Morphic Computing

Germano Resconi(1)

and Masoud Nikravesh(2)

(1)Catholic University, Brescia , Italy, Email resconi@numerica.it

(2)Lawrence Berkeley National Lab, University of California,

Berkeley, CA 94720, US, Email: MNikravesh@lbl.gov

Abstract: In this paper, we introduce a new type of computation called

“Morphic Computing”. Morphic Computing is based on Field Theory and more

specifically Morphic Fields. Morphic Fields were first introduced by Rupert

Sheldrake [1981] from his hypothesis of formative causation that made use of the

older notion of Morphogenetic Fields. Rupert Sheldrake [1981] developed his

famous theory, Morphic Resonance, on the basis of the work by French philoso-

pher Henri Bergson. Morphic Fields and its subset Morphogenetic Fields have

been at the center of controversy for many years in mainstream science and the

hypothesis is not accepted by some scientists who consider it a pseudoscience.

We claim that Morphic Computing is a natural extension of Holographic Compu-

tation, Quantum Computation, Soft Computing, and DNA Computing. All natural

computations bonded by the Turing Machine can be formalised and extended by

our new type of computation model – Morphic Computing. In this paper, we in-

troduce the basis for this new computing paradigm and its extensions such as

Quantum Logic and Entanglement in Morphic Computing, Morphic Systems and

Morphic System of Systems (M-SOS). Its applications to the field of computation

by words as an example of the Morphic Computing, Morphogenetic Fields in

neural network and Morphic Computing, Morphic Fields - concepts and Web

search, and agents and fuzzy in Morphic Computing will also be discussed.

Keywords: Morphic Computing, Morphogenetic Computing, Morphic Fields;

Morphogenetic Fields, Quantum Computing, DNA Computing, Soft Computing,

Computing with Words, Morphic Systems, Morphic Network, Morphic System

of Systems

1. Introduction

Inspired by the work of the French philosopher Henri Bergson, Rupert Sheldrake

[1981] developed his famous theory of Morphic Resonance. Sheldrake’s work on

Morphic Fields, based on Morphic Resonance Theory, was published in his well

known book “A New Science of Life: The Hypothesis of Morphic Resonance”

(1981, second edition 1985). Rupert Sheldrake’s [1981] theory of Morphic Fields

is based on his hypothesis of formative causation that makes use of the older no-

tion of Morphogenetic Fields. Morphic Fields and it‘s subset Morphogenetic

Fields have been at the center of controversy for many years in mainstream sci-

ence and the hypothesis is not accepted by some scientists who consider it a

pseudoscience. Morphogenetic Fields is a hypothetical biological fields and it has

been used by environmental biologists since 1920's which deals with living

things. However, Morphic Fields are more general than Morphogenetic Fields

and are defined as universal information for both organic (living things) and ab-

stract forms. Sheldrake defined Morphic and Morphogenetic Fields in his book,

The Presence of the Past [1988] as follows:

“The term [Morphic Fields] is more general in its meaning than Morphogenetic

Fields, and includes other kinds of organizing fields in addition to those of mor-

phogenesis; the organizing fields of animal and human behaviour, of social and

cultural systems, and of mental activity can all be regarded as Morphic Fields

which contain an inherent memory.” – Sheldrake [1988].

We claim that Morphic Fields reshape multidimensional space to generate local

contexts. For example, the gravitational fields in general relativity are the

Morphic Fields that reshape space-time space to generate local context where

particles move. Morphic Computing reverses the ordinary N input basis fields of

possible data to one field in the output system. The set of input fields form a N

dimension space or context. The N dimensional space can be obtained by a de-

formation of an Euclidean space and we argue that the Morphic Fields is the

cause of the deformation. In line with Rupert Sheldrake [1981] our Morphic

Fields is the formative causation of the context.

Now the output field of data is the input field X in Morphic Computing. To com-

pute the coherence of the X with the context, we project X into the context. Our

computation is similar to a quantum measure where the aim is to find coherence

between the behaviour of the particles and the instruments. So the quantum

measure projects the physical phenomena into the instrument as a context. The

logic of our projection is the same as quantum logic in the quantum measure. In

conclusion, we compute how a context can implement desired results based on

Morphic Computing.

Our new computation paradigm – Morphic Computing- is based on Field Theory

and more specifically Morphic Fields. We claim that Morphic Computing is a

natural extension of Holographic Computation, Quantum Computation, Soft

Computing, and DNA Computing. We also claim that all the natural computation

bonded by the Turing Machine can be formalised and extended by our new type

of computation model – Morphic Computing.

In this paper, we will first introduce the basis for our new computing paradigm –

Morphic Computing based on Field Theory. Then we will introduce its exten-

sions such as Quantum Logic and Entanglement in Morphic Computing, Morphic

Systems and Morphic System of Systems (M-SOS). Then Morphic Computing’s

applications to the field of computation by words will be given. Finally, we pre-

sent Morphogenetic Fields in the neural network and Morphic Computing,

Morphic Field - Concepts and Web search, and Agents and fuzzy in Morphic

Computing.

2. Morphic Computing and Field Theory: Classical and Modern Approach

2.1 Fields

In this paper, we assume that computing is not always related to symbolic entities

as numbers, words or other symbolic entities. Fields as entities are more complex

than any symbolic representation of the knowledge. For example, Morphic Fields

include the universal database for both organic (living) and abstract (mental)

forms. In classical physics, we represent the interaction among the particles by

local forces that are the cause of the movement of the particles. Also in classical

physics, it is more important to know at any moment the individual values of the

forces than the structure of the forces. This approach considers that the particles

are independent from the other particles under the effect of the external forces.

But with further development of the theory of particle physics, the researchers

discovered that forces are produced by intermediate entities that are not located in

one particular point of space but are at any point of a specific space at the same

time. These entities are called “Fields”. Based on this new theory, the structure of

the fields is more important than the value itself at any point. In this representa-

tion of the universe, any particle in any position is under the effect of the fields.

