The goal of this paper is twofold. Firstly, to survey in a systematic and uniform way the main results regarding the way membranes can be placed on processors in order to get a software/hardware simulation of P-Systems in a distributed environment. Secondly, we improve some results about the membrane dissolution problem, prove that it is connected, and discuss the possibility of simulating this property in the distributed model. All this yields an improvement in the system parallelism implementation since it gets an increment of the parallelism of the external communication among processors. Also, the number of processors grows in such a way that is notorious the increment of the parallelism in the application of the evolution rules and the internal communica-tionsstudy because it gets an increment of the parallelism in the application of the evolution rules and the internal communications. Proposed ideas improve previous architectures to tackle the communication bottleneck problem, such as reduction of the total time of an evolution step, increase of the number of membranes that could run on a processor and reduction of the number of processor

Bravo García, Ginés

Mingo López, Luis Fernando de

Peña Camacho, Miguel Ángel

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Membrame dissolution in distributed architectures of P-Systems
MIGUEL ANGEL PEN˜A
Universidad Polite´cnica de Madrid
Facultad de Informa´tica
28660 Madrid
SPAIN
m.pena@upm.es
GINES BRAVO, LUIS FERNANDO DE MINGO
Universidad Polite´cnica de Madrid
Escuela Universitaria de Informa´tica
Crta. De Valencia km. 7, 28031 Madrid
SPAIN
{gines,lfmingo}@eui.upm.es
Abstract: The goal of this paper is twofold. Firstly, to survey in a systematic and uniform way the main re-
sults regarding the way membranes can be placed on processors in order to get a software/hardware simulation
of P-Systems in a distributed environment. Secondly, we improve some results about the membrane dissolution
problem, prove that it is connected, and discuss the possibility of simulating this property in the distributed model.
All this yields an improvement in the system parallelism implementation since it gets an increment of the paral-
lelism of the external communication among processors. Also, the number of processors grows in such a way that
is notorious the increment of the parallelism in the application of the evolution rules and the internal communica-
tionsstudy because it gets an increment of the parallelism in the application of the evolution rules and the internal
communications. Proposed ideas improve previous architectures to tackle the communication bottleneck problem,
such as reduction of the total time of an evolution step, increase of the number of membranes that could run on a
processor and reduction of the number of processors.
Key–Words: Membrane Computing, P-Systems Architectures, Distributed Communication, Membrame Dissolu-
tion
1 Preliminaries
Natural sciences, and especially biology, represented
a rich source of modelling paradigms. Well-defined
areas of artificial intelligence (genetic algorithms,
neural networks), mathematics, and theoretical com-
puter science (L systems, DNA computing) are mas-
sively influenced by the behaviour of various bio-
logical entities and phenomena. In the last decades
or so, new emerging fields of so-called natural com-
puting identify new (unconventional) computational
paradigms in different forms. There are attempts to
define and investigate new mathematical or theoreti-
cal models inspired by nature, as well as investigations
into defining programming paradigms that implement
computational approaches suggested by biochemical
phenomena.
Membrane computing, inspired in “basic features
of biological membranes”, was introduced by Ghe-
orge Paun [10] to solve NP-Complete problems in
polynomial time. As original model –Transition P
System– as remaining models emerging from its; they
are an abstract representation of hierarchical structure
and non-deterministic behavior of biological mem-
branes. A membrane is a region compounds by
other membranes and chemicals (objects) that uses
chemical reactions (evolution rules) generated another
chemicals. Each membrane has a permeability capac-
ity that enable chemicals (objects) to move between
membranes (communication). Chemicals reactions
produced in membranes can dissolve it. This process
implies that contained object and membranes to be-
come part of parent membrane.
In base to this behaviour, P-System are systems
that can be executed on-line that is, in vitro or simu-
lated, using hardware implementations (Petreska [11],
Fernandez [6] or Martinez [9]), using software sim-
ulations(Suzuki [12] o Arroyo [1]) or even in a real
cluster of processors (Ciobanu [5] o Syropoulos [13]).
Currently, researchers focused on simulations by dis-
tributed software, to alleviate the sequential nature of
processors, to obtain lower running time. Must be dis-
tinguish two main steps in this system evolution: ap-
plying rules (with proper selection of it), and object
communication between membranes. To apply rules,
Frutos [7] proposed to create decision trees to deter-
mine possible rules to apply according to membrane
context, and Gil [8] proposed algorithms to distribute
objects among rules and its application. Objects com-
munication between membranes depends on used dis-
