a new type of structured artificial neural networks

The Woodlands, Texas, United States Abstract – The recently introduced Turing-complete Matrix Model of Computation MMC is a con-nectionist, massively parallel, formal mathematical model

A new type of Structured Artificial Neural Networks

based on the Matrix Model of Computation

Sergio Pissanetzky Research Scientist Member, IEEE The Woodlands, Texas, United States

Abstract – The recently introduced Turing-complete

Matrix Model of Computation (MMC) is a

con-nectionist, massively parallel, formal mathematical

model that can be set up as a network of artificial

neurons and represent any other ANN The model is

hierarchically structured and has a natural ontology

determined by the information stored in the model

The MMC is naturally self-organizing and

dynami-cally stable The Lyapunov energy function is

inter-preted as a measure of biological resources, the

attrac-tors correspond to the objects in the natural ontology

The Scope Constriction Algorithm (SCA) minimizes

the energy by systematically switching the network

connections and reveals the ontology In this paper

we consider the MMC as a modeling tool for

applica-tions in Neuroscience We prove as a theorem that

MMC can represent ANNs We present a new, more

efficient version of SCA, discuss the advantages of

MMC ANNs, and illustrate with a small example

Keywords: neural networks, dynamic systems, ontologies,

self-organizing systems, artificial intelligence, semantic web

Work

The Matrix Model of Computation was introduced as

a natural algorithmic form of mathematical notation

amenable to be operated upon by algorithms expressed

in that same notation It is formally defined as a pair of

sparse matrices, the rows of which are tuples in a

rela-tional database Since MMC models can be easily

cre-ated by a parser from existing computer programs, and

then refactored by algorithm, the MMC was proposed as

a virtual machine for program evolution [1] Subsequent

work [2] proved that any finitely realizable physical

sys-tem can be modeled by the MMC, and showed that the

model is naturally self-organizing by way of an

algo-rithm that organizes the information categorically into

weakly-coupled classes of strongly-cohesive objects, an

ontology [3] Finally, applications to very diverse fields

such as theoretical Physics, business and UML models,

and OO analysis and design, were discussed and

illus-trated with small examples [4] Relations have been

applied for the analysis of programs and a relational

model of computation has been proposed [5] and

re-cently characterized by investigating its connection with the predicate transformer model [6]

In this paper we consider the MMC as a structured, massively parallel, generalized, self-organizing, artificial neural network In Section 2 we define the MMC, in-troduce terminology, discuss the hierarchical organiza-tion and parallelism, examine combinaorganiza-tions and con-versions between artificial neurons or ANNs and MMC models, training issues, and dynamics, and briefly com-pare ANNs and MMC with humans In Section 3 we present prove that any ANN can be described as an MMC model, and in Section 4 we present a new, more efficient and biologically plausible version of the Scope Constriction Algorithm, which gives the MMC its abil-ity to self-organize We close with a small example

of Computation

2.1 Definition The MMC is simple, yet very rich in features It is defined [1] as a pair of sparse matrices [7] M = (C, Q), where C is the matrix of services and

Q is the matrix of sequences The rows of C are the services, and the columns of C are the variables used

by the services A domain is the set of values allowed for a variable, and there is a domain associated with each variable Each variable plays a certain role in the service, indicated by A for an input variable or argu-ment, C for an output variable or codomain, and M for

a modifiable variable or mutator The roles A, C and

M are the elements of C in that service’s row

The concept of service is very general A service can represent a neuron, a neural network, a primitive math-ematical or logical operation in a standard computer, a method in a class, or an entire MMC Services can also have their own memory visible only to the service (e.g

a synaptic weight), and their own firing mechanisms Variables are also very general A numerical variable represents a value, a categorical variable represents an instance of an object in a class See Eq (2) below for

a small example of a matrix C, or previous publications [1, 2, 4] for more complete examples

