Catching Transjunctions

Steps towards an emulation of polycontextural transjunctions in memristic systems

Rudolf Kaehr Dr. phil@

ThinkArt Lab Glasgow ISSN 2041-4358



How to understand Gunther’s idea of transjunctional operators? And how to implement transjunctions in memristive systems?

1.  Bifunctorial approach

1.1.  Distribution of junctions

          <br />       Example : M ...                                                                                                  3

          <br />       Example : M ...                                          3                              2.1                      1

       <br />         Distributivity of et a ...                                                                                                  3

1.2.  Transjunctions

   <br />      (X <> ∧ ∧ Y) - scheme : [<&g ...                                          3                              2.1                      1

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 <br />           val (X <> ∧ ∧ ...                           3            3                               3.1           1           1

 <br />      false (X <> ∧ ∧ Y) : <br />   &nb ...      3.1            1           1                  3.1           1           1                   1

1.3.  Discussion of functorial formalization of transjunctions

Distribution (A par (A seq B) ) seq (B par (A seq B) ) = (A seq B) par ((A seq B) seq (A seq B ...  : X seq (Y par Z) = (X seq Y) par Z <br /> R2 .2 : (X par Z) seq Y = (Z seq Y) par Z <br /> Null

Bifunctoriality   for   f  _ 1 ( <> ∧ ∧) : <br /> R1 : (a par c) seq ...  nil) = (a seq b) par (nil seq nil)  a seq b = a seq b f  _ 1 X et f  _ 1  Y Null

Bifunctoriality   for   f  _ 2 ( <> ∧ ∧) : <br /> How to get the rul ... les, while the tableaux rules are intepretations of the value matrix of logical functions . <br />

Tableaux for    f  _ 2 ( <> ∧ ∧) : <br /> (f  _ 2  ... par ((A seq B) seq (A seq B) '), with a = A c = (A seq B), <br /> b = B, <br /> c = (A seq B) ' .

 R1 (Bifunctoriality) : (a par c) seq (b par d) = (a seq b) par (c seq d) . <br />    ... ) <br />         ∐     )/(C o D)) . <br />

Bifunctoriality   for   f  _ 3 ( <> ∧ ∧) : <br /> (a par c) seq (b p ...  1 X et f  _ 1 Y))  sig  _ i X seq sig  _ j Y = sig (X seq Y), i = j Null

               &nbs ...                           3            3                               3.1           1           1

This approach doesn't imply that transjunctions are definable by junctions and negations only but that transjunctions are distributed and mediated over different logical loci and that at those loci junctional parts are placed.
Hence, under the condition of discontexturality and the use of the discontextural operator “transposition” transjunctions are ‘junctionally’ constructible.

Because memristive systems are able to emulate logical and arithmetical operations, a memristive transjunction or transjunctional memristive systems, i.e. memristive systems with transjunctional behaviors are constructible in the framework of presumed discontextural systems.

Hence, the challenge is to construct discontextural fields of memristivity and to establish its specific operators.

As for neurobiology, cognitive actions are not determined by isolated synapses only but by assemblies of neural activities. It is not a big departure form the tradition to postulate that assemblies are distinguishable what means that they are different. Hence, the whole system to work needs interactions between different neural assemblies. Interactions between neural assemblies are of a different kind, at least in their functionality, than the operators insid assemblies, which have to be “bridged”  by interactions. Hence, interactions are different to the intra-systemic operations, i.e. logical operators, like conjunction or disjunction and negation. It might be reasonable to opt for polycontextural transjunctions to deal with the interactions between assemblies of neural activity.

Memristivity is a general property of nano-physical systems and occurs in different forms.
Even nano-electronic memristivity is definable under different conditions. All those differences between memristive systems might be used to define discontextural conditions for the realization of memristive transjunctions.

The technical equivalent for the set of logical signatures {True, False} are the couple {On, Off}.
In an earlier paper I postulated the concept of poly-layered crossbar systems as polycontextural opposites of mono-layered, i.e. multi-layered crossbar systems of mono-contextural memristic systems.

The double functionality, typical for memristors, is necessary and sufficient to construct mediation between discontextural domains.

