Memristics: Memory is more than Storage

Memristive memory is able to surpass the conceptual limits of computational storage methods

Rudolf Kaehr Dr.phil.@

Copyright © ThinkArt Lab ISSN 2041-4358



The relationship between memory and computation was not always a happy one. Once fixed by John von Neumann’s conceptualization of the practice of engineering solutions for practical computer architectures, it has become the ultimate paradigm of architecture but ending now into its permanent bottleneck foreclosing the former interactivity of memory and computation. Insights into mnemonics of the Ancient solutions to the process of memorization are slowly recovering from the military hierarchy of commander and commanded, and the reduction of memory to storage. Memristive systems are prepared to re-dynamize the interplay of memory and computation again. Some orientation towards conceptual generalizations of memristive approaches is given with the use of poly-categorical methods.

1.  "Memory without Record” (Heinz von Foerster)

Memristors as Logic

"The biggest new news about memristors, though, came in a paper in Nature last week, in which HP announced that the devices can also perform logic functions. In other words, Wiliams said, a memristor can act as both a storage element and a logic element, or "a lock as well as a gate."

"There's nothing else I'm aware of that performs both of those functions simultaneously," he said.”

Williams said there is an
intriguing possibility that if you could use the same structure to do actual computing as well as storage, you could send the program to where the data is and execute the problem where the data is stored. Of course, that all depends on what the performance of memristor-based devices ends up being, compared with traditional CPUs and memory systems.

Memory is more than storage

"Note the double closure of the system which now recursively operates not only on what it “sees” but on its operators as well.” (Heinz von Foerster, On constructing a Reality, in: Observing Systems, p. 305, 1984)

Storage implementation by flip-flops based on NAND or NOR gates are first-order concepts realizing storage and computation with the help of an “external” timer.
Memristive realizations are of second-order, they are not genuinely implemented by NAND-derivatives build by IMP but by a new kind of second-order construction. Because of their second-order status they are not primarily emulating storage but memory. Memory, in this generalized sense, is a self-referential construct, allowing to change the memorized object while memorizing, hence the object is not simply stored as a record, but is accessible to re-interpretation.

This observation goes beyond the famous “stateful logic"-concept (Williams), re-unionizing memory and logic, because the formulation “memory and logic” is first re-institutionalizing the old concepts of logic, i.e. the binary 2- or many-valued logics for digital or analog computation, and memory as an a-temporal, non-directional and non-reinterpretable way of storage albeit in a new way of a close interaction between both omitting loss by resource-consuming data-transfers between both, logic and memory.

Therefore, Chua’s statement that memristors are synapses has to be reframed and reconsidered as a second-order statement.

Hence, the highly abstract conceptual interpretation of memristive flip-flops as sketched in my “Memristic flip-flop” -paper gets some crucial concretization towards the difference of the first-order concept “storage” based on NAND flip-flops, or equally implicational implementations, and the second-order memristive “chiasm” (flip-flop) based on distributed memristive (IMP, F)-constructions.

"What has to be saved by the memristive device are in fact not the first-order data of FF but the conditions of the possibility of the data, i.e. the ‘data’ of the matching condition of the composition of the master and the slave flip-flop morphisms (processes). From those conditions, as second-order constructs, the first-order data might be reconstructed on the base of the second-order data saved by the memristive device.

In other words, the history saved by the memristor are not the primary data but the data of the history. Historical data are data of data. Those second-order data might then be used to continue processing on the first-order level of the flip-flop.

As a metaphor, the data of an observer of a data processing system are not the data of the observed system. But such observer-depending data of second-order might be given ‘back’ to the observed, i.e. first-order system to continue its game. Hence, the memristor is playing the game of an observer which is lending or giving away his data to the observed system.
The memristive system is primarily storing the rules of the observed game and only secondarily the data involved.”

