Using the incidence matrix in an evolutionary algorithm for Computing minimal siphons in Petri net models Fernando Tricas1 Jos´ e Manuel Colom1 Juan Juli´ an Merelo2 Depto de Inform´ atica e Ingenier´ ıa de Sistemas Universidad de Zaragoza {ftricas,jm}@unizar.es Depto. ATC/CITIC Universidad de Granada [email protected] 14 de octubre de 2014 ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 1

Outline Petri Nets, Siphons and Liveness The genetic algorithm The proposal Some experiments Conclusions ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 2

Petri Nets, Siphons and Liveness ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 3

The Genetic Algorithm 1. (Start) Generate random population 2. (Fitness) Evaluate the fitness of each chromosome in the population 3. (New population) Create a new population 3.1 (Selection) 3.2 (Crossover) 3.3 (Mutation) 3.4 (Accepting) 4. (Replace) 5. (Test) If the end condition is satisfied, stop 6. (Loop) Go to step 2 ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 4

The Proposal P1_1 P1_2 P2_1 P2_2 R1 R2 P1_3 P1_0 P2_0 T7 T6 T5 T4 T1 T2 T3 _3 _2 _5 T1 T2 T3 T4 T5 T6 T7 P1 1 + − 0 0 0 0 0 1 P1 2 0 + − 0 0 0 0 2 P1 3 0 0 + − 0 0 0 3 P2 1 0 0 0 0 + − 0 4 P2 2 0 0 0 0 0 + − 5 P1 0 − 0 0 + 0 0 0 6 P2 0 0 0 0 0 − 0 + 7 R1 − + 0 0 0 − + 8 R2 0 − − + − + 0 9 ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 5

The Proposal P1_1 P1_2 P2_1 P2_2 R1 R2 P1_3 P1_0 P2_0 T7 T6 T5 T4 T1 T2 T3 _3 _2 _5 T1 T2 T3 T4 T5 T6 T7 P1 1 + − 0 0 0 0 0 1 ⊕R1 − + 0 0 0 − + 8 ± ± 0 0 0 − + ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 6

The Proposal P1_1 P1_2 P2_1 P2_2 R1 R2 P1_3 P1_0 P2_0 T7 T6 T5 T4 T1 T2 T3 _3 _2 _5 T1 T2 T3 T4 T5 T6 T7 P1 1 + − 0 0 0 0 0 1 ⊕R1 − + 0 0 0 − + 8 ± ± 0 0 0 − + ⊕P2 2 0 0 0 0 0 + − 5 ± ± 0 0 0 ± ± ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 7

The Proposal P1_1 P1_2 P2_1 P2_2 R1 R2 P1_3 P1_0 P2_0 T7 T6 T5 T4 T1 T2 T3 _3 _2 _5 T1 T2 T3 T4 T5 T6 T7 P1 1 + − 0 0 0 0 0 1 ⊕R1 − + 0 0 0 − + 8 ± ± 0 0 0 − + ⊕P2 2 0 0 0 0 0 + − 5 ± ± 0 0 0 ± ± •{P1 1, P2 2, R1} = {P1 1, P2 2, R1}• ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 8

The Proposal We need to look for a combination with no + signs For each individual the variables will represent places (rows of the sign incidence matrix) The fitness function returns: the number of places of the siphon when the restrictions are met (minimizing). (big) negative number for: empty or ‘full’ siphons sets with a ‘+’ sign in the result of the sum ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 9

The experiments Initial population Other parameters ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 10

The experiments Initial population Size: We start with a population of 8 individuals and run the experiment 30 times. If it fails more than once, we double the size until the experiment does 30 runs with at most one failed. ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 11

The experiments Initial population Size: We start with a population of 8 individuals and run the experiment 30 times. If it fails more than once, we double the size until the experiment does 30 runs with at most one failed. =⇒ Less than 3.3 % of probability of not finding a solution And: Elitism and rank-based selection. Mutation: bitflip operation with 33 % of probability Two-point crossover operator with probability of 66 %. Initial population is random ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 12

The experiments. FMSxD ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 13

The experiments Times for siphon computation with the proposed method Size Population Time Evaluations FMSAD 3 16 0,03 (0,01) 247 (83,95) 3 32 0,05 (0,01) 479 (76,77) 3 64 0,08 (0,01) 834 (141,33) 4 16 0,04 (0,01) 226 (80,94) 4 32 0,09 (0,02) 565 (145,35) 4 64 0,18 (0,03) 1179 (215,43) 5 16 0,05 (0,02) 205 (98,40) 5 32 0,11 (0,04) 483 (209,11) 5 64 0,27 (0,08) 1225 (395,43) 6 32 0,14 (0,08) 457 (268,78) 6 64 0,31 (0,12) 1036 (463,36) 6 128 0,85 (0,28) 2964 (955,49) 7 32 0,16 (0,08) 407 (228,09) 7 64 0,34 (0,15) 909 (443,33) 7 128 0,93 (0,42) 2474 (1.205,99) 7 256 1,84 (0,77) 5016 (2.368,35) 8 64 0,40 (0,19) 822 (463,71) 8 128 0,88 (0,44) 1842 (1.025,05) 8 256 2,04 (1,20) 4248 (2.669,90) 9 64 0,41 (0,12) 664 (214,16) ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 14

The experiments FMSAD Example ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 15

The experiments Times for siphon computation with the proposed method Size Population Time Evaluations FMSLD 3 8 0,01 (0,00) 107 (29,72) 3 16 0,01 (0,00) 208 (50,96) 3 32 0,03 (0,00) 405 (64,78) 4 8 0,01 (0,00) 118 (31,09) 4 16 0,02 (0,00) 228 (35,98) 4 32 0,04 (0,01) 462 (75,54) 5 16 0,03 (0,01) 272 (51,70) 5 32 0,05 (0,01) 491 (74,13) 6 16 0,04 (0,01) 270 (68,93) 6 32 0,07 (0,01) 535 (105,53) 7 16 0,05 (0,01) 323 (74,41) 7 32 0,10 (0,01) 637 (84,94) 8 16 0,07 (0,02) 336 (106,79) 8 32 0,13 (0,01) 661 (83,84) 8 64 0,23 (0,04) 1222 (208,84) ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 16

The experiments FMSLD Example ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 17

A philosopher forkR_i phil1Waiting_i philForkR_i philForkL_i forkL_i philEating_i T5_i T6_i T3_i T2_i T1_i ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 18

The experiments Times for siphon computation with the proposed method Size Population Time Evaluations Phil 3 16 0,02 (0,01) 200 (60,89) 3 32 0,05 (0,01) 445 (99,29) 3 64 0,08 (0,01) 834 (141,33) 4 32 0,06 (0,02) 391 (118,45) 4 64 0,13 (0,04) 887 (232,26) 5 64 0,16 (0,06) 790 (271,09) 5 128 0,38 (0,11) 1876 (528,81) 6 64 0,19 (0,06) 707 (205,14) 6 128 0,42 (0,15) 1559 (587,77) 7 64 0,23 (0,05) 661 (137,34) 7 128 0,44 (0,07) 1281 (167,60) 8 64 0,27 (0,01) 625 (0,00) 8 128 0,58 (0,14) 1325 (285,92) ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 19

The experiments Philosophers Example ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 20

Conclusions and future work We can compute siphons (but not all!) It is not clear how good the method is (but the problem size can increase...) Testing other variants of the GA Adapting methods that use siphons ICSTCC 2014. October. Sinaia, Romania Fernando Tricas, Jos´ e Manuel Colom, Juan Juli´ an Merelo 21