1. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 1 Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Abraham Gutiérrez Rodríguez Natural Computing Group. Universidad Politécnica de Madrid, abraham@eui.upm.es

2. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 2 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

3. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 3 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.

4. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 4 Distributed Architectures Parallel Application/ Parallel Communication 1 2 3 4 7 8 1112 5 6 910 a--> c bc2 --> (d, in2) a x b y c z 1 2 4 3 56 78 1112 910 n membrane processor data bus

5. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 5 UNFEASIBLE! Nowadays, no technology exists that permit M (  ∞) communication lines per processor Distributed Architectures Time of an evolution step: T = T apl + T com T apl is the maximum time used by the slowest membrane in applying its rules T com is the maximum time used by the slowest membrane for communication Parallel Application/ Parallel Communication T APL T COM P1P1 P2P2 P3P3

6. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 6 Distributed Architectures Parallel Application/ Sequential Communication 1 2 3 4 7 8 1112 5 6 910 a--> c bc2 --> (d, in2) a x b y c z n membraneprocessordata bus communication interface 1 2 4 3 56 7 8 1112 9 10 parent-child membrane relationship

7. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 7 UNFEASIBLE! network congestion Ciobanu: “the response time of the program has been acceptable. There are however executions that could take a rather long time due to unexpected network congestion” Distributed Architectures Implementations with a cluster of PC’s –Message Passing Interface (MPI), Ciobanu –Java Remote Method Invocation (RMI), Syropoulos Time of an evolution step: T = T apl + 2.(M-1)·T com Parallel Application/ Sequential Communication P1P1 P2P2 P3P3 T APL T COM

8. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 8 Distributed Architectures Application & communication partially parallel 1 2 3 4 7 8 1112 5 6 910 a--> c bc2 --> (d, in2) a x b y c z n membraneprocessorExternal communication Internal communication Virtual communication proxy P1P1 3 P2P2 P4P4 P3P3 1 2 5 7 11 12 9 10 4 8 6

9. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 9 FEASIBLE! Evolution step times are acceptable and costs moderated Permits a certain level of parallelism both in application rules phase and communication Distributed Architectures Minimum evolution time obtain with the optimum number of membranes by processor : Application & communication partially parallel Text COM T APL Tint COM P1P1 P2P2 P3P3

10. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 10 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.

11. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 11 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.

12. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 12 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.

13. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 13 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 P- System evolution c)therefore, abandon entropy limit to be able to maintain parallelism level and evolution time reached in previous research works.

14. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 14 Proposed compression schema is presented here in three consecutive steps using the next P-System that generates n 2, n>1 [Gh. Păun, 2000]: Compression Schema M1 M2 M4 M3

15. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 15 Compression Schema Step 0 - Parikh's Vector over P System alphabet M1 M2 M4M3 M2 M1 b’ 0w 1 =0000 abcf M1

16. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 16 Compression Schema Step 0 - Parikh's Vector over P System alphabet M1 M2 M4M3 M2 b’ 0 1r 1 =0000→0100 abcfabcf r 3 > r 4 b’ 0w 2 =0000 abcf b’ 0 0r 2 =0100→0100 abcfabcf b’ 00010 in 4 abcf b’ 0 0r 3 =0002→1001 abcfabcf here b’ 0 0r 4 =0001→1000,δ,δ abcfabcf here M2 here

17. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 17 Compression Schema Step 0 - Parikh's Vector over P System alphabet M1 M2 M4M3 M2 M3

18. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 18 Compression Schema Step 0 - Parikh's Vector over P System alphabet M1 M2 M4M3 M2 M4 b’ 0w 4 =0000 abcf M4 This codification requires 95 storage units for the multiplicities present at the multisets and the evolution rules.

19. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 19 Step 1 - Parikh's Vector over each membrane alphabet Compression Schema M1 M2 M4M3 M2 M1 b’ 0w 1 =000 abf M1 Considers only the alphabet subset for the P-System that may exist in each of the regions for the membrane system. This subset may be calculated by a static analysis, previous to P-System evolution time

20. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 20 Step 1 - Parikh's Vector over each membrane alphabet Compression Schema M1 M2 M4M3 M2

21. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 21 Step 1 - Parikh's Vector over each membrane alphabet Compression Schema M1 M2 M4M3 M2 M3

22. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 22 Step 1 - Parikh's Vector over each membrane alphabet Compression Schema M1 M2 M4M3 M2 M4 w 4 =0 c M4 This codification requires 63 storage units for the multiplicities present at the multisets and the evolution rules.

23. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 23 Step 2 - Parikh's Vector without null values Compression Schema M1 M2 M4M3 M2 M1 b’ 0w 1 =000 abf M1 Is an alteration over the Run Length Encoding (RLE) algorithm. The goal is eliminate all the null values in Parikh’s vector

24. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 24 Step 2 - Parikh's Vector without null values Compression Schema M1 M2 M4M3 M2

25. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 25 Step 2 - Parikh's Vector without null values Compression Schema M1 M2 M4M3 M2 M3 2 1r 1 =1→1 here 11 b’ 0w 3 =11 af M3 here 2 1r 2 =1→,δ,δ 1 r 3 =1→2 here 33

26. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 26 Step 2 - Parikh's Vector without null values Compression Schema M1 M2 M4M3 M2 M4 w 4 =0 c M4 Requires 46 storage units for the multiplicities present at the multisets and evolution rules. This codification reduces information size until a 51.1% from the initial Parikh's Vector codification.

27. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 27 Compression Schema Step 3 - 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.

28. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 28 Analysis of Results Attenuate the storage problem Not penalized with compression, decompression processes. Compression Schema Analysis

29. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 29 Analysis of Results Primitive operations will decrease its execution time approximately until a 26.7%. Evolution rules application time will be approximately 3.75 times faster. Impact Analysis for Evolution Rules Application Time

30. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 30 Analysis of Results 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. Impact Analysis for Communication among Membranes Time

31. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 31 Analysis of Results Impact Analysis over Distributed Architecture Parameters According to the previous empirical data, we get: T app 3.75 times faster –increment of a 93.5% K opt –a reduction until a 51.6% P opt T com 1.80 times faster –increment of a 32.4% P opt –a reduction until a 74.5% K opt

32. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 32 Analysis of Results Impact Analysis over Distributed Architecture Parameters Taking in account previous analysis –a reduction of a 69.3% P opt –an increment of a 44.3% K opt –a reduction until a 38.5% T mim

33. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 33 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)

34. Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Workshop on Membrane Computing Eighth page 34 Optimizing Membrane System Implementation with Multisets and Evolution Rules Compression Abraham Gutiérrez Rodríguez Natural Computing Group. Universidad Politécnica de Madrid, abraham@eui.upm.es