HP’s construction is using the flip-flop channel as a memristive buffer, and is not yet exploiting the possibilities of the interactivity of the master/slave-relationship by the involvement of the memristor.
The concepts of complementarity, simultaneity and antidromicity are not yet used in the construction of HP’s memristive FF device.”
(Flip-Flop, p.17)

Hence, my first critical analysis, gets concretized by the introduction of the difference between first-order “storage” and second-order “memory".

Memories are interpreted data. Data without interpretations are records. Data cannot be changed as data, memories are changed by contexts, i.e. by the change of their operators. Memories without contexts don’t exist; for stored records contexts don’t exist.

How are second-order memories, based on memristors, working?

2.  Second-order flip-flops

A transition from first-order flip-flops to second-order flip-flops is not mixed with the strategy to enlarge the range of decisions from bi-stable to tri-stable or multi-stable devices of first-order. This goes together with the statement that a transition from mono-contextural to polycontextural logics is not in any complicity with the change from 2-valued to ternary and multiple-valued logics. Such enlargements or precision of logical parameters don’t surpass the limits of mono-contextural logics and concept developments.

Interpretation of data is computing data. Hence, memories are data of data; and therefore a unit of data and computation (logic).

If data have to be computed, before they eventually could be stored, second-order memories don’t have records.

A closer look at the structure of computed, i.e. constructed data for non-storage memories, would have to go into the theory of “Eigenbehavior’, the cybernetician von Foerster introduced to understand the “operational closure of cognitive systems”. In other words, the study of computation and memory for memristive systems should be connected with the results of the studies towards a theory of living systems as it was developed by the approaches of second-order cybernetics. Because of Chua’s interpretation of synapses as memristors, such a connection with cybernetics could be of mutual productivity.

The whole interplay of memory and computation might be radically reduced to an operational interplay of operators and operands in a contexture. With operators representing computation and operands representing memory as stored data.

- first-order category theory (CAT): objects, morphisms, composition and yuxtaposition
- second-order category theory: dissemination of first-order CAT and new rules between CATs.

It could be said, that von Foersters insistence on recursivity and self-referentiality is based on the fact that there is no memory-function explicitly involved in his mathematical modeling strategies. Self-referentiality, say as self-repair, is thought by von Foerster as “repair of repair”, i.e by an iteration/accretion of the term. And obviously, but not necessarily, the first and the second term have to be at least similar if not identical. A concept of self-x as ” x of x” is not working with a difference in the terms, say “x of y”, like self-repair as repair of selling, instead, repair of repair.

In contrast, Gunther’s philosophical “proemial" (chiastic) modeling was fundamentally involved with memory-functions, albeit this was not always explicitly stated as such, and his formalism is not operative enough to be seriously applied in the construction of second-order devices.

On the other hand, recursive functions and contemplations on it are still more descriptive than operative. For recursive function the address of re-entry in a circular calculation is pre-defined, and the question, how can a function miss its re-entry-address, appears to be strictly absurd.

For proemial operativity, the interchange of operator and operand has to be installed with the help of a ‘coincidence’ relations, otherwise the process of self-application might be confused and, because of the complexity of the construction, might easily miss the intended re-entry-point for another destination (slot). Therefore, the concepts of proemial relationship got a concretization as a chiasm with the introduction of the coincidence or similarity relations. A further formalization of proemiality was introduced with the construction of poly-category-theoretical diamonds.

It seems to be reasonable to interpret and implement such constituents like ‘coincidence’ relations in chiasms as memory functions.

Are memristors non-trivial machines?

Non-Trivial Machines
Non-trivial machines had been introduced by Heinz von Foerster.

Non-trivial machines have the following properties:
(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/Memory is mor than Storage_1.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'.
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.‘’
History dependence
"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)

Memristors are in fact  “assemblies of nanoparticles" (Kim, 2009), therefore, their behavior is not “analytically” pre-defined, their behavior, depending on the context of the system, 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.

History-dependence is not yet a criteria for memristive behavior if the kind of dependency is not properly characterized. Classical finite state machines are behaving “history-dependent” too. The dependence on the internal state of a machine and its history of events are typical for finite state machines albeit they are not time-dependent.

