On Minimality and Equivalence of Petri Nets Annegret K. Wagler, - - PowerPoint PPT Presentation

On Minimality and Equivalence of Petri Nets

Annegret K. Wagler, Jan-Thierry Wegener

Laboratoire d’Informatique, de Mod´ elisation et d’Optimisation des Syst` emes (LIMOS) UMR 6158 CNRS Universit´ e Blaise Pascal Clermont-Ferrand, France

MoVeP 2012 / CIRM, Marseille December 06, 2012

• J. Wegener

December 06, 2012 1 / 15

Outline

1

Petri nets

2

Classification of X ′-Deterministic Extended Petri Nets

3

Conclusion

• J. Wegener

December 06, 2012 2 / 15

Outline

1

Petri nets

2

Classification of X ′-Deterministic Extended Petri Nets

3

Conclusion

• J. Wegener

December 06, 2012 3 / 15

Petri nets: The networks

The networks

are graphs G = (P, T, A, w) with P set of involved components (“places” ), T set of involved reactions (“transitions” ), interconnected by directed links in A (“arcs” →). Each place p ∈ P can be marked with an integral number xp of tokens. A state can be represented as a vector x ∈ N|P| with entries xp for all p ∈ P. A transition t ∈ T is enabled at a state x if there are enough tokens available on the pre-places of t. Switching t transfers x into a new system state, denoted by x → x′. x =   3 1  

H2 O2 H2O 2 2

⇒

H2 O2 H2O 2 2

  1 2   = x′

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December 06, 2012 4 / 15

Deterministic extended Petri nets

Deterministic extended Petri nets

A triple (P, cap, O) extended Petri net P = (P, T, AS ∪ AC, w) capacity on places cap : P → N set of priorities O (e.g., partial order)

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December 06, 2012 5 / 15

Deterministic extended Petri nets

Deterministic extended Petri nets

A triple (P, cap, O) extended Petri net P = (P, T, AS ∪ AC, w) capacity on places cap : P → N set of priorities O (e.g., partial order)

t2 t3 t1 t4 t5

O = {t1 < t3, t2 < t3, t5 < t3}

• J. Wegener

December 06, 2012 5 / 15

Deterministic extended Petri nets

Deterministic extended Petri nets

A triple (P, cap, O) extended Petri net P = (P, T, AS ∪ AC, w) capacity on places cap : P → N set of priorities O (e.g., partial order)

(2) (2) t2 t3 t1 t4 t5

O = {t1 < t3, t2 < t3, t5 < t3}

• J. Wegener

December 06, 2012 5 / 15

Deterministic extended Petri nets

Deterministic extended Petri nets

A triple (P, cap, O) extended Petri net P = (P, T, AS ∪ AC, w) capacity on places cap : P → N set of priorities O (e.g., partial order)

(2) (2) t2 t3 t1 t4 t5

O = {t1 < t3, t2 < t3, t5 < t3}

• J. Wegener

December 06, 2012 5 / 15

Network reconstruction algorithm

Input:

a set P of components (considered to be crucial for the studied phenomenon) a capacity cap for each component experimental time-series data X ′ = (x1, . . . , xm) obtained by stimulating a biological system and observing how its states change over time

Output:

all deterministic extended Petri nets explaining X ′ (so called X ′-deterministic extended Petri nets): all generated nets have the same set of places P and capacity cap there are enough transitions to simulate all observed state changes AC and O are so that the experiments can exactly be reproduced

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December 06, 2012 6 / 15

Output: Minimal solution sets

To keep the solution set small while guaranteeing its completeness, return only “minimal” X ′-deterministic extended Petri nets.

How to obtain “minimal” solutions?

Easy in standard Petri nets: remove unnecessary transitions. Difficult in deterministic extended Petri nets: we can also remove unnecessary priorities and/or control-arcs, replace priorities by control-arcs (or vice versa).

Idea:

consider equivalence classes of deterministic extended Petri nets having the same places and transitions, select representatives being minimal w.r.t. priorities and control-arcs.

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December 06, 2012 7 / 15

Output: Minimal solution sets

To keep the solution set small while guaranteeing its completeness, return only “minimal” X ′-deterministic extended Petri nets.

How to obtain “minimal” solutions?

Easy in standard Petri nets: remove unnecessary transitions. Difficult in deterministic extended Petri nets: we can also remove unnecessary priorities and/or control-arcs, replace priorities by control-arcs (or vice versa).

Idea:

consider equivalence classes of deterministic extended Petri nets having the same places and transitions, select representatives being minimal w.r.t. priorities and control-arcs.