Therefore, the fields are used to connect all the particles of the universe in one

global entity. However, if any particle is under the effect of the other particles,

every local invariant property will disappear because every system is open and it

is not possible to close any local system. To solve this invariance problem, scien-

tist discovered that the local invariant can be conserved with a deformation of the

local geometry and the metric of the space. While the form of the invariant does

not change for the field, the action is changed. However, these changes are only

in reference to how we write the invariance. In conclusion, we can assume that

the action of the fields can be substituted with deformation of the space. Any par-

ticle is not under the action of the fields, the invariance as energy, momentum,

etc. is true (physical symmetry). However, the references that we have chosen

change in space and time and in a way to simulate the action of the field. In this

case, all the reference space has been changed and the action of the field is only a

virtual phenomena. In this case, we have a different reference space whose geom-

etry in general is non Euclidean. With the quantum phenomena, the problem be-

comes more complex because the particles are correlated one with one other in a

more hidden way without any physical interaction with fields. This correlation or

entanglement generates a structure inside the universe for which the probability to

detect a particle is a virtual or conceptual field that covers the entire Universe.

2.2 Morphic Computing: Basis for Quantum, DNA, and Soft Computing

Gabor [1972] and H. Fatmi and Resconi [1988] discovered the possibility of

computing images made by a huge number of points as output from objects as a

set of huge number of points as input by reference beams or laser (holography). It

is also known that a set of particles can have a huge number of possible states that

in classical physics, separate one from the other. However, only one state (posi-

tion and velocity of the particles) would be possible at a time. With respect to

quantum mechanics, one can have a superposition of all states with all states pre-

sented in the superposition at the same time. It is also very important to note that

at the same time one cannot separate the states as individual entities but consider

them as one entity. For this very peculiar property of quantum mechanics, one

can change all the superpose states at the same time. This type of global computa-

tion is the conceptual principle by which we think one can build quantum com-

puters. Similar phenomena can be used to develop DNA computation where a

huge number of DNA as a field of DNA elements are transformed (replication) at

the same time and filtered (selection) to solve non polynomial problems. In addi-

tion, soft-computing or computation by words extend the classical local definition

of true and false value for logic predicate to a field of degree of true and false in-

side the space of all possible values of the predicates. In this way, the computa-

tional power of soft computing is extended similar to that which one can find in

quantum computing, DNA computing, and Holographic computing. In conclu-

sion, one can expect that all the previous approaches and models of computing

are examples of a more general computation model called “Morphic Computing”

where “Morphic” means “form” and is associated with the idea of holism, geom-

etry, field , superposition, globality and so on.

2.3 Morphic Computing and Conceptual Fields – Non Physical Fields

Morphic Computing change or compute non physical conceptual fields. One ex-

ample is in representing the semantics of words. In this case, a field is generated

by a word or a sentence as sources. For example, in a library the reference space

would be where the documents are located. At any given word, we define the

field as a map of the position of the documents in the library and the number of

the occurrences (values) of the word in the document. The word or source is lo-

cated in one point of the reference space (query) but the field (answer) can be lo-

cated in any part of the reference.

Complex strings of words (structured query) generate a complex field or complex

answer by which the structure can be obtained by the superposition of the fields

of the words as sources with different intensity. Any field is a vector in the space

of the documents. A set of basic fields is a vector space and form a concept. We

break the traditional idea that a concept is one word in the conceptual map. Inter-

nal structure (entanglement) of the concept is the relation of dependence among

the basic fields. The ambiguous word is the source (query) of the fuzzy set (field

or answer).

2.4 Morphic Computing and Natural Languages – Theory of Generalized Constraint

In a particular case, we know that a key assumption in computing with words is

that the information which is conveyed by a proposition expressed in a natural

language or word may be represented as a generalized constraint of the form “X

isr R”, where X is a constrained variable; R is a constraining relation; and r is an

indexing variable whose value defines the way in which R constrains X. Thus, if

p is a proposition expressed in a natural language, then “X isr R” representing the

meaning of p, equivalently, the information conveyed by p. Therefore, the gener-

alised constraint model can be represented by field theory in this way. The mean-

ing of any natural proposition p is given by the space X of the fields that form a

concept in the reference space or objective space, and by a field R in the same

reference. We note that a concept is not only a word, but is a domain or context X

where the propositions p represented by the field R is located. The word in the

new image is not a passive entity but is an active entity. In fact, the word is the

source of the field. We can also use the idea that the word as an abstract entity is

a query and the field as set of instances of the query is the answer.

2.5 Morphogenetic and Neural Network

A neural network is a complex structure that connects simple entities denoted as

neurons. The main feature of the neural network is the continuous evolution in

complexity of the interconnections. This evolutionary process is called morpho-

genesis. Besides the evolution in the structure, we also have an evolution in the

biochemical network inside the neurons and the relations among neurons. The bi-

ochemical morphogenesis is useful for adapting the neurons to the desired func-

tionality. Finally, in a deeper form, the biochemical and structural morphogene-

sis is under the control of the gene network morphogenesis. In fact, any gene will

control the activity of the other genes through a continuous adaptive network.

Morphogenesis is essential to brain plasticity so as to obtain homeostasis or the

invariant of the fundamental and vital functionality of the brain. The predefined

vital functions interact with the neural network in the brain which activity is ori-

ented to the implementation or projection of the vital function into the neural ac-

tivities. In conclusion, morphogenetic activity is oriented to compensate for the

difference between the vital functions and the neural reply. The field nature of the

designed functionality as input and of the implemented functionality in the neural

network as output suggest the holistic nature of the neural network activity. The

neural network with its morphogenesis can be considered as a prototype of

morphic computing.

2.6 Morphic Systems and Morphic System of Systems (M-SOS) We know that System of System or SoS movement study large scale systems in-

tegration. Traditional systems engineering is based on the assumption that if

given the requirements the engineer will give you the system. The emerg-

ing system of systems context arises when a need or set of needs are met with a

mix of multiple systems, each of which are capable of independent operation in

order to fulfil the global mission or missions.