What is the difference to memristic systems?
The internal states of a finite state machine are first-order states. They are states of the machines and not states of states of the machine. Without this difference, memristic systems collapse to ordinary finite state machines, and the proclaimed paradigm change is lost.

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 between 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?

Because functionally 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.

Further Notes
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 because 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”.
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.

Further analysis will be published as “Towards Abstract Memristic Machines".

3.  Are memristors morphograms?        

3.1.  Physical model

3.1.1.  Physical realization

The study of the behavior of memristors has been focussed on the abstract and non-located functionality of memristors.
The definition of memristors, physically as doped/undoped nanodevices, and logically as material implication with negativity, had been restricted to a context-free, i.e. non-located design and understanding.

But it seems, that memristors, in contrast to classical logical devises, are not necessarily connected to a single physical or logical place to develop their functionality.

"x is the width of the doped region, referenced to the total length D of the TiO2 layer, and Roff and Ron are the limit values of the memristor resistance for w = 0 and w = D. The ratio of the two resistances is usually given as 102 - 103.”

Phenomenological Description
Strukov, Stewart, Snider & Williams, Nature 453 80 (2008)
[Graphics:HTMLFiles/Memory is mor than Storage_2.gif]

Formula R _ mem (x) = R _ on x + R _ off (1 - x) = R _ off - (R _ off - R _ on ) x , <br /> x = w/D ∈ (0, 1) .

"x is the width of the doped region, referenced to the total length D of the TiO2 layer, and Ron and Roff are the limit values of the memristor resistance for w = 0 and w = D. The ratio of the two resistances is usually given as 102 - 103."

3.1.2.  Physical diagrams, the pinged-hysteresis loop

What is the category-theoretical modeling for mathematicians is the data-representation of the behavior of memristors under specific physical conditions by curve-diagrams.
It is stated that such a modeling of a data-function with a purely structural method like polycontextural monoidal categories is opening up not only a conceptual insight into the mechanism of the pinched-hysteresis loop but also interesting generalizations in a structural and domain-specific sense.

It seems that such an approach had its justification in the historical example of the understanding of physical events as an emulation of the abstract figuration of Chua’s definition of a memristor.

Medha Haridas et al
"That’s when Greg Snider, who had worked with Kuekes on the Teramac, brought the Chua memristor paper from the September 1971 IEEE Transactions on Circuits Theory. Chua’s paper had included a graph that looked suspiciously similar to the experimental data that were being collected.
The graph described the current-voltage (I - V) characteristics that Chua had plotted for his memristor. Chua had called them “pinched-hysteresis loops”, I - V characteristics for the platinum device were called “bow ties.” A pinched hysteresis loop looks like a diagonal infinity symbol with the center at the zero axis, when plotted on a graph of current against voltage. The voltage is first increased from zero to a positive maximum value, then decreased to a minimum negative value and finally returned to zero. The bow ties were nearly identical. That’s not all.”

[Graphics:HTMLFiles/Memory is mor than Storage_4.gif] |

Translation (diagram above) Possible translation of the pinched - hysteresis loop for voltage  ...  _ 2) | <br /> corresponds : full hysteresis loop    with pinch ([-0, 0.0], [0, 0.0]) .

Kim’s simplified diagram
It seems to be only an easy step to accept a category-theoretic formualtion considering Kim's simplification of the data-loop to a directed arrow diagram.

typeset structure
h-loop = loop1 ∪ loop2.

This step to conceptual modeling is not yet concerning a categorization of the mathematical formulas describing the physical behaviors.

[Graphics:HTMLFiles/Memory is mor than Storage_7.gif]

3.1.3.  Pinched-hysteresis loops as interchangeability

As it was shown in previous papers, there is a close similarity between the conceptual structure of the definition of a memristor and diamond category-theoretical constructions of chiasms and interchangeability.
Hence, it doesn’t comes as a surprise to see a chance to model the data-flow schemes of memristors with the help of diamond categories. It gives probably some more insight into the structure of the temporal behavior of memristors if this behavior is modeled as an interchangeability of its constituents, which are “positive maximum”, “negative minimum” and “positive zero”, “negative zero” of voltage (V) and current (mA) as a “bow tie” movement.

