
               (Σ, S, s0, δ, F), where:
    • Σ is the input alphabet (a finite, non-empty set of symbols).
    • S is a finite, non-empty set of states.
    • s0 is an initial state, an element of S.
    • δ is the state-transition function
    • F is the set of final states, a (possibly empty) subset of S.

The state-transition function is ”“time- and history-independent” as it should be by definition. It is a morphisms from the initial to the final states based on the input alphabet delivering its out-put.

Concatenation in poly-categories
The simplest concatenation system is the system of natural numbers NN. Category-theoretically, this system is introduced as a commutative graph with (0, s) for the numbers and (a, h) for the model. Such a system is based or anchored on an initial object “0” as the starting point of the linear succession. This is stated in the recursive formula:

f (0 ) = a f(n + 1) = h(f(n))

A categorification of the recursive formula is introduced by the following steps.
For short, if 0: 1 --> N, s: N --> N and a: 1 --> A, h: A --> A are morphism and are producing a diagram that commutes then the object defined with this graph are the natural number system NN. The unique morphism, which make the diagram commute is the morphism f.

<br />    NN :           1 Overscript[- ... bsp;        A     Overscript[-->, h] A <br />

This definition or construction is categorical because it is defining the interactions, i.e. morphisms, between the objects, and is not defining natural numbers as elements of a set with specific properties. It defines the natural number system, NN, as a correlation (morphism) between the numerical system, (N, s), and a model  (a, h) of the numerical system ”(N, s)".

The question arises: How can we compare two natural number systems NN? Each NN is categorically defined as an interaction of a number system, (0, s), and as a model (a, h) of this number system. Hence, a comparison of NN1and NNtypeset structurehas to consider both of each system, the number system and the model system. Obviously, this kind of comparison is a comparison between the number systems as such and not a comparison of special cases belonging to the ultimate, i.e. “the one and only one” NN.

This is naturally generalized from number systems “NN” to sign systems “SS” in general.
Therefore, a simple modeling of the comparison shall happen in a polycontextural monoidal context. If we are considering the result of the comparison of SS1and SS2 the comparison is producing super-additively a new SignSystem SS3 with the inscription of the comparison as such. Obviously, the mechanism of comparison is not an isomorphism between a sign system and its ‘model’ or a functorial mapping but a mechanism of mediation with its super-additivity between contexturally different systems.

What’s the matter with this tedious formula? If we want to compare two machines, for convenience, two trivial machines based on the concatenation of signs, it is the first thing we should know: How are they defined in regard to their behavior, in contrast to their set-theoretical properties? The answer, at least a first step to it, is constructed with the help of polycontextural monoidal categories as shown below.

All that is, without doubt, simply a beginning to get some orientation and directions for further studies.

     <br />        U ^(3) = (U  _ 1 ...                              3                                                                   3

1.2.2.  Mediated trivial machines

Also the scheme of interchangeability is rather simple, it allows a quite complex combinatorics in its applications. A few example shall be exposed. Nevertheless, these exercises and the examples above, are just a first step towards a diamond-theoretic formalization and are not yet demonstrating much of this new approach to modeling and formalization as prerequisite of implementation and emulation of new ways of computing. Especially the symmetric relation between system and model has to be questioned and extended to an interplay of symmetric and asymmetric relations. The reasons to study such abstract relationships is motivated by the requirements of a theory of multi- and poly-layered crossbar constructions for memristive systems.

     <br />     Composition of two sign systems <br />   &n ...                                                                   1   o                          2

      <br />     Mutual cross - exchange   of   sign   systems   and  ...                                                                                o                 2

  <br />      Cross - Iteration <br />     Composition of three  ...                   o                                                          2 o                 3

   <br />     Accretion <br />     Composition of two  ...                                                                                                1.1

1.2.3.  Additivity vs. Super-additivity of compositions

A composition of finite state-machines is additive, linear and associative.
A combination of memristic systems, designed as polycontextural and morphogrammatic machines, is super-additive, tabular, and poly-associative.

Deterministic finite-state machine (DFA)

"DFAs are one of the most practical models of computation, since there is a trivial linear time, constant-space, online algorithm to simulate a DFA on a stream of input. Given two DFAs there are efficient algorithms to find a DFA recognizing:
    * the union of the two DFAs
    * the intersection of the two DFAs
    * complements of the languages the DFAs recognize.

DFAs are equivalent in computing power to nondeterministic finite automata.” (WiKi)

         Polycategorical concatenation of automata   <br /> ...                             3.1                                                                3.2

1.2.4.  Products of DFAs

Products of DFAs are of special interests for a theory of atomata.

1.3.  Memristors as non-trivial machines

1.3.1.  History-dependence

Obviously, memristors are not representing trivial machines because their “resistance depends on the internal state of the system”, while trivial machines, mono-contextural as well as ‘polycontextural’, are “history independent".
Also memristors have a strict distinction of input and output functionality, their out-put is depending not only on the input and the definition of the function but also on the history of the former input/output relation. Hence they are “history dependent”.
Are memristors therefore non-trivial?
Trivial machines have additionally to their history-independence a “predictable” behavior, non-trivial machines behave “unpredictable”. Is this true for memristors? If yes, in which sense is the behavior of memristors unpredictable?
The behaviors of memristors is interpretable in two distinct ways: 1. as digital, and 2. as analog.

"Ionized atomic degrees of freedom define the internal state of the device.‘’
"New possibilities in the understanding of neural processes using memristive memory devices whose response depends on the whole dynamical history of the system.” (Kim, New Scientist, 2009)

Further characterizations of “time- and history-dependence” of memristive behavior and its modeling by kenomic operators gets strong support by my previous studies towards a new paradigm of computation and a “Theory of Living Systems”, making use of retro-grade iter/alterability. 4

In other words, repetition, i.e. iterability as retrograde recursivity is involved in self-referentiality, transparency, memory and history, and evolution of objects (morphograms). Until now, only the retro-grade and self-referential aspect of time- and memory-depending actions in memristive systems had been in focus.

Di Ventura et al, Putting memory into circuit elements: memristors, memcapacitors and meminductors
"Equivalently, the memristor relates the current to the voltage, but unlike its traditional counterpart, its resistance, upon turning off the power source, depends on the integral of its entire past current waveform. In other words, it has memory of past states through which the system has evolved. [...]
A simple reason for this is that at the nanoscale the dynamical properties of electrons and ions strongly depend on the history of the system, at least within certain time scales. Therefore, many devices at these length scales retain partial memory of the electron and ion dynamics.”

It seems that a lot of the wordings and conceptual developments for ‘memristive’ systems has been repeatedly done long ago. What is new today is a better mathematical apparatus (polycontextural category and diamond theory) and, crucially, the discovery of memristive behaviors in nanoelectronics (memristor, memdevices).

Memristive properties are depending, according to Di Ventura, “on the integral of its entire past current waveform”, “has memory of past states through which the system has evolved”, “history of the system, at least within certain time scales” and therefore “retain partial memory of the electron and ion dynamics”.

