Discrete Parameters in Petri Nets SYNCOP2015 Based on a Paper - PowerPoint PPT Presentation

Discrete Parameters in Petri Nets SYNCOP2015 Based on a Paper accepted in PN2015 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux April 22, 2015 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 1 / 34

1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 2 / 34

1 Introducing Parameters Preliminaries On the Use of Parameters Parametric Properties 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 3 / 34

Why Introducing Parameters ? modeling arbitrary large amount of processes (markings) modeling unspecified aspect of the environement ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 4 / 34

Classic Model ? a marked Petri Net (PPN) p 1 p 4 t 1 t 2 3 2 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

Classic Model ? a marked Petri Net (PPN) p 1 p 4 t 1 t 2 3 2 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

Classic Model ? a marked Petri Net (PPN) p 1 p 4 t 1 t 2 3 2 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 5 / 34

Generalization toward Parametric marked Petri Net (PPN) p 1 p 3 λ 2 t 1 t 2 λ 1 λ 2 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

Generalization toward Parametric marked Petri Net (PPN) p 1 p 3 2 t 1 t 2 3 λ 3 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

Generalization toward Parametric marked Petri Net (PPN) p 1 λ p 3 2 t 1 t 2 3 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

Generalization toward Parametric marked Petri Net (PPN) p 1 λ p 3 λ 2 t 1 t 2 λ 1 λ 2 λ 3 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 6 / 34

Some concrete examples Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 7 / 34

Some concrete examples p 1 p 2 t λ Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 7 / 34

Some Concrete Examples Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 8 / 34

Some Concrete Examples p 1 p 2 p 1 p 2 t t λ 1 λ 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 8 / 34

Instantiation p 1 p 3 ν 1 ( λ ) = 2 t 1 t 2 ν 1 ( λ 1 ) = 3 ν 1 ( λ 2 ) = 1 3 2 ν 1 ( λ 3 ) = 2 p 3 p 2 p 1 λ p 3 λ 2 t 1 t 2 λ 1 λ 2 λ 3 p 1 p 3 p 2 p 3 ν 2 ( λ ) = 3 2 t 1 t 2 ν 2 ( λ 1 ) = 1 ν 2 ( λ 2 ) = 2 2 ν 2 ( λ 3 ) = 1 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

Instantiation p 1 p 3 ν 1 ( λ ) = 2 t 1 t 2 ν 1 ( λ 1 ) = 3 ν 1 ( λ 2 ) = 1 3 2 ν 1 ( λ 3 ) = 2 p 3 p 2 p 1 λ p 3 λ 2 t 1 t 2 λ 1 λ 2 λ 3 p 1 p 3 p 2 p 3 ν 2 ( λ ) = 3 2 t 1 t 2 ν 2 ( λ 1 ) = 1 ν 2 ( λ 2 ) = 2 2 ν 2 ( λ 3 ) = 1 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

Instantiation p 1 p 3 ν 1 ( λ ) = 2 t 1 t 2 ν 1 ( λ 1 ) = 3 ν 1 ( λ 2 ) = 1 3 2 ν 1 ( λ 3 ) = 2 p 3 p 2 p 1 λ p 3 λ 2 t 1 t 2 λ 1 λ 2 λ 3 p 1 p 3 p 2 p 3 ν 2 ( λ ) = 3 2 t 1 t 2 ν 2 ( λ 1 ) = 1 ν 2 ( λ 2 ) = 2 2 ν 2 ( λ 3 ) = 1 p 3 p 2 Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 9 / 34

Reminders... Definition (Reachability) Let S = ( N , m 0 ) = ( P , T , Pre , Post , m 0 ) and m a marking of S , S reaches m iff m ∈ RS ( S ) . Definition (Coverability) Let S = ( N , m 0 ) = ( P , T , Pre , Post , m 0 ) and m a marking of S , S covers m if there exists a reachable marking m ′ of S such that m ′ is greater or equal to m i.e. ∃ m ′ ∈ RS ( S ) s.t. ∀ p ∈ P , m ′ ( p ) ≥ m ( p ) (1) Decidability studied in [2] and [1] Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 10 / 34

Some Examples p 1 1 p 3 t 1 p 2 RS = { (2 , 1 , 0) , (1 , 0 , 1) } CS = { m | m ≤ (2 , 1 , 0) ∨ m ≤ (1 , 0 , 1) } Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 11 / 34

