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SSYYSSTTEEMM

PROJECT PREPARED BY:

Atit Gaonkar

XII A

SESSION: 2013-2014

KENDRIYA VIDYALAYA

NAVAL BASE

ARGA, KARWAR

OBJECTIVE

A comparison between Internet based

computing and SN P system and use Internet

computing Pebble game for SN P system.

ACKNOWLEDGEMENT I wish to express my deep gratitude and

sincere thanks to Principal Mrs.

BHANUMATHY H.D for her

encouragement and facilities that she

provided for this project. I shall remain

indebted to her.

It would be my utmost pleasure to express

my sincere thanks to my computer science

teacher for providing a helping hand in

this project.

I can’t forget to offer my sincere thanks to

my classmates who helped me to carry out

this project.

My thanking will be incomplete without

thanking my parents who where always

with me and supported me in doing this

project. It is their help which made the

project to attain its present form.

CONTENTS

(i) INTRODUCTION.

(ii) COMPUTATION DAGS

(iii) SPIKING NEURON P SYSTEM

(iv) IC PEBBLE GAME.

(v) THEOREM.

(vi) REFERENCE

INTRODUCTION

A number of basic life processes can be considered as

computation. Natural computing studies these

computations. Mainly there are three types of Computing.

DNA, Membrane, & Quantum Computations.

Membrane Computing inspired by computational

processes in living cells, was introduced by Gheorghe Paun

in 1998 and they are called P systems. P systems are a

class of distributed parallel computing devices of a

biochemical type which can be seen as general computing

architecture where various types of objects can be

processed by various operations.

Now SN P Systems are proved to be number computing

devices. A neuron (node) sends signals (spike) along its

outgoing synapses (edges).

We use a reserved symbol or letter ‘a’ to represent a spike.

Each neuron has its own rule for either sending Spike

(firing rules) or for internally consuming spike (forgetting

rules). The spiking can take place at the moment of time or

a later moment. In this paper a comparison between this

graphical model of SN P system and Internet based

computing is done. Then we use Internet Computing

pebble game to make SN P system a computational device.

COMPUTATIONAL DAGS

A directed graph given by a set of nodes NG and a set of

arcs or directed edges AG each having the form ( u → v),

where u, v NG. A path in G is a sequence of arcs that

share adjacent end points, as in the path from node u1 to

node un: ( u1 → u2 ), ( u2 → u3 ), …, ( un-1 → un ).

A dag or a directed acyclic graph is a digraph that has no

cycles; that is G cannot contain a path from ( u1 → u2 ), (

u2 → u3 ), …, ( un-1 → un ) wherein u1 = un. When a dag is

used to model a computation it is called a computation

dag.

Each node v NG represents a tasks of the computation An arc (u → v) AG represents the dependence of task v on

task u; v cannot be executed until u is.

For each arc (u → v) AG . ‘u’ & ‘v’ are related as parent &

child respectively in G. Every dag has at least one

parentless node (which is called the source). Every finite

dag has at least one childless node called sink. Here we use

this model of dags to represent the SN P system.

SPIKING NEURON P

SYSTEM

The brain is a vastly complicated signal system with

neuron forming the basis of the system. Electrochemical

signals flow in one direction in neurons. The majority of

neurons receives input on the cell body and dendrite tree,

and transmits output via the axon. The connection

between the ends of axons and dendrites or cell body of

the neurons is the specialized structures called synapses

In SN P system electric impulses are passes through the

synapses .Such a pulse or impulse is called spike or action

potential .Sequences of such impulses which occur at

regular or irregular intervals are called spike trains .Since

all spikes of a given neuron look alike the form of action

potential does not carry any information .Rather it is the

number and timing of spike what matter.

Definition 2.1 Mathematically we represent a Spiking

Neural P system (in short, an SN P system), of degree m

1.

It is expressed in the form

=( O,1,…,m,syn,i0 )

Where:

1. O= {a} is the singleton alphabet (a is called Spike)

2. 1,…,m, are neurons, of the form

i = (ni, Ri), 1 i m

Where:

a) ni 0 is the initial number of spikes contained by the cell ;

b) Ri is a finite set of rules of the following two forms : (1) E/ar a; t, where E is a regular expression over O, r

1, and t 0;

(2) asλ for some s 1, with the restriction that asL(E)

for any rule. E/ ar a; t, of type (1) from Ri.

(3) Syn {1, 2, …, m} X {1, 2, …., m} with (i,i) syn for 1 i

m (synapses among cells);

(4) i0 {1,2,..,m} indicates the output neurons.