For example, design optimisation strategies focus on minimizing or maximizing

an objective while meeting several constraints. These objectives and constraints

typically characterize the performance of the individual system for a typical de-

sign mission or missions. However, these design strategies rarely address the im-

pact on the performance of a larger system of systems, nor do they usually ad-

dress the dynamic, evolving environment in which the system of systems must

act. A great body ? of work exists that can address “organizing” a system of sys-

tems from existing single systems: resource allocation is currently used in any

number of fields of engineering and business to improve the profit, ? other sys-

tems optimisation. One reason for this new emphasis on large-scale systems is

that customers want solutions to provide a set of capabilities, not a single specific

vehicle or system to meet an exact set of specifications. Systems engineering is

based on the assumption that if given the requirements the engineer will

give you the system “There is growing recognition that one does not nec-

essarily attack design problems solely at the component level.” The cardinal

point for SoS studies is that the solutions are unlikely to be found in any

one field There is an additional consideration: A fundamental characteristic

of the problem areas where we detect SoS phenomena will be that they are

open systems without fixed and stable boundaries and adapt to changing

circumstances. In SoS there is common language and shared goals, commu-

nications helps form from? communities around a confluence of issues,

goals, and actions. Therefore, if we wish to increase the dimensionality we

need tools not to replace human reasoning but to assist and support this

type of associative reasoning. Important effects occur at multiple scales, in-

volving not only multiple phenomena but phenomena of very different

types. These in turn, are intimately bound to diverse communities of ef-

fected individuals. Human activity now yields effects on a global scale.

Yet we lack the tools to understand the implications of our choices. Given

a set of prototype systems, integrations (superposition ) of these systems generate

a family of meta-systems. Given the requirement X the MS ( Morphic System )

reshape in a minimum way or zero the X. The reshaped requirements belong to

the family of meta-systems and can be obtained by an integration of the proto-

type systems. In conclusion, the MS is the instrument to implement the SoS. To

use MS we represent any individual system as a field the superposition with the

field of the other systems is the integration process. This is similar to the quan-

tum computer where any particle ( system ) in quantum physics is a field of un-

certainty (density of probability ). The integration among particles is obtained by

a superposition of the different uncertainties. The meta-system or SoS is the result

of the integration. Any instrument in quantum measure has only a few possible

numbers of integration. So the quantum measure reshape the given requirement or

physical integration in a way to be one of the possible integrations in the instru-

ment. So the quantum instrument is a particular case of the SoS and of the MS. In

MS we also give suitable geometric structure ( space of the fields of the prototype

systems ) to represent integration in SoS and correlation among the prototype

systems. With this geometry we give an? invariance property for transformation

of the prototype systems and the requirement X. So we adjoin to the SoS integra-

tion a dynamical process which dynamical law is the invariant. Since the geome-

try is in general a non Euclidean geometry, we can define for the SoS a morpho

field ( M F ) which action is to give the deformation of the geometry from Eu-

cliden ( independent prototype system ) to a non Euclidean ( dependent prototype

system ) . The Morpic Field is comparable to the gravity field in general relativi-

ty. So the model of quantum mechanics and general relativity are a special case of

MS and are also the physical model for the SoS.

2.7 Morphic Computing and Agents – Non Classical Logic

In the agent image, where only one word (query) as a source is used for any

agent, the field generated by the word (answer) is a Boolean field (the values in

any points are true or false). Therefore, we can compose the words by logic oper-

ations to create complex Boolean expression or complex Boolean query. This

query generates a Boolean field for any agent. This set of agents creates a set of

elementary Boolean fields whose superposition is the fuzzy set represented by a

field with fuzzy values. The field is the answer to the ambiguous structured query

whose source is the complex expression p. The fields with fuzzy values for com-

plex logic expression are coherent with traditional fuzzy logic with a more con-

ceptual transparency because it is found on agents and Boolean logic structure.

As points out [Nikravesh, 2006] the Web is a large unstructured and in many cas-

es conflicting set of data. So in the World Wide Web, fuzzy logic and fuzzy sets

are essential parts of a query and also for finding appropriate searches to obtain

the relevant answer. For the agent interpretation of the fuzzy set, the net of the

Web is structured as a set of conflicting and in many case irrational agents whose

task is to create any concept. Agents produce actions to create answers for am-

biguous words in the Web. A structured query in RDF can be represented as a

graph of three elementary concepts as subject, predicate and complement in a

conceptual map. Every word and relationship in the conceptual map are variables

whose values are fields which their superposition gives the answer to the query.

Because we are more interested in the meaning of the query than how we write

the query itself, we are more interested in the field than how we produce the field

by the query. In fact, different linguistic representations of the query can give the

same field or answer.

In the construction of the query, we use words as sources of fields with different

intensities. With superposition, we obtain the answer for our structured query. We

structure the text or query to build the described field or meaning. It is also possi-

ble to use the answer, as a field, to generate the intensity of the words as sources

inside a structured query. The first process is denoted READ process by which

we can read the answer (meaning) of the structured query. The second process is

the WRITE process by which we give the intensity or rank of the words in a que-

ry when we know the answer. As an analogy to holography, the WRITE process

is the construction of the hologram when we know the light field of the object.

The READ is the construction of the light field image by the hologram. In the ho-

lography, the READ process uses a beam of coherent light as a laser to obtain the

image. Now in our structured query, the words inside of text are activated at the

same time. The words as sources are coherent in the construction by superposi-

tion of the desired answer or field. Now the field image of the computation by

words in a crisp and fuzzy interpretation prepares the implementation of the

Morphic Computing approach to the computation by words. In this way, we have

presented an example of the meaning of the new type of computation, “Morphic

Computing”.

2.8 Morphic Computing: Basic Concepts

Morphic Computing is based on the following concepts:

1) The concept of field in the reference space

2) The fields as points or vectors in the N dimension Euclidean space of the ob-

jects ( points )

3) A set of M N basis fields in the N dimensional space. The set of M fields

are vectors in the N dimensional space. The set of M vectors form a non Eu-

clidean subspace H ( context ) of the space N. The coordinates S

in M of the

field X are the contro-variant components of the field X. The components of

X in M are also the intensity of the sources of the basis field. The superposi-

tion of the basis field with different intensity give us the projection Q of X or

Y = QX into the space H When M < N the projection operator of X into H

define a constrain or relation among the components of Y.