With (typeset structuretypeset structure) for the (f, g)-left loop-part,  (typeset structure) for the (f, g)-right loop-part of the distributed double loop with “zero crossing property” the pinched-hysteresis loop gets a structural model. The involved domains typeset structure and typeset structure here are the positive and the negative value-space of voltage (V) and current (mA), hence might be modeled as belonging to a common domain typeset structure

 Cross - interchangeability of two internal memristive domains <br />  <br />   &nbs ...                                                                      1   o                       1

 Bracket reading <br /> [B    B ] <br /> vertical : " serial " compo ...                            A            A                           1                    2

    Generalized notation <br /> <br />     U  _ 1 = U &# ...                                                                      1   o                       1

<br /> w  _ 1 --> D _ 1 : right - loop = (w _ 1, D _ 1, D&# ...  --> w _ 2    : left - loop = (w _ 2, D _ 1, D _ 2) .

Yuxtaposition: parallel (typeset structure)
A mono-contextural modeling of the On-Off-mechanism, depending on two different temporal situations of the common domain, producing two sub-domains typeset structure,  is possible too. Instead of a polycontextural mediation of the two disjunct domains, a yuxtaposition typeset structure” is defined over the objects of the common domain typeset structure

It seems to be natural to suppose a unity for the two domains typeset structurei.e. typeset structure. This is presupposed by the definition of the memristive behavior as a pinched-hysteresis loop. But in more complex constellation such a unity has to be created by the system and has to be differentiated from other pinched-hysteresis loops from neighbor systems. Epistemologically, it also has to be distinguished between an internal and an external description of the behaviors. Both ways of description have to be harmonized towards an acceptable final characterization of memristive behaviors. All behaviors being involved together in the creation of a complex memristive system.

   Yuxtaposition :   parallel   (⊗ ) <br />     U  _ 1  ...                              on         1.1    o                                   off         1.1

<br />    Yuxtaposition :   cross - interchange   (◇) <br />   &nbs ...                               on         1.1    o                                   on         1.2

Considering the temporal change between On and Off in a single domain as On1.1--> Off1.1 and Off1.2 --> On1.2,  a cross-interchange (typeset structure) between On and Off might be a more concrete modeling. The cross-interchange operation might be defined in monoidal categories with yuxtapositiontypeset structure” and permutation ”σ”. Thus, the cross-interchange operator "typeset structure” is used in this mono-contextural context as an abbreviation for "(typeset structure, σ).”

3.1.4.  Pinched hysteresis loops as a braided loops

“‘Hysteresis is a tell-tale manifestation of the fourth circuit element--the memristor’,said Chua.
Chua predicts that electrical engineers will soon begin discovering all types of new materials that manifest the hysteresis relationship between flux and charge.”

The general interchangeability conceptualization of the phenomenon of “pinched-hysteresis loops” is open for 3 different readings:
1. as a pinched loop in a two-dimensional coordinate system,
2. as left braided loop in a topology, and
3. as a right braided loop in a topology.

Hence, a pinched-hysteresis loop in the sense of memristic behavior has to be properly specified by its pinch, not only as a common interchanging point but also as a braided intersection.

This is a generalization of the idea that different hysteresis loops coinciding at the pinch might not occupy the identical pinch as a crossing point but are just similar and not identical.

Therefore, a single pinched-hysteresis loop might topologically be a braided loop without a common identical pinch.

A pinched-hysteresis loop which is left- and right-braided is a pinched-hysteresis loop with identical pinch.

This speculation is thus abstracting from a fixed coordinate system, and measuring has to be realized in a more flexible manner.
All three loops meet at the same pinch, but this sameness is an overlapping or coincidence of their pinch and not an identity like in the previous modeling.
Knot theory is opening up new aspects of memristic behaviors.

