Embedded Systems 4. Petri nets. Introduced in 1962 by Carl Adam Petri in his PhD thesis. Different Types of Petri nets known

Transcription

1 Embedded Sysems Peri nes Inroduced in 962 by Carl Adam Peri in his PhD hesis. Differen Tyes of Peri nes known Condiion/even nes Place/ransiion nes Predicae/ransiion nes Hierachical Peri nes, - 2 -

2 Used for Modelling, Analysis, Verificaion of Disribued Sysems (oher) alicaion areas: auomaion engineering business rocesses Focus on modeling causal deendencies; no global synchronizaion assumed (message assing only) Examle : The four seasons - 4-2

3 Key Elemens Condiions Eiher me or no me. Condiions reresen local saes. Se of condiions describes he oenial sae sace. Evens May ake lace if cerain condiions are me. Even reresens a sae ransiion. Flow relaion Relaes condiions and evens, describes how an even changes he local and global sae. Tokens Assignmens of okens o condiions secifies a global sae. Condiions, evens and he flow relaion form a biarie grah (grah wih wo kinds of nodes) Examle 2: Synchronizaion a single rack rail segmen muual exclusion: here is a mos one rain using he rack rail Precondiions of x fulfilled x - 6-3

4 Playing he oken game : dynamic behavior x Playing he oken game : dynamic behavior x Poscondiion of x fulfilled - 8-4

5 Playing he oken game : dynamic behavior Conflic for resource rack : wo rains comeing - 0-5

6 Peri Nes Def.: N=(C,E,F) is called a Peri ne, iff he following holds. C and E are disjoin ses 2. 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 - - Boolean marking and comuing changes of markings A Boolean marking is a maing M: C {0,}. Firing evens x generae new markings on each of he condiions c according o he following rules: a ransiion a x can be fired, iff x, i.e. all recondiions of x are marked and x is no marked, afer firing x is unmarked and x is marked M M, iff M resuls from M by firing exacly one ransiion - 2-6

7 Comeing Trains Examle: Comeing Trains Examle: Conflic for resource rack - 4-7

8 Comeing Trains Examle: Boolean marking and comuing changes of markings Comeing Trains examle Consider he maing sym: C C wih sym(c)=-c for all c=0,,2,3,-,-2,-3 We call wo markings M, Ms symmeric, iff M can be ransformed o Ms by changing he marks from a node c o node -c, i.e Ms(sym(c)):= M(c). I follows easily: M M2 iff Ms M2s Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M M2 M3 Sli. M4 M5 M6 Sli 4. Sli

9 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M6 M Sli Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M6 M7 M2s Sli

10 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M5 M6s Sli Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M4 M8 M6 Sli

11 Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M M2s Sli Comeing Trains Examle: Boolean marking and comuing changes of markings Reachable markings M8 M M2 M3 M4 M5 M6 M7 M2s M3s M4s M5s M6s M7s M8s

12 Basic srucural roeries: 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 Srucural roeries: Simle nes Def.: A ne is called simle, iff [x,y (C E) ( x = y ) (x = y )] x = y Examle (no a simle ne):

13 Condiion/even nes (C/E nes) Def.: A Peri ne N=(C,E,F) ogeher wih a se of Markings M is called condiion/even ne (C/E ne), iff N is simle and has no isolaed elemens M is closed w.r.. firing and inverse firing wo markings in M can be ransformed ino each oher by firing and inverse firing for each even e E, here exiss a marking in M, ha allows firing a e Comeing Trains Examle is C/E ne: M8 M M2 M3 M4 M5 M6 M7 M2s M3s M4s M5s M6s M7s M8s

14 Proeries of C/E Def.: Marking M is reachable from marking M, iff here exiss sequence of firing ses ransforming M ino M (No.: M M ) A C/E ne is cyclic, iff any wo markings are reachable from each oher. A C/E ne fulfills liveness, iff for each marking M and for each even e here exiss a reachable marking M ha acivaes e for firing Exressiveness: basic examles concurrency of ransiions alernaive or conflic synchronizaion

15 Examle Thalys rains: more comlex Thalys rains beween Cologne, Amserdam, Brussels and Paris. Synchronizaion a a Brussels and Paris s Realisic scenarios need more general definiions More han one oken er condiion, caaciies of laces weighs of edges sae sace of Peri nes may become infinie! k= ready roduce 2 idle k= 2 send Sorage k=5 acce 3 k=2 acceed 4 4 consume 5 ready k=2 Producer Consumers

16 Place/ransiion nes Def.: (P, T, F, K, W, M 0 ) is called a lace/ransiion ne (P/T 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 = Comuing changes of markings Firing ransiions generae new markings on each of he laces according o he following rules:

17 Acivaed ransiions Transiion is acivaed iff Acivaed ransiions can ake lace or or fire, bu don have o. o. The order in in which acivaed ransiions fire is is no fixed (i (i is is non-deerminisic) Shorhand for changes of markings Firing ransiion: Le ( ) W (, )if \ + W (, )if \ = W (, ) + W (, )if 0 P: M () = M()+ () M = M+ +: vecor add

18 Marix N describing all changes of markings 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. + + = 0 )if, ( ), ( \ )if, ( \ )if, ( ) ( W W W W Examle: N = s

19 Comuaion of Invarians We are ineresed in subses R of laces whose number of labels remain invarian under ransiions, e.g. he number of rains commuing beween Amserdam and Paris (Cologne and Paris) remains consan Imoran for correcness roofs, e.g. he roof of liveness Comeing Trains Examle: Place Invarian 2,0,