In this study I will restrict myself to the aspects of self-referentiality and morphogrammatic evolution of minimal time- and history-dependent memristive systems as it appears as retro-grade and evolutive patterns of morphogrammatics, therefore omitting aspects of positionality, locality and transparency. Aspects of transparency are connected with the autonomy of a system as a whole. Transparency of a computational system is not reasonably accessible for information processing systems. It needs an additional abstraction from the complexity of informational processes offered by a morphic abstraction.
Together with the mentioned features, the fact that nano-devices are localized, i.e. are taking a position in a positional matrix, has to be emphasized too. System-theoretic notions are not covering features (or principle) of localization and positionality.

Memristors are in fact  “assemblies of nanoparticles" (Kim, 2009), therefore, their behavior is not “analytically” pre-defined, their behavior, depending on the contextual history of the system, i.e. the position in a memristive matrix, has to be interpreted. This might happen as an abstraction of identity, producing a predictable binarity of values, or it might be interpreted analogously, producing a non-predictable set of values.
The concept of a single or a collection of memristors as nano-technological devices has therefore its machine-theoretic description by the concept of a non-trivial machine.


Because functional complete logical functor-sets are representing trivial machines, the characterization of memristors as non-trivial machines might be in conflict with the understanding of memristors as material implications together with the functional completeness of a logic with implication and a negative constant, {IMP, F}. Complete junctional sets in logic theories are decidable, hence, their behavior is predictable.  
This characterization by logically complete sets is only halve the story of the possible behaviors of memristors.

Notes from the Variety
A finite state machine has a state but not a memory of a state.
A memristive machine has a state of a state, i.e. a meta-state as a memory, therefore a memristic machine is not a finite state machine.
A meta-state always can be taken as a simple state in the sense that a reduction from an as-abstraction to an is-abstraction is directly possible because the necessary informations are stored in the meta-state. From “x as y is z” there is an easy way to reduce it to “x is x”. Such a reduction of a second-order system to a first-order system is nevertheless losing the essential features of the reduced system.

A memristive machine, then, is a machine with a tensed time, while finite-state machines are not tensed machines. Their temporality is of first-order, memristic time is of second-order, i.e. an interpretation of a state of a state.
Todays interpretation of memristors as memory devices in an ANN is reducing the possibility of second-order learning to simple first-order learning as trained adaption.

Memristors and states
In a further analysis the conceptual levels of  the constructions become more clear:
1. Zero-level : the RIC-elements are of zero-level because they don't have a state.
2. First-level as machines: States of a state-machine based on RIC-elements are elementary states.
3. First-level as elements: Memristors, or mem-elements, are devices with a state. Hence they replace first-level
    finite-state machines.
4. Second-level: States of memristive machines are states that have a state, i.e. a memrory of a state.

Hence, RIC-elements have no state, mem-elements have a state and mem-machines have a state of a state.

Memristive iterability
Therefore, an iteration of an electronic action for non-memristive devices is a history-independent action. While a repetition of an action for a memristitive device is history-dependent and therefore changing its character in time. Non-,meristive repetitions are conceptually well modeled by arithmetic or semiotic successor operations, i.e. by concatenation based on a pre-given alphabet. Repetition in non-memristive systems is stable, i.e. monoton.
Memristive iterability is modeled by kenomic operators of change based on monomorphies. Here again, iterability is alterability too.
The complexity of time- and history-dependence for memristors is very minimal but significantly different from semiotic concatenation. Each repetition or each new run might change the dynamics between first- and second-order characteristics in memristive devices.


Morphogrammatics offers a complex theory of history-dependend changes of morphograms, i.e. of iter-/alterability.6

1.3.2.  Kenomic modeling

In contrast to semiotic concatenation, kenomic evolution has to be considered as retro-grade, depending on the history of its occurrences. As developed before in several papers, kenomic evolutions are not framed by initial and final (terminal) objects.

"In contrast to morphogramatic evolution, kenogrammatic ‘concatenation’ still relies to some degree on the linear order of its kenoms. But there is no need anymore for a pre-given alphabet, and concatenation itself is only one of the elementary operations of change. Further operations are chaining and different kinds of fusion. Without a pre-given alphabet the risk has to be taken to develop change out of the encountered kenogram sequences only. With that the abstractness of the semiotic concatenation is surpassed. There is not only no alphabet given, but the kenoms involved are semiotically indistinguishable. The operation of concatenation is defined by an interaction with the encountered kenogram sequence. Its range is determined by the occurring kenoms of the sequence which remains itself still untouched by the process of concatenation. Hence, kenogrammatic concatenation is not defined in an abstract way but retro-grade to the encountered kenomic pattern.” (Kaehr, Morphogrammatics of Change)

Such retro-grade recursion to develop kenomic progressions is a necessary condition to develop a “history-dependent” mechanism of change on a pre-semiotic level of inscription without the presumption of a pre-given sign-repertoire and its restrictions of atomicity, linearity and identity.

Hence, what is repeated and involved at new into a calculation is not a data from an external (re)source (sign repertoire, environment) but the result of a former mark (activity), which is remembered (retained) at that locus. Therefore, the repeated mark is a memorized constellation of a former activity. It is thus a mark of a mark what defines a history-dependent mark.
As a consequence, an external observer can’t predict the outcome of a kenomic concatenation. The concepts of input and output are losing their relevance. Hence, the lack of predictability is not to confuse with a gain of stochastics or propabilistics of non-deterministic machines.

1.3.3.  Morphogrammatic prolongations

            <br />    &n ... p;  Underscript[-->,     progression      ] <br />

<br /> <br />       semiotic concatenation   <br />     ... nbsp;            morphogram     

<br />    chiastic change of morphograms and monomorphies <br /> <br />    ... script[<--,      change     ]    monomorphy

     arithmetic <br />     initial repetition <br /> <br /> &nbs ... m    Overscript[-->, h  _ j] gm Overscript[-->,  h  _ j] gm <br />

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 si. To each prolongation a further prolongation is defined by the iterated application of the operator si.
The morphogrammatic succession typeset structure is founded by its model typeset structure 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.

<br />    Diamond of the MG - successor rule     <br /> <br />    &nb ... 0; _ -] Overscript[<--,  Overscript[h, -]] Overscript[MG   ,  _ -], _] <br />

<br /> <br />    Diamond of the MG - successor system <br />    <br /> &nb ... p;,  _ -]            <br />   

<br /> <br />    Diamond    MG - successor system <br />    <br  ... p;,  _ -]            <br />   

2.  Morphogrammatics

2.1.  A new framework for memristics

A new framework of design and analysis shall be proposed.

Properties of retro-gradeness (antidromicity), self-referentiality, simultaneity and locality (positionality) of operations as they occur in kenogramamtic and morphogrammatic basic operations, like the successor operation (function) have to be realized on all levels of operativity in memristive systems.

Hence, the tiny memristive properties of time- and history-dependence for kenomic successors have to be developed for all further operations, like “addition" (coalition), "multiplication”, “reflection”, etc.

Morphogrammatics, Diamond Category Theory, Diagrammatics, Memristics, Nanotechnology.

2.1.1.  Electronics


Each activity in a non-memory-dependent system happens, iterates, successively without being reflected by prior activities in a linear monotony.
A run through a RCL-circuit might iteratively be repeated, ideally, without changing its predefined conditions.

