Symbolic concurrent semantics of safe petri nets
This presentation is the property of its rightful owner.
Sponsored Links
1 / 28

Symbolic Concurrent Semantics of Safe Petri nets PowerPoint PPT Presentation


  • 72 Views
  • Uploaded on
  • Presentation posted in: General

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?.

Download Presentation

Symbolic Concurrent Semantics of Safe Petri nets

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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


Sequential behaviours marking graph finite state automaton

Sequential behaviours: marking graph finite state automaton

Exhaustive

simulation


Sequential behaviours marking graph finite state automaton1

Sequential behaviours: marking graph finite state automaton

Exhaustive

simulation


Sequential behaviours marking graph finite state automaton2

Sequential behaviours: marking graph finite state automaton

Exhaustive

simulation


Sequential behaviours marking graph finite state automaton3

Sequential behaviours: marking graph finite state automaton

Exhaustive

simulation


Sequential behaviours marking graph finite state automaton4

Sequential behaviours: marking graph finite state automaton

Exhaustive

simulation


Concurrent semantics processes

Concurrent semantics: processes

  • v and w can be executed concurrently

Processes

(partially ordered

executions):


Unfolding union of all the processes

Unfolding: union of all the processes

Prefix (*≤1):


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 game

Unfolding: the puzzle game


Unfolding the puzzle game1

Unfolding: the puzzle game

v


Unfolding the puzzle game2

Unfolding: the puzzle game

w


Unfolding the puzzle game3

Unfolding: the puzzle game

u


Unfolding the puzzle game4

Unfolding: the puzzle game

u


Unfolding the puzzle game5

Unfolding: the puzzle game

v


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


Representation as a set of events event structure

Representation as a set of events: event structure

e=(e,e,Me)


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


Symbolic concurrent semantics of safe petri nets

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

Note:

Conflict(abw,abv)

Conflict(bcw,abv)


Symbolic concurrent semantics of safe petri nets

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.


  • Login