4) With the tensor calculus with the components S of the vector X or the com-

ponents of more complex entity as tensors , we can generate invariants for

any unitary transformation of the object space or the change of the basis

fields.

5) Given two projection operators Q1 , Q2 on two spaces H1 , H2 with dimension

M1 and M2 we can generate the M = M1 M2 , with the product of Y1 and Y2

or Y = Y1 Y2 . Any projection Q into the space H or Y = QX of the product of

the basis fields generate Y. When Y Y1 Y2 the output Y is in entanglement

state and cannot separate in the two projections Q1 and Q2 .

6) The logic of the Morphic Computing Entity is the logic of the projection op-

erators that is isomorphic to the quantum logic

The information can be coded inside the basis fields by the relation among the ba-

sis fields. In Morphic Computing, the relation is represented by a non Euclidean

geometry which metric or expression of the distance between two points shows

this relation.

The projection operator is similar to the measure in quantum mechanics. The pro-

jection operator can introduce constrains inside the components of Y.

The sources are the instrument to control the image Y in Morphic Computing.

There is a strong analogy between Morphic Computing and computation by ho-

lography and computation by secondary sources ( Jessel ) in the physical field.

The computation of Y by X and the projection operator Q that project X into the

space H give his result when the Y is similar to X. In this case, the sources S are

the solution of the computation. We see the analogy with the neural network

where the solution is to find the weights wk at the synapse. In this paper, we show

that the weights are sources in Morphic Computing.

Now, it is possible to compose different projection operators in a network of

Morphic Systems. It is obvious to consider this system as a System of Systems.

Any Morphic Computation is always context dependent where the context is H.

The context H by the operator Q define a set of rules that are comparable with the

rules implemented in a digital computer. So when we change the context with the

same operations, we obtain different results. We can control the context in a way

to obtain wanted results. When any projection operator of X or QX is denoted as

a measure, in analogy with quantum mechanics, any projection operator depends

on the previous projection operator. In the measure analogy, any measure de-

pends on the previous measures. So any measure is dependent on the path of

measures or projection operators that we realise before or through the history. So

we can say that different projection operators are a story ( See Roland Omnès in

quantum mechanics stories).

The analogy of the measure also gives us another intuitive idea of Morph Compu-

ting. Any measure become a good measure when gives us an image Y of the real

phenomena X that is similar, when the internal rules to X are not destroyed. In the

measure process, the measure is a good measure. The same for Morphic Compu-

ting, the computation is a good computation when the projection operator does

not destroy the internal relation of the field in input X.

The analogy with the measure in quantum mechanics is also useful to explain the

concept of Morphic Computing because the instrument in the quantum measure is

the fundamental context that interferes with the physical phenomena as H inter-

feres with the input field X.

A deeper connection exists between the Projection operator lattice that represents

the quantum logic and Morphic Computing processes (see Eddie Oshins).

Because any fuzzy set is a scalar field of the membership values on the factors

(reference space) (Wang and Sugeno). We remember that any concept can be

viewed as a fuzzy set in the factor space. So at the fuzzy set we can introduce all

the processes and concepts that we utilise in Morphic Computing.

For the relation between concept and field, we introduce in the field theory an in-

trinsic fuzzy logic. So in Morphic Computing, we have an external logic of the

projection or measure (quantum logic) and a possible internal fuzzy logic of the

fuzzy interpretation of the fields.

In the end, because we also use agents’ superposition to define fuzzy sets and

fuzzy rules, we can again use Morphic Computing to compute the agents incon-

sistency and irrationality. Thus, fuzzy set and fuzzy logic are part of the more

general computation denoted as Morphic Computing.

3. Reference Space, Space of the objects, Space of the fields in the Morphic Computing

Given the n dimensional reference space ( R1 , R2 ,…,Rn ), any point

P = ( R1 , R2 ,…,Rn ) is an object.

Now we create the space of the objects which dimension is equal to the number

of the points and the value of the coordinates in this space is equal to the value of

the field in the point. We call the space of the points “space of the objects”.

Any field connect all the points in the reference space and is represented as a

point in the object space. The components of this vector are the value of the field

in the different points. We know that each of the two points connected by a link

assume the value of one of the connections. All the other points assume zero val-

ue. Now any value of the field in a point can be considered as a degree of connec-

tion of this point with all the others. Therefore, in one point where the field is ze-

ro, we can consider this point as non-connected to the others. In fact, because the

field in this point is zero the other points cannot be connected by the field to the

given point. In conclusion, we consider the field as a global connector of the ob-

jects in the reference space. Now inside the space of the objects, we can locate

any type of field as vectors or points. In field theory, we assume that any complex

field can be considered as a superposition of prototype fields whose model is well

known.

The prototype fields are vectors in the space of the objects that form a new refer-

ence or field space. In general, the field space is a non Euclidean space. In con-

clusion, any complex field Y can be written in this way

Y = S1 H1 ( R1 ,…,Rn ) + S2 H2 ( R1 ,…,Rn )+ ....+

Sn Hn ( R1 ,…,Rn ) = H( R ) S (1)

In equation (1) , H1 , H2 ,…, Hn are the basic fields or prototype fields and S1 ,

S2 ,…, Sn are the weights or source values of the basic fields. We assume that

any basic field is generated by a source. The intensity of the prototype fields is

proportional to the intensity of the sources that generates the field itself.

F

F

3.1 Example of the basic field and sources

In Figure 1, we show an example of two different basic fields in a two dimen-

sional reference space ( x, y). The general equation of the fields is

2 2(( ) ( ) )0 0

( , ) [ ]h x x y y

F x y S e

(2)

the parameters of the field F1 are S=1 h =2 and x0 = -0.5 and y0 = -0.5,

the parameters of the field F2 are S=1 h =2 and x0 = 0.5 and y0 = 0.5

Figure 1. Two different basic fields in the two dimensional reference space (x,y).

For the sources S1 = 1 and S2 = 1 the superposition field F that is shown in Figure

2 is F = F1 + F2. For the sources S1 = 1 and S2 = 2, the superposition field F that is

shown again in Figure 2 is F = F1 + 2 F2 .

F F

F = F1 + F2. F = F1 + 2F2.