Composition of networks
Behaviors of RCL-circuit networks are structurally based on serial and parallel compositions.

An important consideration in non-linear analysis is the question of uniqueness. For a network composed of linear components there will always be one, and only one, unique solution for a given set of boundary conditions. This is not always the case in non-linear circuits.

2.1.2.  Memristics

Antidromic iteration
Each activity in a memory-dependent system happens, iterates or accreates, successively in strict dependence of the  prior activities of the whole circuit.
A run through a RCL-circuit is iteratively reflecting or learning, ideally, its new and changing conditions, i.e. states of procedurality.
Hence, the retro-grade activity of the progression might be well modeled with the basic structures of kenomic and morphogrammatic operations.

Combinations, compositionality
Behaviors of RCL-circuits are structurally based on serial and parallel compositions. What are the memristic analogies? Coalitions, combinations and products of morphograms might model features of memristive compositionality. Such compositionality is understood as radically memristive and shall not be reduced to the topic of applications of memristors as elements in classical circuits.
Thus, the logical interpretation of memristors as a material implication is just a start and is not yet reflecting on the holistic patterns of memristive behaviours.

If memristors are intepreted in the memristic framework as second-order elements, a change of metaphors could help to develop a simple theory of series and parallel compositions for memristors. Instead of “second-order’ elementsis shall be understood as 2-layered basic memristive elements.

Hence, by diamondization, the rules for series and parallel compositions for memristors might be added to the classical rules as their second layer, i.e. as their structural environment. The daisy chain-coupled circuits gets coupled with its neighbors.

Structured antidromicity
As a new step in the modeling of memristive activities by the application of morphogrammatics it seems to be necessary to understand the change of memristance (memductance, memcapacitance) as a structured process. Hence, patterns of second-order features have to be studied in memristics. Until know, this appears as a speculation because there are not yet any experimental results and descriptions available about such a topic.

Presupposing the existence of structured memristive antidromicity as a pattern of second-order activities, a morphogrammatic modeling in the framework of monomorphies follows quite naturally as a consequence of the retro-grade and holistic structure of morphograms.

Framework of a memristic research program

typeset structure

2.2.  A very first approximation for non-structured memristive elements

By diamondization, the rules for series and parallel compositions of memristors might be added to the classical rules of series and parallel compositions as their second layer, their structural environment, covered formally by diamond-theoretic notions. Again, these are conceptual speculations, first attempts to understand memristive systems and their possible technology.

Again, a memristor has a state,that means, a memristor is not a state.
This says it clearly enough that a memristor is no a state. Its behavior is not characterized to be a state but to have a state. This as a result of an interaction with the memristor.
What means having a state in the case of a memristor?
It was shown clearly enough that a memristor is a specific resistor having a state. That’s why it is described as resistor with memory, called memristor.

If a memristor is to be described or defined by its resistance and in parallel by its memory of that resistance, we obviously have to deal with a device that has two functionalities. One as a resistor, another as a memory of that resistance.

It is clear too, that such a characterization as a two-fold functionality is not properly understood as a superposition of one function over the other. It is not simply a memory of a resistance or a resistance of a memory. Such a successive and hierarchical interpretation is useful for a simulation of the behavior of a memristor only. But it is not adequate to describe the characterization of the behavior of a memristor as such.

A proper understanding of a memristor as having a state has to take into account the simultaneity of both functions: the resistance and the memristance of the memristor. Both are defined as two levels or layers of a second-order device. Such a simultaneity and parallelism puts both levels into a heterarchical yuxtaposition.

'Memristance is a property of an electronic component. If charge flows in one direction through a circuit, the resistance of that component of the circuit will increase and if charge flows in the opposite direction in the circuit, the resistance will decrease. If the flow of charge is stopped by turning off the applied voltage, the component will ‘remember’ the last resistance that it had, and when the flow of charge starts again the resistance of the circut will be what it was last active.”
"In other words, a memristor is ‘a device which bookkeeps the charge passing its own port'" (Stanley Williams)

"v = R[q(t)]i
The meaning of this equation is that the charge flowing through the memristor dynamically changes the internal state of the memristor making it a nonlinear element."

Again, bookkeeping is a parallel activity, happening simultaneously to the incoming ‘bookings’.

2.2.1.  Phenomenological description

change of flow (resistor)  _ 1    --> recording    new state (s ...  of flow    (resistor)  _ 3 <-- retrieving kept state, (memory)  _ 2

<br /> memristor = chiasm(change, resistance, recording, retrieving)  memristor = [ (change)   ...                                                   <--                                 (storage)

The phenomenological result corresponds the structuration pattern " proemiality " (o ...        record --> retrival     <br />   

2.2.2.  Physical description

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b .    <br />     d | ud    -    <br />  &nb ... r />     d | ud    -    <br />    ud | d   

"FIG. 2:
a: Schematic of a memristor of length D as two resistors in series. The doped region (TiOtypeset structure) has resistance Rtypeset structurew/D and the undoped region (TiO2) has resistance ROFF(1−w/D). The size of the doped region, with its charge +2 ionic dopants, changes in response to the applied voltage and thus alters the effective resistance of the memristor.
b: Two memristors with the same polarity in series. d and ud represent the doped and undoped regions respectively. In this case, the memristive effect is retained because doped regions in both memristors simultaneously shrink or expand.
c: Two memristors with opposite polarities in series. The net memristive effect is suppressed.”
Yogesh N. Joglekar and Stephen J. Wolf, The elusive memristor: properties of basic electrical circuits, 2009

b. M + M = M
typeset structure ==> typeset structure

c. M + M = R
typeset structure ==> typeset structure

A physical description of a memristor is not yet the description of its behavior. A memristor defined as a programmable non-volatile double-resistor device is not yet describing its behavioral history-dependency. This quality is mostly mentioned after the physical description of a memristor as an appendix, additional interpretation of the working of a physical memristor.


"The primary property of the memristor is the memory of the charge that has passed through it, reflected in its effective resistance M(q)." (Joglekar)

2.2.3.  Some more concrete electronic modeling

1. Different polarity η M  _ T (q) = (R  _ (0, 1) -    η ( ... sp;                (1)

 M  _ T (q) = (R  _ (0, 1) + R  _ (0, 2)) -    η (Δ ... bsp;                (2)

where : <br /> -    R  _ (0, i) for i = 1, 2 is the effectie memrist ... 16; R  _ i  =  R  _ (off, i) - R  _ (on, i) ~= R  _ (off, i) Null

M = (R | Overscript[r, <-]) ≡ (R, Δ R)  M  _ T (q) = M  _ 1 + M  ... 0, 1)) -    η (Δ R  _ 2 - Δ R  _ 1) q (t)/Q  _ 0

" It is clear that the overall behaviour of these memristors  depends on the relationship ...  (Δ R  _ 1)/(Δ R  _ 2) " <br /> (F . Merrikh - Bayat, 24.08 .2010)

 Δ R  _ i  =  R  _ (off, i) - R  _ (on, i) ~= R  _ (off,  ... A0; _ 2)/(Δ R  _ 1) , i . e .    α  != Overscript[α, -] .