The rows of Q are the sequences The columns of Q include the actors that initiate sequences, the links be-tween services, and the control variables that activate

or inhibit the links

2.2 The data channel The scope of a variable is the

vertical extent between the C or M where the variable

is first initialized and the terminal A where it is used

for the last time, in that variable’s column The set

of scopes represents a data channel where data carried

by the variables flows from its source, the initializing

services, to its destinations, the services that use the

data The sum of all scopes happens to be equal to the

vertical profile of C, immediately suggesting the use of

profile minimization techniques to make the data

chan-nel narrow, a highly desirable feature discussed below

2.3 MMC algebra and transformations MMC

has a rich algebra, which includes matrix operations

such as permutations, partitioning and submatricing,

relational operations such as joins, projections,

normal-ization and selection, and graph and set operations [1]

Algorithms can be designed based on these operations

to induce transformations on the MMC Of particular

interest are refactorings, defined as invariant

transfor-mations that preserve the overall behavior of the model

This is a general definition and it applies to all

sys-tems The MMC has been proposed for that purpose

[1] Algorithms can also be designed for training or for

self-organization One of them is discussed below

2.4 Control flow graph, linear submatrices, and

canonical submatrices A control flow graph (CFG)

is a directed graph G = (V, E) where a vertex v ∈ V

corresponds to each service in matrix C and an edge

e ∈ E corresponds to each tuple in matrix Q A path

in the CFG represents a possible flow of control The

path is said to be linear if its vertices have no

addi-tional incoming or outgoing edges except for the end

vertices, and the linear path is maximal if it can not be

enlarged without loosing the linear property Given a

set of services S, a submatrix of services can be defined

by deleting from matrix C all rows with services not in

S and all columns with variables not used by the

ser-vices in S A linear submatrix is a submatrix of serser-vices

based on the services contained in a linear path Linear

submatrices are very common in a typical MMC model

A service in a general MMC can initialize or modify

several variables at once, and a variable can be

repeat-edly re-initialized or modified As a result, a submatrix

of services can contain many C’s and M ’s in each row or

column However, the following simple refactoring can

convert any submatrix of services to a form without M ’s

and exactly one C in every row and every column: (1) if

a service has n > 1 codomains C, expand it into n

simi-lar services that initialize one variable at a time, and (2)

if a variable is mutated or assigned to more than once,

introduce a new local variable for each assignment or

mutation The resulting submatrix is square, and, since

there is only one C in every row and every column, a suitable (always legal) column permutation can bring it

to a canonical form, where all the C’s are on the diag-onal, the upper triangle is empty, the lower triangle is sparse and contains only A’s, and the lowermost A in each column is the terminal A in that column Canoni-cal submatrices correspond to the well-known single as-signment representation, a connectionist model directly translatable into circuits Examples of canonical matri-ces have been published ([4], figures 1, 2)

2.5 Ontologies The roles A, C and M in a row of matrix C establish an association between the service

in that row and the variables in the columns where the roles are located Since variables represent attributes and can take values, and services represent the pro-cesses and events where the variables participate, the association represents an object in the ontological sense [3] We refer to this object as a primitive object, and

we say that matrix C defines a primitive ontology of which the primitive objects are the elements and the domains are the classes Domains can be joined to form super-domains, of which the original domains are the subdomains Super-domains inherit the services and attributes of their subdomains Multiple-inheritance

is possible, and a subdomain can be shared by many super-domains In the ontology, the super-domains are subclasses and the subdomains are super-classes, and the super-classes subsume the subclasses The sub-domains of a super-domain can be replaced in matrix

C with a categorical variable representing that super-domain, and similarly, the associated services can be replaced with a “super-service” declared in an MMC submodel in terms of the subservices, thus reducing the dimension of C by submatricing The process can be continued on the simplified C, creating a hierarchy of models and submodels that represents an inheritance hi-erarchy These features have been previously discussed [1, 4] Primitive objects do in fact combine sponta-neously to form larger objects when the profile is mini-mized, giving rise to the self-organizing property of the MMC discussed below In a biological system an ob-ject could represent a cell, a neuron, a neural clique, an organ, or an entire organism

