Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Optimizing Membrane System Implementation
with Multisets and Evolution Rules Compression
Abraham Gutiérrez Rodríguez
Natural Computing Group.
Universidad Politécnica de Madrid,
[email protected]
Eighth Workshop on Membrane Computing
page 1
Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Introduction
”the next generation of simulators may be
oriented to solve (at least partially) the
problems of information storage and
massive parallelism by using parallel
language programming or by using
multiprocessor computers”
G. Ciobanu, Gh. Păun y M. Pérez-Jiménez
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Goals
• Present an algorithm for compressing information
of multisets and evolution rules stored in
membranes.
– In particular, without penalizing evolution rules
application and communication times with complex
processes;
– and keeping the same parallelism degree obtained in
P-Systems implementation over distributed
architectures.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Parallel Application/ Parallel Communication
ax bycz
1
2
1
4
7
8
2
11
3
12
4
5
6
3
5
7
6
8
9
9
10
10
11
12
a--> c
bc2 --> (d, in2)
Eighth Workshop on Membrane Computing
n
membrane
processor
data bus
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Parallel Application/ Parallel Communication
P1
P2
UNFEASIBLE!
TAPL
TCOM
P3
Nowadays,
no technology exists that
permit M (∞) communication lines
• Time of an evolution
T = Tapl + Tcom
per step:
processor
Tapl is the maximum time used by the slowest membrane in applying its rules
Tcom is the maximum time used by the slowest membrane for communication
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Parallel Application/ Sequential Communication
ax bycz 1
1
2
2
4
3
7
8
4
11
12
5
9
8
7
6
10
3
5
11
n membrane
processor
6
12
9
10
a--> c
bc2 --> (d,communication
in2)
data bus
interface
Eighth Workshop on Membrane Computing
parent-child
membrane relationship
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Parallel Application/ Sequential Communication
P1
P2
UNFEASIBLE!
TAPL
T
COM
Ciobanu:
“the
response
time
of the
P3
program has been acceptable. There
are however
that could
• Implementations
withexecutions
a cluster of PC’s
– Message
Passing
Interface
(MPI),
Ciobanu
take
a rather
long
time
due to
– Java
Remote Method
Invocation congestion”
(RMI), Syropoulos
unexpected
network
• Time of an evolution step: T = Tapl + 2.(M-1)·Tcom
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Application & communication partially parallel
P1 ax bycz
1
2
1
4
72
8
P2
7
4
3
11
12
5
P4
6
9
3
10
5
P3
6
9
11
n membrane
processor
8
10
12
a--> c
Internal bc2 --> (d, in2)External
communication
communication
Eighth Workshop on Membrane Computing
Virtual
communication
page 8
proxy
Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
Application & communication partially parallel
P1
FEASIBLE!
P2
TAPL
TintCOM
P3
Text
 Evolution step times are acceptable
and costs moderated
• Minimum
evolution
time obtain
the optimum
Permits
a certain
levelwith
of parallelism
number of membranes by processor :
both in application rules phase and
T  2 communication
2 M T T  2T
COM
min
apl
com
Eighth Workshop on Membrane Computing
com
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Distributed Architectures
To reach minimum times over distributed architectures,
there should be a balance between the time dedicated to
evolution rules application and the time used for
communication among membranes.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Implementation Technologies Context
• Tejedor and Bravo distributed architectures are
independent of specific technology.
• Thus, in specific hardware implementations (FPGA’s and
microcontrollers) and solutions based upon cluster of
microprocessors, the amount of information that has to
be stored and transmitted is very important.
– In the first case, the main problem is due to their low storage
capacity.
– In the second case, the main problem is due to the bottleneck in
processor communication.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Requirements
1. there should be no information loss;
2. it should use the lowest amount of space for
storage and transmission;
3. it should not penalize time for rules application
phase and communication among membranes
while processing compressed information.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Requirements
•
Thus, this means that the compression
schema should:
a) encode information for a direct manipulation in both
phases without having to use encoding/decoding
processes.
b) do the compression in a previous stage to the PSystem evolution
c) therefore, abandon entropy limit to be able to
maintain parallelism level and evolution time
reached in previous research works.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
• Proposed compression schema is presented here in
three consecutive steps using the next P-System that
generates n2, n>1 [Gh. Păun, 2000]:
M1
M3
M2
M4
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 0 - Parikh's Vector over
P System alphabet
M1
M1
w1 =
0
0
0
0
0
a
b
b’
c
f
M1
M2
M3
Eighth Workshop on Membrane Computing
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 0 - Parikh's Vector over
P System alphabet
w2 =
r1 =
r2 =
r3 =
r4 =
0
0
0
0
0
a
b
b’
c
f
0
0
1
0
0
a
b
b’
c
f
0
1
0
0
0
a
b
b’
c
f
0
0
0
0
2
a
b
b’
c
f
0
0
0
0
1
a
b
b’
c
f
r3 > r4
→
→
→
→
M2
0
1
0
0
0
a
b
b’
c
f
0
1
0
0
0
a
b
b’
c
f
0
0
0
1
0
a
b
b’
c
f
1
0
0
0
1
a
b
b’
c
f
1
0
0
0
0
a
b
b’
c
f
M1
here
here
M2
M2
in4
here
M3
here
M4
,δ
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 0 - Parikh's Vector over
P System alphabet
w3 =
r1 =
r2 =
r3 =
1
0
0
0
1
a
b
b’
c
f
1
0
0
0
0
a
b
b’
c
f
1
0
0
0
0
a
b
b’
c
f
0
0
0
0
1
a
b
b’
c
f
M1
M3
→
→
→
1
0
1
0
0
a
b
b’
c
f
0
0
1
0
0
a
b
b’
c
f
0
0
0
0
2
a
b
b’
c
f
here
M2
here
,δ
here
Eighth Workshop on Membrane Computing
M3
M3
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 0 - Parikh's Vector over
P System alphabet
M1
• This codification requires 95
M4
storagewunits
=
0 0for
0 the
0 0
multiplicities present at the
multisets and the evolution rules.