F1 F2

Figure 2. Example of superposition of elementary fields F1 , F2

.

3.2 Computation of the sources

To compute the sources Sk, we represent the prototype field Hk and the input field

X in a Table 1 where the objects are the points and the attribute are the fields.

Table 1. Fields values for M points in the reference space

H1 H2 … HN Input Field X

P1 H1,1 F1,2 … H1,N X1

P2 H2, 1 H2,2 ... H2,N X2

… … … … … ...

PM HM,1 HM,2 … HM,N XM

The values in Table 1 is represented by the following matrices

...1,1 1,2 1,N 1

...2,1 2,2 2,N 2H = , X =

...... ... ... ...

... MM,NM,1 M,2

H H H X

H H H X

XH H H

The matrix H is the relation among the prototype fields Fk and the points Ph. At

this point, we are interested in the computation of the sources S by which they

give the best linear model of X by the elementary field values. Therefore, we

have the superposition expression

1,1 1,2 1,

2,1 2,2 2,...1 2

... ... ...

,,1 ,2

H H H

H H H

H H H

n

nY S S S HSn

M nM M

(3)

Then, we compute the best sources S in a way the difference Y X is the mini-

mum distance for any possible choice of the set of sources. It is easy to show that

the best sources are obtained by the expression

1( )

T TS H H H X

(4)

Given the previous discussion and field presentation, the elementary Morphic

Computing element is given by the input-output system as shown in Figure 3.

Figure 3. Elementary Morphic Computing

Sources

S = (HTH)

-1 H

T X

Field X Field Y = H S = QX

Prototype

fields H(R)

Figure 4. Shows the Network of Morphic Computing.

Figure 4 shows network of elementary Morphic Computing with three set of

prototype fields and three type of sources with one general field X in input and

one general field Y in output and intermediary fields from X and Y.

When H is a square matrix, we have Y = X and

-1S = H X and Y = X = H S (5)

Now for any elementary computation in the Morphic Computing, we have the

following three fundamental spaces.

1) The reference space

2) The space of the objects ( points )

3) The space of the prototype fields

Figure 5 shows a very simple geometric example when the number of the objects

are three ( P1 , P2 , P3 ) and the number of the prototype fields are two ( H1 , H2 ).

The space which coordinates are the two fields is the space of the fields.

S2 S3

S1

H1

H2 H3

X Y

Figure 5. The fields H1 and H2 are the space of the fields. The coordinates of the

vectors H1 and H2 are the values of the fields in the three points P1 , P2 , P3 .

Please note that the output Y = H S is the projection of X into the space H

Y = H ( HT H )

-1 H

T X = Q X

With the property Q2 X = Q X

Therefore, the input X can be separated in two parts

X = Q X + F

where the vector F is perpendicular to the space H as we can see in a simple ex-

ample given in Figure 6.

P1

P2

P3

H2

H1

P1

P2

P3

F2

F1

X

QX = Y

F

Figure 6. Projection operator Q and output of the elementary Morphic Compu-

ting. We see that X = Q X + F , where the sum is the vector sum.

Now, we try to extend the expression of the sources in the following way

Given G ( ) = T and G ( H ) = H

T H and

S* = [ G ( H ) + G ( ) ]-1

HT X and

[ G ( H ) + G ( ) ] S* = HT X

So for S* = ( HT H )

-1 H

T X +

= S

+

we have

([ G ( H ) + G ( ) ] ( S

+ ) = H

T X

( HT H ) ( H

T H )

-1 H

T X + ([ G ( H ) + G ( ) ]

= H

T X

G ( ) S

+ [ G ( H ) + G ( ) ] = 0

and

where for is function of S by the equation

G ( ) S + [ G ( H ) + G ( ) ] = 0

For non-square matrix and/or singular matrix, we can use the generalized model

given by Nikravesh [] as follows;

Where we transform by the input and the references H.

The value of the variable D ( metric of the space of the field) is computed by the

expression (6)

D2 = ( H S )

T ( H S ) = S

T H

T H S = S

T G S = ( Q

X )

T Q X (6)

For the unitary transformation U for which, we have UT U = I and H’ = U H the

prototype fields change in the following way

H’ = U H

1

*-T T TS S H H + Γ Γ H X

1 1

* ( ) ( ) ( )- -T T T T T TS H Λ Λ H H Λ Λ X ΛH Λ H ΛH Λ X

G’ = ( U H )T ( U H ) = H

T U

T U H = H

T H

And

S’= [ ( U H )T ( U H ) ]

-1 ( U H )

T Z = G

-1 H

T U

T Z = G

-1 H

T ( U

-1 Z )

For Z = U X we have S’ = S and

the variable D is invariant. for the unitary transformation U .

We remark that G = HT H is a quadratic matrix that gives the metric tensor of the

space of the fields. When G is a diagonal matrix the entire elementary field are

independent one from the other. But when G has non diagonal elements, in this

case the elementary fields are dependent on from the other. Among the elemen-

tary fields there is a correlation or a relationship and the geometry of the space of

the fields is a non Euclidean geometry.

4. Quantum Logic and Entanglement in Morphic Computing

In the Morphic Computing, we can make computation on the context H as we

make the computation on the Hilbert Space. Now we have the algebras among the

context or spaces H. in fact we have

H = H1 H2 where is the direct sum. For example given

H1 = ( h1,1 ,h1,2 ,…..,h1,p ) where h1,k are the basis fields in H1

H2 = ( h2,1 ,h2,2 ,…..,h2,q ) where h2,k are the basis fields in H2

So we have

H = H1 H2 = ( h1,1 ,h1,2 ,…..,h1,p , h2,1 ,h2,2 ,…..,h2,q)

The intersection among the context is

H = H1 H2

The space H is the subspace in common to H1 , H2 . In fact for the set V1 and V2

of the vectors

V1 = S1,1 h1,1 + S1,2 h1,2 + …..+ S1,p h1,p

V2 = S2,1 h2,1 + S2,2 h2,2 + …..+ S2,q h2,q

The space or context H = H1 H2 include all the vectors in V1 V2 .

Given a space H , we can also built the orthogonal space H of the vectors that

are all orthogonal to any vectors in H.