2.6 Parallelism A service declaration is the root of

a tree, where only the external interface is declared in

a row of C but links present in matrix Q progressively expand it in terms of more and more detailed declara-tions, down to the deepest levels where declarations are expressed in terms of services provided by the hardware

or wetware To accommodate traditional computational language, we say that services in a level invoke or call those in the lower levels The service declaration tree also functions as a smooth serial/parallel interface as

well as a declarative/imperative interface The services

near the top are sequentially linked by the scopes of the

variables, but as the tree expands, many new local

vari-ables are introduced and the interdependencies weaken,

allowing parallelism to occur It is in this sense that

the MMC is considered as a massively parallel model

The smooth transition between the two architectures is

a feature of MMC models

2.7 ANN/MMC conversions and combinations

Structured models entirely based on artificial neurons

can be formulated for any system by creating an initial

MMC model with serial services down to the level where

parallelism begins to appear, and continuing with

tradi-tional ANNs from there on The services in the higher

levels are already connected in a network, and the

in-vocations of the lower level services involve only

eval-uations of conditionals Conditionals can, in turn, be

translated to ANN models, and at least one example of

such translations has been published [8] In this way, a

homogeneous model consisting entirely of artificial

neu-rons is obtained, where collective behavior and

robust-ness are prevalent in the ANNs while a higher level of

functional and hierarchical organization is provided by

the underlying MMC Another exciting possibility is to

combine the robustness and efficiency of ANNs with the

mathematical rigor and accuracy of traditional

comput-ers and the interoperability of the MMC by

implement-ing some services as ANNs and the rest as CPUs The

theorem presented in the next Section clarifies some

as-pects of these conversions

2.8 Training MMC operations can be used to design

algorithms that add or organize MMC data SCA is an

example SCA does not add data but it creates new

information about data and organizes it into structure

As such, it should be considered training Direct

train-ing is another example A modified parser can

trans-form a computer program into an MMC Conversions

from other sources such as business models or theories

of Physics are possible [4] There has been a recent

resurgence of interest in connectionist learning from

ex-isting information structures and processes [8, 9] In

addition, the ANNs in the MMC support all traditional

modes of training Conversely, a trained MMC network

will have a high ability to explain its decision-making

process, an important feature for safety-critical cases

2.9 Self-organization Under certain circumstances,

row and column permutations can be applied to C to

rearrange the order of the services and variables The

permutations can be designed in such a way that they

constrict the data channel by reducing the scopes of

the variables, and at the same time cause similar

prim-itive objects to spontaneously come together and

coa-lesce into larger, more cohesive, and mutually uncou-pled objects This process is called scope constriction, and is performed by the Scope Constriction Algorithm discussed below The transformation is also a refactor-ing because it preserves the behavior of the model The process can continue with the larger objects, progres-sively creating even larger objects out of the smaller ones The resulting hierarchical structure is the natural ontology of the model The natural ontology depends

on and is determined by the information contained in the model, and is therefore a property of the model Definitions and properties of cohesion and coupling are well established [10]

2.10 Dynamics It is possible to imagine a scenario where (1) new information keeps arriving, for example from training or sensory perception, (2) the scope con-striction process is ongoing, (3) the resulting natural ontology evolves as a result of the changes in the body