 2. Same polarity η <br /> M  _ T (q) = (R  _ (0, 1) + R  _ (0, 2 ... gt;    M  _ 1 + M  _ 2 !=    M  _ 2 + M  _ 1 .

In the following I will give a first complementary approach to the physical approach to define conceptually behavioral traits of memristors and memristive systems.

2.2.4.  Memristive series and parallel circuits

Series M = (R | Overscript[r, <-]) : Memristor as a 2 - layered resistor with resistance R  ... ;-]  _ n)  <br /> Memristive crossbars are constructed by series of memristors .

<br /> Parallel <br /> 1/(R  _ total) = 1/(R  _ 1) + 1/(R  _ 1) + ...  ... 1 + R  _ 2 | Overscript[r, <-]  _ 1 + Overscript[r, <-]  _ 2, _]    <br />

<br /> Diamondization of serial composition <br /> <br /> diam (R  _ 1 + R  _ 2) = R & ... ipt[r, <-]  _ 1 + Overscript[r, <-]  _ 2) | Overscript[r  _ 4, <-] <br />

Series with saltitions
Numbering of diamond subsystems with jump-operation (saltisition || ) between subsystem 8 and 4. Number of subsystem (k, j) = 9

[Graphics:HTMLFiles/Abstract Memristic Machines_54.gif]

<br /> diam (M  _ 1 + M  _ 2 + M  _ 3) = (R  _ 1 + R  _ 2 + R  _ 3 |   ... lt;-] + Overscript[r, <-]  _ 8 ) || Overscript[r  _ 9, <-]     <br />

<br /> Diamodization of parallel composition <br />  <br />      diam  ... p;                

2.2.5.  Arithmetic of linear electronic elements

It sounds probably strange to speak about an arithmetic of electronic elements.
Formally, series and parallel circuits, but also mixed combinations, are behaving arithmetically in the kind they are adding elements together. This is not surprising because electronic circuits are logically equivalent to restricted Boolean algebras.

Addition is linear (monotone)
(((R1+ R1)+ .... )+ Rn)

R1+ R1 = R1

Succession of R’s are following abstractly like natural number one the other without any need to relay on the previous succession. The same holds for a modeling with the concatenation operation on signs.

Addition is commutative
R1 + R2 = R2 + R1

Addition is associative and is not in any case involved into superadditivity:
(R1 + R2) + R3 = R1 + (R2 + R3)

2.2.6.  Mono-structured memristive addition

Things are changing dramatically for non-linear additions of electronic elements, like memristors and other memdevices. Memristive addition is based on a retrograd reliance of the preceding events.

This might be modeled and studied as an addition of memristors or as a repeted application of a single memristor. What is of interest is the “history-dependence” of iteration, i.e. the retrograde character of succession.
This kind of addition is super-additive and its value is retro-grade depending on the preceding state of the previous event.


"The memristor came later because it's inherently nonlinear. Why? A linear memristor is just a linear resistor, since we can differentiate the linear relationship p = Mq and get p' = Mq'. But if p' = Mq' for a nonlinear function f we get something new:
              p' = f'(q) q'
So, we see that in general, a memristor acts like a resistor whose resistance is some function of q. But q is the time integral of the current q'. So a nonlinear memristor is like a resistor whose resistance depends on the time integral of the current that has flowed through it! Its resistance depends on its history. So, it has a "memory" - hence the name "memristance”.” (John Baez)

Heterarchization of superposition <br /> p ' = f ' (q) q ’ : <br /> p ' = ((f ' (q) &nbs ... q ’    ==> ((f ' (M)        )/(f ' ((M) r)))

Non - Monotony (((M  _ 1 + M  _ 1) + ... .) + M  _ n) ==> (((M  ...   _ 1 = :    M  _ 1 + M  _ 1 r  _ 1    <br />

Hence, commutativity is resolved and abolished for memristive addition.

typeset structure

In words:
For M1 + M2:
M1is having a value “r1”, Mtypeset structureis having its value “rtypeset structuretoo, but it depends additionally on the value of M1.
M1r1 + Mtypeset structure = M1r1 + (M2rtypeset structurer1

For M2 + M1:
Mtypeset structure+ M1r1 = Mtypeset structure + (M1rtypeset structurer2

Therefore, a combination of memristors is replicating (retrieving, fetching) the inner state of the repeated memristor with the succiding memristor or the succeding memristive behavior. This reflects the history-dependence of memristive behaviours.

     non - commutativity for memristive addition <br /> <br />     M  ...   _ 1 != M  _ 2 r  _ 1 + (M  _ 1 r  _ 1) r  _ 1 .

Non - Associativity <br />    (M  _ 1 + M  _ 2) + M  _ 3 !=  ... 3A0; _ 3) + ((M  _ 2 r  _ 2 ) r  _ 3) + (M  _ 3 r  _ 3 ))

2.2.7.  Categorical modeling as yuxtaposition and as mediation

        Bifunctoriality of memristive addition with iteration

   <br />    BIFUNCT ^(2)  _ [r     r  ] & ... xF3A0;                                                                             1             2

 M  _ 1 r  _ 1 + M  _ 2 r  _ 2 = M  _ 1 r  _ 1 ... ) o    (M  _ 2  ⊗ (r  _ 2 o r  _ 1)) : Bif  _ asym

<br /> <br />    (M  _ 1 ⊗ r  _ 1 )    [o O]  & ...                                                                                    1             2

Interchangeability of memristive addition with replication
The formula of combination of memristors, typeset structuregets a direct modeling with categorical notions.

A combination of memristors, (typeset structure, is replicating (retrieving, fetching),  typeset structure, the inner state r1 of the repeated memristor (Mtypeset structure with the succeeding memristor (Mtypeset structurer2), i.e.  the succeeding memristive behavior of (Mtypeset structurer2), resulting in the memristive combination  (typeset structure).  

This is category-theoretically modeled with the composition operation (typeset structure) for the combination of memristors and the yuxtaposition (typeset structure) for the two-leveled constitution of the memristors as (M r), while the retrieving (fetching) operation is modeled by the replication operation (typeset structure) for typeset structure.

The modeling might happen in two modi of strictness:
- first a bifunctoriality with the yuxtaposition of two domains of a multi-sorted category with a single universe typeset structure, and
- second by a mediation of discontextural domains typeset structuretypeset structure of a polycontextural category with a mediated polyverse typeset structuretypeset structure  and super-additivity between typeset structure1and typeset structure2.

<br />    1 - category with yuxtaposition for memristors <br /> <br />     ...                                                                                    1             1

A polycontextural modeling of the behavior of memristive devices is of importance, if the domains are not belonging to a common universe of objects. Instead, the discontectural domains are mediated, keeping the difference of both domains.

<br />  <br />       Super - additivity of a 3 - category with replic ...                                                                                                  3

"The primary property of the memristor is the memory of the charge that has passed through it, reflected in its effective resistance M(q)." (Joglekar)

The conceptual modeling with chiasms is emphasizing the possibility of a continuation on a second-order level with the memorized value of the memristor. Hence, the double character of memristance as resistance and stored resistance, typeset structure,  gets involved into a double functionality as a calculated and a calculating aspect.