No we have this logic structure

Q ( H1 H2 ) = Q1 Q2 = Q1 OR Q2

where Q1 is the projection operator on the context H1 and Q2 is the projection op-

erator on the context H2 .

Q ( H1 H2 ) = Q1 Q2 = Q1 AND Q2

Q ( H

) = Q – ID = Q = NOT Q

In fact we know that Q X – X = ( Q – ID ) X is orthogonal to Y and so orthog-

onal to H. In this case the operator ( Q – ID ) is the not operator.

Now it easy to show [ 9 ] that the logic of the projection operator is isomorphic to

the quantum logic and form the operator lattice for which the distributive low (

interference ) is not true. In Figure 7, we show an expression in the projection

lattice for the Morphic Computing

Figure 7. Expressions for projection operator that from the Morphic Computing

Entity

Now we give an example of the projection logic and lattice in this way :

Given the elementary field references

Sources

S

Field X QX =[ ( Q1 Q2 ) Q3 ] X = Y

Prototype fields

H= ( H1 H2 ) H3

1

1 0 2 , ,

1 2 30 1 1

2

H H H

For which we have the projection operators

1 01( )

1 1 1 1 1 0 0

0 01( )

2 2 2 2 2 0 1

1 1

1 2 2( )3 3 3 3 3 1 1

2 2

T TQ H H H H

T TQ H H H H

T TQ H H H H

With the lattice logic we have

T -1

1,2 1 2 1 2 1,2 1,2 1,2 1,2

T -1

1,3 1 3 1 3 1,3 1,3 1,3 1,3

T -1

2,3 2 3 2 3 2,3 2,3 2,3 2,

1 0 1 0H = H H = , Q Q = H (H H ) H =

0 1 0 1

11

1 02H = H H = , Q Q = H (H H ) H =

1 0 10

2

10

1 02H = H H = , Q Q = H (H H ) H =

1 01

2

1

And

1 2 1 3 2 3

1 2 1 3 2 3

0 = =

0

0 0 Q Q =Q Q =Q Q =

0 0

H H H H H H H

and

And in conclusion we have the lattice

We remark that

( Q1 Q2 ) Q3 = Q3 but ( Q1 Q3 ) ( Q2 Q3 ) = 0 0 = 0

When we try to separate Q1 from Q2 in the second expression the result change.

Between Q1 and Q2 we have a connection or relation ( Q1 and Q2 generate the

two dimensional space ) that we destroy when we separate one from the other. In

fact Q1 Q3 project in the zero point. A union of the zero point cannot create the

two dimensional space . the non distributive property assume that among the pro-

jection operators there is an entanglement or relation that we destroy when we

separate the operators one from the other.

Given two references or contexts H1 , H2 the tensor product H = H1 H2 is the

composition of the two independent contexts in one.

We can prove that the projection operator of the tensor product H is the tensor

product of Q1 , Q2 . So we have

Q = H ( HT H )

-1 H

T = Q1 Q2

The sources are S

= S

1 S

2

Q1 Q2 = Q1 Q3 = Q2 Q3

Q1 Q2

Q3

Q1 Q2 = Q1 Q3 = Q2 Q3= 0

So we have

Y = Y1 Y2 = ( H1 H2 ) S

The two Morphic System are independent from one another. The output is the

product of the two outputs for the any Morphic System. Now we give some ex-

amples

1 11 1

2 2

1 11 , 0

1 22 2

1 1 11

2 2 2

1 11 1

2 2

1 1 11 0 0

2 2 2

1 1 1 1

2 2 2 2

1 1 11 1

2 22 2

1 1 1, ,0 0 1

1 2 1 2 2 22 2 2

1 1 11

2 22 2 2

H H

H H

H H H H H H

H H

12

1 , , ,0

2

1 1

2 2

1 11 1

2 2

1 1 11 0 0

2 2 2

1 1 1 1

2 2 2 2

H

At every value of the basis fields we associate the basis field H2 multiply for the

value of the basis fields in H1. Now because we have

1

1 1 1 1 1

1

2 2 2 2 2

2 2 2

1 2 2 2 2

2 2 2

1 10

2 2

( ) 0 1 0

1 10

2 2

2 1 1

3 3 3

1 2 1( )

3 3 3

1 1 2

3 3 3

1 10

2 2

0 1 0

1 10

2 2

T T

T T

Q H H H H

Q H H H H

Q Q Q

Q Q Q Q Q Q

Q Q Q

2

1 2 1 2 2

2

1 1

1 1 1 1 1 2 2 2 2 1

1 11

3 34

0 , 0 , 0

1 1 1

3 4 3

4 3

9 2( ) , ( )

42

39

T T T T

X

X X X X X X

X

S H H H X S H H H X

And

2

1 2

2

4

9

2

9

S

S S S

S

For

1 1 1 2 2 2

2

1 2 2

2

71

123

20 ,

31

13

12

1

3

0

1

3

Y H S Y H S

Y

Y Y Y Y

Y

In conclusion we can say that the computation of Q , S and Y by H and X can be

obtained only with the results of Q1 , S1 , Y1 and Q2 , S2 , Y2 independently.

Now when H and X cannot be write as a scalar product of H1 , X1 and H2 , X2 ,

the context H and the input X are not separable in the more simple entities so are

in entanglement.

In conclusion with the tensor product, we can know if two measures or projection

operators are dependent or independent from one another.

So we have for the tensor product of the context that all the other elements of the

entity are obtained by the tensor product (Figure 8.)

Figure 8. tensor product for independent projection operators or measures.

5. Field theory , concepts and Web search

To search in the web, we use the term-document matrix to obtain the information

retrieved. In Table 2, we show the data useful in obtaining the desired infor-

mation in the Web.

Table 2. Term (word), document and complex text G

Word1 Word2 … WordN Concept X

Document1 K1,1 K1,2 … K1,N X1

Document2 K2, 1 K2,2 ... K2,N X2

… … … … … ...

