Hybrid CA-Systems: Coupling Cellular Automata with Artificial Neural Nets

Hybrid CA-Systems: Coupling Cellular Automata with Artificial Neural Nets

Christina Stoicawww.cobasc.de

Institute for Computer Science and Business Information Systems

University of Duisburg-Essen

Germany

Hybrid Systems

The nearly universal usability of cellular automata (CA) is well known.

Models become still more powerful by coupling CA with artificial neural nets (NN).

Such CA-NN-models may be called "hybrid systems" that contain certain characteristics of

learning,

adaptability and

flexibility.

First example

SIMULATION OF TRAFFIC FLOWS

Kohonen Feature Map

The cells of the CA represent different types of cars

regulates the special traffic lights

Second Example

NEURAL CA-SYSTEM

The cells of the CA consist of Bi-directional Associative Memory-Nets (BAM) and a Kohonen Feature Map (KFM).

models individual learning processes in dependency of a certain social milieu

simulation of traffic flows - CA

The cells of the CA represent different types of cars, i.e. different with respect to velocity and type of driving.

These artificial cars move on different lanes of a highway.

Because of the different velocities and types of driving accidents and other problems will occur that lead to backups.

In particular, high density of traffic will increase the probability of accidents.

simulation of traffic flows - CA

Obstacle: If possible: overtake, else slow down

Speed limit: adapt the speed

The state of the cells is defined as: S = {0,1,2}

0 = speed up; 1 = overtake; 2 = adapt the speed

”Enlarged" Moore neighborhood i.e. two additional cells outside of the Moore neighborhood are taken into consideration.

Is the cell on the right line, then consider only the cells at the front, on the left and the additional two cells on the left side.

If car = right line then compute the state St+1:

If R = set of all cells on the right line, then Sir R: If Sir > 4, then the car speeds up.

If L = set of all cells on the left line, then apply for all Sir R: If Sil > 4, then the car can overtake

sir t 1

0,if skr t 4k,r R ir

1,if skr t 4 sk,l t 4k,l

k,r R ir

2,if sir t 4 sk,l t 4k,l

k,r R ir

simulation of traffic flows - CA

simulation of traffic flows

In order to regulate the traffic and to avoid too many accidents the access roads to the highways are regulated by special traffic lights.

These traffic lights stop the access if the density of traffic is too high and/or if there are already accidents with according backups.

simulation of traffic flows

In the CA-model the traffic lights are regulated by a Kohonen feature map, which belongs to the type of non supervised learning nets.

The net is trained to certain critical values of traffic density.

simulation of traffic flows

Assumptions:

There exists a station of measurement, e.g. 1 km before and after an access road.

The numbers of cars, the distance and the speed is measured.

The collected data from the CA are the training data for the KFM.

simulation of traffic flows - KFM

Traffic light offNormalphase of

traffic light

Trafficlight

flashes x 45 X - -

45 x 25 - X -Distance

(in m) x 25 - - X x 90 X - -

90 x 50 - X -Speed

(in km/h)x50 - - Xx 20 X - -

20 x 33 - X -Number of carsx 33 - - X

simulation of traffic flows - KFM Learning rule: Winner-take-all:

The amount the units learn will be governed by a neighbourhood kernel h, which is a decreasing function of the distance of the units from the winning unit on the map lattice

wij

t 1 wij

t oi w

jt

x wz min

ix w

jX=(w1,....xn), wj={wi,...wnj}

X=InputvectorW=connection strength x w

ij x

i w

ij 2

i

1

2

hiz exp j z 2 / 2 z2

wij t 1 wij t hiz x t w ij t , j The weights will be changed according to the formula:

j-z= Distance of neuron j to the kernel

z=radius within the units will be changed

simulation of traffic flows - KFM

(t1) max* ( min/ max)t / t max

Radius:max

maxminmax/)/(*)1( tttz

t : actual iteration step max1 tt maxt : number of iteration steps

max : size of the radius at the beginning (in this case 40)min : limit of the radius (in this case 2)

For )(tz is valid: maxmin )( t

Learning rate (for this model):

t : actual iteration step max1 tt maxt : number of iteration stepsmax : learning rate at the beginning (in this case 1)min : minimum of the learning rate (in this case 0,001)

For )(t is valid: maxmin )( t

simulation of traffic flows

simulation of traffic flows

The practical use of such a system is the possible optimization of the real regulating systems that already exist on the Autobahnen in the German Rhein-Ruhr Region.

Second Example:

The Evolution of Neural Networks in a "Social" Context

The Evolution of Neural Networks in a "Social" Context

A computational model as a possibility to analyse some important concepts of cognitive development embedded in a social context.

The model consists of

Neural Networks (BAM / SOM)

individual level

Cellular Automaton Boolean Network social level

Theoretical descriptions of cognitive ontogenesis have a long and famous tradition;

in the last years research changed the focus of descriptions by including the interdependency between social context and cognitive development.