It becomes clear that memristive behavior is not to be thematized on single memristors and their combinations but on memristive systems, i.e. memristive or memristic complexions, offering the conceptual and physical space for interchanging functionalities.
The following conceptual hint, which is ab/using some electronic formulas, shouldn't be confused with an engineering approach.
The chiasm is mediating two functionalities of a 2-complexion of interacting memristors.
One functionality is: typeset structure<==> typeset structure <-> typeset structure,
the other is:               typeset structure<-> typeset structure <==> typeset structure,
both together are defining a feedback loop conceptualized as a chiasm between resistance and memristance of two instances.

This might be the place to promote the idea, again, that memristics should start with memristive complexions.
A single memristor is then a special case, separated and isolated from the memristive complexion.
Series and parallel circuits of memristors are not yet defining memristive complexions.

Hence, the minimal, still conceptual, conditions for memristive complexions might be defined by at least 3 memristive functionalities, 2 for the relation--> and 1 for relationtypeset structure. It might be necessary to implement the cross-relation with 2 more memristors.

χ(M, r)  ==> typeset structure  ==>  typeset structure

M  _ T (q) = (R  _ (0, 1) + R  _ (0, 2)) -    η (Δ ... _ 2) q (t)/Q  _ 0             

MemR  _ 1 =  (M  _ 1 | r  _ 1) MemR  _ 2 =  (M  _ 2 |  ... _               <br /> <br />

Diamondization <br /> diam (M  _ 1 + M  _ 2) = (M  _ 1 + M  _  ...    M  _ 3           M  _ 6

2.3.  Poly-structured memristive compositions

Not only memristors are in fact  “assemblies of nanoparticles" (Kim, 2009) but also memristive behavior is observable as structured “assemblies”. Therefore, construction and study of single memristors is selective and just the beginning of the adventure. Structured assemblies of memristive behaviors might be understood as complexions of memristors and their memristances. Hence, memristance of a memristive system is conceived more as a field of interacting memristive agents then as a circuit of memristive elements.

This paragraph intends to give a very first glance into the idea of compositions of poly-structured memristive complexions.

As the leading formal approach to modeling, monomorphy based morphogrammatics is applied.

Structuration: Iter/alterability of monomorphies
A combination of the retro-grade mechanism of prolongation (succession) with the complexity of the retro-grade prolongated memristive complexion demand  for the monomorphy-based morphogrammatics.
Instead of kenomic iteration/accretion, prolongation is retro-grade defined on the monomorphies of morphograms.
Combinations of structured memristive systems are demanding for morphogrammatics and its study of monomorphies.

Technically, this seems to be a step further in the concretization of memristics, i.e. a step from the memristor as an element to memristive systems as complexions.

This topic will be developed in a next paper ‘Morphogrammatics of Memristic Machines'.

2.3.1.  Monomorphic evolvement

The morphogram [aa] gets a memristic interpretation by [M|r1r1] or [M|r1.1], [ab] then corresponds to [M|r1r2].
The same holds for [aba] ==> [M|r1r2r1] and [aab] ==> [M|r1r1r2].

<br />      evol([abb]) => ( [[abb], [a]] <br />    &nbs ...                                                                       1.2 .3                    cc

The morphogram [abb] is build by two monomorphies [a] and [bb]. Hence, the operation of evolvement “evol” has to be applied kenogrammatically on both monomorphies, i.e. evol(mg1, mg2) = (evol(mg1), evol(mg2))

The result of an evolution of [abb] is not considered as a set of results but as a simultaneity of resulting evolutions.
The patterns [[abb] [a], [[abb] [b]] and [[abb] [c]] are seen as disjunct and simultaneously produced. They represent 3 different patterns, morphograms, i.e. [abba], [abbb] and [abbc] as mono-form resuslts of the evolvement of the morphogram [abb] with the monomorhy [a]. A further evolvement of [abb] is based on the second monomorphy [bb] of [abb] and is delivering the 3 morphograms [abbaa], [abbbb] and [abbcc].
There are no other constellations possible in this framework of morphogrammatic modeling of the evolvement of the morphogram [abb]. Other specifications of evolvement are defining different results.

Recall, morphograms are invariant patterns of kenograms. Therefore, semiotic representations of the morphogram [abb] as [baa] or as [acc] or as [#%%] etc. are morphogrammatically equivalent.
Again, this shows the retro-grade definition of operations in morphogrammatics, i.e. morphograms are defined as evolvements from themselves, and are not depending on a external alphabet.

The simultaneity of the results holds for the monomorphic modeling of the complex memristor typeset structure.

Thus, an evolvement of complex memristance produces simultaneous results.

What is offered by an evolvement as a structural multitude of possibilities has not always to be realized at once. Depending on other criteria, say from a context, a decision for a single or a subsystem of possibilities might be realized.
From a conceptual point of view, all possibilities have to be developed formally. Because of the holistic character of morphograms, changes of morphograms are always inherently finite.

   <br />              e ...                                                                                                3.3

2.3.2.  Monomorphic multiplicative coalitions

Also morphograms    MG  _ 1 = [ab], MG  _ 2 = [ab]    are  ... e possible, one , a multiplication of    MG  _ 1 with MG  _ 2 is shown .

        <br />       MG  _ 1 =  ... p;          ∐ <br /> mul ([ab], [cd] ) = [abcd]

<br /> M ultiplication   tables   for   kmul ([a, b], [a, b] )   <br />    &nbs ...   d                                                    b  b             b   a  b   c  b   a  b   d

 Context - Rule for monomorphic multiplication <br />    <br />    ∀ ... er(MG) = 1 < 2 < 3 < ... <br /> alphabetical lexical order(MG) = a < b < c < ...

Context   rules : kmul ([a, b], [a, b] )   1. Identity : [a] x [a, b] = [a, b] : head - and bo ... mposed, Dec, into its monomorphies mg  _ i wich get placed at the loci loc  _ i .

Position MG^(m, n) <br />           (m)                                   MG   ...                                   (m)                                   Ken(MG   )   kenom

<br /> Positionality of morphograms : < Position, Locality, Place > . <br /> Position of ... ation . <br /> Place of a kenom in a monomorphy depending on the length of the monomorphy . <br />

      a   b                                                           ...    y          z       u

<br /> Non - commutativity <br /> kmul ([ab], [aaa)] != kmul([aaa], [ab]) <br /> kmul ([ab], [ ...  kmul([aaa], [ab]) = [aaabbb] <br /> But reflection holds for [aaabbb] = refl([aaabbb]) = [bbbaaa]

<br /> Multiplicative coalitions of memristive systems

        <br />       MEM  _ 1 = ...        2                   3           4                     1           2           3           4

<br /> The total memristance of the complexion kmul(MEM  _ 1, MEM  _ 2) is thu ... xF3A0; _ 3 r  _ 1], [ r  _ 1 r  _ 2 r  _ 3 r  _ 4]) <br />

 <br />    Total   memristance   of    MEM _  _ (( r  _ 1  ...                                                                                3                4

<br /> with   [ [r   r  ]                 [ r   r  ] ...                                                                                 3                4

    <br />     Total   memristance   of    MEM _ ó ...                                                                                3           3     O

<br /> [ r   r   r  ] [r   r   r  ] [ r ༺ ...   2                          2           3           3              1           2           2    O