tributed architecture. Since Ciobanu detected that net-
work congestion produce higher response time, fu-
ture studies have focused on searching architectures
to eliminate network collision.
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 229
(a) Peer to Peer architecture (b) Hierarchical Peer to Peer architecture
(c) Master-Slave architecture (d) Hierarchical Master-Slave architecture
Figure 1: Different distributed P-System architectures including the object communication order.
2 P-Systems Architectures
The first distributed architecture to implement P-
Systems that eliminate collisions is proposed like
“partially parallel evolution with partially parallel
communication” [14]; namely Peer-to-Peer (P2P )
as opposed to other architectures. In this architec-
ture (Figure 1(a)), processors are connected with a
tree topology following the inner scheme in P-System
structure: k membranes are placed in every processor
P and sequentially executed applying rules. On com-
munication step, following a post order way of pro-
cessor tree (Figure 1(a), proxies situated on each pro-
cessor send generated objects in its membranes, using
only one message, to proxies of processors that are
target membranes. Total time used in each evolution
of P-System is:
T = KsTapl + 2s(P − 1)sTcom (1)
Based on the same idea and same tree topology,
Bravo in [2] adds parallelism to transmission of ob-
jects, so simultaneously multiple processors can be
communicating with others processors with no colli-
sion at all (Figure 1(b)). This architecture cited like
Hierarchical Peer-to-Peer (HP2P ) reduces, in a noto-
rious way, time used in evolution steps:
T =
M(A− 1)Tapl
AL − 1 + 2sTcomsA(L− 1) (2)
Where A is amplitude or processor number of
children and L is number of processors tree levels, tak-
ing root like L=1.
Third architecture reduces time and eliminates
complicated restrictions of tree topology of previ-
ous models, it is Master-Slave architecture (MS) [3].
With this architecture, also based on unique simul-
taneously communication between proxies (Figure
1(c)), can make the communication step while other
processors are in their communication step. Time
used by every evolution phase is:
T = ksTapl + Tcom(Ps + 1) (3)
Communications on Master-Slave architecture
are sequentials and it produces that time of this step
is linear. To address these issues arises Hierarchical
Master-Slave architecture (HMS) [4] groups advan-
tages of HP2P architecture with MS architecture
(Figure 1(d)), obtaining a time of:
T =
MsTapl
AL−1
+ Tcom(2L+A+AL−2 − 4) (4)
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 230
3 Behavior of distributed architec-
tures
All this architectures are designed to reduce P-System
running time, considering two execution steps of it. In
addition to this stages, in different types of P-System
may exists membrane dissolution or membrane divi-
sion. There are to make changes to include these
new steps in these architectures. P2P and MS ar-
chitectures may work with only the addition of a im-
provement proxy that permit dissolution. HP2P and
HMS architectures have to modify message flow to
permit, through intermediate processors, communica-
tions among nodes even though it is not connected di-
rectly. This new communication sequence are shown
in figures 2(a) (to HP2P ) and 2(b) (to HMS).
HP2P architecture starts its communication on
root processor, which communicates with its children
sequentially. Each one of these can starts communica-
tion with its children, when received father communi-
cation (root). In the same way, children communicate
with it corresponding children. Reverse process, from
leaves to root, is similar considering the restrictions
that one processor only can have one communication
to avoid collisions. When root received the last com-
munication from it child, starts next evolution on ap-
plication phase.
HMS architecture starts with communication
with all slaves processors. Communications of all
slaves are made in sequential, because there are many
slaves processor connected to same master. But, two
slaves of different master can starts to communicate
in parallel, because there are not collisions. With the
same idea, master can communicate with its masters-
father, and so, to reach master-root. When it have re-
ceived from whole its masters-children, made its pro-
cess and sequentially communicate to whole masters-
children. Each one of them, after receive the commu-
nication, it can communicate with its children (simi-
larly to make to HP2P architecture). When a master
has terminated to communicate with its slaves, these
can execute the application step of its next evolution
phase, and it starts communication step when finished
application step, avoiding collisions.
Except for the first evolution of P System, HMS
and HP2P architectures require the same time to
communicate (as shown in figures 2(a) and 2(b)).
Knowing that communication time between two pro-
cessors (only one way) is Tcom, total communication
time for each evolution of the P System is:
T = Tcom(AL+ L− 2) (5)
In both architectures, communications number
behaves same, but its application time is different, be-
cause in HP2P architecture exists membranes in ev-
ery processor; and in HMS only exists in processors
slaves. Existing k membranes on each processor, to
take Tapl to apply, application time would be:
T = kTapl (6)
