fakultät für informatik informatik 12 technische universität dortmund Petri Nets Peter Marwedel TU Dortmund, Informatik /10/10

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1 2 Peri Nes Peer Marwedel TU Dormund, Informaik /0/0 Grahics: Alexandra Nole, Gesine Marwedel, 2003

2 Generalizaion of daa flow: Comuaional grahs Examle: Peri nes Inroduced in 962 by Carl Adam Peri in his PhD hesis. Focus on modeling causal deendencies; no global synchronizaion assumed (message assing only). Key elemens: Condiions Eiher me or no me. Evens May ake lace if cerain condiions are me. Flow relaion Relaes condiions and evens. Condiions, evens and he flow relaion form a biarie grah (grah wih wo kinds of nodes). 2,

3 Examle: Synchronizaion a single rack rail segmen Precondiions 2,

4 Playing he oken game 2, 2008 use normal view mode! - 4 -

5 Conflic for resource rack 2,

6 More comlex examle () Thalys rains beween Cologne, Amserdam, Brussels and Paris. [h:// 2,

7 More comlex examle (2) Slighly simlified: Synchronizaion a Brussels and Paris, using saions Gare du Nord and Gare de Lyon a Paris s 2, 2008 use normal view mode! - 7 -

8 Condiion/even nes Def.: N=(C,E,F) is called a ne, iff he following holds 2. C and E are disjoin ses 3. F (C E) (E C); is binary relaion, ( flow relaion ) Def.: Le N be a ne and le x (C E). x := {y y F x} is called he se of recondiions. x := {y x F y} is called he se of oscondiions. Examle: x x x 2,

9 Loos and ure nes Def.: Le (c,e) C E. (c,e) is called a loo iff cfe efc. Def.: Ne N=(C,E,F) is called ure, if F does no conain any loos. 2,

10 Simle nes Def.: A ne is called simle if no wo ransiions and 2 have he same ses of inu and ouu laces. Examle (no a simle ne): Def.: Simle nes wih no isolaed elemens meeing some addiional resricions are called condiion/even nes (C/E nes). 2,

11 Place/ransiion nes Def.: (P, T, F, K, W, M 0 ) is called a lace/ransiion ne iff 2. N=(P,T,F) is a ne wih laces P and ransiions T 3. K: P (N 0 {ω}) \{0} denoes he caaciy of laces (ω symbolizes infinie caaciy) 4. W: F (N 0 \{0}) denoes he weigh of grah edges 5. M 0 : P N 0 {ω} reresens he iniial marking of laces W (Segmen of some ne) M 0 defauls: K = ω W = 2,

12 Comuing changes of markings Firing ransiions generae new markings on each of he laces according o he following rules: 2,

13 Transiion is acivaed iff Acivaed ransiions Acivaed ransiions can ake lace or fire, bu don have o. We never alk abou ime in he conex of Peri nes. The order in which acivaed ransiions fire, is no fixed (i is non-deerminisic). 2,

14 Shorhand for changes of markings Slide 2: Le ( ) W (, )if \ + W (, )if \ W (, ) + W (, )if 0 = P: M () = M()+ () M = M+ +: vecor add 2,

15 Marix N describing all changes of markings W (, )if \ + W (, )if \ ( ) = W (, ) + W (, )if 0 Def.: Marix N of ne N is a maing N: P T Z (inegers) such ha T: N(,)=() Comonen in column and row indicaes he change of he marking of lace if ransiion akes lace. For ure nes, (N, M 0 ) is a comlee reresenaion of a ne. 2,

16 - 6-2, 2008 Examle: N = s

17 Place - invarians Sandardized echnique for roving roeries of sysem models For any ransiion j T we are looking for ses R P of laces for which he accumulaed marking is consan: Examle: R j ( ) = 0 2,

18 Characerisic Vecor R j ( ) = 0 Le: c R ( ) = if 0 if R R 0 = ( ) = j R P j ( ) c R ( ) = j c R Scalar roduc 2,

19 Condiion for lace invarians R j ( ) = ( ) c ( ) = c = j j R P R 0 Accumulaed marking consan for all ransiions if n... c c R R = = 0 Equivalen o N T c R = 0 where N T is he ransosed of N 2,

20 - 20-2, 2008 More deailed view of comuaions = ) (... ) ( ) ( ) ( )... (... ) ( )... ( ) ( )... ( n R R R n m m n n c c c Sysem of linear equaions. Soluion vecors mus consis of zeros and ones. Equaions wih mulile unknowns ha mus be inegers called diohanic ( Greek mahemaician Diohanos, ~300 B.C.). Diohanic linear equaion sysem more comlex o solve han sandard sysem of linear equaions (acually NP-hard)) Differen echniques for solving equaion sysem (manual,..)