A separation of topological aspect of the loops and the metrics of the loops as representations of data-functions enables more direct conceptual analysis of the basic characteristics of memrstive behaviors.

3.1.5.  Example of a non-pinched hysteresis loop

[Graphics:HTMLFiles/Memory is mor than Storage_30.gif]

    Yuxtaposition :   parallel   (⊗ ) <br />      U & ...                                                              1.1    o                          1.1

Karl Steinuch’s sharp hysteresis loop of a Schmitt trigger from the ‘60s, representing a ferro-magnetic hysteresis, is an ideal example of a non-pinched hysteresis loop. It might be formalised as a parallelism of (M, T)- and (-M, -T)-components of the loop (M, -T, -M, T).

Hence, there are some strong and simple distinctions to draw, which are offering interesting classifications and orientations in the field of memristive systems.
Non-meristive systems are studying non-pinched hysteresis loop behaviors, while memristive systems are focused on pinched hysteresis loops.
A further difference is introduced with the distinction of single and multiple pinched hysteresis loops with common or mediated pinch (crossing point).

3.1.6.  Iterations of pinched-hysteresis loops
Parallel iteration

 <br />    (h     h  ) ∈ U : <br />    <br />   ...  2           2                                    2                         2                    2

Two partly-overlapping pinched-hysteresis loops

   <br />     loop _ 1 ∈ (U  _ 1.1 ∪ U &# ...           2           2                        2            2            2           2           2

[Graphics:HTMLFiles/Memory is mor than Storage_34.gif]

[Graphics:HTMLFiles/Memory is mor than Storage_35.gif]

Two pinched-hysteresis loops with common pinch

<br /> h - loops : brown - loop ∈ U  _ 1.1, blue - loop ∈ U  _ 1.2 ... ) h - loop _ 2) = pinch (h - loop _ 1) ∩ pinch (h - loop _ 2) <br />

 <br />     U  _ 1 = U  _ (1.1 .1) ∪ U  _ (1.1 .2 ... .2 .1              on         1.2 .1    o                                       off         1.2 .2

Three pinched-hysteresis loops

[Graphics:HTMLFiles/Memory is mor than Storage_38.gif]

    <br />    brown - loop ∈ U  _ 1.1 = (r - loop༺ ...            1.2 .0                               2           0.2 .3                               2

Accretive modeling of three pinched-hysteresis loops

There might be a reason to thematize the three different pinched-hysteresis loops as being in some way incommensurable albeit their common and overdetermined pinch. In this case, a polyconcextural modeling with disjunct but mediated domains would be appropriate. All three loops meet at the same pinch, but this sameness is an overlapping or coincidence of their pinch and not an identity like in the previous modeling.

    <br />       brown - loop ∈ U  _ 1 = (r - ...                       1.2 .0                    1           1.0 .0                               1

3.2.  Distribution model

3.2.1.  Distribution of the Rtypeset structure

Based on the insight of the contextural locality of the occurrence of memristors a distribution and mediation of the basic concepts as they had been introduced by Stan Williams at the HP Lab, it might be opted for a dissemination of those constructs. First as a strict parallelism of the very same constructs for Rmem at three discontextural loci.

( R   (x)   ∈ (0, 1)                                            ) <br /> ...  -  x _ i = w _ i/D _ i ∈ (0, 1)  _ i, i = 1, 2, 3 | <br />

             ----•---- <br / ...            : (0, 2) = (0, 1)  _ 3 <br />

Memristors are depending on their material realizations. To conceptualize the behavior of memristive systems, the context in which such behavior is realized has to be considered too. Hence, memristors, logically modeled as material implications need a conceptual context to localize their logical distinctions.

