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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 Claude.Jard@bretagne.ens-cachan.fr. with Thomas Chatain, currently postdoc in Aalborg.
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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 Claude.Jard@bretagne.ens-cachan.fr with Thomas Chatain, currently postdoc in Aalborg
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
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 Net Its finite complete prefix
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
Firing Sequences 0 • (p1p2,00,0) 0
Firing Sequences 0 • (p1p2,00,0) -t2-> (p1p4,01.3,1.3) 1.3
Firing Sequences 0 • (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3) 3 1.3
Firing Sequences 3 • (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3) -t0-> (p1p2,33,3) 3 3 1.3
Firing Sequences 3 • (p1p2,00,0) -t2-> (p1p4,01.3,1.3) -t1-> (p3p4,31.3,3) -t0-> (p1p2,33,3) -t1-> (p2p3,33,3) 3 3 1.3
Firing Sequences 3 • (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 3 5
Firing Sequences 3 • (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 5
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)]
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
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)
Unfolding • Defined as the superimposition (union) of all the time processes (share the common prefixes)
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
Event duplication • Introduction of read arcs • (to preserve concurrency) • Efficient concurrent • semantics by considering • local firing rules
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”) _
Local Enabling • ) To minimize the number of read arcs: _
Local Firing Rule • (M, dob, lrd) -t,,L-> (M’, dob’, lrd’)
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
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
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…)