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Symbolic Concurrent Semantics of Safe Petri nets. Application to Time Petri Nets. Claude Jard, ENS Cachan / IRISA, Rennes, France & Thomas Chatain, ENS Cachan / LSV, Cachan, France. Why are we interested in PN and unfoldings?.

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symbolic concurrent semantics of safe petri nets

Symbolic Concurrent Semantics of Safe Petri nets

Application to Time Petri Nets

Claude Jard, ENS Cachan / IRISA, Rennes, France

&

Thomas Chatain, ENS Cachan / LSV, Cachan, France

why are we interested in pn and unfoldings
Why are we interested in PN and unfoldings?
  • Supervision and diagnosis: inferring causal dependencies from observations in a distributed system (guided unfolding) -> already in use on Alcatel platforms
  • Composition of QoS contracts in WS orchestrations (need a partial order view of the behaviours) -> concurrent semantics for ORC

In such application domains, we do not need strong decidability results and thus consider extensions of PN with data, time, probas, …

focus of the talk
Focus of the talk
  • Generalize our symbolic approach about unfoldings of time Petri nets
  • Better understand time specificities in a concurrent setting
    • Safe colored PN with linear real constraints
    • Concurrent semantics for such nets
    • Translations of Time PN
background pns
Background: PNs
  • Places P = {a,b,c}, Transitions T = {u,v,w}

Consumed (pre(p,t)), read (cont(p,t)) or written (post(p,t)) by transitions

  • Marking: p M(p)  {0,1},
    • initially: M0(a)= M0(b)=1, M0(c)=0
  • t fireable iff ppre(t)cont(t), M(p)=1
  • Sequential move by firing t:
    • p, M(p):=M(p)-pre(p,t)+post(p,t)
why do we need read arcs
Why do we need read arcs?
  • To be able to test the presence of tokens without serialisation
concurrent semantics processes
Concurrent semantics: processes
  • v and w can be executed concurrently

Processes

(partially ordered

executions):

notion of conflict
Notion of conflict
  • fg = (f ≤ g)  (cont(f)  pre(g)  )
  • Conflict(F) =
    •  f,g  F, pre(f)  pre(g)  

or

    •  (fi)i[1,n]  F, fn=f1  i[1,n-1] fi fi+1
unfolding the puzzle game6
Unfolding: the puzzle game

Maximal co-sets of places

correspond to markings

-> notion of

finite complete prefix

-> bounded in space by the

size of the marking graph

(can be exponentially smaller)

-> but the time complexity

can be exponential

(size of the prefix to the power

of the degree of concurrency)

w

our safe colored pns
Our Safe Colored PNs
  • Places P: finite set of real variables
  • Transitions T: labeled (G(t)) with linear expressions over pre(t)+cont(t)+post(t)’
  • Initial expression: ζ0
concurrent semantics
Concurrent semantics
  • Set of events:
  • U={e=(e,e,Ce,Me)}
  • ⊥=(∅ζ0[x/x⊥]x∈M0, M0) ∈ U
  • pre(e)  cont(e)  f∈e Mf
  • Me=post(e)
  • Ce=G(e)[x/xe]x∈pre(e)cont(e) [x’/xe]x∈post(e)
  • e is conflict-free

f∈e Cf  Ce satisfiable

 e∈ U

unfolding process trace
Unfolding / Process / Trace
  • Unfolding is the union of processes
  • Processes are the conflict-free and downward-causally-closed subsets of the unfolding
  • Linear extensions of processes are the sequential traces
  • No hope to obtain in general a complete finite prefix
safe time pns
Safe Time PNs
  • Syntax:
  • TPN=(P,T,pre,post,efd,lfd,M0)
  • efd: T|R
  • lfd: T |R{}
  • Sequential semantics:
  • dob: P|R
  • (M,dob) -t,-> (M’,dob’) iff
  • - pre(t)M
  • - maxppre(t) dob(p) + efd(t) ≤ 
  • - t’T, pre(t’)M   ≤ maxppre(t’) dob(p) + lfd(t’)
  • - maxpPdob(p) ≤ 
  • M’=(M\pre(t))  post(t)
  • dob’(p)= if ppost(t), dob(p) otherwise
slide26

PE(u) = {bc}, PE(v) = {a,ab}, PE(w)={b,ab,bc}

Note:

Conflict(abw,abv)

Conflict(bcw,abv)

slide27

TPN to CPN : read arcs are added to take into account

the time dependencies

-> duplication of transitions

-> try to minimize the number of read arcs

short term perspectives
Short term perspectives
  • Experiments
  • Existence of finite complete prefixes ? OK
  • Coding of some TPN extensions ? Stopwatches, parametric PNs…
  • Study a similar approach for networks of Timed Automata. Experiment with different semantics for time.
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