This idea of a construction is not properly working without memristors. If transjunctional behavior gets modeled with CMOS devices and NOR or NAND logic, the crucial difference between domains is not realizable because all subsystems belong to the same systematic locus, i.e. all are defined by the very same kind of logical values and physically by the same electronic conditions.

The principle of localization of memristive operations is of crucial importance.

Where enters the crucial feature of memrsitors, its retrograde recursiveness, into the mechanism of mediation?
Retro-gradeness was analyzed as a chiastic structure which occurs in the process of continuation (concatenation, prolongation) of morphograms. Obviously, the mediation of logical functions, like in (AND, AND, AND), is a case of chaining (composing) functions and morphograms.

Mediation of ANDtypeset structuremight category-theoretically be modeled by cod(AND1)= dom(AND2), with the commuting morphism (AND3)and the properties dom(AND1)= dom(AND3)and cod(AND2)= cod(AND3) and the presumption of a single universe of objects and morphisms.

But this construction gets a chiastic interpretation and formalization in a polycontextural setting with diffferent, i.e. discontextural universes and morphisms.
Chiasms are retro-grade constructions working as an interplay of ‘memory'- and ‘computing'-functions of memristors in different roles.

Example of some combinations of mediation of junctors

typeset structure typeset structure typeset structure

Each logical AND is emulated by memristive devices, memristors only or mixed with other mem-capacitators, mem-inductors, as Di Ventra et al defined and simulated it. Such memristive constructions are not yet focused on the time- and history dependence of the used memristors.

A new use of memristors enters the theater whith the interactions between memristors of different domains. This mechanism of connecting different domains happens in the mode of chiasms which are defined by their retro-grade properties.
Hence, memrisor-based logical junctions are mediated by memristive elements exploiting their memristive features of retro-gradeness. Hence, the focus on the roles of memristors are different, on a first-level interpretation, memristors are defining junctors without any involvement in retro-gradeness. On a second-order level, the memristive property of retro-gradeness o the behavior of memristors is in the focus and utilized for the mediation of memristive structures of different loci (domains).

Short remainder of the internal structure of chiastic mediation <br />     ^ ... > With α as domain, and ω as codomain of the mediation of the levels (domains) 1 and 2.

2.  Modeling and emulating logical operators

2.1.  Modeling and emulating material implication

Lehtonen’s example for the emulation of material implication with two memristors and 1 resistor only.
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2.2.  Memristive modeling and emulating of logical functions

2.2.1.  Di Ventra’s physical model

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"Fig. 6. (Color online) Electronic circuit for Boolean logic and arithmetic operations.
In this circuit, an array of N
memristors Mtypeset structure, memcapacitor C and resistor Rtypeset structure are connected to a common (horizontal) line. The circuit operation involves charging the memcapacitor C through input memristors and discharging it through the output ones.
Because of the circuit symmetry, each memristor can be used as input or output.
From the opposite (top) side, memristors are driven by 3-state (0V, V
typeset structure, not connected) drivers.
The right 4-state (0V, Vtypeset structure/2, Vtypeset structure, not connected) driver DR connected through a resistor to the common line is used to initialize memristors and memcapacitor. This scheme employs threshold-type bipolar memristors assuming that the application of positive voltage to the top memristor electrode and 0V to the bottom electrode switches memristor from low resistance state (1 or ON) to the high resistance state (0 or OFF). It is also assumed that the memristors’ threshold voltage Vtypeset structure is between Vtypeset structure/2 and Vtypeset structure.” (Di Ventra)

Memristive simulation of the logical junctors AND, OR, NOT
[Graphics:HTMLFiles/Catching Transjunctions_31.gif] [Graphics:HTMLFiles/Catching Transjunctions_32.gif]