"dependency of social context"

cognitive developmentof a learning system

gets inform

ation from

its enviro

nment

organizes its own evolution by constructing cognitive representations

The factual development of the system is dependent on:

its particular developmental logic, i.e., the cognitive dynamics that governs its evolutionary path.

environment or context respectively determines the development by orientating the system into certain directions and by slowing or fastening the whole process.

Referring to cognitive ontogenesis, the fact must be taken into consideration that intelligent actors "construct" actively the concepts and cognitive categories they use for world representation.

Even learning processes by which people take over concepts from other people are no simple imitation processes but rather complex constructive ones whose results are dependent on the individual learning biography of the learners and the social context in which they take over the new concepts.

Concept Building Analogy

Conceptual learning (supervised vs. unsupervised learning)

Social learning

Supervised vs. unsupervised learning

Supervised learning means that the learner gets an immediate response (valuation) after solving a problem.

Non supervised learning means that the cognitive task has to be fulfilled by applying particular schemas that the learner has learned before.

Usually theses processes are done without immediate responses or valuation respectively by the environment

Bi-directional Associative Memory (BAM)Hetero-associative network

The network gets pairs of vectors e.g.:

X1 = (x11,x12,....,x1n)T Y1= (y11,y12,.....,y1m)T

X2 = (x21,x22,....,x2n)T Y2= (y21,y22,.....,y2m)T

X3 = (x31,x32,....,x3n)T Y3= (y31,y32,.....,y3m)T

(Contains the features) (contains the concepts for the features)

(x,y)1,-1

„Learning rule“:

The weight matrix is computed by the following algorithm:

BAM

Each cognitive system has often the task to create concepts by its own. This creative operation is not done arbitrarily but mainly by formation of analogy:

If a learner has to create new concepts by himself - without supervising - (s)he will rather often (perhaps not always) do so by applying the logic (s)he has learned before.

Creating new concepts

Creating new concepts -> Analogy

If (X,-) is a new vector with no according Y-part, then Y is calculated:

XW = Y withX‘W = Y‘

W is the weight matrix of X‘ and Y‘, with

H(X,X‘) = min for all X‘.

H(X,X‘) is the Hamming distance of X and X‘.

Building semantical networks

The second type that is used to model the generation of semantic networks is a "Kohonen Feature Map" (KFM), which is able to learn in an unsupervised way.

KFM is the best known example of unsupervised learning.

Its task is the collecting and ordering of singular concepts, that is the forming of concept clusters. Learning occurs in this type conforming to the following learning rule:

Kohonen Feature Map (KFM)

Self-Organising Map (SOM)

Learning rule: Winner-take-all:

The amount the units learn will be governed by a neighbourhood kernel h, which is a decreasing function of the distance of the units from the winning unit on the map lattice

wij

t 1 wij

t oi w

jt

x wz min

ix w

jX=(w1,....xn), wj={wi,...wnj}

X=InputvectorW=connection strength x w

ij x

i w

ij 2

i

1

2

hiz exp j z 2 / 2 z2

wij t 1 wij t hiz x t w ij t , j The weights will be changed according to the formula:

j-z= Distance of neuron j to the kernel

z=radius within the units will be changed

Building semantical networks

The resulting ensemble of clusters is a formal representation of a semantic network.

The KFM gets the information directly from the different BAM networks. The Y-vectors represent the concepts that shall be clustered according to the X-vectors, which consist of the respective features of perceptions.

The KFM clusters only the concepts and not the features, so it is not always evident why the clusters are generated this way (this fact can be observed in human interactions as well).

Each learner A (a cell in the CA) can be represented as theaccording set of concepts

CA ={c1,....,cn} with ci = (Xi,Yi).

If B N(A) (the Moore neighbourhood of A) has a set CB

with

CB | | CD for all D N(A) and

If ck CB and ck CA andIf (Xk) is presented to A, then

CA ={c1,....,cn,ck} in the next time step

Learning in a social milieu

Reproduction

Two actors who are placed together on the grid and who have reached a sufficient age can „get“ a child, i.e., a new actor is generated with an age of 0.

The relations between the „parents“ and the „child“ become asymmetrical, that is one-directional from the parents to the child.

Basically the actors (learners) are placed on the grid according to the topology of a cellular automaton (CA). This means that the relations between the actors are symmetrical:

R(a,b) = R(b,a)

If two artificial actors became parents, then the CA is transformed into a Boolean net (BN) with asymmetrical or anti-symmetrical relations.

R(a,b) R(b,a) R(a,b) R(b,a)

Transformation

The Program

Conclusions

The differences of individual developments are often (although not always) due to the temporal order in which learners get acquainted with new concepts.

Therefore it is not enough to analyse the difference of learning milieus in terms of the number of concepts they offer to the learners but it is nearly as important to observe the temporal order of informational processes.

In this sense culture as ordered sets of concepts must be taken into regard when analysing learning processes.

Conclusions

A social milieu that forces the learner to learn everything the social environment offers can be counterproductive for the learner: he has to spend all his time to take over knowledge already known and can not unfold his own innovative capability.

Therefore a cognitive development that allows the learner to unfold his creativity must rely upon an environment that allows "social forgetting", i.e., ignoring some knowledge that has been achieved by elders.

Thank You