4
a
b
b’
c
M2
f
M3
Eighth Workshop on Membrane Computing
M4
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 1 - Parikh's Vector over
each membrane alphabet
• Considers only the alphabet
subset for the P-System that
may exist in each of the regions
M1
w =
0 0 0 0
for the membrane
system.
• This subset may be calculated
by a static analysis, previous to
P-System evolution time
M1
M1
1
a
b
b’
M2
f
Eighth Workshop on Membrane Computing
M3
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
w2 =
r1 =
r2 =
0
0
0
0
a
b
b’
f
0
0
1
0
a
b
b’
f
0
1
0
0
a
b
b’
f
r3 > r4
→
→
0
1
0
0
a
b
b’
f
0
1
0
0
a
b
b’
f
1
in4
M2
Step 1 - Parikh's Vector over
each membrane alphabet
M1
here
here
M2
M2
c
r3 =
r4 =
0
0
0
2
a
b
b’
f
0
0
0
1
a
b
b’
f
→
→
1
0
0
1
a
b
b’
f
1
0
0
0
a
b
b’
f
here
here
M3
M4
,δ
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 1 - Parikh's Vector over
each membrane alphabet
w3 =
r1 =
r2 =
r3 =
1
0
1
a
b’
f
1
0
0
a
b’
f
1
0
0
a
b’
f
0
0
1
a
b’
f
M1
M3
→
→
→
1
1
0
a
b’
f
0
1
0
a
b’
f
0
0
2
a
b’
f
here
M2
here
,δ
here
Eighth Workshop on Membrane Computing
M3
M3
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 1 - Parikh's Vector over
each membrane alphabet
M1
• This codification requires 63
storage
for the M4
w =units
0
multiplicities present at the
multisets and the evolution rules.
4
M2
c
M3
Eighth Workshop on Membrane Computing
M4
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 2 - Parikh's Vector
without null values
• Is an alteration over the Run
Length Encoding (RLE)
algorithm.
w1 =
0
0
0
0
a
b
b’
f
M1
M1
M1
• The goal is eliminate all the null
values in Parikh’s vector
Eighth Workshop on Membrane Computing
M2
M3
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
w2 =
r1 =
0
0
0
0
a
b
b’
f
1
→
3
r2 =
1
1
r3 > r4
M2
Step 2 - Parikh's Vector
without null values
here
M1
2
→
2
1
here
2
1
M2
M2
in4
1
r3 =
2
→
4
r4 =
1
4
→
1
1
1
4
1
here
here
M3
M4
,δ
1
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 2 - Parikh's Vector
without null values
w3 =
r1 =
1
0
1
a
b’
f
1
→
1
r2 =
1
→
1
r3 =
1
3
M3
1
1
1
2
1
here
M1
here
M2
,δ
2
→
2
here
M3
M3
M4
3
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 2 - Parikh's Vector
without null values
• Requires 46 storage units for
the multiplicities present at the
multisets and evolution rules.
M4
w =
0
• This codification reduces
information size until a 51.1%
from the initial Parikh's Vector
codification.
M1
4
M2
c
Eighth Workshop on Membrane Computing
M3
M4
M4
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Compression Schema
Step 3 - Storage Unit Compression
Depending on the storage unit size (measured in bits), we
will be able to codify a greater or smaller range of values.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Analysis of Results
Compression Schema Analysis
• Attenuate the storage problem
• Not penalized with compression, decompression
processes.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Analysis of Results
Impact Analysis for Evolution Rules Application Time
• Primitive operations will decrease its execution time
approximately until a 26.7%.
• Evolution rules application time will be approximately
3.75 times faster.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Analysis of Results
Impact Analysis for Communication among Membranes Time
• A reduction until a 55.6% of the information to transmit
among membranes may be reached in the worst case.
• Communication time among membranes will be
approximately 1.80 times faster.
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Analysis of Results
Impact Analysis over Distributed Architecture Parameters
According to the previous empirical data, we get:
• Tapp 3.75 times faster
– increment of a 93.5% Kopt
– a reduction until a 51.6% Popt
• Tcom 1.80 times faster
– increment of a 32.4% Popt
– a reduction until a 74.5% Kopt
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Analysis of Results
Impact Analysis over Distributed Architecture Parameters
• Taking in account previous analysis
– a reduction of a 69.3% Popt
– an increment of a 44.3% Kopt
– a reduction until a 38.5% Tmim
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Conclusions
• The compression schema presented:
– reduce degrees of compression varying from
51.1 % to 18.1% depending on the size in
bits needed to store objects multiplicities
– does not penalize evolution rule application
nor communication times during P System
evolution
– does not required compression
decompression process during P System
evolution (static analysis)
Eighth Workshop on Membrane Computing
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Optimizing Membrane System Implementation with Multisets and
Evolution Rules Compression
Optimizing Membrane System Implementation
with Multisets and Evolution Rules Compression
Abraham Gutiérrez Rodríguez
Natural Computing Group.
Universidad Politécnica de Madrid,
[email protected]
Eighth Workshop on Membrane Computing
page 34