• J. Wegener

December 06, 2012 7 / 15

Outline

1

Petri nets

2

Classification of X ′-Deterministic Extended Petri Nets

3

Conclusion

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Equivalence and Inclusion

X ′-equivalence

Two X ′-deterministic extended Petri nets (P, cap, O), ( ˆ P, cap, ˆ O) are X ′-equivalent if they only differ in the set of priorities and/or control-arcs. Consider an X ′-deterministic extended Petri nets (P, cap, O). A sequence of transitions t1, . . . , tk is an O-feasible switching sequence for x1 in P if for all j xj+1 is the successor state of xj switching tj, there does not exist a t ∈ T enabled in xj with (tj < t) ∈ O.

Inclusion relation

Consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O). We say P is included in ˆ P, denoted by P ⊆ ˆ P, if and only if for all states x ∈ X, every ˆ O-feasible switching sequence for x in ˆ P is an O-feasible switching sequence for x in P.

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December 06, 2012 9 / 15

Inclusion relation: Example

A B C D D C B A t1 t2 t2 t1 P = ˆ P = O = {t1 < t2} ˆ O = ∅ ⇒ P ⊂ ˆ P

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December 06, 2012 10 / 15

Inclusion relation: Example

A B C D D C B A t1 t2 t2 t1 P = ˆ P = O = {t1 < t2} ˆ O = ∅ ⇒ P ⊂ ˆ P

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December 06, 2012 10 / 15

Classification of X ′-Deterministic Extended Petri Nets

Minimal X ′-deterministic extended Petri net

Among all X ′-equivalent extended Petri nets, (P, cap, O) is minimal if and only if (P, cap, O) does neither have unnecessary elements nor another X ′-deterministic extended Petri net ( ˆ P, cap, ˆ O) being X ′-equivalent to (P, cap, O) is included in P. In order to classify X ′-equivalent extended Petri nets for inclusion we consider (P, cap, O) and ( ˆ P, cap, ˆ O) and distinguish the following four cases:

1

O ⊂ ˆ O and AC = ˆ AC,

2

O = ˆ O and AC ⊂ ˆ AC,

3

O ⊂ ˆ O and AC ⊂ ˆ AC,

4

ˆ O ⊂ O and AC ⊂ ˆ AC.

• J. Wegener

December 06, 2012 11 / 15

Classification of X ′-Deterministic Extended Petri Nets

Minimal X ′-deterministic extended Petri net

Among all X ′-equivalent extended Petri nets, (P, cap, O) is minimal if and only if (P, cap, O) does neither have unnecessary elements nor another X ′-deterministic extended Petri net ( ˆ P, cap, ˆ O) being X ′-equivalent to (P, cap, O) is included in P. In order to classify X ′-equivalent extended Petri nets for inclusion we consider (P, cap, O) and ( ˆ P, cap, ˆ O) and distinguish the following four cases:

1

O ⊂ ˆ O and AC = ˆ AC,

2

O = ˆ O and AC ⊂ ˆ AC,

3

O ⊂ ˆ O and AC ⊂ ˆ AC,

4

ˆ O ⊂ O and AC ⊂ ˆ AC.

• J. Wegener

December 06, 2012 11 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC AC = ˆ AC

• ˆ

AC ⊂ AC

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC AC = ˆ AC P ⊆ ˆ P

• ˆ

AC ⊂ AC

Theorem (Case 1)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊂ ˆ O and AC = ˆ

• AC. Then P ⊆ ˆ

P holds.

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC

Theorem (Case 1)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊂ ˆ O and AC = ˆ

• AC. Then P ⊆ ˆ

P holds.

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC

Remark (Cases 2 and 3)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊆ ˆ O and AC ⊂ ˆ

• AC. Then, in general, neither P ⊆ ˆ

P nor ˆ P ⊆ P follows.

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC ? ?

Remark (Cases 2 and 3)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊆ ˆ O and AC ⊂ ˆ

• AC. Then, in general, neither P ⊆ ˆ

P nor ˆ P ⊆ P follows.

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC ? ?

Conjecture (Case 4)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent Petri nets with ˆ O ⊂ O and AC ⊂ ˆ

• AC. Furthermore, let every control-arc in ˆ

AC be necessary. If for all (t < t′) ∈ O \ ˆ O the following properties hold: there exists a “specific” control-arc (p, t) ∈ ˆ AC \ AC, (t < t′) is strictly necessary in O, there does not exist a transition t′′ with (t′′ < t) ∈ ˆ O, t and t′ are not both enabled in any state in ˆ P, then P ⊆ ˆ P follows.