of information, and (4) an ability to “reason” in terms

of the new objects rather than from the raw information

is developed In such a scenario, some objects will stabi-lize, others will change, and new objects will be created This scenario is strongly reminiscent of human learn-ing, where we adapt our mental ontologies to what we learn about the environment It is also consistent with recent work on neural cliques [11], suggesting that in-ternal representations of exin-ternal events in the brain do not record exact details but are instead organized in a categorical and hierarchical manner, with collective be-havior prevalent inside each clique and a higher level of organization and functionality at the network level The scenario can find other important applications, such as semantic web development Some of these ideas are further discussed in Section 4 These ideas are not very well supported by traditional ANNs For quick refer-ence, Table 1 shows some of the features of ANN and MMC models that we have rated and compared with humans The comparison suggests that MMC models, particularly MMC/ANN hybrids, may be better suited

as models of the brain than ANNs alone, and may help

to develop verifiable hypotheses

Table 1 Ratings of ANN and MMC features com-pared with humans 1 = poor, 2 = good, 3 = best Supported feature humans ANN MMC

3 Describing Artificial Neural

Networks with MMC models

The Theorem of Universality for the MMC states that

“Every finitely realizable physical system can be perfectly

represented by a Matrix Model of Computation” [2] In

this Section we prove the following theorem:

Any ANN, consisting of interconnected artificial

neu-rons, can be equivalently described by an MMC model

where the neurons correspond to services and the

con-nections to scopes in the matrix of services

This theorem follows from the theorem of universality

However, in order to make the correspondence more

ex-plicit, we present the following proof by construction

In the ANN model, a nonlinear neuron is described by

the following equation:

yk= ϕ

m

X

i=1

wkixki+ bk

!

(1)

where k identifies the neuron, m is the number of inputs,

xki are the input signals, wki are the synaptic weights,

ϕ is the activation function, bkis the bias, and yk is the

output signal Service neuron k (nr k) in the following

MMC matrix of services C describes equation (1):

C =

serv ϕ {xki} {wki} bkyk{x`i} − x`1{w`i} b`y`

(2) where x`1 ≡ yk, and set notation is used Sets,

func-tions, etc, are considered objects in the ontological

sense, meaning for example that {xki} stands not only

for the elements of the sets but also their respective

car-dinalities and other properties they may possess

Ser-vice neuron ` (nr `) in eq (2) represents a second

neuron that has the output signal yk from neuron k

connected as its first input x`1 The scope of variable

yk, extending from the C to the A in that column,

rep-resents the network connection The rest of the proof is

by recurrence To add neurons, the same construction

is repeated as needed, and all connections to previous

neurons in the model are represented in the same way

This completes the proof

Algorithm (SCA)

In this Section, we present a new version of the SCA

algorithm with a lower asymptotic complexity than the

original version [2] The algorithm narrows the data

channel (§2.2) and reveals the natural ontology of the

model (§2.5) by minimizing the profile of the matrix

of services C SCA operates on a canonical submatrix (§2.4) of C, but for simplicity in presentation we shall assume that the entire C is in canonical form If N is the order of C and j is any of its columns, then Cjj= C

If there are any A’s in that column, then the downmost

A, say in row Dj, is the terminal A, and the length of the scope of the corresponding variable is Dj − j If there are no A’s, the variable is an output variable and

Dj= j The vertical profile of C is:

p(C) =

N

X

j=1

The variable in column j is initialized by the C in that column Then, the data travels down the scope to the various A’s in column j, and then horizontally from the A’s to the C’s in the corresponding rows, reaching as far as the C in column Dj, which corresponds to the terminal A in column j New variables are initialized

at the C’s, and the process repeats itself The “conduits

of information” that carry the traveling data constitute the data channel, and the lengths of the scopes are a measure of its width The maximum width Wm(C) and the average width Wa(C) of the data channel are defined

as follows:

Wm(C) = max

SCA’s goal is to reduce the lengths of the scopes and the width of the data channel by minimizing p(C)

In the canonical C, services are ordered the same

as the rows Matrix Q still applies, but is irrelevant because it simply links each service unconditionally to the service below it Commuting two adjacent services means reversing their order without affecting the overall behavior of the model The lengths of the scopes and the value of the profile p(C) depend on the order of the services, hence SCA achieves its goal by systematically seeking commutations that reduce the profile Since a behavior-preserving transformation is a refactoring, a commutation is an element of refactoring and SCA is a refactoring algorithm