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"Fig. 7. (Color online) Experimental realization of the basic logic gates with memristors based on the circuit shown in Fig. 6 with a capacitor C = 10µF instead of a memcapacitor.
In our experimental circuit, the role of memristors is played by memristor emulators [...] governed by the threshold type model of Eqs. (3) and (4) with the following set of parameters: V
typeset structure = 4V,  α = 0, β = 62MOhms/(V·s).
Each plot shows several measurements of the voltage between the top plate of the capacitor (the common line defined in the caption of Fig. 6) and ground taken for different possible states of input memristors.
In (a) we also show an example of voltages applied to the drivers to generate the top voltage plot. The voltages on the drivers in (a) are in absolute values while the other curves were vertically displaced for clarity.”
Yuriy V. Pershin and Massimiliano Di Ventra, Neuromorphic, Digital and Quantum Computation with Memory Circuit Elements

The simulation runs with 3 memristors, 1 memcapacitor and 1 resistor. If two memristors are enough to simulate logical function, a distribution of logical functions over 3 contextures needs therefore not more than 9 memristors. But with 9 memristors only separated and not mediated logical functions could be simulated in their specific domain. Additionally to the definition and realization of 3 different domains, which will be represented by different voltage-domains, at least 3x2 additional memristors are demanded to realize the mediation (connection) between the 3x3 distributed memristors.

2.2.2.  Mediation of memristors

Types of combinations of memristors
Classical combinations: serial and parallel,
Transclassical combinations: mediations, i.e. interactional, reflectional and interventional mediations.

How are mediator defined?

In more abstract terms, mediators are realizing the matching conditions for the composition of morphisms (functions, operations).

typeset structure

MEMtypeset structureare memristors with an intra-contextural selective switch-function, basic for the emulation of logical and arithmetical operations. In contrast, the role of memristor MEM3 is an elective trans-contextural function, realizing the functor ‘mediation' (typeset structure).

funct(MEM) = (election, selection)
                   selection = [ON/OFF] for digital and [ON,..., OFF] for analog operations in a contexture,
                   election = switch of contextures in the mode of transpositions, reflections and interventions.
Selection is realized by the first-order functionality of memristors in a crossbar system,
Election is realized by the second-order functionality of memristors between different crossbar systems.

First-order states of a memristive configuration are defining the type of primary action, i.e. logical junctions and basic arithmetic operations.

Second-order states of a memristive configuration are states of states, memorizing the first-order states of precessing states and as a new functionality, the context of those first-order states. Because states are always realized in a context, the memorization function of the state of the state delivers both, the former first-order state and its context (contexture).

"Because of the graphematical tabularity of ConTextures there are not only two possibilities to perform a selection. For each contexture there are intra-contexturally only two possibilities to perform a selection. But between contextures, transcontexturally, there are as many new selectors as neighbor contextures. These new "selectors" should be called electors. Electors are electing the election for selectors to perform mutually each its selection.
In other words, such a general a selector as any other successive or procedural the same contexture or trans-conA selector as a complexion can be realized at once in different contextures. It could therefore be called a "poly-selector". Such a poly-selector can be defined as an overlapping of an intra-contextural selector "sel" and a trans-contextural selector, called elector ‘elect'." (ConTeXtures, p. 20, 2005)

A prolongation of such a tupel of (state, context) is a double-function, realized by ‘selection’ and ‘election’.
Hence, a further logical or arithmetical step always has first to select the context, i.e. to chose to stay in the actual context or to chose another context for prolongation. Both at once might happen too, to change and to stay in context. After this contextural decision (election), the well known logical and arithmetical operations might continue by selection.

2.3.  Dissemination

2.3.1.  Superoperators in action

Boiled down to the simplest structure avaible the whole game of dissemination (monoidal) categories and memristive crossbar systems too, is to introduce an additional operator to the known operators, i.e composition and yuxtaposition.
This operator, called disseminatior, consisting of the two compelemtary aspects “distribution” and “mediation”, is abstracting the whole box of the theory of monoidal categories as a new unit and is distributing it with the operator of “election” over a kenomic matrix.
A specification of the distribution operator is given by the so called “super-operators”:

id: identity
perm: permutation,
red: reduction,
repl: replication,
bif: bifurcation.

DISS ((o, ⊗), 3, 3) =                     (4, 4)                                         ...                                      3.1                          3.2                          3.3

Some typical constellations for m = n ,   DISS ((o, ⊗), 3, 3) <br /> Identity : (id, id, ...             -                              -                                                  3.3

2.4.  Distribution of conjunction

typeset structure =

1. typeset structure :

conceptual mediation (typeset structure) of the distributed implications typeset structure.