• J. Wegener

December 06, 2012 12 / 15

Classification of X ′-Deterministic Extended Petri Nets

We consider two X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC “ ˆ P ⊆ P” ? ?

Conjecture (Case 4)

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent Petri nets with ˆ O ⊂ O and AC ⊂ ˆ

• AC. Furthermore, let every control-arc in ˆ

AC be necessary. If for all (t < t′) ∈ O \ ˆ O the following properties hold: there exists a “specific” control-arc (p, t) ∈ ˆ AC \ AC, (t < t′) is strictly necessary in O, there does not exist a transition t′′ with (t′′ < t) ∈ ˆ O, t and t′ are not both enabled in any state in ˆ P, then P ⊆ ˆ P follows.

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December 06, 2012 12 / 15

Outline

1

Petri nets

2

Classification of X ′-Deterministic Extended Petri Nets

3

Conclusion

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December 06, 2012 13 / 15

Conclusion

The network reconstruction algorithm creates Petri nets with the following extensions (so called X ′-deterministic extended Petri nets): control-arcs, capacities on places, priorities on transitions. We defined an inclusion relation on sets of X ′-equivalent extended Petri nets, as well as the notion of a minimal representative of this set. We examined X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC “ ˆ P ⊆ P” ? ? Future work: prove the conjecture, fill gaps in table and integrate results in reconstruction algorithm.

• J. Wegener

December 06, 2012 14 / 15

Conclusion

The network reconstruction algorithm creates Petri nets with the following extensions (so called X ′-deterministic extended Petri nets): control-arcs, capacities on places, priorities on transitions. We defined an inclusion relation on sets of X ′-equivalent extended Petri nets, as well as the notion of a minimal representative of this set. We examined X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC “ ˆ P ⊆ P” ? ? Future work: prove the conjecture, fill gaps in table and integrate results in reconstruction algorithm.

• J. Wegener

December 06, 2012 14 / 15

Conclusion

The network reconstruction algorithm creates Petri nets with the following extensions (so called X ′-deterministic extended Petri nets): control-arcs, capacities on places, priorities on transitions. We defined an inclusion relation on sets of X ′-equivalent extended Petri nets, as well as the notion of a minimal representative of this set. We examined X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC “ ˆ P ⊆ P” ? ? Future work: prove the conjecture, fill gaps in table and integrate results in reconstruction algorithm.

• J. Wegener

December 06, 2012 14 / 15

Conclusion

The network reconstruction algorithm creates Petri nets with the following extensions (so called X ′-deterministic extended Petri nets): control-arcs, capacities on places, priorities on transitions. We defined an inclusion relation on sets of X ′-equivalent extended Petri nets, as well as the notion of a minimal representative of this set. We examined X ′-equivalent extended Petri nets (P, cap, O) and ( ˆ P, cap, ˆ O): O ⊂ ˆ O O = ˆ O ˆ O ⊂ O AC ⊂ ˆ AC ? ? “P ⊆ ˆ P” AC = ˆ AC P ⊆ ˆ P

• ˆ

P ⊆ P ˆ AC ⊂ AC “ ˆ P ⊆ P” ? ? Future work: prove the conjecture, fill gaps in table and integrate results in reconstruction algorithm.

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December 06, 2012 14 / 15

Thank You

Thank You

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December 06, 2012 15 / 15

Case 2) & 3) O ⊆ ˆ O and AC ⊂ ˆ AC

Remark

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊆ ˆ O and AC ⊂ ˆ

• AC. Then, in general, neither P ⊆ ˆ

P nor ˆ P ⊆ P follows.

A C D E E D C A t1 t2 t2 t1 ˆ O = {t1 < t2} O = {t1 < t2} ˆ P = P = B B

• J. Wegener

December 06, 2012 16 / 15

Case 2) & 3) O ⊆ ˆ O and AC ⊂ ˆ AC

Remark

Let (P, cap, O) and ( ˆ P, cap, ˆ O) be two X ′-equivalent extended Petri nets with O ⊆ ˆ O and AC ⊂ ˆ

• AC. Then, in general, neither P ⊆ ˆ

P nor ˆ P ⊆ P follows.

A C D E E D C A B B t1 t2 t2 t1 ˆ O = {t1 < t2} O = {t1 < t2} ˆ P = P =

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December 06, 2012 16 / 15

Input: Experimental time-series data

We consider experimental time-series data, where set of observed states X ′ stimulations to the network (dashed arcs)

• bserved responses (solid arcs)

                  1             1 1             1             1             1 1 1             1 1             1             1       FR R R

• J. Wegener

December 06, 2012 16 / 15