Petri Nets. Peter Marwedel TU Dortmund, Informatik /05/13 These slides use Microsoft clip arts. Microsoft copyright restrictions apply.

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1 2 Peri Nes Peer Marwedel TU Dormund, Informaik 2 Grahics: Alexandra Nole, Gesine Marwedel, /05/3 These slides use Microsof cli ars. Microsof coyrigh resricions aly.

2 Models of comuaion considered in his course Communicaion/ local comuaions Undefined comonens Communicaing finie sae machines Daa flow Peri nes Shared memory SaeChars (No useful) Message assing Synchronous Asynchronous Plain ex, use cases (Message) sequence chars SDL Kahn neworks, SDF C/E nes, P/T nes, Discree even (DE) model Von Neumann model VHDL*, Verilog*, SysemC*, C, C++, Java Only exerimenal sysems, e.g. disribued DE in Polemy C, C++, Java wih libraries CSP, ADA * Classificaion is based on imlemenaion of VHDL, Verilog, SysemC wih cenral queue - 2 -

3 Inroducion 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)

4 Examle: Synchronizaion a single rack rail segmen Precondiions - 4 -

5 Playing he oken game use normal view mode! - 5 -

6 Conflic for resource rack - 6 -

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

8 s More comlex examle (2) Slighly simlified: Synchronizaion a Brussels and Paris, using saions Gare du Nord and Gare de Lyon a Paris use normal view mode! - 8 -

9 Condiion/even nes Def.: N=(C,E,F) is called a ne, iff he following holds. C and E are disjoin ses 2. F (C E) (E C); is binary relaion, ( flow relaion ) - 9 -

10 Pre- and os-ses Def.: Le N be a ne and le x (C E). x := {y y F x} is called he re-se of x, (or recondiions if x E) x := {y x F y} is called he se of os-se of x, (or oscondiions if x E) Examle: x x x - 0 -

11 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. - -

12 Simle nes Def.: A ne is called simle if no wo nodes n and n2 have he same re-se and os-se. Examle (no simle nes): Def.: Simle nes wih no isolaed elemens meeing some addiional resricions are called condiion/even nes (C/E nes)

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

14 Comuing changes of markings Firing ransiions generae new markings on each of he laces according o he following rules: - 4 -

15 Acivaed ransiions Transiion is acivaed iff 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)

16 Shorhand for changes of markings Slide 4: Le W (, ) if \ + W (, ) if \ ( ) W (, ) + W (, ) if 0 P: M () = M()+ () = M = M + +: vecor add - 6 -

17 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

18 - 8 - Examle: N = s

19 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 ( ) = j 0-9 -

20 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

21 Condiion for lace invarians R j = ( ) ( ) c ( ) = c = P j R j 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 -

22 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,..)

23 Alicaion o Thalys examle N T c R = 0, wih N T = rows =0 linear deendency among rows rank = 0- = 9 Dimension of soluion sace = 3 - rank = 4 Soluions? Educaed guessing

24 Manually finding generaing vecors for 4-dimensional sace Se of 4 comonens = ohers = 0 o obain generaing vecors Find indeenden ses comonens in he marix afer deleing one row (7, 8, 2 easy o idenify) s or assume ha a aricular lace is included in R (e.g. 6 ), whereas laces along he ah of oher objecs (e.g., 2, 3 ) are no. 6=delee s row

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

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

27 2 nd basis vecor Se one of comonens (6,, 2, 3) o, ohers o 0. 2nd basis b 2 : b 2 ( 6 )=0, b 2 ( )=, b 2 ( 2 )=0, b 2 ( 3 )=0 0 ( 0 ) b 2 ( 0 ) + 0 ( ) b 2 ( ) + 0 ( 3 ) b 2 ( 3 ) = 0 b 2 ( 0 ) = 9 ( 9 ) b 2 ( 9 ) + 9 ( 0 ) b 2 ( 0 ) = 0 b 2 ( 9 )= b 2 = (0,-,-,-,0,0,0,0,,,,0,0) b = (,,,,,,0,0,0,0,0,0,0) b 2 no a characerisic vecor, bu c R,2 =b +b 2 is c R,2 = (,0,0,0,,,0,0,,,,0,0)

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

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

30 Alicaions Modeling of resources; modeling of muual exclusion; modeling of synchronizaion

31 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

32 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 o model conflic for for forks? How o o guaranee avoiding sarvaion?

33 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 o exend o o more hilosohers. Normal view mode!

34 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 f 2 f 3 Tokens: individuals. Semanics can be be defined by by relacing ne by by equivalen condiion/even ne

35 Predicae/ransiion model of he dining hilosohers roblem (2) 2 3 f f 2 f 3 Model can be be exended o o arbirary numbers of ofeole. use normal view mode!

36 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. back o full screen mode

37 UML Aciviy diagrams: Exended Peri nes Include decisions (like in flow chars). Grahical noaion similar o SDL. Cris Kobryn: UML 200: A Sandardizaion Odyssey, CACM, Ocober, 999 swimlane

38 Summary Peri nes: focus on causal deendencies 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 Aciviy diagrams in UML are exended Peri nes