The rules of type (1) are firing (we also

say spiking) rules, and they are applied as follows. If the

neurons i contains k spikes, and ak L (E), k r, then the

rule E/ ar a; d can be applied. The application of this rule

means consuming or removing r spikes and thus only k-r

remain in i the neuron is fired, and it produces a spike

after d time units. If d=0, then the spike is emitted

immediately, if d=1 then the spike is emitted in the next

step etc. If the rule is used in step t and d 1, then in steps

t, t + 1, t + 2 ,…,t+d-1 the neuron is closed, so that it cannot

receive new spikes .

In step t + d, the neuron spikes and becomes again open,

so that it can receive spike which can be used in step

t+d+1.

The rule of type (2) is forgetting rules, and they are applied

as follows: if the neurons i contains exactly s spikes, then

the rule asλ from Ri can be used, meaning that all s

spikes are removed from i .

(i) If a neuron spike with times t1, t2,…then either the set of numbers t1,t2,…can be considered as computed by , that is ST ( ) or Spike train of where ST ( ) = < t1, t2….>.

(ii) The set of intervals between spikes ts –t s-1 , s 2 can be the set computed by Nk( ), where Nk( ) ={ n / n = ti –t i-1 , 2 i k }.

IC PEBBLE GAME

As soon as a spike enters in a neuron as input in the initial

configuration or from another neuron through synapses, it

makes it active altogether with the synapses that it

establishes with other neurons. That is if a neuron uses the

rule of the form ar a; 0 one spike is sent out at the time of

spike and the neuron is active. If a neuron uses the

forgetting rule as λ the neuron will be inactive. In a self

activating SN P system we have an arbitrary large number

of neurons which differ by the number of spikes and the

rules they contain. Some of these will be active and others

inactive.

If there is a spike from (u → v) that means u is the parent

node and v is the child of u. In SN P system there can be

more than one parent node, all neuron which contain

spike in the first step are parent nodes. One output neuron

that sends a spike to the environment is known as the sink

node.

In this section we explain the IC Pebble game and what are

the similarities of this game with the working of SN P

system. The basic idea underlying pebble game is to use

tokens called pebbles to model the progress of a

computation on a dag. The placement or removal of

pebbles of various types- which is constrained by the

dependencies modeled by the dag’s arcs – represents the

changing status of the task represented by the dag’s node.

The IC Pebble game on a computation dag G involves one

player S, the server, who has access to unlimited supplies

of two types of pebbles:

ELIGIBLE pebbles: whose presence indicates a task’s eligibility for execution.

EXICUTED pebbles: whose presence indicates a task’s having been executed.

The rules of IC Pebble games are

1. S begins by placing an ELIGIBLE pebble on each unpebbled

source of G.

2. At each step, S

Select a node that contain an ELIGIBLE pebble Replace that pebble by an EXICUTED pebble

Places an ELIGIBLE pebble on each unpebbled node of G all of whose parents contain EXICUTED pebbles.

3. S’s goal is to allocate in such a way that every node v of G

eventually contains an executable pebble.

For each step t of the play IC pebble game on the

a dag G, let X(t) denote the number of EXICUTED pebbles

on G’s nodes at step t and let E(t) denotes the number of

ELIGIBLE pebbles. X(t) = t is the idealized version of the

game this is not true in the original version. The aim of the

game is to maximize E(t) as possible.

A Pebble game for SN P System

Consider SN P system in the form of a directed

acyclic graph.

1. The players of IC pebble game in SN P system are Neurons that is all the neurons which contains spikes initially will start the game or all the neurons of such type are known as the sources.

2. A finite or infinite set of clients, more than one neuron is connected to the source neuron

Can be ELIGIBLE neuron ( EL neurons) whose presence indicates a spike eligible for execution.

EXICUTED neuron (EX neurons)whose presence indicates a neuron already executes the task or already a spike is send out from the neuron.

Places an ELIGIBLE neuron on each unpebbled node of G at least one of whose parents contain EXICUTED pebble

THEOREM The Theorems Are :

a) For any schedule that allocates nodes sequentially along successive diagonal levels of the mesh E(t)

= n whenever nC2 t < (n + 1)C2 .

b) For any schedule for the mesh, if t lies in the preceding range, then E(t)

can be as large as possible as n.

Conclusion

In this paper we introduced IC Pebble game,

which incorporated the features of SN P system. Also

present methodologies to derive a mesh structure for SN P

system using different rules inside the neurons. This

shows the computational completeness of SN P system.

REFERENCES

1) G. Malewicz, A.L.Rosenberg, On batch scheduling dags for Internet based Computing. Euro- Par 2005, In Lecture notes in Computer Science 3648, Springer – Verlag, Berlin, PP 262-271.

2) G. Malewicz, A.L.Rosenberg, A Pebble game for Internet – Based Computing, CIRM, Marseille, France, 29 May – 2 June 2006.

3) Gh.Paun: Membrane computing-An introduction. Springer – Verlag, Berlin 2002.

TTHHAANNKK YYOOUU