On HP2P architecture there will be P proces-
sors:
P =
AL − 1
A− 1 (7)
On HMS architecture there will be the same number
of total processors, of whom, processors slaves and
masters, will be:
Ps = AL−1 (8)
Pm =
AL−1 − 1
A− 1 (9)
Knowing total number of membranes M is results
of multiply membranes of each processor by the pro-
cessors number that contain membranes, total times of
each execution step toHP2P andHMS respectively
are:
THP2P =
TaplM(A− 1)
AL − 1 +(LA+L−2)Tcom (10)
THMS =
MTapl
AL−1
+ (LA+ L− 2)Tcom (11)
Knowing the time lasting each evolution step, it
would be determined what is the optimum value of A
and L to obtain minimum time. Also knowing that A
and L must be integers. Figures 3(a) and 3(b) shown
the better combination of A and L to be the minimum
time, according relationship between Tapl and Tcom,
and number of membranes.
4 Results and Future Work
Entering membrane dissolution and membrane divi-
sion step modifies the existing architectures behavior.
The communications order of this architecture must
also make these changes. Hierarchical Peer to Peer
Architecture and Hierarchicla Master-Slave acquired
like behavior. However, the total time and the distri-
bution varies because in HP2P are membranes on all
processors and HMS only on slaves. Moreover, this
change reduces the execution time.
To achieve a minimum time, should make other
distributions different than when they took into ac-
count the dissolution of membranes. With this new
configuration the number of children of a processor
increases to 3 with less than 1000 membranes. The
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 231
(a) Hierarchical Peer to Peer Architecture (b) Hierarchical Master-Slave Architecture
Figure 2: Chronogram of different P-System architectures with 4 depth levels and amplitude equal to 3.
optimal configuration for the number of children and
the height of the tree of processors, depending on the
number of membranes and the ratio of the time it takes
to apply membranes and that take in communicating
processors.
References:
[1] F. Arroyo, C. Luengo, A.V. Baranda and L.F. de
Mingo, A software simulation of transition P
System in Haskell,Lecture Notes in Computer
Science 2597, Springer-Verlag, Berlin 2003, pp.
19-32.
[2] G. Bravo, L. Fernandez, F. Arroyo and J.A. Fru-
tos, A Hierarchical Architecture with parallel
communication for implementing P Systems, In-
formation Theories and Applications, Vol 14.
Varna 2007.
[3] G. Bravo, L. Fernandez, F. Arroyo and J. Teje-
dor, Master-Slave Distributed Architecture for
Membrane System Implementation, 8th WSEAS
Int. Conf. on Evolutionary Computing (EC’07),
June 2007, Vancouver, Canada.
[4] G. Bravo, L. Fernandez, F. Arroyo and
M.A. Pen˜a, Hierarchical Master-Slave Archi-
tecture for Membrane Systems Implementation,
Proceeding of the 13th Int. Symposium on Arti-
ficial Life and Robotics, Jan 31-Feb 2, Beppu,
Oita, Japan.
[5] G. Ciobanu and W. Guo, P System Running on a
Cluster of Computers, Workshop on Membrane
Computing, LNCS 2933, 2004, pp. 123-2004.
[6] L. Fernandez, V.J. Martinez, F. Arroyo and
L.F. Mingo, A Hardware Circuit for Selecting
Active Rules in Transition P Systems, First In-
ternational Workshop on Theroy and Applica-
tion of P System, September 2005, pp. 45-48.
[7] J.A. de Frutos, L. Fernandez and F. Arroyo
(2009), Decision Trees for Obtaining Active
Rules in Transition P System,Proceedings of the
Tenth Workshop on Membrane Computing, Au-
gust 24-27, Romania.
[8] F.J. Gil, J.A. Tejedor and L. Fernandez, Fast
Linear Algorithm for active rules application in
Transition P Systems, Sixth International Con-
ference Information Research and Applications,
Varna 2008.
[9] V. Martinez, L. Fernandez, F. Arroyo and
A. Gutierrez, Implementation HW of an En-
closed Algorithm to Rules Applications in a
Transition P System,8th International Sympo-
sium on Syumbolic and Numeric Algorithms for
Scientific Computing, 2006.
[10] G. Paun, Computing with membranes, Journal
of Computer and System Sciences, 1(61). 2000,
pp. 108-143.
[11] B. Petreska and C. Teuscher, A hardware mem-
brane system,Preproceedings of the Workshop
on Membrane Computing, 2003, pp. 242-255.
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 232
(a) HP2P Architecture
(b) HMS Architecture
Figure 3: Amplitude and depth for the minimum time in P-Systems Architectures.
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 233
[12] Y. Suzuki and H. Tamaka, On a Lisp implemen-
tation of a class of P Systems,Romanian Jounal
of Information Science and Technology, 2000,
pp. 173-186.
[13] A. Syropoulos, E.G. Mamatas, P.C. Allilomes
and K.T. Sotiriades, A distributed simulation of
P Systems,Preproceedings of the Workshop on
Membrane Computing, 2003, pp. 455-460.
[14] J. Tejedor, L. Fernandez, F. Arroyo and
G. Bravo, An architecture for attacking the bot-
tleneck communication in P systems, Proceed-
ings of the 12th Int. Symposium on Artificial Life
and Robotics, 2007, pp. 500-505.
Recent Researches in Applied Computer and Applied Computational Science
ISBN: 978-960-474-281-3 234


Membrane Dissolution in Distributed Architectures of P-Systems

Archivo Digital UPM

http://oa.upm.es/12178/1/INVE_MEM_2011_87341.pdf

Membrane Dissolution in Distributed Architectures of P-Systems

Abstract

Similar works

Full text

Available Versions

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Archivo Digital UPM