Related Works: Viable Sw Architecture (I) Bravo[2007], Tejedor[2007]

2. Related Works: Viable Sw Architecture (I)Bravo[2007], Tejedor[2007] • The evolution steptime is of the order of square root of the number of membranes that the P system has. • With a rules application algorithm N times faster, the evolution step time is divided by square root of N. • It is needed an evolution rules application algorithm which maximum execution time can be known beforehand. Tevo ≈ #membranes

3. Related works: [Tejedor2007]Active Rules Elimination Algorithm • Maximal applicability benchmark: the maximum number of times that a rule can be applied. • To eliminate one by one the rules of the active rule set. • Each step for elimination of a rule A requires the sequential execution of 2 actions: • Any rule other than A belonging to the set of active rules is applied a random number of times between 0 and its maximal applicability benchmark. • The rule A is applied a number of times that is equal to its maximal applicability benchmark.

4. Related works: [Tejedor2007]Active Rules Elimination Algorithm (II) • Execution trace of the algorithm { r1r2r3 r4 } { r1r2r3 r4 } { r1r2r3 } { r1r2} { r1} = = = = = R R R R R Rule Elimination forr4 Rule Elimination forr3 Rule Elimination forr2 Rule Elimination forr1

5. Related works: [Tejedor2007]Active Rules Elimination Algorithm (III) • Execution time of the algorithm max max max max [r1] [r2] [r3] [r4] #rules · (#rules - 1) 0 0 0 #operations = max max max [r1] [r2] [r3] 2 0 0 max max [r1] [r2] 0 max [r1] Complexity = O(#rules2) #operations =10

6. Related works: [Gil2007]Delimited Massively Parallel Algorithm • To eliminate one by one the rules of the active rule set. • Each step for elimination of a rule A requires the sequential execution of 2 actions: • Any rule other than Aproposes in an independent manner, a multiset to be consumed from the membrane multiset. If the addition of all the proposed multisets by the rules is smaller than the membrane multiset, then the proposed multiset is subtracted from the membrane multiset. • The rule A is applied a number of times that is equal to its maximal applicability benchmark. • Complexity = O(#rules·log(#rules))

7. Goals • To develop a faster rules Application algorithm using the ideas of Active Rules Elimination Algorithm and taking into account the rules competitiveness. • This new algorithm must have sequential and parallel versions.

8. Active Rules Elimination Improvement • Competitiveness Graph a r1:ab… r1 r2 r2:a2… r3:cd3… c, d r3 r4 r4:c2d …

9. Active Rules Elimination 1st Improvement (I) Parallel Sequential max max [r1] [r2] w = {axby} + {cwdz} 0 w = {axby} max [r1] + max max max max [r1] [r2] [r3] [r4] max max [r3] [r4] 0 0 w = {cwdz} max 0 max [r3] [r1] max [r3] #operations = 3 #operations = 6

10. Active Rules Elimination 2nd Improvement (II) • A rule is an articulation if and only if the subgraph resulting from the elimination of this rule has more connected components than the competitiveness graph. r4 r4 r1 r3 r1 r2 r2

11. Active Rules Elimination 2nd Improvement (III) Parallel Sequential max max max max max max max max [r1] [r2] [r4] [r3] [r1] [r2] [r4] [r3] 0 0 0 0 0 0 max max [r1] [r2] max max max [r1] [r2] [r4] 0 0 max [r1] max [r1] max [r4] #operations = 7 #operations = 8

12. Active Rules Elimination 2nd Improvement (IV)

13. Active Rules Elimination 3rd Improvement (I) • In the elimination step for rule A, another rule B may be also eliminated. There are two reasons for this: • Rules applied before B in the elimination step of A consume the objects necessary for B to be no longer applicable. • The random value which defines the number of times that rule B is applied is equal to its maximal applicability benchmark.

14. Active Rules Elimination 3rd Improvement (II) • Do not execute the elimination step for rule B that has been eliminated • Rule B is not to be applied • To change the composition and order of the next elimination steps max max max max max [r5] [r4] [r3] [r2] [r1] r4 r3 0 0 0 0 max max max [r3] [r2] [r5] 0 0 max r1 [r3] r2 max r5 [r2]

15. Active Rules Elimination 3rd Improvement (III) • Execution of the algorithm of application of competitiveness rules is a loop that ends when it reaches a state with no active rules. In each iteration, there are 3 steps: • The elimination steps associated to the state are executed. • Active rules are calculated. • The state represented by active rules is transited.

16. Active Rules Elimination 3rd Improvement (IV) • Experimental data r5 r1 r2 r8 r7 r3 r4 r6 r9 r10

17. Active Rules Elimination 3rd Improvement (V) • Experimental data

18. Conclusions • Based on this concept of a competitiveness relationship, a new way of parallelism has been opened towards the massively parallel character needed in rules application in P systems. • The sequential version of this algorithm is the fast until now. • Both the sequential and the parallel versions of the algorithm thus allowing for determination of execution time beforehand.