Commutation is legal if and only if it does not reverse the order of initialization and use of any variable More specifically, a service in row i initializes the variable in column i, because Cii = C Since this is the only C

in that row, the service in row i and the service in row

i + 1 are commutative if and only if Ci+1,i is blank In other words, commutations are legal provided the C’s stay at the top of their respective columns For exam-ple, the two services in eq (2) are not commutative because of the presence of the A under the C in column

y Commutation preserves the canonical form of C

Repeated commutation is possible If service S in

row i commutes with the service in row i − 1, the

com-mutation can be effected, causing S to move one row up,

and the service originally in row i − 1, one row down

If S, now in row i − 1, commutes with the service in

row i − 2, that commutation can be effected as well,

and so on How high can S go? Since there are no A’s

above the C in column i of S, all commutations will

be legal until the rightmost A in row i, say in column

Ri, gets to row Ri + 1 and encounters the C in row

Ri of that column Thus, service S can go upwards as

far as row Ri+ 1 by repeated commutation Similarly,

service S in row i can commute with the service in row

i + 1, then with the service in row i + 2, and so on, until

the C in column i of S encounters the uppermost A in

that column, say in row Ui, namely all the way down to

row Ui− 1 The range (Ri+ 1, Ui− 1) is the range of

commutation for service S in row i

Repeated commutation of services amounts to a

per-mutation of the rows of C To preserve the canonical

form, a symmetric permutation of the columns must

follow Thus:

where P is a permutation matrix The symmetric

per-mutation is also behavior-preserving, and it is a

refac-toring SCA can be formally described as a procedure

that finds P such that p(C) is minimized The

mini-mization of p(C) is achieved by systematically

examin-ing sets of legal permutations and selectexamin-ing those that

reduce p(C) the most However, SCA does not

guar-antee a true minimum In the process, p(C) decreases

smoothly, but individual scopes behave in a complicated

way as they get progressively constricted against the

constraints imposed by the rules of commutation The

refactoring forces related services and variables to

co-alesce into highly cohesive, weakly coupled clusters, a

phenomenon known as encapsulation The clusters are

recognized because few or no scopes cross intercluster

boundaries, they correspond to objects, and the term

constriction is intended to convey all these ideas The

original version of the algorithm, known as SCA2,

op-erates as follows:

(1) Select a row i of C in an arbitrary order

(2) Determine the range of commutation Ri, Uifor the

service in that row

(3) For each k, Ri < k < Ui, calculate p(Ck), where Ck

is obtained from C by permuting the service from

row i to row k, and select any k that minimizes p

(4) Permute the service to the selected row

(5) Repeat (1-4) until all rows are exhausted

(6) Repeat the entire procedure until no more

reduc-tions are obtained

To calculate the asymptotic complexity of SCA2 we

as-sume that C, being sparse, has a small, fixed number of off-diagonal nonzeros per row Assuming the roles are indexed by service, the calculation of Ri, Ui requires a small, fixed number of operations per row, or O(N ) op-erations for step (2) in total The calculation of the profile, eq 3, requires the calculation of Dj for each column j, which takes a small, fixed number of oper-ations per column, or O(N ) in total In a worst case scenario, the range for k in step (3) may be O(N ), so step (3) will require O(N2) operations per row, or a total of O(N3) for the entire procedure The rest of the operations is O(N ) or less Thus, the asymptotic complexity of SCA2 is O(N3), caused by the repeated calculation of the profile The new version of SCA dif-fers from SCA2 only in step (3), as follows:

(3) (new version) Calculate ∆i,k(C) for each k, Ri <

k < Ui, and select the smallest

∆i,k(C) is the increment in the value of the profile when the service in row i is reassigned to row k, and can be calculated based on the expression:

∆i,i+1(C) = qi+ pi− qi+1 (7) Let ni be the number of terminal A’s in row i, mj

be the number of terminal A’s in column j (0 or 1), and qi = ni − mi be the excess of terminal A’s for row/column i Also let pi be the number of terminal pairs in row i We say that a terminal pair exists in row i, column j when Ci,j = A and Ci+1,j is a terminal

A Equation 7 follows, and ∆i,kis obtained by repeated application of that equation

Assuming as we did before that the roles are indexed

by service, and the services by row and column, the cal-culation of Ri, Ui, qi, pi and ∆i,i+1 each takes a small, fixed number of operations, and the calculation of ∆i,k

for all k takes O(N ) operations Thus, the new step (3) takes O(N ) operations, and the asymptotic complex-ity of SCA is O(N2) The improvement in complexity

is due to the fact that actual values of the profile are never calculated The new SCA is a second order al-gorithm because the neutral subsets are properly taken care of as part of the range of commutation [2] SCA is a natural MMC algorithm in the sense that it modifies the MMC itself and is universal As such, and since the MMC is a program [1], SCA can be installed

as a part of MMC itself, making the MMC a dynamical system, a self-refactoring MMC where the energy (Lya-punov) function is the profile p(C) and the attractors are the objects that SCA converges to Since SCA is behavior-preserving, it can run in the background with-out affecting the operation of the MMC The dynamical operation is characterized by two well-differentiated but coexisting processes: (1) new information arrives as the

result of some foreground training process and is

ap-pended to C, resulting in a large profile, and (2) SCA

minimizes the profile and updates the natural ontology

by creating new objects or modifying the existing ones

in accordance with the information that arrives The

objects are instated as new categorical variables and

op-eration continues, now in terms of the new objects Such

a system allows higher cognition such as abstraction and

generalization capabilities, and is strongly reminiscent

of the human mind, particularly if the creation of

ob-jects representing the natural ontology is interpreted as

“understanding”, and the recognition of objects for

fur-ther processing as “reasoning” These views offer a new

interpretation of learning and meaning

The term “energy” used above refers to resources

in general, including not just physical energy but also

building materials, or some measure of the physical

re-sources needed to implement the system When neurons

form their axons and dendrites they must maximize

in-formation storage but minimize resource allocation [12]

The correspondence between the scopes and the

net-work connections discussed in Section 3 suggests a

cor-respondence between their respective lengths as well, in

which case there should be a biological SCA-type

pro-cess that rewires the network by proximity or migrates

the neurons to shorten their connections Either way,

the net result is that neurons close in the logical

se-quence become also geometrically close, creating an

as-sociation between function and information similar to

an OOP object These observations are consistent with

the minimum wiring hypothesis, as well as with Horace

Barlow’s efficient coding hypothesis, Drescher’s schemas

[13], and Gell-Mann’s schemata [14] Similar

observa-tions may apply to other biological structures such as

organs, or to entire organisms

In comparison with other algorithms such as MDA,

we note that SCA uses no arbitrary parameters, is

ex-pandable in the sense that new elements and new classes

can be added and the model can grow virtually

indef-initely, both in size and refinement, and is biologically

plausible because it uses local properties, likely to be

available in a cell or an organ MDA, instead, uses

mathematical equations, very unlikely to exist in a

bio-logical environment

Applications for SCA can be found in many domains

An example in theoretical Physics was published [4],

where the model consists of 18 simple equations with

30 variables, and SCA constructs an ontology consisting

of a 3-level multiple-inherited hierarchy with 3 objects

in the most specialized class, that describes an

impor-tant law of Physics Here we consider classification

For classification, associations must be established be-tween some property of the objects to be classified and