DocumentM KM,1 KM,2 … KM,N XM

Where Ki,j is the value of the wordj in the documenti . The word in Table 2 is a

source of a field which values are the values in the position space of the docu-

ments. Any document is one of the possible positions of the word. In a Web

search, it is useful to denote the words and complex text in a symbolic form as

queries, the answers are the fields generated by the words or text as sources. Any

ontology map is a conceptual graph in RDF language where we structure the que-

ry as a structured variable. The conceptual map in Figure 9 is the input X of a

complex field obtained by the superposition of the individual words in the map.

Sources

S = X1 X2

Field X = X1 X2

Prototype fields

H= H1 H2

QX =Q(X1 X2 ) = Y1 Y2 =Y

Figure 9. Conceptual map as structured query. The map is a structured variable

whose answer or meaning is the field G in the documents space located in the

Web

With the table two we have that the library H is the main reference for which we

have

1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

...

...

... ... ... ...

...

N

N

M M M N

K K K

K K KH

K K K

Where M > N.

For any new concept in input

1

2

...

M

X

XX

X

we can found the suitable sources S by

which we can project X into the knowledge inside the library H.

The library can be represented in a geometric way in this form

Figure 10. Geometric representation of the Library ( Documents ) and concepts

( words ).

Recently much work has been done to realise conceptual maps of lexicon for

natural language. This work has a foundation in Description Logic. With a con-

ceptual map of natural language and Description Logic, we can create a semantic

web. The problem with a semantic web is the integration of different concepts in

a geometric frame. We are in the same situation as the quantum computer where

the Geometric Hilber space is the geometric tool to show the global properties of

the classical logic and quantum logic. We propose in this paper to give a geomet-

ric image of knowledge. The meaning of the words and concepts are not located

in the axiomatic description logic, in the conceptual map or in the lexical struc-

ture. The meaning is inside the space of the fields of instances of the words or

documents. A prototype set of concepts, inside the space of the documents as ob-

ject, generate an infinite number of other concepts in a space. Any concept inside

the space of the concepts is obtained by an integration process of the prototype

concepts. Now any new concept can be compared with the concepts that belong

to the space of the concepts. The MS can reshape, with the minimum change the

new concept X into another concept Y. Concept Y belongs to the set of concepts

obtained by the integration of the prototype concepts. Y is inside the space of the

prototype fields that represent prototype concepts.

MS can compute the part of X that cannot be integrated in the space of the con-

cepts. Transformation of the space of the concepts generate a set of new prototype

concepts by which we can represent the dynamical process of the concepts and

the possible conceptual invariance. In this way a huge set of concepts can be re-

duced to a more simple set of prototype concepts and invariance. In the semantic

web based on description logic and the conceptual map, we cannot reduce the

concepts to a simple set of basic concepts which. With the description of invari-

ance conceptual frame and the type of geometry ( metric , Euclidean and non -

Document1

Document2

Document3

Concept2 = Word2

Concept1 = Word1

Concepì in input X

QX = Y = Projection of X

Eucliden ) we believe we can improve the primitive semantic structure base on

the lexical description of the concepts.

6. Morphogenetic Field in Neural Network and Morphic Com-

puting: Morphic Systems and Morphic System of Systems (M-

SOS)

We know that in traditional neural network one of the problems was the realisa-

tion of the Boolean function XOR. The solution of this problem was realised by

the well known back propagation. Now the physiological realisation in the brain

of the Boolean functions are obtained by the genetic change on the synaptic form.

The morphogenetic field is the biochemical system that change the synaptic struc-

ture. The solution of the XOR problem and other similar problems is relate to the

hidden layer of neurons which activity is not used directly to the output computa-

tion but are an intermediary computation useful to obtain the wanted output. In

Figure 11, we show the neural network that realise the function

Z = X XOR Y = X Y

Figure 1.1 Neural network for the Boolean function XOR

For the previous neural network we have

2 2 2( ) ( )3 3 3

Z X Y X Y X Y

The weights for the neural network can be obtained by different methods as back

propagation or others.

X

Y

-1

-1

2 2/3

2/3 = 2/3

Z =XY

Now we study the same Boolean function by the Morphic Computing network or

System of System. In fact at the begin we consider the simple Morphic System

with

0 0 0 0 0

0 1 0 1 1H = X Y = = , Input = Z = X Y =

1 0 1 0 1

1 1 1 1 0

where the input of the Morphic System is the field of all possible values for Z.

The field reference H or the context is made by the fields one are all possible val-

ues for the input X and the other are all possible values for Y.

With the context H ( field references ) and the input Z we compute the sources

and the output. So we have

0

11 0 0

31 10 1T -1 T 3 1S = (H H) H Input = , Output = H S = + =1 1 03 3

31 13 2

3

we remark that the space generate by the fields X and Y in H is given by the su-

perposition

2

00 0

S0 1 2Output = S + S =

1 2 S1 0 11 1 S S

1

Now we know that the Q projection operator for

0 0 0 0

0 1 0 1H = X Y = =

1 0 1 0

1 1 1 1

(7)

is

1)

0 0 0 0

2 1 10 -

3 3 3T T

1 2 1Q = H(H H H0 -

3 3 3

1 1 20

3 3 3

Because the determinant of Q is equal to zero , the colon vectors are not inde-

pendent so for

2 1 1-

3 3 3

1 2 1H = - , X =

3 3 3

1 1 2

3 3 3

we have

11) and y

4 1 2 2 3 2 31

T TS = (H H H X S y S y y y

(8)

So we have the constrain by the projection operator that any Y must have this

form.

3

0

y2

Y =y3

y y2

(9)

Now because the output that we want must have this form

y > 0 , y > 0 , y + y < min(y , y )2 3 2 3 2 3

but this is impossible so the projection operator and reference H that we use put a

limitation or a constraint that the XOR Boolean function cannot satisfy.