In other words, it is not the case that the concept of material implication involved with memristors gets an implementation on a physical level as it happens for NAND-gates, which get a physical inscription into silicon on a microelectronic level, rather it is the nano-physical event of memristive behavior on nanoscale which enables a logical interpretation as a material implication. Again, “the phenomena associated with memory are ubiquitous at the nanoscale” (Di Ventra). This is not in contradiction to the fact that memristive devices too have to be artificially constructed.

Until now the modeling was focused on pinched-hysteresis loops as parallel or mediated loops with a common albeit not necessarily identical pinch-point. That is, the specification of the pinch-points with (0, 0) is not supposing a conceptual and physical identity but rather a kind of sameness, which allows an interactive mediation of the different loops.

A step further is introduced by the concept of complementarity, crucial for quantum physics.
Therefore, the next step of modeling will consider the poly-categorical approaches with distributed pinch-points.

3.2.2.  Complementarity and disseminated pinched-loops

As it is mentioned before, time- and history -dependent events and features are ubiquitous on the nanosclae of quantum physics. What is also know quite well is that quantum physical events are not descriptable by classical logics. Especially the phenomenon of complementarity is demanding for a complex formalisms, i.e. logics, which are able to map complementarity. Hans Primas has developed those aspect with his concept of non-commensurable Boolean logics.

On the other hand it seems that memristic studies are neutral to such insights into complementarity. There is no mention about complementarity and no idea therefore, how complementarity and non-commensurability of different Boolean logics could have an influence on the definition of memristive systems.

The physical behavior of memristive devices is well described by pinched-hysteresis loops.
The question is: What happens with the pinched-hysteresis loops in a complementary setting?
Obviously, a complementarity demands for at least two different, not only disjunct but discontextural domains. In this sense, complementarity is different to duality. Therefore, the mono-contextural modeling in the frame of a single universe typeset structurewith different domains typeset structure, i∈N, has to be extended to a polycontextural modeling with mediated universes typeset structure

To elude the idea of disseminated “pinched-hysteresis loops” in a poly-contextural modeling, some citations from the eminent quantum theoretician Hans Primas might be of support by hinting to the background knowledge of a non-reductionist understanding of quantum theory.

"Two Boolean descriptions are said to be complementary if they cannot be embedded into a single Boolean description.” (Primas, p. 17)

Complementarity of tensed and tenseless time
"All known fundamental principles of physics refer to laws which are invariant under time translations. That is, the fundamental laws of physics do not contain any tensed notions. In contrast, one of the most the distinguishing qualities of consciousness is the Now.” (Primas, p. 29)

"Since tensed and tenseless time refer to different domains, the still prevailing discussion whether the tensed or the tenseless theory of time is “true” makes no sense. Physical time is a crucial element in theoretical physics, but the experienced time cannot be dismissed as irrelevant for the understanding of physics. These two concepts of time are not contradictory but complementary. None of them is sufficient, none can replace the other, both are necessary.” (Primas, p.33)

Dressed electron
"The state representing an electron as actually observed in the laboratory is called a dressed electron,
it has a very complicated structure. Without an appropriate concept of an environmental background the concept of an individual quantum object makes no operational sense. Heuristically, a dressed electron can be thought of as consisting of the bare electron, interacting with its own radiation field by emitting and reabsorbing virtual photons. The presence of a virtual cloud does not only modify the properties of the bare elementary system and its bare environment, but also its dynamics. A dressed object is not only adapted to its environment. Its individuality emerges as by the interplay with the environment.”
(Primas, p. 33 )

3.2.3.  Categorical descriptions of distributed interchangeability

Parallel distributed interchangeability with distributed pinch - points      <b ...                      1                              1   |                      1                1

The interchangeability formula rules the parallel distribution of the w-D-distinction for two distributed memristors in poly-layered crossbar systems. The domains typeset structureand typeset structurecontaining the different w-D-distinctions, might be defined by two distinctive ranges of TiO2 layers or two different physical domains. Thus, the two domains of the distributed memristors are physically and conceptually disjunct,  typeset structure∩  typeset structure= ∅. The relationship of w and D, typeset structure, gets a categorization as a composition of morphisms, typeset structure and typeset structure, and the distribution gets a categorization by the typeset structure-functor for dissemination.
As a first consequence of this mediated parallelism of memristive functionalities, one domain may represent digital events while the other may, simultaneously, represent analog events.