<br />    Total   memristance   of    MEM _  _ ((r  _ 1.2) ...                                                 4                  O                             O

<br /> MEM _  _ ((r  _ 1.2), (r  _ (1.2 .3)))^^(2, 3) | [ r &# ...  2           4                 O                                                                 O

     Total memristance of    MEM _  _ ((r  _ (1. ...                    O                    O                    O                              5.6 .6

 Example for (I) :       (1)       &nbs ...                    1.2 .2             3.4 .4                 5.6 .6                         1.2 .2

2.4.  Interactivity in meristive systems

A study of coalitions in memristive systems is focused on the interactvity of memristive agents. The two layers of memristors are then interpreted as system and environment. A memristor, M, as an interacting agent, has an inner and an outer environment. The inner environment of a memristor, env, corresponds to its state, r, the outer environment corresponds to an another agent with an inner environment (M, r). A full modeling of the interaction corresponds the diamond pattern of structuration, while a reduced modeling, focusing on the intrinsic structure only, corresponds the chiasm pattern of structuration.
This point wasn’t yet considered in the definitions of prolongation and addition.

<br />     A  _ 3 - A  _ 1 --> B  _ 1 - B  _ ...      env  _ 3 - env  _ 2 <-- Sys  _ 2 - Sys  _ 4

inner environment of Sys1is env1
inner environment of Sys2is env2
outer environment of Systypeset structure, env2)
outer environment of Systypeset structure, env1)
inner environment of Sys1 and Sys2 is (Sys3, env3)
outer environment of Sys1 and Sys2 is (Sys4, env4)

The inner state of Mtypeset structurethe environment env1of the memristor Mtypeset structuremodeled as system Systtypeset structurefunctions as system Sys2 with an inner environment env2.  
Hence, env1<==> Sys2--> env2.
The same holds for the inner environment of Mtypeset structureas envtypeset structureSys1and env1.
Hence,  Sys2--> env2 <==> Sys1.

This two paths are connected together, and the coincidence relations, represented as X, are guaranteeing their correspondence.
The inner environment of Sys1 and Sys2, (Sys3, env3), is representing the results of the interaction between the two systems at the place 3, also called “acceptance”.
In contrast, the outer environment of Sys1 and Sys2, (Sys4, env4), is localized at place 4, and representing the “matching conditions” of the interaction between the two systems, also called “rejectance".

Reflexive forms

Reflexive forms are predominant in the theory of second-order cybernetics.
Louis H. Kauffman , Reflexivity and Eigenform  -The Shape of Process

Left distributivity
A[b * c] = A[b] * A[c].  
"We can ask of a domain that every element of the domain is itself a structure preserving  mapping of that domain."

Interestingly, in all those highly elaborated structures of second-order cybernetics, bifunctoriality and interchangeability doesn't appear. Interchangeability is reduced to distributivity.

2.5.  Memristors in classic service

A complementary approach is studied by Lehtonen with the aim to simulate Boolean functions with the help of memristors only .

Lehtonen et al ., Two memristors suffice to compute all Boolean functions " Thus, if p an ...  m  _ 1 -> m  _ 2   =   (¬ m  _ 1) ∨ m  _ 2 . "

"In the following we study how a given Boolean function f on a set of input memristors can be computed using a set of work memristors. The states of the input memristors correspond to the input of the Boolean function and are not to be altered, while the work memristors are used for the computation of the function. The result of the computation will be stored in one of the work memristors. In [4] it was shown how any conjunctive normal form can be synthesised using three work memristors, thus allowing universal computation. In this Letter, we show that regardless of the number of input memristors, two work memristors suffice to compute all Boolean functions."

Everything which is able to have two states might be used to model Boolean logical functions. Therefore, the specifics of memristors, like history-dependence, are disappearing when modeled as a 2-state device. What makes memristors interesting in contrast to other devices with the ability of having two states?

What is shown with approaches like that is that Boolean function might be realized by memristors.
What is not said is that this approach reduces the behavior of memristors to linear, non-time-dependent, binary devices. Hence, everything interesting memristors are performing is eliminated to reproduce well-known Boolean functions.
Therefore, memristors are treated as first-order electronic elements and their second-order quality is omitted.

Hence, the classical Boolean properties of first-order devices, like of commutativity, associativity, idempotence and monotony, completeness and decidability, are restored, making memristor systems trivial machines.

It might nevertheless be of interest to model Boolean functions with memristors plus resistors and to answer the question “How many memristors are necessary to model implication logic”, and others, but there are still some open questions with this approach too.
Where does the distinction of “input memristors” and “work memristors” enter the game?

"In this paper computation with memristors is studied in terms of how many memristors are needed to perform a given logic operation. It has been shown that memristors are naturally suited for performing implication logic (combination of implication and false operation) instead of Boolean logic.”
Lehtonen, E.  Laiho, M. , Stateful implication logic with memristors

To identify a memristor with its states, and to state “Identifying memristors and their states we may thus write..." as it happens with Lehtonen’s approach is highly misleading. It might make sense as an abstraction and separation of one functionality from the other functionality of a memristor as a 2-layered nano-electronic device to define classical Boolean logic. But it tells nothing about the nonlinear behavior of a memristive device.

Triadic constellations
From a holistic point of view it is more reasonable to understand the elementary electronic elements not in their separation but as building a triad.
Hence, there is a resistance of a capacitor and an inductor, as well as a capacitance and a resistance of a resistor.

Leon Chua stated that the textbooks of electronics have to be re-written. Hence, way not start with the basics?

2.6.  Levels of memristivity

Idealizations applied in electronics
linear vs. non-linear
analog vs. digital
separation of resistor, inductor and capacitor

   Electronic triad <br />    resistor --- capacitor    <br /> & ... bsp;              inductor <br />

resistance of a resistor
resistance of a capacitor  
resistance of an inductor
resistance of a capacitor and an inductor

"In reality, all capacitors have imperfections within the capacitor's material that create resistance. This is specified as the equivalent series resistance or ESR of a component.” (WiKi)

The solution, again, is based on separated elements, put in parallel or series to correct the behavior and to eliminate the disturbance.

Levels of memristivity

[Graphics:HTMLFiles/Abstract Memristic Machines_139.gif]

   Chuatronic    triadic system <br />        ... ;    meminductor <br /> nonlinear          

From a more functional point of view, elements or components are parts of a functional triad and their realizations as concrete elements always has to consider the functionality of its neighbor parts of the triad.
Hence, what is called “imperfection” is positively a part of its triadic constitution and the negative behavior is negative only in respect to the idealization of the behavior of an abstractly conceived element.

Idealized series and parallel circuits are therefore special series and parallel combinations of triads where the value of two elements are practically zero. In general, classical electronics of elementary devices might be seen - in the jargon of “just a special case” - as a reduction of diamond triads, i.e. as elements with their functionality reduced to single elements and the possibility of combining them. Hence, without a possibility of metamorphosis, of environments and its specific combinations.
More precisely, classical electronic elements are simple, time-less, history-independent and not localized.

typeset structure

From a conceptual point of view it seems that this are the necessary elementary types of nano-electronic elements. Are they also sufficient?
Considering the two-level constitution of nano-electronic elements it might be argued that all higher level-elements are compositions of the 3 elementary levels.

