INVARIANT Priese/Wimmel 2008 ANALYSIS Lautenbachs miracle - - PowerPoint PPT Presentation

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 I NDEX ❑ Starke 90, p.121 exponential number of invariants ❑ Starke 90, p. 111 CPI & CTI, but not live INVARIANT ❑ Priese/Wimmel 2008 ANALYSIS ❑ Lautenbach‘s miracle - ❑ EXAMPLES pathway analysis T-invariants - elementary modes - extreme pathways ❑ carbon oxidation P/T-invariants and their interpretation Z:\Documents\teaching\nl\nl_skript_fm\nl06a_mutexAnalysis.sld.fm 7 - 1 / 28 monika.heiner@b-tu.de 7 - 2 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 [ STARKE 121] [ STARKE , P . 111] p2a p3a pka p1a p1 p2 tk t1 t1 t2 t3 (2) pkb p1b p2b p3b (2) t3 t2 t4 (2) k = 4, 2 4 = 16 minimal P-invariants: ❑ p3 (p1a, p2a, p3a, p4a), (p1b, p2a, p3a, p4a), (p1a, p2a, p3a, p4b), (p1b, p2a, p3a, p4b), (p1a, p2a, p3b, p4a), (p1b, p2a, p3b, p4a), (p1a, p2a, p3b, p4b), (p1b, p2a, p3b, p4b), t-invariant= {t1, t2, t3, t3} (p1a, p2b, p3a, p4a), (p1b, p2b, p3a, p4a), p1 t1 p2 (p1a, p2b, p3a, p4b), (p1b, p2b, p3a, p4b), ? p2 (p1a, p2b, p3b, p4a), (p1b, p2b, p3b, p4a), (p1a, p2b, p3b, p4b), (p1b, p2b, p3b, p4b) generally 2 k P-invariants -> p1 p3 t3 p1 t1 p2 t2 ❑ p2 p3 t3 p1 analogously for T-invariants INA: ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES N N N N Y N Y Y N N N N N N N N N DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S ? Y Y Y Y N N ? N N N N N monika.heiner@b-tu.de 7 - 3 / 28 monika.heiner@b-tu.de 7 - 4 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 [ STARKE , P . 111], [ PRIESE 2003, P . 80] INVARIANTS ❑ ❑ incidence matrix the system p1 t1 p4 t4 -1 -1 1 0 2 1 -1 0 0 C = 2 -1 0 0 p2 t2 p5 -> side conditions, here p2 for t4, are not reflected in C 2 ❑ CPI -> the only P-invariant (p1, p2, p3) covers the net p3 t3 ❑ CTI ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES -> T-inv1: (1, 1, 2, 0) -> (t1, t2, 2 t3) -> {t1, t2, t3, t3} N Y N Y N Y Y N N Y N N N N Y Y Y DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S -> T-inv2: (t4) ? N Y N N ? ? ? N Y ? ? N ❑ but not live ❑ its basic steps -> t4 - the only live transition ❑ state equation, counter example p1 p4 p1 p3 t1 -> m0 = (1, 0, 0), m1 = (0, 0, 1) p5 p2 m1 = m0 + Cx t2 p3 Cx = m1 - m0 p4 p2 p5 Cx = (-1, 0, 1) -> x = (t1, t2, t3) -> BUT, no permutation of t1, t2, t3 can be fired p3 p5 t3 p4 t4 (t2 needs two tokens) monika.heiner@b-tu.de 7 - 5 / 28 monika.heiner@b-tu.de 7 - 6 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 [ PRIESE 2003, P . 80], [ PRIESE 2003, P . 80], T - INVARIANTS T - INVARIANTS ❑ ❑ The net is covered by one minimal T-invariants. possible runs, short notation t-inv = (1, 1, 2, 2) = {t1, t2, t3, t3, t4, t4} t4 t4 ❑ two possible runs t1 t3 t1 t3 t2 t4 t2 t4 t3 t3 p1 p1 p4 t4 t1 ❑ p1 required terminus ?! p3 p5 t3 -> “maximally unordered” p2 t2 p2 p5 t3 p3 p4 t4 p5 p1 p1 p4 t4 t1 p5 p3 p2 p5 t3 p1 p5 t3 p3 p2 t2 p4 t4 monika.heiner@b-tu.de 7 - 7 / 28 monika.heiner@b-tu.de 7 - 8 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 [ PRIESE 2003, P . 80], LAUTENBACH ’ S P - INVARIANTS MIRACLE ❑ The net is NOT covered by semipositive P-invariants. ❑ the system Covered places: p1, p2, p3, p5, -> semipositive place invariants = 1 | p2 : 2, t1 t3 t5 p1 t1 p4 t4 | p3 : 1, p1 (2) 2 2 2 3 ta tb | p5 : 1 2 p2 p3 p4 t2 t4 t6 2 p2 t2 p5 (2) ORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 pF0 MG SM FC EFC ES N N N Y N N Y N Y Y N N N N Y Y Y p3 t3 DTP CPI CTI B SB REV DSt BSt DTr DCF L LV L&S ? N Y N N ? N ? N ? Y Y N 2 | p1 : 2, 2 p1 t1 p4 t4 | p3 : 1, (2) ❑ its basic steps | p5 : 1 p2 p2 p2 p2 t2 p5 p1 p3 p3 p2 t1 t3 t5 (2) p4 p2 p2 p1 p4 p3 