Because of the disjunctivity of the two domains typeset structure and typeset structure, the objects of the categories are disjunct too. Therefore, the conditions for an interpretation of distribution as yuxtaposition in the sense of monoidal categories are not given. Monoidal categories are mono-contextural and the objects of composition and yuxtaposition belong to the same common domain (universe). Hence, monoidal composition and yuxtaposition are referring to the same universe of objects, and are therefore mono-contextural.

Considering the distribution of the Rtypeset structure and Rtypeset structureaspects of the two disjunct domains, the interchangeability takes the following  formulation.

3.2.4.  Super-additivity of categorical descriptions

         balanced 3 - contextural interchangeability with super  ...                                                                                                  3

The interchangeability formula rules the parallel distribution of the w-D-distinction for two memristors. The domains typeset structuretypeset structure2 and typeset structurecontaining the different w-D-distinctions, might be defined by three distinctive ranges of TiO2 layers. The relationship of w and D, typeset structure, (w | D), gets a categorization as a composition of morphisms, typeset structure, typeset structure and typeset structure.

(                                                                                      ...                                                3                                                 3

This approach to modeling might be slightly too abstract but fits well into the conceptual modeling language and the concrete behaviors to be modeled. It also serves as steps towards a generalization of the concrete approaches for applications to other domains.

It follows naturally from the interchangeability formula, that it holds independently from specific domains. Hence, instead of voltage for nano-electronic devices, like memristors, capacity or inductivity might be involved, producing mem-capacitors or mem-inductors. But as Leon Chua made it clear, mem-behaviors appear in memristive systems under many different definitions and physical domains (universes).

It seems not to be too wrong to apply this kind of thinking to cognitive, mental, psychic, social and other domains too. This might be understood as a hint to analyze the different applications of second-order cybernetic constructions, like autopoiesis and re-entry under the focus of memristics. Obviously, such generalizations have not to be restricted to two and only two domains at all.

3.2.5.  General cross-interchangeability

Cross-interchangeability for  2-contextures

    <br />    ∀ f  _ i, g  _ i ∈ Universe &nbs ...                                                                      1   o                       1

The topics studied now on this level of abstraction are the loops and their pinch-points. Hence, the distribution and mediation of full loops and their corresponding pinches has to be thematized. Before, the internal structure of the loops as left- and right-loops with pinch-point had been in focus.
Hence, what has to be brought into interaction now, are the distributed loops and the distributed pinches of different loci. A model to think about such a dissemination is intended by the concept to be developed of poly-layered crossbar systems.

The aim of such tedious formalisms, like the poly-categorical interchangeability formulas, is to give orientation to understand the interplay of different kinds of memristive devices in a complex constellation in the framework of a theory of living systems.

      <br />     Mutual cross - exchange of loops and pinches <br />  ...                                                                                   o              1

     <br />    Mutual cross - exchange    ofloops   and   pinches <br /> & ...                                                                                                  3

3.2.6.  Diamond interchangeability

Diamond interchangeability enters the game if the fact of the importance of environments of operations are taken seriously and modeled in a specific formalism, like the proposed diamond category theory.

In-sourcing as memorization
In another turn of the argumentation for diamond-theoretic combinations, i.e. composition and saltisition, the role of memory as the “in-sourcing” of the matching conditions of compositions, which are inscribed as saltisitions, might be emphasized. It might become clear that this mechanism of in-sourcing is possible only if the mutual interaction of memory as saltisition and computation as composition is considered. Therefore, diamond combination might be realized in a memristive system as an interaction of memory and computation.  Diamond systems are physically realizable as an interaction of memristors as memory and memristors as computation. Or in another setting, as an interaction between memristors and classical emulations, like NAND-gates.