2. (typeset structure typeset structure:

conceptual mediation of conjunction and realization of conjunction typeset structure by two memristors per contexture.
Rtypeset structureis omitted.

3. typeset structure

realization of conjunction AND and mediation by two memristors per contexture and
one memristive processor between contextures (procm).

More concretely, the interaction of ANDtypeset structure and ANDtypeset structure is ruled by the operator of mediationtypeset structure which is the ‘diagonal’ mediation “typeset structure” of the planar kenomic matrix.

This together constitutes a memristive system of distributed and mediated memristors in their double role as memory for the realization of the conjunction AND and in the role as processors for the mediation “typeset structure” of the contextures of the distributed conjunctions.

The same wording holds for the distribution of disjunction OR.

Because of the lacalization of operators in memristic systems not all operators are placed on the “diagonal” of the kenomic matrix. For a distribution and mediation of implication further operators, like replication have to be applied.

Coincidence relation
For typeset structurea coincidence relation between v(ON1) and  v(ON3) holds.
Also V1and V3are disjunct, V1∩ Vtypeset structure= ∅, they coincede at typeset structurev(ON1) ≡  v(ON3) and at OFF2.3 with
v(OFF2) ≡  v(OFF3).

Because ON1and ON3 are mediated by the operator typeset structure, which is realized by a processing memristor, there is no logical or electronic contradiction or conflict involved in this mediating mechanism. The same holds for OFF2 and OFF3.

This is nothig mysterious. For the classical case, a logical interpretation of the physical values happens too. In many respects, electronics and computer hardware is a question of interpretation, i.e. hemeneutics, and not of naked physics.

"In this type of application, each memristor is used in the “digital” mode of operation, namely only one bit of information is encoded in the memristor’s state. We call “1” (ON) the state of lower resistance and “0” (OFF) that of higher resistance. The operation of the circuit sketched in Fig. 6 relies on charging a memcapacitor through input memristors and subsequent discharging through the output ones.” (Di Ventra)

What’s definitively new and not yet tested is the mix of the voltage intepretation with the new domain interpretation.
Again, a value in a disconctxtural system is always a value of a contexture (domain), hence it has a double characteristics, one for the internal physical value (voltage) and one for the domain it belongs, i.e. its place-value marked by the place-designator.

Hence, “0” as “OFF” and “1” as “ON” has to be specified, i.e. indexed by its domain (contexture).

For distributed conjunction (AND AND AND)we get: ((AND typeset structure1.2 AND) typeset structure AND)

typeset structure

Again, there might be some obstacles. The interpretation ON1 ≅ ON3 and OFF2 ≅ OFF3 might have some plausibility. It seems to be more difficult to accept the interpretation: OFF1 ≅ ON2.
But that’s not much more than the matching conditions between contexture1 and contexture2, with
cod(contexture1) ≅ dom(contexture2).

2.5.  Formal modeling

A simplified categorical formulation for distributed memristors and implications based on memristors is shown with the following two formulas.

typeset structure

If we accept this categorical construction for AND and OR for experimental reasons we might continue with the more intriguing situation of interactions between contextures ruled by transjunctions.

On the base of the proposed sketches, here and in the paper “Poly-layered crossbars”, an implementation of transjunctions on the base of memristive devices in the framework of discontexturality shouldn’t be an impossible task.

Hence the logical interpretation of transjunctions in cooperation with conjunctions, as studied before, are easily transposed into an electronic setting.

Trivially, True is ON, False is OFF

2.5.1.  Interchangeability for (<> typeset structure

typeset structure

typeset structure

<br /> CASE for ON <br /> ON  _ 1 (<> ∧ ∧) : <br /> ON  _ 1  ... ;  = (ON  _ 3 X et ON Y  _ 3) par (ON  _ 1 X et ON  _ 1 Y) .