Now for the reference or context

0 0 0

0 1 1H =

1 0 1

1 1 1

That we can reduce to

0 1 1

H = 1 0 1

1 1 1

Now because H is a quadratic matrix

, in the projection we have no limitation or constrain so when in input we have

0

1 Input = Z = X Y =

1

0

by the projection operator Q we have that the output = input. In this case we can

compute the sources S in a traditional way for the context

0 1 1

H = 1 0 1

1 1 1

The sources are

1) 1T T

-1

S = (H H H X

2

Now given the reference (4) we have the constrain (5). Now for the input

0

1Input =

1

1

(10)

The output is (6) that is coherent or similar to the input. In fact we have that for

3

2

3

2

0

yY =

y

y y

because in input we have the property

x = 0,x > 0 ,x > 0 , x > 04 2 31

and in output we have the same property

0,1

y y > 0 , y > 0 , y = y + y > 04 2 3 2 3

For the context (4) and the input (7) the sources are

2

1 3)

2

3

T TS = (H H H X

In conclusion we found the same weights as in

the Figure 12 with the Morphic System

Figure 12. Network of Morphic Systems or System of Systems.

We can conclude that in the ordinary back propagation we compute the weights

in neuron network by a recursive correction of the errors in output. In Morphic

Computing we are interested in the similarity between the input and output. So

we are interested more in the morphos ( form or structure ) that in the individual

values in input and output.

7. Agents and Fuzzy in Morphic Computing

Given the word “warm” and three agents that are named Carlo , Anna and Anto-

nio, we assume that any agent is a source for a Boolean fields in Figure 13.

0

1X =

1

1

2

3S =

2

3

0

2/3Y =

2/3

4/3

0 0

0 1H =

1 0

1 1

0

1Z =

1

0

-1

S = -1

2

0

1Y =

1

0

0 0 0

0 1 1H =

1 0 1

1 1 1

Figure 13. Three Boolean fields F1(A,B) , F2(A,B) , F3 (A,B) for three agents.

The superposition of the three Boolean fields in Figure 13 is the fuzzy set

F (A,B) + F (A,B) + F (A,B)1 1 2 2 3 3

μ(A,B) = S S S (11)

At the word “warm” that is the query we associate the answer ( , )A B that is a

field. With the superposition process, we can introduce the logic of the fuzzy

fields in this way.

When the logic variables values for p agents assume the usual 1 and 0 ( boolean

fields ), we have the operations

( , , ..., )( , , ..., ) ( , , ..., )1 2 1 2 1 1 2 2

( , , ..., ) ( , , ..., ) ( , , ..., )1 2 1 2 1 1 2 2

( , , ..., ) ( , , ..., )1 2 1 2

X X X Y Y Y Y X Y X Y Xp p p p

X X X Y Y Y Y X Y X Y Xp p p p

X X X X X Xp p

(12)

That are the vector product , the vector sum and the vector negation.

For any vector we can compute the scalar product

...1 1 2 2

( , , ..., ) ( )1 2

...1 2

.....1 1 2 2

m X m X m Xp pX X X Xp

m m mp

S X S X S Xp p

where is the membership function and mk where k = 1 ,.., p are the weights of

any component in the Boolean vector. So the membership function is the

weighted average of the 1 or 0 value of the variables Xk

We can show that

CARLO ANNA ANTONIO

( , ..., ) ( , ..., ) we have1 1

( , ..., ) min[ ( ), ( )] ( ) (13)1 1

( , ..., ) max[ ( ), ( )] ( ) (14)1 1 1

when Y Y X Xp p

Y X Y X X Y Y Xp p

Y X Y X X Y Y Xp p

In fact we have X Y + X Y = Yk k k k k

and X Y = Y - X Yk k k k k

in the same

way we have

X + Y = X + X Y + X Y = X (1+ Y ) + X Y = X + X Yk k k k k k k k k k k k k k

because in the fuzzy set for the Zadeh rule we have

[( , ..., )( , ..., )] min[ ( , ..., ), ( , ..., )] 011 1 1

[( , ..., )( , ..., )] max[ ( , ..., ), ( , ..., )] 111 1 1

X X X X X X X Xp p p p

X X X X X X X Xp p p p

the negation in (11) is not compatible with the other operations. So in the decom-

position of the vector

( , , ..., ) ( ( ), ( ), ...., ( ))1 2 1 2

X X X F X F X F Xp p (15)

at the negation Xk

we substitute the negation

( ) and is the XOR operationF X Xk k k

We remark that when k = 1 we have ( )k k

F X X when k = 0 we have

( )k k

F X X For the negation F we have

( ( )k k

F F X X ..

In fact

F(F(X)) = F(X (X)) = (X (X)) (X) = ( X (X) + X (X)) (X) =

[X + (X)][X + (X)] (X) + ( X (X) + X (X)) (X) = X (X) + X (X) = X

So the function F is an extension of the negation operation in the classical logic.

We remark that in (15) we introduce a new logic variable Sk. The vector

( , ,..., )1 2 p is the inconsistent vector for which the Tautology is not

always true and the contradiction is not always false as in the Zadeh rule. In fact

we have for the contradiction expression

C = ( ) ( )F X X X X Xk k k k k k k

For the previous case the fuzzy contradiction C is not always equal to zero. We

remember that the classical contradiction is 0X Xk k

always. When

( , ,..., ) (0,0,..,0)1 2 p we came back to the classical negation and

Ck = 0.

For the tautology we have

T = ( ) ( )F X X X X X Xk k k kk k

The reference space is the space of all possible propositions and any agent is a

Boolean field or prototype field. Given a Boolean field X in input, external agent,

by the projection operator we generate the fuzzy field Y in output by the sources

S.

Figure 14. because in the projection operation we loss information the original

boolean field for the input agent X is transformed in the fuzzy field

( ) .....1 1 2 2

QX Y X S X S X S Xp p

8. Conclusion

In this paper we introduced a holistic interpretation of the computation denoted

Morphic Computing. We show the possibility of going beyond the traditional

computation based on a step-by-step process. Our type of computation is similar

to a quantum measure where the instrument is the context with proper rules. Input

fields are introduced in the instrument as a context and write in internal parame-

ters or sources of internal prototype fields. Morphic Computing projects the ex-

ternal fields to the internal context. In the projection process, the input is reshaped

in a way to select part of the input that is coherent with the context. Because the

projection operator uses the internal prototype fields and the computed sources,

we can give a context dependent model of the external input. This is the meaning

of the new computation.

Reference

Sources S

weights

Agent X ( boolean

field )

Prototype agents and

Boolean fields H

QX =Y Fuzzy fields

for the agent

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