A new type of Structured Artificial Neural Networks

based on the Matrix Model of Computation

Sergio Pissanetzky

The Matrix Model of Computation (MMC) consists of two sparse matrices:

M = (C, Q)

C = Matrix of Services

Q = Matrix of Sequences

The MMC is simple, yet very rich in features

Universal Self-organizing

Mathematically formal Natural ontology

Turing – equivalent Dynamic mode

Quantum-equivalent Connectionist

Relational database Massively parallel

Computer program Data channel

Object – oriented Transformations, refactoring

Algebra, formal algorithms

Training modes

w1

w2

Σ

x1

x2

b1

w3

w4

Σx3

x4

b2

φ1v1 x5φ2v2

equation serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2v1=w1x1+w2x2+b1 ILF1 A A A A A Cx3=φ1(v1) φ1 C Av2=w3x3+w4x4+b2 ILF2 A A A A A Cx5=φ2(v2) φ2 C A

Converting a neural network to MMC

equation serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2entire network ntw A A A C A A A A A A

equation serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2v1=w1x1+w2x2+b1 ILF1 A A A A A Cx3=φ1(v1) φ1 C Av2=w3x3+w4x4+b2 ILF2 A A A A A Cx5=φ2(v2) φ2 C A

MMC as new type of neural network

● MMC is equivalent to the equations● MMC supports all ANN features.● In addition, MMC supports global features such as network structure and organization, objects and ontologies.

The Scope Constriction Algorithm(SCA)

SCA organizes information into objects and reveals the natural ontology of the system.

CC

CC

CC

CC

CA C

A A CA C

A CA A C

A CA A C

A CA C

tc = a * fz

tj = b * fx

tf = d * vz

tk = b * fy

tb = a * fy

te = d * vy

tl = b * fz

ta = a * fx

td = d * vx

wz = vz + tl

tg = ta + td

wx = vx + tj

sx = rx + tg

th = tb + te

wy = vy + tk

ti = tc + tf

sz = rz + ti

sy = ry + th

tc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz syPROGRAM DATA

a, fz

b, fx

d, vz

b, fy

a, fy

d, vy

b, fz

a, fx

d, vx

vz

vx

rx

vy

rz

ry

CC

CC

CC

CC

CA C

A A CA C

A CA A C

A CA A C

A CA C

Profiletc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy

CC

CC

CC

CC

CA C

A A CA C

A CA A C

A CA A C

A CA C

Data channel

tc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy

“Turbulent” flow

CC

CC

CC

CC

CA C

A A CA C

A CA A C

A CA A C

A CA C

Service Commutativity

tc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy

CC

A A CA C

CA C

CC

A A CA C

CA C

CC

A A CA C

CA C

td ta tg sx tj wx te tb th sy tk wy tf tc ti sz tl wztd = d * vx

ta = a * fx

tg = ta + td

sx = rx + tg

tj = b * fx

wx = vx + tj

te = d * vy

tb = a * fy

th = tb + te

sy = ry + th

tk = b * fy

wy = vy + tk

tf = d * vz

tc = a * fz

ti = tc + tf

sz = rz + ti

tl = b * fz

wz = vz + tl

PROGRAM DATAd, vx

a, fx

rx

b,fx

vx

d, vy

a, fy

ry

b, fy

vy

d, vz

a, fz

rz

b, fz

vz

“Laminar” flowG

H

H

H

G

G

Where do objects come from?

● SCA only minimizes the scopes.● In nature, scopes are resources.● Processes compete for resources.● Scopes are naturally minimized.● Objects and inheritance arise naturally.

● New information keeps arriving.● SCA keeps forming new objects.● Larger objects grow out of smaller objects. ● Objects evolve with time. Some stabilize.● The process continues indefinitely.

DYNAMICS

Lyapunov Dynamics vs. MMC Dynamics

Lyapunov MMCstate variables X(t) row indicesstate space continuous discretedynamics dX(t)/dt = F[X(t)] SCAattractors points, orbits classes of objectsequilibrium depends stable and convergentenergy function V[X(t)] profilebiologically viable

no yes

universal no yes

CONCLUSIONS AND OUTLOOK

● Objects and inheritance arise naturally when concurrent processes compete for resources. They are the solution to the optimum resource allocation problem.

● Impact on Computer Science, Software Engineering,Refactoring, the Semantic Web, Artificial Intelligence, Biology, Neuroscience, Linguistics, Jurisprudence.

● Conjecture: Our mind uses a similar process tomake ontologies.