2.5.2.  Memristic emulation of transjunction and junctions

 <br />         ON (X <> ∧ ∧ Y) :   <br / ...                          3            3                               3.1            1           1

 <br />      OFF (X <> ∧ ∧ Y) : <br /> <br />  &nbs ...                       3.1             1           1                  3.1             1           1

  <br />      OFF (X <> ∧ ∧ Y) : index - free <br /> <br ...                                    3.1             1                             3.1             1

<br /> <br />  <br />         ON (X  ∨ ∧ ∧ ...                                                                                     3            3

<br />  <br />      OFF (X ∨ ∧ ∧ Y) : index - free <br / ...                                                                                                  3

2.6.  Formal modeling negation

Negation is a fundamental operator for logic, and therefore for memristive implementations too.
From a formal point of view both aspects have to be considered, the permutation and the mediation of negators in a mediated complexion.

typeset structuretypeset structure typeset structure ; typeset structuretypeset structure

       Monoidal permutation: A1 typeset structure.

 <br />      (neg  _ 1) : <br />     (  neg  ...                                                                                                  2

2.6.1.  Speculative emulation for negations

RowBox[{GridBox[{{Cell[GraphicsData[PICT, LWh000000>01400A0_l<0?on0000B00004P000000000h0 ... 75, 180.625}, ImageMargins -> {{0., 0.}, {0., 0.}}, ImageRegion -> {{0., 1.}, {0., 1.}}]}, {O}}]}]

 <br />    ON    (neg  _ 1) : <br />    (  neg   ...                                                                                                  2

 <br />    ON    (neg  _ 2) : <br />    (  perm  ...                                                                                                  1

2.7.  Speculative emulation for (AND AND AND)

FormBox[RowBox[{vol (AND  _ 1), =,  , RowBox[{V  _ (t  _ 1) is between ...                            3                                 2                                 2.3

2.7.1.  Speculative modeling of transjunctions

<br /> (X <> ∧ ∧ Y) - scheme : [<>       <>      ...                                  -                      -                                      3.3

FormBox[RowBox[{<br />, RowBox[{RowBox[{Mod  _ elect (X <> ∧ ∧ Y),   ...                                                                                                3.3

3.  Morphogrammatics of transjunctions

3.1.  Morphic abstraction of typeset structure

<br /> val (X <> ∨ ∧ Y) :    (1   3   3) = [T              ...                                            3                         2                         2.3

 Sys  _ 2 (∨) : morph (f  _ 2 ([A  _ 2] ∨  _ 2.2 [ ... nbsp;            = [∧  _ 3.3]

Sys  _ 1 (<>) : morph (TRANS) = [ [frame  _ 1.1], [core  _ 2.1], ... p;                

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[Graphics:HTMLFiles/Catching Transjunctions_89.gif]
The Abacus of Universal Logics

3.2.  General schemes for transjunctions and reflectors

3.2.1.  Transjunction schemes

Transjunction scheme for typeset structure:

[⊕  _ (i, j)] = [o         O      ]  _ (i, j)     : <br />                                 O         ∧

[[S _ i^i], [S _ i^(i + 1)], [S _ i^(i + 2)]] = [[o ∧], [o o ∧], [o O O]] <br />

               &nbs ...                                                                          3                 3.1   1

               &nbs ...                                                                                                  3

<br /> Transjunction as bifurcation <br /> ⊕ = bifurcation " bif " . <br /> <b ... ) = cod ([S _ 1^2]) = dom ([S _ 2^2]) <br />    cod ([S _ 2^2]) = cod ([S _ 3^3]) <br />

 <br />         [⊕  _ 1.1 MG  _ 2.2   ...                                                3                  3.1   3                  3.2   3

3.2.2.  Reflector

   Refl  _ 1 ([[⊕  _ 1.1], [MG  _ 2.2 ], [MG   ...                                  3                 3.1              1  1                    3.3

[[o ∧] 1, [o o ∧] 2, [o O O] 3]         &n ...    ∧   O                                                              O         ∧   O

<br /> Refl  _ 1 ( [⊕ ∨ ∧]) = [⊕ ⊕ ∧] ((           ...  [MG     ]                 3                 3.1              1  1                    3.3