21 Alicaion o Thalys examle N T c R = 0, wih N T = Soluions? Educaed guessing: rows =0 linear deendency among rows rank = 0- = 9 Dimension of soluion sace = 3 - rank = 4 4 comonens of (6,, 2, 3) of c R are indeenden se one of hese o and ohers o 0 o obain a basis for he soluion sace 2,

22 s basis Se one of comonens (6,, 2, 3) o, ohers o 0. s basis b : b (s 6 )=, b (s )=0, b (s 2 )=0, b (s 3 )=0 0 (s 0 ) b (s 0 ) + 0 (s ) b (s ) + 0 (s 3 ) b (s 3 ) = 0 b (s 0 ) = 0 9 (s 9 ) b (s 9 ) + 9 (s 0 ) b (s 0 ) = 0 b (s 9 )=0 b = (,,,,,,0,0,0,0,0,0,0) All comonens {0, } c R = b 2,

23 Inerreaion of he s invarian c R, = ( ) Characerisic vecor describes laces for Cologne rain. We roved ha: he number of rains along he ah remains consan. C R, s 2,

24 2 nd basis Se one of comonens (6,, 2, 3) o, ohers o 0. 2nd basis b 2 : b 2 (s 6 )=0, b 2 (s )=, b 2 (s 2 )=0, b 2 (s 3 )=0 0 (s 0 ) b 2 (s 0 ) + 0 (s ) b 2 (s ) + 0 (s 3 ) b 2 (s 3 ) = 0 b 2 (s 0 ) = 9 (s 9 ) b 2 (s 9 ) + 9 (s 0 ) b 2 (s 0 ) = 0 b 2 (s 9 )= b 2 = (0,-,-,-,0,0,0,0,,,,0,0) b = (,,,,,,0,0,0,0,0,0,0) 2, 2008 b 2 no a characerisic vecor, bu c R,2 =b +b 2 is c R,2 = (,0,0,0,,,0,0,,,,0,0-24 -

25 Inerreaion of he 2 nd invarian c R, 2 = (,0,0,0,,,0,0,,,,0,0) We roved ha: None of he Amserdam rains ges los (nice o know ). C R,2 s 2,

26 Seing b 3 (s 2 ) o and b 4 (s 3 ) o leads o an addiional 2 invarians c R, = c R, 2 = c R, = 3 ( ) ( ) ( ) C R,2 C R,3 C R, c R, 4 = ( ) s We roved ha: he number of rains serving Amserdam, Cologne and Paris remains consan. he number of rain drivers remains consan. C R,4 2,

27 Alicaions Modeling of resources; modeling of muual exclusion; modeling of synchronizaion. 2,

28 Predicae/ransiion nes Goal: comac reresenaion of comlex sysems. Key changes: Tokens are becoming individuals; Transiions enabled if funcions a incoming edges rue; Individuals generaed by firing ransiions defined hrough funcions Changes can be exlained by folding and unfolding C/E nes, semanics can be defined by C/E nes. 2,

29 Examle: Dining hilosohers roblem n> hilosohers siing a a round able; n forks, n laes wih saghei; hilosohers eiher hinking or eaing saghei (using lef and righ fork). 2 forks needed! How o model conflic for forks? How o guaranee avoiding sarvaion? 2,

30 Condiion/even ne model of he dining hilosohers roblem Le x {..3} x : x is hinking e x : x is eaing f x : fork x is available Model quie clumsy. Difficul o exend o more hilosohers. 2, 2008 Normal view mode!

31 Predicae/ransiion model of he dining hilosohers roblem () Le x be one of he hilosohers, le l(x) be he lef soon of x, le r(x) be he righ soon of x. 2 3 f f2 f3 Tokens: individuals. Semanics can be defined by relacing ne by equivalen condiion/even ne. 2,

32 Predicae/ransiion model of he dining hilosohers roblem (2) x 2 3 x v l(x) r(x) f f f2 f3 l(x) r(x) u x e x Model can be exended o arbirary numbers of eole. 2, 2008 use normal view mode!

33 Evaluaion Pros: Aroriae for disribued alicaions, Well-known heory for formally roving roeries, Iniially a quie bizarre oic, bu now acceed due o increasing number of disribued alicaions. Cons (for he nes resened) : roblems wih modeling iming, no rogramming elemens, no hierarchy. Exensions: Enormous amouns of effors on removing limiaions. 2, 2008 back o full screen mode

34 Summary Peri nes: focus on causal relaionshis Condiion/even nes Single oken er lace Place/ransiion nes Mulile okens er lace Predicae/ransiion nes Tokens become individuals Dining hilosohers used as an examle Exensions required o ge around limiaions 2,