t3 ta tb p1 p4 p1 ❑ Karp-Milller graph p2 p3 p4 p2 p3 -> 18 nodes t2 t4 t6 p1 -> capacities needed: p1 p1 p2 p3 p4 p5 2 1 3 oo 3 monika.heiner@b-tu.de 7 - 9 / 28 monika.heiner@b-tu.de 7 - 10 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 LAUTENBACH ’ S LAUTENBACH ’ S MIRACLE MIRACLE ❑ two minimal T-invariants ❑ -> not realizable under the empty marking a non-minimal T-invariant -> not reproducing the empty marking -> covering the net -> reproducing the empty marking ❑ unique runs t-inv1 = {ta, t2, t2, t4, t4, t6, t6, tb} ❑ T-inv3 = {ta, t1, t2, t3, t4, t5, t6, tb} -> T-inv3 = (T-inv1 + T-inv2) / 2 p1 p3 t6 -> non-negative linear combination of minimal ones p2 p4 tb t2 t4 ta p1 p1 p3 p4 t6 ❑ two possible runs t2 t4 p1 p1 p2 p2 p3 p4 trapped token t4 t5 p3 p2 p1 t6 tb t1 ta t2 t4 tb p1 t2 t4 t6 p2 ta t3 t6 p3 p4 p1 t2 p2 p2 t-inv2 = {ta, t1, t1, t3, t3, t5, t5, tb} p1 p1 p3 p4 p2 t1 t3 t5 p3 p1 p4 ta tb p1 p2 p2 p4 t1 t3 t5 p2 p2 p3 tb ta t2 t4 p2 p2 t5 p2 p3 p1 p2 trapped token t1 t3 t6 p3 p1 p4 t1 p2 ta t3 t5 t5 t3 tb t1 monika.heiner@b-tu.de 7 - 11 / 28 monika.heiner@b-tu.de 7 - 12 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 LAUTENBACH ’ S PATHWAY ANALYSIS MIRACLE ❑ substances involved ❑ short notation of the two possible runs -> input substance A -> output substance C t5 ta t1 t4 t3 t6 tb -> auxiliary substance B t2 gA t5 ta t2 t4 t3 t6 tb t1 A ❑ comparison ? ab ac ❑ guess gB rC -> no chance for uniqueness of rB B bc C non-minimal T-invariants’runs ❑ steady state substance flows -> T-invariants ❑ all flow behaviour under the steady state assumption -> non-negative linear combination of minimal T-invariants monika.heiner@b-tu.de 7 - 13 / 28 monika.heiner@b-tu.de 7 - 14 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 T - INVARIANTS AND CARBON OXIDATION , EXTREME PATHWAYS BASIC REACTIONS inv3 B inv2 B C gB r1) 2 C + O 2 --> 2 CO C O 2 gB bc rC rB 2 r1 gA gA gA inv1 inv5 inv4 2 carbon monoxide CO A A A ab ab ac O2 oxygen r2) C + O 2 --> CO 2 C O 2 ab rb rB rC B B C C r2 gA inv4 = inv2 + inv5 - inv3 carbon dioxide CO 2 A r3) C + CO 2 <--> 2 CO ab C CO 2 CO rB B + 2 bc rC gB B C r3a r3b - gB 2 rB B C CO 2 CO monika.heiner@b-tu.de 7 - 15 / 28 monika.heiner@b-tu.de 7 - 16 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 CARBON OXIDATION , CARBON / BND , INCIDENCE MATRIX COMPOSITION BASIC MODEL O 2 C start init 2 3 2 r1 r3a r3b r2 O 2 C 2 2 2 2 CO CO 2 r3a r2 r1 r3b 2 2 2 SYSTEM ’ S TOTAL EQUATION CO CO 2 1) 2 C + O 2 -> 2 CO 2 2) C + O 2 -> CO 2 repeat 3) C + CO 2 <-> 2 CO 3 C + 2 O 2 -> 2 CO + CO 2 T r1 r2 r3a r3b start repeat P O 2 -1 -1 0 0 +2 0 MODEL OF THE SYSTEM ’ S TOTAL EQUATION C -2 -1 -1 +1 +3 0 C CO +2 0 2 -2 0 -2 CO 2 3 CO 2 0 +1 -1 +1 0 -1 2 2 sum O 2 CO init 0 0 0 0 -1 +1 monika.heiner@b-tu.de 7 - 17 / 28 monika.heiner@b-tu.de 7 - 18 / 28

dependability engineering & Petri nets November 2013 dependability engineering & Petri nets November 2013 CARBON / BND , P - INVARIANTS CARBON / BND , T - INVARIANTS 1, 2 T-inv1 = (r3a, r3b) -> inner cycle P-inv1 = (3 init, C, CO, CO 2 ) -> carbon preservation s1 start init 3 X start 2 3 start init 2 3 C O 2 s2 C O 2 2 2 r1 r3a r3b r2 r1 r2 r1 r3a r3b r2 2 r3b 2 s3 s4 2 2 r3a 2 CO CO 2 2 r2 r1 r2 CO CO 2 2 r3b repeat 2 repeat s6 s5 r3a repeat T-inv2 = (start, 2 r1, r3b, repeat) -> input/output cycle P-inv2 = (4 init, 2 O 2 , CO, 2 CO 2 ) -> oxygen preservation s1 start 4x start init init 2 3 start 3 2 C O 2 2x C O 2 s2 2 2 r1 r3a r3b r2 r1 r2 2x r2 r1 r3a r3b r3b 2 2 2 s4 s3 2 2 r3a 2 CO CO 2 2x CO CO 2 r2 r1 r2 2 r3b repeat 2 repeat s5 s6 repeat r3a monika.heiner@b-tu.de 7 - 19 / 28 monika.heiner@b-tu.de 7 - 20 / 28