1. A new type of Structured Artificial Neural Networks based on the Matrix Model of Computation
I am presenting a new type of structured ANNs based on the MMC.
Sergio Pissanetzky

2. The Matrix Model of Computation (MMC) consists of two sparse matrices: M = (C, Q) C = Matrix of Services Q = Matrix of Sequences
The matrix model consists of two matrices, the matrix C of services and the matrix Q of sequences.
This particular form of the MMC is called the imperative form.
In this talk I will explain only matrix C.

3. 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
In this talk I will explain only the most important of these features.

4. Converting a neural network to MMCb1 x1 w1 b2 Σ φ1 w3 v1 x3 x2 w2 Σ φ2 x5 v2 x4 w4
equation
serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2
v1=w1x1+w2x2+b1
ILF1 A C
x3=φ1(v1) φ1
v2=w3x3+w4x4+b2
ILF2
x5=φ2(v2) φ2
(ILF=Induced local field or activation potential. Activation function, synaptic weights. )
Here is an example of a matrix of services for a neural network.
On the left, the equations that govern the system. On top, the variables used in the equations.
The matrix contains a set of relations, each row is a tuple in one of the relations, called a service.
Explain the roles, the service names, the variables. A column displays the life-cycle of a variable.
I can also write a model for the entire network. Explain models and submodels.
equation
serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2
entire network
ntw A C

5. MMC as new type of neural network
equation
serv x1 x2 x3 x4 x5 w1 w2 w3 w4 b1 b2 v1 v2
v1=w1x1+w2x2+b1
ILF1 A C
x3=φ1(v1) φ1
v2=w3x3+w4x4+b2
ILF2
x5=φ2(v2) φ2
● 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 matrix model is equivalent to the equations that govern the network.
The matrix model supports all the features of the neural network.
For example, the matrix model can be trained the same as the network.
In addition, the matrix model supports global features such as structure,
organization, objects and ontologies, which are difficult to deal with using ANNs.
It is in this sense that I say the MMC is a new type of ANN.
The rest of my talk concentrates on the higher level structural abilities of the MMC.

6. The Scope Constriction Algorithm
(SCA)
SCA organizes information into objects and reveals the natural ontology of the system.
I will now discuss the Scope Constriction Algorithm SCA, which organizes information
into objects and reveals the natural ontology of the system.

7. C A PROGRAM tc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy DATA
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 C A
a, fz
b, fx
d, vz
b, fy
a, fy
d, vy
b, fz
a, fx
d, vx vz vx rx vy rz ry
I need a bigger example. On the left, a set of equations, or the statements of a computer program.
On top, the variables used in the equations. Columns permuted in such a way that all C’s fall on the diagonal.
This is called the canonical form of the MMC. Explain the scopes of the variables.
The conversion from any computer program can be done by a parser.
A column displays the life-cycle of a variable. Explain the pink column, the scope.

8. Profile C A tc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy
Here is one motivation for the SCA. The profile is the union of all scopes.
This profile is too big. There is no reason to initialize variables so far ahead of the point where they are used.

9. Data channel “Turbulent” flow C A tc tj tf tk tb te tl ta td wz tg wxsx th wy ti sz sy
Data channel C A
“Turbulent” flow
Here is another motivation for SCA. A different view of the matrix: a channel where data flows.
Data flows in the variables, from the point where a variable is initialized to the point where it is used.
This channel is very wide and disorganized. Data flows are too long and entangled.
We want to make the channel narrower, and better organized.

10. Service Commutativitytc tj tf tk tb te tl ta td wz tg wx sx th wy ti sz sy
Service Commutativity C A
Service commutativity is the tool used to reduce the profile and narrow the data channel.
Every variable must be initialized before it is used. Otherwise, the order of the services is irrelevant.
The profile of the matrix changes when the order of the services is changed.
The green services are commutative, but commuting them would increase the profile.
The pink services are not commutative.
A service such as the blue one can be shifted by repeated commutation. Doing so reduces the profile.
Systematic repeated commutation is used by SCA to minimize the profile.

11. “Laminar” flow G H G H G H C A PROGRAM td ta tg sx tj wx te tb th sytk wy tf tc ti sz tl wz
DATA
td = 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 C A
d, vx
a, fx rx
b,fx vx
d, vy
a, fy ry
b, fy vy
d, vz
a, fz rz
b, fz vz G
“Laminar” flow H G H
Narrow, well-organized channel. The scopes are short, no entanglement  Laminar flow.
More important: the matrix is partitioned into diagonal blocks. No flows cross the dotted lines  encapsulation.
Both the data and the functions that use it have been encapsulated  objects.
There are 6 objects, but only two classes  SCA has revealed the natural ontology.
The objects inherit from previously existing objects  SCA has revealed the inheritance hierarchy.
The program has also been “refactored”. The objects can be extracted as submodels. G H

12. 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.
Scopes are resources. A variable in scope is information that must be kept somewhere.
In a biological system, concurrent processes compete for resources, energy, space, construction materials.
SCA occurs naturally. The scopes are naturally minimized, objects and inheritance arise naturally.

13. DYNAMICS ● 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.
I can imagine a process where new information keeps arriving to the model.
SCA keeps organizing the information into objects, and forms larger objects out of the smaller ones.
As a result, objects evolve with time, and the process continues indefinitely.
This is strongly reminiscent of the way our brains work.
MMC is a mathematical model, not intended to be a model of the brain.
The brain is an implementation of the MMC, therefore similarities are to be expected.

14. Lyapunov Dynamics vs. MMC Dynamics
state variables
X(t)
row indices
state space
continuous
discrete
dynamics
dX(t)/dt = F[X(t)]
SCA
attractors
points, orbits
classes of objects
equilibrium
depends
stable and convergent
energy function
V[X(t)]
profile
biologically viable no
yes
universal
Lyapunov uses a vector of state variables. MMC uses the row indices assigned to the services.
The state space is continuous here, but discrete for the MMC.
The dynamics is a first-order nonlinear total differential equation with a vector function of state. For the MMC, it is the SCA algorithm.
The attractors could be points or orbits in the state space. They are objects for the MMC.
Can I manipulate the attractors? You bet. We do it every day when we write OO code.
The equilibrium depends on the system. For MMC, it is always stable and convergent.
The potential energy is a scalar function of state. In the MMC, it is the profile.
MMC is biologically viable and universal, Lyapunov is not.

15. 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 to
make ontologies.
This is an extraordinary result, but I conjecture that our mind uses the same process to make objects.