![Page 1: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/1.jpg)
Concurrent Operational Semantics of Safe Time
Petri Nets
Claude Jard European University of Brittany,
ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170 Bruz,
France [email protected]
cachan.fr
with Thomas Chatain, currently postdoc in Aalborg
![Page 2: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/2.jpg)
Time & Concurrent Semantics
• Seems to be a bit contradictory:– Time is a global notion– Concurrency is implied by independent actions, which are locally decided
• Is difficult: – even for the simple case of safe time Petri net, the definition of unfoldings and finite complete prefixes was left open
![Page 3: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/3.jpg)
What we have done…
Ingredients:– Mixed of graphical and symbolic constraints
– Duplication of events to capture different temporal constraints for a same event
– Introduction of reads arcs
Safe Time Petri NetIts finite completeprefix
![Page 4: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/4.jpg)
Time Petri Nets [Merlin 1976]
• Syntax: <P,T, pre, post, efd, lfd, M0>• Sequential semantics:
Global state (M,dob,)(M,dob,) -t-> (M’,dob’,’) iff
![Page 5: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/5.jpg)
Firing Sequences
• (p1p2,00,0)
0
0
![Page 6: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/6.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3)
0
1.3
![Page 7: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/7.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3)
0
1.3
3
![Page 8: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/8.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3)
-t0-> (p1p2,33,3)
3
1.3
3
3
![Page 9: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/9.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3)
-t0-> (p1p2,33,3) -t1-> (p2p3,33,3)
3
1.3
3
3
![Page 10: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/10.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3)
-t0-> (p1p2,33,3) -t1-> (p2p3,33,3)
-t2-> (p3p4,35,5)
3
5
3
3
![Page 11: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/11.jpg)
Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3)
-t0-> (p1p2,33,3) -t1-> (p2p3,33,3)
-t2-> (p3p4,35,5) -t3-> (p4p5,55,5)
3
5
3
3
5
![Page 12: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/12.jpg)
Symbolic Firing Sequences
• (p1p2,00,0) -t2-> (p1p4,01,1) -t1-> (p3p4, 21,2) -t0-> (p1p2,33,3) -t1-> (p2p3,34,4) -t2-> (p3p4,45,5) -t3-> (p4p5,56,6)
where [1≤1≤2][0≤2][max(2,1)≤3≤max(2,1)][3≤2+2][3+2≤4][3+1≤5≤3+2][5≤4+2] [6=4+2][6≤max(4,5)]
![Page 13: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/13.jpg)
Time processes
• First attempt [Aura, Lilius 1997]– Time processes are those of the underlying (untimed) Petri net that are consistently dated
– Can be defined inductively from the firing sequences
![Page 14: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/14.jpg)
![Page 15: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/15.jpg)
t2(1.3)
![Page 16: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/16.jpg)
t2(1.3) t1(3)
![Page 17: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/17.jpg)
t2(1.3) t1(3) t0(3)
![Page 18: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/18.jpg)
t2(1.3) t1(3) t0(3) t1(3)
![Page 19: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/19.jpg)
t2(1.3) t1(3) t0(3) t1(3) t2(3)
![Page 20: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/20.jpg)
t2(1.3) t1(3) t0(3) t1(3) t2(3) t3(5)
![Page 21: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/21.jpg)
Unfolding• Defined as the superimposition (union) of all the time processes (share the common prefixes)
t2(1.3) t1(3) t0(3) t1(3) t2(3) t3(5)
t1(1) t3(3)
![Page 22: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/22.jpg)
Unfolding• Defined as the superimposition (union) of all the time processes (share the common prefixes)
![Page 23: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/23.jpg)
Problem with the symbolic
representation
• Consider the processes built from the sequences t1(1).t3(3) and t1(1).t2(2).t3(3)
• Constraints on t3 are:– [1≤2] [3=1+2] (first case)– [1≤2] [1≤2≤2] [2≤1+2] [3=1+2] [3≤max(1,2)]
• The superimposition will provide a disjunction, which will make difficult the extraction of processes in the general case
• Except for the particular case of Time Extended Free Choice Nets
![Page 24: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/24.jpg)
• Event duplication• Introduction of read arcs (to preserve concurrency)• Efficient concurrent semantics by considering local firing rules
![Page 25: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/25.jpg)
Local States
• Assume a partition {(Pi)} in mutually exclusive places (simplifies the test of token absence) (pPi, p = Pi \ {p})
• {{p1,p3,p5},{p2,p4,p6}} in our example • A maximal (partial) marking is a set of places with
one place per each subset of the partition• A local state is <L, dob, lrd> (lrd stands for “latest
reading date”)
_
![Page 26: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/26.jpg)
Local Enabling
• ) _
To minimize the number of read arcs:
![Page 27: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/27.jpg)
Local Firing Rule
• (M, dob, lrd) -t,,L-> (M’, dob’, lrd’)
![Page 28: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/28.jpg)
(Extended) Time Processes
![Page 29: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/29.jpg)
t2(1.3,p2)
![Page 30: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/30.jpg)
t2(1.3,p2) t1(3,p1)
![Page 31: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/31.jpg)
t2(1.3,p2) t1(3,p1) t0(3,p3p4)
![Page 32: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/32.jpg)
t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1)
![Page 33: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/33.jpg)
t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1) t2(3,p2)
![Page 34: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/34.jpg)
t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1) t2(3,p2) t3(5,p3p4)
![Page 35: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/35.jpg)
Unfolding• Defined as the union of all the time processes
• Properties:– Unfolding of Time Extended Free Choice Nets is the standard unfolding
– Unfolding of Time Nets limited to [0,+[ is the untimed usual unfolding
– The unfolding of disconnected nets remains disconnected
![Page 36: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/36.jpg)
Process extraction
![Page 37: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/37.jpg)
Complete Finite Prefixes
• Two maximal states are considered equivalent if they have the same marking and the same reduced ages in the places
• The set of expressions defining the reduced ages in the maximal states of the processes is finite
• Prefixes are computed incrementally provided a substitution in the firing date expressions
![Page 38: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/38.jpg)
Substitution
![Page 39: Concurrent Operational Semantics of Safe Time Petri Nets Claude Jard European University of Brittany, ENS Cachan Bretagne, IRISA Campus de Ker-Lann, 35170](https://reader036.vdocuments.net/reader036/viewer/2022062712/56649c765503460f9492a3a1/html5/thumbnails/39.jpg)
Conclusion
• An original notion of unfolding of Time PN and its finite representation
• Tries to code the maximum of concurrency in the graphical part
• Complexity issues are to be worked
• Other models? (non-safe, networks of Timed Automata, local semantics of time…)