Parametric Properties Given a class of problem P (coverability, reachability,...), SP a PPN and φ is an instance of P Definition ( P -Existence problem) ( E - P ): Is there a valuation ν ∈ N Par s.t. ν ( SP ) satisfies φ ? Definition ( P -Universality problem) ( U - P ): Does ν ( SP ) satisfies φ for each ν ∈ N Par ? Definition ( P -Synthesis problem) ( S - P ): Give all the valuation ν , s.t. ν ( SP ) satisfies φ . Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 12 / 34

Existence ν 1 ( SP ) ν 1 ν 2 ( SP ) ν 2 classic models ν 3 parametric SP ν 3 ( SP ) model ... ν n ν n ( SP ) ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

Existence ν 1 ( SP ) ν 1 ν 2 ( SP ) ν 2 classic models ν 3 parametric SP ν 3 ( SP ) ν 3 ( SP ) � φ model ... ν n ν n ( SP ) ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

Existence ν 1 ( SP ) ν 1 ν 2 ( SP ) ν 2 classic models ν 3 parametric SP ν 3 ( SP ) ν 3 ( SP ) � φ model ... ν n SP � E φ ν n ( SP ) ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 13 / 34

Universality ν 1 ( SP ) ν 1 ν 2 ( SP ) ν 2 classic models ν 3 parametric SP ν 3 ( SP ) model ... ν n ν n ( SP ) ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

Universality ν 1 ( SP ) ν 1 ( SP ) � φ ν 1 ν 2 ( SP ) ν 2 ( SP ) � φ ν 2 classic models ν 3 parametric SP ν 3 ( SP ) ν 3 ( SP ) � φ model ... ν n ν n ( SP ) ν n ( SP ) � φ ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

Universality ν 1 ( SP ) ν 1 ( SP ) � φ ν 1 ν 2 ( SP ) ν 2 ( SP ) � φ ν 2 classic models ν 3 parametric SP ν 3 ( SP ) ν 3 ( SP ) � φ model ... ν n SP � U φ ν n ( SP ) ν n ( SP ) � φ ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 14 / 34

Mixing Properties and Parameters... ( U -cov) asks: ”Does each valuation of the parameters implies that the valuation of the PPN covers m ?” i.e. � ∀ ν ∈ N Par , ∃ m ′ ∈ RS ( ν ( SP )) m is U -coverable in SP ⇔ s.t. m ′ ≥ m (2) Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 15 / 34

1 Introducing Parameters 2 Undecidability of the General Case 3 Toward Decidable Subclasses 4 Conclusion Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 16 / 34

Results Theorem (Undecidability of E -cov on PPN) The E -coverability problem for PPN is undecidable. Theorem (Undecidability of U -cov on PPN) The U -coverability problem for PPN is undecidable. Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 17 / 34

2-Counters Machine two counters c 1 , c 2 , states P = { p 0 , ... p m } , a terminal state labelled halt finite list of instructions l 1 , ..., l s among the following list: increment a counter decrement a counter check if a counter equals zero Counters are assumed positive. Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 18 / 34

Example of 2-Counters Machine p 1 . C 0 := C 0 + 1; goto p 2 ; p 2 . C 1 := C 1 + 1; goto p 1 ; instructions sequence: ( p 1 , C 1 = 0 , C 2 = 0) → ( p 2 , C 1 = 1 , C 2 = 0) → ( p 1 , C 1 = 1 , C 2 = 1) → ( p 2 , C 1 = 2 , C 2 = 1) → ... Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 19 / 34

Undecidability halting problem (whether state halt is reachable) is undecidable counters boundedness problem (whether the counters values stay in a finite set) is undecidable proved by Minksy [3] Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 20 / 34

Why ? simulation of a counter machine E -cov can be reduced to halting problem U -cov can be reduced to counter boundedness Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 21 / 34

Simulation of Instructions: m ( C 1) + m ( ¬ C 1) = λ π π π θ θ θ λ λ λ λ C 1 λ ¬ 0 0 ¬ C 1 C 1 ¬ C 1 C 1 ¬ C 1 C 1 + + C 1 − − p i λ p j p i p j p i p j error p k incrementation decrementation zero test of of a counter of a counter a counter Nicolas David, Claude Jard, Didier Lime, Olivier H. Roux Discrete Parameters in Petri Nets 22 / 34