Modular Processings based on Unfoldings

1
Modular Processings based on Unfoldings

• Eric Fabre Agnes Madalinski
• DistribCom Team
• Irisa/Inria

UFO workshop - June 26, 2007
2
Outline

• Assembling Petri nets products, pullbacks,
  unfoldings and trellises
• Modular computationson a constraint graph an
  abstract viewpoint
• Application 1 modular diagnosisor modular
  computation of a minimal product covering
• Application 2 modular prefixesor how to compute
  a FCP directly in factorized form
• Conclusion

3
Nets as Products of Automata

• Caution in this talk, for simplicity
• we limit ourselves to safe Petri nets, although
  most results extend to ½ weighted nets
• we represent safe nets in complemented form,
  i.e. their number of tokens remains constant

4
Nets as Products of Automata (2)

• Composition of variables by product
• disjoint union of places
• transitions with shared labels are glued
• transitions with private labels dont change

S V1 V2 V3

• This product yields a safe (labeled) nets, and
  extends to safe nets

5
Interest of Product Forms

• The 1st interests are
• a natural construction method starting from
  modules
• the compactness of the product form
• on this example, the expanded product contains
  mn transitions, instead of mn in the factorized
  form

6
Composition by Pullback

• Generalizes the product
• allows interactions of nets by an interface
  (sub-net)
• outside the interface, interactions are still by
  shared labels

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Graph of a Product Net

• Interaction graph of a net
• shared labels define the local interactions
• but its better to re-express interactions
  under the form of shared variables (or sub-nets).

S V1 Vn
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Unfoldings in Factorized Form

• The key Universal Property of the unfolding of
  S
• Let denote the unfolding of S, and
  its associated folding
  (labeling)

U(S)
fSU(S)?S
8O, 8ÁO?S, 9! ÃO?U(S), Á fS Ã
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Unfoldings in Factorized Form (2)

• Example

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Important properties

• The category theory approach naturally provides
• an expression for operators (and )
• recursive procedures to compute them (as for
  unfoldings)
• notions of projections associated to
  products/pullbacks

O
ÆO
Si U(S) ? U(Si)
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Important properties

• The category theory approach naturally provides
• an expression for operators (and )
• recursive procedures to compute them (as for
  unfoldings)
• notions of projections associated to
  products/pullbacks

O
ÆO
Si U(S) ? U(Si)
12
Important properties

• The category theory approach naturally provides
• an expression for operators (and )
• recursive procedures to compute them (as for
  unfoldings)
• notions of projections associated to
  products/pullbacks

O
ÆO
Si U(S) ? U(Si)
13
Important properties

• The category theory approach naturally provides
• an expression for operators (and )
• recursive procedures to compute them (as for
  unfoldings)
• notions of projections associated to
  products/pullbacks

O
ÆO
Si U(S) ? U(Si)
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Important properties (2)

• Factorized forms of unfoldings are often more
  compactbut they can however contain useless
  parts.

• Thm
• let Oi be an occ. net of component Si,
• then is an occ. net
  of
• define then
• and this is the minimal product covering of O

OO1OO On
SS1Sn
Oi Si(O) v Oi
OO1OO On
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Trellises in Factorized Form

• The trellis of net S is
• obtained by merging conditions of with
  identical height
• a close cousin of merged processes (Khomenko et
  al., 2005)

T(S)
U(S)
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Trellises in Factorized Form

• The trellis of net S is
• obtained by merging conditions of with
  identical height
• a close cousin of merged processes (Khomenko et
  al., 2005)
• enjoys exactly the same factorization properties
  as unfoldings

T(S)
U(S)
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Outline

• Assembling Petri netsproducts, pullbacks,
  unfoldings and trellises
• Modular computationson a constraint graph an
  abstract viewpoint
• Application 1 modular diagnosisor modular
  computation of a minimal product covering
• Application 2 modular prefixesor how to compute
  a FCP directly in factorized form
• Conclusion

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Abstract Constraint Reduction

• Ingredients
• variables
• systems or components Si defined by
  (local) constraints on

Vmax V1,V2 ,
Vi µ V1,,Vn
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Abstract Constraint Reduction (2)

• Reductions
• for , reduces constraints
  of S to variables V
• reductions are projections

VµVmax
V(S)
V1 V2 V1ÅV2
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Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

21
Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

22
Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

23
Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

24
Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

25
Modular reduction algorithms

• Problem
• Given where Si operates
  on Vi
• compute the reduced components
• i.e. how does Si change once inserted into the
  global S ?

S S1Æ Æ Sn
Si Vi(S)

• This can be solved by Message Passing Algorithms
  (MPA)
• always converges
• only involves local computations
• exact if the graph of S is a (hyper-) tree

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What about systems with loops ?