a suitable discriminant or classifier Then, SCA finds patterns and classifies the objects dynamically For example, if the objects are points in some space, then the discriminant is a mesh of cells of the appropriate dimensionality and desired resolution, points are associ-ated with the cells that contain them, and the resulting classes are clusters of points If the objects are neurons that fire at different times, the discriminant is a mesh

of time intervals, neurons are associated with the time intervals where they fire, and the classes would be neu-ral cliques [11] Table 2 summarizes these observations Table 2 Parameters used for SCA classification objects property discriminant class

points position mesh of cells cluster of points neurons firing event time mesh neural clique Our classification example involves a set of 167 points defined by their coordinates in some space In the ex-ample, the space is two-dimensional, but the number of dimensions is irrelevant In Figure 1, the points are at the center of the symbols The discriminant consists of

4 overlapping meshes, simulating the continuity of the space The basic mesh consists of cells of size 1 × 1, and 3 more meshes are superposed with relative shifts

of (0.5, 0), (0, 0.5), and (0.5, 0.5), respectively The resulting matrix of services C is of order 1433, and is already in canonical form

××× × × ×

×× ×× ×× ××

× × ×

×

×

× ×

× × ×

× ×

+ + ++

+ + + + + + + + + + + + + + +

+ + + + + + + + + + + +++ +++ + + + +++ + + + + + + + + + ++ + + +++ ++ + + + + + + + +++ + ++ + + + ++ + ++ + ++ + ++ + ++ + + + + + +

+

△

△

△ △

△ △

△△

△△△ △△△△

△△△△△△ △ △△△ △

△

△ △

△ △

Figure 1 The set of points for the example The given points are at the center of the symbols, the symbols in-dicate the resulting classes

The initial 167 services initialize the points (assuming each service knows where to initialize them from, which

is irrelevant for our purpose) The next 345 services initialize all the necessary cells The last 921 services establish the point/cell associations Each service takes

one point and one cell as arguments (indicated with an

“A” in that row and the corresponding columns), and

initializes one association (a “C” in that association’s

column) The initial profile is 299,565 and the data

channel’s average width is 209.1 and maximum width

is 1266 SCA converges in two passes, leaving a final

profile of 15,642 and a data channel with an average

width of only 10.9 and a maximum width of 705 The

points are classified into three clusters as indicated by

the symbols in Figure 1 The ontology for this system

consists of just one class with three objects, the clusters

MMC is a form of mathematical notation designed to

express our knowledge about a domain Any ANN can

be represented as an MMC, and ANN/MMC

combina-tions are also possible The models are formal, have a

hierarchical but flexible organization, and are

machine-interpretable Algorithms can be designed to induce

transformations, supported by a rich algebra of

opera-tions All modes of training are inherited In addition,

ANN/MMC models can be directly constructed from

existing ontologies such as business models, computer

programs or scientific theories

We believe that the MMC offers an excellent

oppor-tunity for creating realistic models of the brain and

nervous system, particularly when used in combination

with traditional ANNs A model can consist of many

submodels representing different subsystems and

hav-ing different degrees of detail, dependhav-ing on the extent

of the knowledge that is available or of interest for each

subsystem It is possible to start small and then grow

virtually indefinitely, or to add fine detail to a particular

submodel of interest, while still retaining

interoperabil-ity Dynamic, self-organizing submodels will find their

own natural ontologies, which can then be compared

with observation, an approach that is radically different

from the more traditional static man-made ontologies,

and has remarkable similarities with human and animal

learning MMC offers a framework for constructing,

combining, sharing, transforming and verifying

ontolo-gies

We conclude that the MMC can serve as an

effec-tive tool for neural modeling But above all, the MMC

will serve as a unifying notion for complex systems, by

bringing unity to disconnected fields, organizing

infor-mation, and providing convergence of concepts and

in-teroperability to tools and algorithms

References

[1] Sergio Pissanetzky “A relational virtual machine

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Acknowledgements To Dr Peter Thieberger (BNL, NY) for his generous and unrelenting support, without which this might not have happened