Obviously, the third-order element is conceived as a kind of a chiasm between the aspects: first-order, mem and 2 levels.
therefore, a further level seems to be based on an iteration or accretion of the first 3 levels. Hence, a higher order construction might not constitute an own new level but just a combination of the basic level of the elements of the table.

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Those questions appear at the time as quite academic and of no special interest. Nevertheless, from a conceptual point of view, they deserve some reflections.
The first two levels are well constructed by the SPICE model.
The third level, unknown today, seems to open up new possibilities to construct memristive systems. One candidate might be a memristive interaction between different discontextural systems in poly-crossbar constructions.
The first two levels are non-interactional, the third is introduced as an interaction between two second-level elements. The first-level element is without any reflection, but the second-level element is reflecting the first-level element in its behavior.

A fourth-level element might be a construction that is reflecting super-additively of the activities of a third-level system.

fourth-level elements : typeset structure

Calculus of triads
Composition of triads into series and parallel superpositions instead of single elements only.

Composition of triadic diamonds
Triadic Diamonds8

3.  Memristive systems as self-organizing machines

Memristive systems as complexions might give a chance to construct self-organizing non-trivial machines, which are realizing different dynamics of chiastic interactions.
There is no need to restrict the concept of self-organizing machines to the apparatus of recursive functions and paradox logical systems as it was emphasized by Heinz von Foerster, Francesco Varela and others, and applied in different disciplines like sociology by Niklas Luhmann.

Conceptually, self-organizing machines are well understood as chiastic figurations. This has been pointed out with much sophistication and aesthetics by the cybernetician Gordon Pask. But this approach never got its proper scientific recognition.

Self-organizing systems in the framework of cybernetics are still struggling with the problem of “re-entry”. How is a function or action re-entering its scope without missing it? This is guaranteed by definition, i.e. it is pre-installed by the external designer of the system. Without such an external regulation, the action would easily miss its re-entry point.

Now, with the ability of memristive systems to store their previous values, a re-entry is well defined as a retro-grade returning to the stored value for further calculations.  Therefore, the temporal gap in classical systems between calculation and re-entry (by feedback loop) is bridged by the retro-grade memristance of the former action.

Self-organizing systems must be able to accept their rules of learning to build meta-learning, i.e. hierarchies of learning of learning.

A full realization of self-organizing systems leads to autopoietic systems as a radicalization of self-referentiality towards their own existence. Such systems are leaving the paradigm of information processing; they are not processing information but are in-formed by interactions.

4.  Memristive systems as co-creative autonomous machines

Because of the ability to realize a complementarity of computation and memory, memristive systems seem to be able to open up the possibility of emulating the proemial relationship between cognition and volition. And therefore for autonomous and co-creative interactions.

The term “objectional” means both: refutation and ‘objectification'
Autonomous systems must be able to reject their rules of meta-learning.

Hence, for the first time, the Gödel-Argument that machines will never be more reflectional (intelligent) than human beings who are constructing the computational machine, fails definitively.
Formal languages, defining the general concept of computation, are not involved in retro-grade monomorphies, i.e. in complex pattern surpassing the limits of identitive signs and marks.

Such considerations are not involved into the discussion of developing “brain-like” machines on the base of memristively conceived and implemented synapses.

Memristics starts a decisive departure from the logico-mathematical understanding of neural networks as they have been conceived and formalized by Warren McCulloch and Walter Pitts.
For the same reasons, memristic systems have to abandon the primacy of chaotic complex and self-organizing systems.

Heinz von Foerster’s principle of “Order from noise”, i.e. order from (order and disorder), is not catching retro-grade recursivity of time-dependent events. Neither happens this with Gotthart Gunther’s “Cybernetic Ontology and transjunctional operators”, which gives conceptual explanation of the principle of “order from noise”.

Recursivity happens in time; the iteration of the recursion, but recursivity is not defined by a time-dependent formalism.
Time for recursivity is measured by a “Schrittzahl”, i.e. the number of the steps of the recursion, and is therefore not involved with time- and history dependence of its iterative steps.

As the ethymology of the terms shows, there is no time-dependence involved in this “run back; return” activity.
From Latin recursio (“the act of running back or again, return”), from recurro (“run back; return”), from re- (“back, again”) + curro (“run”).

Hence, the recurrence of recursivity (or of recursive functions) is not depending on retro-grade time- and history-dependence.

"From McCulloch's "experimental epistemology," the mind - purposes that ideas - emerged out of the regularities of neuronal interactions, or nets. That science of mind thus became a science of signals based on binary logic with clearly defined units of perception and precise rules of formation and transformation for representing mental states. Aimed at bridging the gulf between body and mind (matter and form) and the technical gulf between things man-made and things begotten, neural nets also laid the foundation for the field of artificial intelligence. Thus this paper also situates McCulloch;'s work within a larger historical trend, when cybernetics, information theory, systems theories, and electronic computers were coalescing into a new science of communication and control with enormous potential for industrial automation and military power in the Cold War era.”

A logic of memristic systems has to be determined by the history-dependence of the events it tries to model logically.
Time- and history-dependence, nevertheless, is not caught by a modal logic of time or time-events, nor from a logic of time-statements like Carl Friedrich von Weizsäcker’s “Logik der Zeitaussagen”.

Obviously, the attempts to connect the behavior of memristors with logic, say by material implication, might be a start but probably also a start into the wrong direction.

It may be said that trivial and non-trivial machines are well known, and self-organizing machines much less, “auto-poietic co-creative machines”, i.e. autonomous machines are probably not known at all.

A new principle shall be added to the list of principles of structuration:

Fourth principle: Order from (order neither disorder).

4.1.  Schemes of structuration

Order-Scheme of structuration:
1. Order from order, McCulloch-Pitts
2. Order from disorder, Chaos theory, Grossberg, ANN
3. Order from (order and disorder), Schroedinger, Heinz von Foerster, Chua, Williams
4. Order from (order neither disorder), Gunther, Derrida, Kaehr

Order-type-1 corresponds deductive axiomatic formalisms, artificial intelligence.
Order-type-2 corresponds inductive classifications, neural networks, learning
Order-type-3 corresponds acceptive self-organizing systems, learning to learn
Order-type-4 corresponds transjunctional rejections, diamond saltisitions. Learning to (learn and reject to learn)

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More information at:

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[Graphics:HTMLFiles/Abstract Memristic Machines_152.gif]

         (RelSyst, Anch) - chiasm : <br />   &nbs ...                              1                         ◇                                Rel

<br />     γ .    cross - polarity    (green (◇ ...                             o                                                                  Rel

Principles of order and levels of structuration

1. Order from order

linear                       polar            monoidal
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2. Order from disorder

typeset structure  typeset structuretypeset structure  typeset structure

3. Order from (order and disorder)

typeset structure   typeset structure

4. Order from (neither order nor disorder)

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1 "Non-trivial machines have internal states. The relation between the inputs and outputs of a
non-trivial machine is anything but invariant. Instead, it is determined by the machine’s
previous operation. Thus the history of the machine’s operations affects its preceding
function. Ashby and von Foerster [8] prove that some of them are in principle, and others in
practice, analytically indeterminable and therefore unpredictable.”

