Concurrent Operational Semantics of Safe Time Petri Nets

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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_at_bretagne.ens-cachan.fr

with Thomas Chatain, currently postdoc in Aalborg
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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

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What we have done
Safe Time Petri Net
Its finite complete prefix

• Ingredients
• Mixed of graphical and symbolic constraints
• Duplication of events to capture different
  temporal constraints for a same event
• Introduction of reads arcs

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Time Petri Nets Merlin 1976

• Syntax ltP,T, pre, post, efd, lfd, M0gt
• Sequential semantics
• Global state (M,dob,?)
• (M,dob,?) -t-gt (M,dob,?) iff

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Firing Sequences
0
0

• (p1p2,00,0)

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Firing Sequences
0
1.3

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3)

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Firing Sequences
0
3
1.3

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3) -t1-gt
  (p3p4,31.3,3)

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Firing Sequences
3
3
3
1.3

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3) -t1-gt
  (p3p4,31.3,3)
• -t0-gt (p1p2,33,3)

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Firing Sequences
3
3
3
1.3

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3) -t1-gt
  (p3p4,31.3,3)
• -t0-gt (p1p2,33,3) -t1-gt (p2p3,33,3)

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Firing Sequences
3
3
3
5

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3) -t1-gt
  (p3p4,31.3,3)
• -t0-gt (p1p2,33,3) -t1-gt (p2p3,33,3)
• -t2-gt (p3p4,35,5)

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Firing Sequences
3
3
5
3
5

• (p1p2,00,0) -t2-gt (p1p4,01.3,1.3) -t1-gt
  (p3p4,31.3,3)
• -t0-gt (p1p2,33,3) -t1-gt (p2p3,33,3)
• -t2-gt (p3p4,35,5) -t3-gt (p4p5,55,5)

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Symbolic Firing Sequences

• (p1p2,00,0) -t2-gt (p1p4,0?1,?1) -t1-gt (p3p4,
  ?2?1,?2)
• -t0-gt (p1p2,?3?3,?3) -t1-gt (p2p3,?3?4,?4)
• -t2-gt (p3p4,?4?5,?5) -t3-gt (p4p5,?5?6,?6)
• where 1?12?0?2?max(?2,?1)?3max(?2,?1)?
• ?3?22??32?4??31?5?32??5?42?
• ?6?42??6max(?4,?5)

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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

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t2(1.3)
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t2(1.3) t1(3)
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t2(1.3) t1(3) t0(3)
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t2(1.3) t1(3) t0(3) t1(3)
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t2(1.3) t1(3) t0(3) t1(3) t2(3)
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t2(1.3) t1(3) t0(3) t1(3) t2(3) t3(5)
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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)
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Unfolding

• Defined as the superimposition (union) of all the
  time processes (share the common prefixes)

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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
• ?12 ? ?3?12 (first case)
• ?12 ? 1?22 ? ?2?12 ? ?3?12 ?
  ?3max(?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

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• Event duplication
• Introduction of read arcs
• (to preserve concurrency)
• Efficient concurrent
• semantics by considering
• local firing rules

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Local States

• Assume a partition (Pi) in mutually exclusive
  places (simplifies the test of token absence)
  (p?Pi, 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 ltL, dob, lrdgt (lrd stands for
  latest reading date)

_
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Local Enabling
To minimize the number of read arcs

• )

_
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Local Firing Rule

• (M, dob, lrd) -t,?,L-gt (M, dob, lrd)

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(Extended) Time Processes
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t2(1.3,p2)
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t2(1.3,p2) t1(3,p1)
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t2(1.3,p2) t1(3,p1) t0(3,p3p4)
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t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1)
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t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1) t2(3,p2)
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t2(1.3,p2) t1(3,p1) t0(3,p3p4) t1(3,p1) t2(3,p2)
t3(5,p3p4)
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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

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Process extraction
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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

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Substitution
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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)