• Message passing algorithms
• converge to a unique fix point (independent of
  message scheduling)
• that gives an upper approximation
• How good are their results ?
• Local extendibility to any tree around each
  component.

Vi(S) v Si v Si
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What about systems with loops ?

• Message passing algorithms
• converge to a unique fix point (independent of
  message scheduling)
• that gives an upper approximation
• How good are their results ?
• Local extendibility to any tree around each
  component.

Vi(S) v Si v Si
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Outline

• Assembling Petri netsproducts, pullbacks,
  unfoldings and trellises
• Modular computationson a constraint graph an
  abstract viewpoint
• Application 1 modular diagnosisor modular
  computation of a minimal product covering
• Application 2 modular prefixesor how to compute
  a FCP directly in factorized form
• Conclusion

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Distributed system monitoring
distributed supervision
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We are already equipped for that !

• Consider the netand move to trajectory sets
  (unfolding or trellis)
• In the category of occurrence nets (for ex.), we
  have
• a composition operator, the pullback
• trajectories of S are in factorized form
• we have projection operators on occ.
  nets,where Vi are the variables of Si
• Thm projections and pullback satisfy the central
  axiom (here we cheat a little however)

S S1 Æ Æ Sm
ÆO
U(S) U(S1) ÆO ÆO U(Sm)
Vi
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A computation example
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A computation example
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A computation example
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A computation example
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A computation example
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A computation example
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Outline

• Assembling Petri netsproducts, pullbacks,
  unfoldings and trellises
• Modular computationson a constraint graph an
  abstract viewpoint
• Application 1 modular diagnosisor modular
  computation of a minimal product covering
• Application 2 modular prefixesor how to compute
  a FCP directly in factorized form
• Conclusion

38
Objective

• Given compute a finite
  complete prefix of in factorized form
• Obvious solution
• compute a FCP of
• then compute its minimal pullback
  coveringwhere

S S1 Æ Æ Sm
U(S)
Us(S)
U(S)
Us(S) v U(S1) Æ Æ U(Sm)
U(Si) Vi(Us(S))

• but this imposes to work on the global
  unfolding we rather want to obtain directly
  the factorized form

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Local canonical prefixes dont work

• Canonical prefix
• defined by a cutting context T ( , ? ,
  ?ee?E )
• equivalence relation on Conf ? set of
  reachable markings
• ? adequate order on Conf ? partial order on
  Conf refining inclusion
• ?ee?E a subset of Conf, ? configurations
  used for cut-off identification

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Extended canonical prefix

• Toy example
• two components, elementary interface (automaton)

S ACB (AC) Æ (CB)
SA Æ SB
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Extended canonical prefix (2)

• extended prefix of w.r.t. its interface C
• restriction of the cutting context TC (,?,
  ?ee?E )to particular configurations ?e
• e cut-off event, corresponding event e
  ?e?e and ?e ? ?e where usually ?e e
• if e is a private event, then PC (?e ? ?e)Ø
• if e is an interface event, then e is also an
  interface event

SA
where ? is the symmetric set difference
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Extended cut-off event
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Summary net

• Summary net
• behaviors allowed by an extended prefix on the
  interface
• obtained by projecting the extended prefix on the
  interface,
• and refolding matching markings

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Distributed computations
augmented prefixes
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Distributed computations
extract summary nets
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Distributed computations
exchange summary nets
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Distributed computations
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Distributed computations
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Distributed computations
Killed in the pullback
Local factors are a little too conservative (not
the minimal pullback covering of the FCP)
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Outline

• Assembling Petri netsproducts, pullbacks,
  unfoldings and trellises
• Modular computationson a constraint graph an
  abstract viewpoint
• Application 1 modular diagnosisor modular
  computation of a minimal product covering
• Application 2 modular prefixesor how to compute
  a FCP directly in factorized form
• Conclusion

51

• A few lessons
• Factorized forms of unfoldings are generally more
  compact.
• One can work directly on them, in an efficient
  modular manner, without ever having to compute
  anything global.
• Optimal when component graphs are trees.
• Sub-optimal, but provide good upper
  approximations otherwise.

• and some questions
• Finite complete prefixes in factorized form
• we need to understand better how to compute them,
• and provide complexity results.
• Can this be useful for model checking?
• Can this be useful for distributed optimal
  planning?
• (see last talk today)

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Factorized forms are more compact
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Augmented branching process

• Standard projections lose information
• important causal links or conflicts may
  disappear.
• We must keep track of them in augmented BP,
• which makes the central axiom valid in all
  cases.