The complementary construction proposed in another paper was to ‘deduce’ memristance from diamond-categorical properties. This is a conceptual approach to memristive systems. The inverse move, to realize diamond-behaviors within the framework of memristive system is the physical approach to a transclassical understanding of technology.

The complementarity of conceptual and physical approaches corresponds the complementarity which Hans Primas is stressing in his research: the complementarity of Mind and Matter. Quantum physics isn’t possible without an understanding of its thematization, i.e. quantum physics is involving observer-positions to be observed (Exo-/Endophysics).

Hence, a further understanding of diamond category theory is possible with the concept and physics of memristive devices. To realize the in-sourcing of the matching conditions while a composition on the base of its matching conditions happens implies a remembrance of it. This remembrance is inscribed in the realm of saltisitions. Again, the interplay between composition and saltisition might be understood as an interaction between computation and memorization.

Second-order states as bi-objects
In more technical terms, following Coecke’s definition of a quantum state, a state is categorically a nil-operation I.

typeset structure

The lack of any history-dependence is clearly stated by “we don't need to know from what system  A has been produced”, only the actualilty of the state counts: “just that it is in state ψ  and available for processing”.

Because such formal systems like monoidal category theory, are strictly determined by their basic notions, a later and secondary introduction on the base of timeless notions is not delivering a fundamental approach to history-dependent memristive systems.

In contrast, diamond category is involved in the complementarity of system and environment, i.e. state and meta-state, from the very beginning. Such an approach seems to be suitable and concretely realizable for memristive systems.

In a mono-contextural world, like it is presumed by category theory, something is something and nothing is nothing. This is well known since Aristotle. In a polycontextural world, something is as something in a context, hence, nothing is not noting but nothing in context. Because, what is nothing in one contexture might turn out to be something in another contexture and vice versa. Therefore, an object of a nil-object or a nil-process is always an object in context, or generally in a contexture. A (nil-)object is or what it or what it not is in context. Thatis, an (nil-)object depends on its history.

A second-order state as it occurs in memristive systems, i.e. a state with its memory, is not simply an iteration of the nil-operation I. It is not I(I), i.e.

typeset structure

According to the diamond-theoretic understanding of processes and morphisms, each process exists with its environment, only. Therefore, an elementary state as a process (morphisms) is embedded complementary with its environment and constitutes the bi-object “A || a”.

Second-order states (objects) are not simply states of states in an iterative sense, but accretively states of states, where the first and second level are not in an iterative but accretive order. Which means, that the first-level gets its environment in the second-level.
A state with “history-depending” memory, i.e. a state with memory, is a two-layered state.
Two-layered states (objects) are called in diamond theory “bi-objects”.
In memristive systems history-dependency is not an obscure single event but the very definition of the memristive medium, i.e. the whole of the memristive system time- and history-depending.

typeset structure

For a mono-contextural setting, or a setting with one and only one context, the diamond-formula collapses into the common formulation.
A typeset structuretypeset structure
typeset structure|| btypeset structuretypeset structure
and ∀a, b: a≡b ==> Atypeset structurea ~= A typeset structureb ==> Atypeset structurea ~= A typeset structurea ==> Atypeset structurea ==> A.
Hence, ∀a, b: a≡b  ==> A typeset structure.

"Therefore it is useful to have an object standing for no system, preparation or state then being of the type I --> A and destruction being of the type A --> I  [...] these respectively are the so-called kets and bras. Clearly, since I stands for ‘no system’ we have: A typeset structure.” (Coecke)

Context depending introduction of ‘nothing is changing’ the system according to the context or environment of nothing.

typeset structure

typeset structure

cod(typeset structure) ≡ dom()

typeset structure

typeset structure

Transposed into a 2-dimensional notation we get the obvious diagram of the interchangeability of mediation (typeset structure) and context-separation (typeset structure).

<br />      π || i         &n ... sp;      ψ          j <br />

   <br />    Interchangeability    [||, ∐ ]   <br /> <br / ...                               2                           2                     2                2