"Let n be the number of inputs to the machine. Let us suppose that the number of outputs is
equal to the number of inputs. The number N of all trivial machines that can be synthesized is
therefore NT (n) = nn, and the number of non-trivial machines is as much as NNT (n) = nnz,
where z represents the number of internal states. In this case, z cannot be greater than the
number of possible trivial machines (z <= nn). Thus, for trivial machines with four possible
inputs, NT (4) = 256 and for non-trivial machines, NNT (4) = 41024, which means
approximately 10620 elements. And we are still dealing with a simple machine operating only
with four variable values, having only 256 internal states at its disposal. Nevertheless, even
the complexity of this system is unthinkable to the point that it is absolutely impossible to
analytically explore its functioning. The problem is transcomputational.” (Urban Kordeš )

2      Peter Selinger, A survey of graphical languages for monoidal categories
    “Abstract. This article is intended as a reference guide to various notions of monoidal categories and their associ    ated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.”

     3.2 Behavior
     “Let A be a sequential algorithm.
     Postulate 1 (Sequential Time). A is associated with
     -a set S(A) whose elements will be called states of A,
     -a subset I(A) of S(A) whose elements will be called initial states of A, and
    -a map  A : S(A) −! S(A) that will be called the one-step transformation of A.

    The three associates of A allow us to de ne the runs of A.

    Definition 3.1. A run (or computation) of A is a  nite or in nite sequence
                                     X0;X1;X2; : : :
    where X0 is an initial state and every Xi+1 =  A(Xi).

    We abstract from the physical computation time. The computation time reflected
    in sequential-time postulate could be called logical. The transition from X0 to X1 is
    the  rst computation step, the transition from X1 to X2 is the second computation
    step, and so on. The computation steps of A form a sequence. In that sense the
    computation time is sequential.”

    4.2 The Abstract State Postulate
    Let A be a sequential algorithm.
    Postulate 2 (Abstract State).
   |States of A are  rst-order structures.
   |All states of A have the same vocabulary.
   |The one-step transformation  A does not change the base set of any state.
   |S(A) and I(A) are closed under isomorphisms. Further, any isomorphism from
   a state X onto a state Y is also an isomorphism from  A(X) onto  A(Y ).
  (Gurevich, p. 7, The Sequential ASM Thesis)
4.5 Inalterable Base Set
While the base set can change from one initial state to another, it does not change during the computation. All states of a given run have the same base set. Is this plausible? There are, for example, graph algorithms which require new vertices to be added to the current graph. But where do the new vertices come from? We can formalize a piece of the outside world and stipulate that the initial state contains an infinite naked set, the reserve. The new vertices come from the reserve, and thus the base set does not change during the evolution.
Who does the job of getting elements from the reserve? The environment. In an application, a program may issue some form of a NEW command; the operating system will oblige and provide more space. Formalizing this, we can use a special external function to  sh out an element from the reserve. It is external in the sense that it is controlled by the environment.
Even though the intuitive initial state may be  nite, in nitely many additional elements have muscled their way into the initial structure just because they might be needed later. Is this reasonable? I think so. Of course, we can abandon the idea of inalterable base set and import new elements from the outside world. Conceptually it would make no di erence. Technically, it is more convenient to have a piece of the outside world inside the state. (p. 13)
Sequential Abstract State Machines, Capture Sequential Algorithms
Yuri Gurevich, September 13, 1999, Revised February 20, 2000, MSR-TR-99-65, Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399

4 "Diese Bestimmung des Begriffs der Wiederholung als retro-grad rekursiv involviert vier neue Aspekte, die der Rekursion als rekurrierender Wiederholung, fremd sind: einen Begriff der Selbstbezüglichkeit, der Transparenz, des Gedächtnisses bzw. der Geschichte und einen Begriff der Evolution im Gegensatz zur abstrakten Konkatenation und Iteration.” (Kaehr, 2003)

”Die Verkettung einer Kenogrammmkomplexion K mit einem einzelnen Kenogramm k kann hingegen nur unter Rückbezug auf die innere Struktur von K geschehen, da k kein wohlunterschiedenes Atomzeichen ist. [...]
Diese Rückbezüglichkeit bei der Verkettung lässt eine Kenogrammkomplexion als Ganzheit oder Gestalt entstehen, die nicht auf eine lineare Verkettung reduzierbar ist.” (Kaehr, Mahler, Morphogrammatik, p.31, 1993)

Evolution: “Synthetische retrograde Ausgliederung" (Kaehr, 1982)

5 Recall
„Es ist auch fraglich, ob der Begriff eines Zustands als Menge von Attributwerten (in
imperativen Sprachen üblicherweise als record implementiert) ausreicht, um alle interessanten
Objekttypen zu erfassen. Aus seinem Zustand soll ja in gewisser Weise die
Identität eines Objektes erschlossen werden. Reichen dazu immer augenblickliche Attributwerte
aus? Wann bestimmt eher die Geschichte des Objektes, d.h. die Folge der
Zustände, die es bisher durchlaufen, seine Identität? Kann die Geschichte immer in
eine endliche Zustandsstruktur hineincodiert werden?
Die klassische Automatentheorie ist inzwischen zu mehreren Theorien kommunizierender
Systeme erweitert worden, wo man gar nicht mehr von Objekten spricht, sondern
nur noch Prozesse, also Objektgeschichten, untersucht und als - manchmal
unendliche - Strukturen darstellt.“ (Peter Padawitz, Vorlesung)

"Die kenogrammatische Operation der Nachfolge dagegen wird nicht durch ein vorgegebenes
Alphabet definiert, sondern geht aus von dem schon generierten Kenogramm
hervor. Jede Operation auf Kenogrammen ist „historisch“ vermittelt. D.h. die Aufbaugeschichte
der Kenogramm-Komplexionen räumt den Spielraum für weitere Operationen
ein. Diese können nicht abstrakt-konkenativ auf ein vorausgesetztes
Zeichenrepertoire zurückgreifend definiert werden, sondern gelten einzig retro-grad
rekursiv bezogen auf die Vorgeschichte des Operanden. Diese Bestimmung des Begriffs
der Wiederholung als retro-grad rekursiv involviert vier neue Aspekte, die der Rekursion
als rekurrierender Wiederholung, fremd sind: einen Begriff der
Selbstbezüglichkeit, der Transparenz, des Gedächtnisses bzw. der Geschichte und einen
Begriff der Evolution im Gegensatz zur abstrakten Konkatenation und Iteration.” (KAehr, SKIZZE-0.9.5, p. 36, 2003)

6 "Iterability alters" (Derrida 1977)
"Iterability is the capacity of signs (and texts) to be repeated in new situations and grafted onto new contexts.  
Derrida's aphorism "iterability alters" (Derrida 1977) means that the insertion of texts into new contexts continually produces new meanings that are both partly different from and partly similar to previous understandings.
(Thus, there is a nested opposition between them.).  The term "play" is sometimes used to describe the resulting instability in meaning produced by iterability.”  (Jack M. Balkin, 1995-1996)