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Modular Processings based on Unfoldings Eric Fabre & Agnes Madalinski DistribCom Team Irisa/Inria UFO workshop - June 26, 2007

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Assembling Petri nets products, pullbacks, unfoldings and trellises Modular computations on a constraint graph : an abstract viewpoint Application 1: modular diagnosis or modular computation of a minimal product covering Application 2: modular prefixes or how to compute a FCP directly in factorized form Conclusion Outline

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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 Building bloc: a site or variable V = labeled automaton labeling of transitions V = (S,T,s 0,, ¸, ¤ ) ¸ : T ¤

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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 don’t change S = V 1 £ V 2 £ V 3 This product yields a safe (labeled) nets, and extends to safe nets

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Interest of Product Forms The 1 st interests are a natural construction method starting from modules the compactness of the product form on this example, the expanded product contains m*n transitions, instead of m+n in the factorized form

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Composition by Pullback Generalizes the product allows interactions of nets by an interface (sub-net) outside the interface, interactions are still by shared labels S = V 1 £ V 2 £ V 3 = (V 1 £ V 2 ) Æ (V 2 £ V 3 ) Main property

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Graph of a Product Net Interaction graph of a net shared labels define the local interactions… … but it’s better to re-express interactions under the form of shared variables (or sub-nets). S = V 1 £ … £ V n V 1 £ V 2 £ V 3 = (V 1 £ V 2 ) Æ (V 2 £ V 3 ) = S 1 Æ S 2 Translation in terms of pullbacks define components S i in order to “cover” the shared labels

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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) 8 O, 8 Á :OS, 9 ! Ã :O U (S), Á = f S ± Ã f S : U (S)S U (S) Consequences: functor has a left adjoint, and thus preserves products, pullbacks, … U U (S) = U (S 1 ) £ O … £ O U (S n )S = S 1 £ … £ S n ) U (S) = U (S 1 ) Æ O … Æ O U (S m )S = S 1 Æ … Æ S m )

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Unfoldings in Factorized Form (2) Example: U (S) = U (V 1 ) £ O U (V 2 ) £ O U (V 3 )S = V 1 £ V 2 £ V 3 )

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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£O ÆOÆO ¦ S i : U (S) U (S i )

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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£O ÆOÆO ¦ S i : U (S) U (S i )

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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£O ÆOÆO ¦ S i : U (S) U (S i )

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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£O ÆOÆO ¦ S i : U (S) U (S i )

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Important properties (2) Thm let O i be an occ. net of component S i, then is an occ. net of define then and this is the minimal product covering of O O=O 1 £ O … £ O O n S=S 1 £ … £ S n O’ i = ¦ S i (O) v O i O=O’ 1 £ O … £ O O’ n The reduced occurrence nets represent the behaviors of component S i that remain once S i is inserted in the global system S or the local view in each component S i of the behaviors of the global system S are interesting objects ! O’ i v O i Factorized forms of unfoldings are often more compact… …but they can however contain useless parts.

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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) time is counted independently in each V i for S = V 1 £ … £ V n

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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) = T (S 1 ) £ T … £ T T (S n )S = S 1 £ … £ S n ) T (S) = T (S 1 ) Æ T … Æ T T (S m )S = S 1 Æ … Æ S m ) T (S) U (S)

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Assembling Petri nets products, pullbacks, unfoldings and trellises Modular computations on a constraint graph : an abstract viewpoint Application 1: modular diagnosis or modular computation of a minimal product covering Application 2: modular prefixes or how to compute a FCP directly in factorized form Conclusion Outline

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S2S2 S3S3 S4S4 “Abstract” Constraint Reduction Ingredients : variables “systems” or “components” S i defined by (local) constraints on V max = {V 1,V 2,…} V i µ {V 1,…,V n } S1S1 V1V1 V5V5 V3V3 V2V2 V7V7 V6V6 V4V4 V8V8 S = S 1 Æ S 2 a composition operator (conjunction)

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“Abstract” Constraint Reduction (2) Reductions: for, reduces constraints of S to variables V reductions are projections V µ V max ¦ V (S) ¦ V 1 ± ¦ V 2 = ¦ V 1 Å V 2 Central axiom: S 1 operates on V 1, S 2 operates on V 2 let then i.e. all interactions go through shared variables V 3 ¶ V 1 Å V 2 ¦ V 3 (S 1 Æ S 2 ) = ¦ V 3 (S 1 ) Æ ¦ V 3 (S 2 )

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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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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Modular reduction algorithms Problem : Given where S i operates on V i compute the reduced components i.e. how does S i change once inserted into the global S ? S = S 1 Æ … Æ S n S’ i = ¦ V i (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. ¦ V i (S) v S’ i v S i

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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. ¦ V i (S) v S’ i v S i

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Assembling Petri nets products, pullbacks, unfoldings and trellises Modular computations on a constraint graph : an abstract viewpoint Application 1: modular diagnosis or modular computation of a minimal product covering Application 2: modular prefixes or how to compute a FCP directly in factorized form Conclusion Outline

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centralized supervizor Distributed system monitoring… ab c b a b caa distributed supervision

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Consider the net and 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 V i are the variables of S i Thm: projections and pullback satisfy the central axiom (here we cheat a little however…) We are already equipped for that ! ÆOÆO S = S 1 Æ … Æ S m U (S) = U (S 1 ) Æ O … Æ O U (S m ) ¦Vi¦Vi

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A computation example

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Assembling Petri nets products, pullbacks, unfoldings and trellises Modular computations on a constraint graph : an abstract viewpoint Application 1: modular diagnosis or modular computation of a minimal product covering Application 2: modular prefixes or how to compute a FCP directly in factorized form Conclusion Outline

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Objective Given compute a finite complete prefix of in factorized form Obvious solution: compute a FCP of then compute its minimal pullback covering where S = S 1 Æ … Æ S m U (S) U s (S) U (S) U s (S) v U ’(S 1 ) Æ … Æ U ’(S m ) U ’(S i ) = ¦ V i ( U s (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 don’t work Canonical prefix defined by a cutting context Θ = ( ~, ⊲, {κ e } eE ) ~ equivalence relation on Conf set of reachable markings ⊲ adequate order on Conf partial order on Conf refining inclusion {κ e } eE a subset of Conf, configurations used for cut-off identification cut-off event

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Extended canonical prefix Toy example : two components, elementary interface (=automaton) S = A £ C £ B = (A £ C) Æ (C £ B) = S A Æ S B interface

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Extended canonical prefix (2) extended prefix of w.r.t. its interface C restriction of the cutting context Θ C = (~, ⊲, {κ e } eE ) to particular configurations κ e e cut-off event, corresponding event e’ : κ e ~κ e’ and κ e’ ⊲ κ e where usually κ e =[e] if e is a private event, then C ( κ e ∆ κ e’ )=Ø if e is an interface event, then e’ is also an interface event SASA where ∆ is the symmetric set difference

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Extended cut-off event e : extended cut-off e’ : interface 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 merge

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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 build pullbacks

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Distributed computations construct prefixes

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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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Assembling Petri nets products, pullbacks, unfoldings and trellises Modular computations on a constraint graph : an abstract viewpoint Application 1: modular diagnosis or modular computation of a minimal product covering Application 2: modular prefixes or how to compute a FCP directly in factorized form Conclusion Outline

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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.

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