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 Petri nets 1 Classification of X ′ -Deterministic Extended Petri Nets 2 Conclusion 3 J. Wegener December 06, 2012 2 / 15
Outline Petri nets 1 Classification of X ′ -Deterministic Extended Petri Nets 2 Conclusion 3 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 x p of tokens. A state can be represented as a vector x ∈ N | P | with entries x p 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 ′ . H 2 H 2 3 1 2 2 2 2 = x ′ x = 1 ⇒ 0 H 2 O H 2 O 0 2 O 2 O 2 J. Wegener December 06, 2012 4 / 15
Deterministic extended Petri nets Deterministic extended Petri nets A triple ( P , cap , O ) extended Petri net P = ( P , T , A S ∪ A C , w ) capacity on places cap : P → N set of priorities O (e.g., partial order) 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 , A S ∪ A C , w ) capacity on places cap : P → N set of priorities O (e.g., partial order) t 1 t 2 t 3 t 4 t 5 O = { t 1 < t 3 , t 2 < t 3 , t 5 < t 3 } 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 , A S ∪ A C , w ) capacity on places cap : P → N set of priorities O (e.g., partial order) t 1 (2) t 2 t 3 (2) t 4 t 5 O = { t 1 < t 3 , t 2 < t 3 , t 5 < t 3 } 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 , A S ∪ A C , w ) capacity on places cap : P → N set of priorities O (e.g., partial order) t 1 (2) t 2 t 3 (2) t 4 t 5 O = { t 1 < t 3 , t 2 < t 3 , t 5 < t 3 } 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 ′ = ( x 1 , . . . , x m ) 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 A C and O are so that the experiments can exactly be reproduced J. Wegener 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. J. Wegener 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 Petri nets 1 Classification of X ′ -Deterministic Extended Petri Nets 2 Conclusion 3 J. Wegener December 06, 2012 8 / 15
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 t 1 , . . . , t k is an O -feasible switching sequence for x 1 in P if for all j x j +1 is the successor state of x j switching t j , there does not exist a t ∈ T enabled in x j with ( t j < 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 . J. Wegener December 06, 2012 9 / 15
Inclusion relation: Example C D C D P = ˆ P = t 1 t 1 t 2 t 2 A B A B O = { t 1 < t 2 } ˆ O = ∅ ⇒ P ⊂ ˆ P J. Wegener December 06, 2012 10 / 15
Inclusion relation: Example C D C D P = ˆ P = t 1 t 1 t 2 t 2 A B A B O = { t 1 < t 2 } ˆ O = ∅ ⇒ P ⊂ ˆ P J. Wegener 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: O ⊂ ˆ O and A C = ˆ A C , 1 O = ˆ O and A C ⊂ ˆ A C , 2 O ⊂ ˆ O and A C ⊂ ˆ A C , 3 O ⊂ O and A C ⊂ ˆ ˆ A C . 4 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: O ⊂ ˆ O and A C = ˆ A C , 1 O = ˆ O and A C ⊂ ˆ A C , 2 O ⊂ ˆ O and A C ⊂ ˆ A C , 3 O ⊂ O and A C ⊂ ˆ ˆ A C . 4 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 = ˆ ˆ O O O ⊂ O A C ⊂ ˆ A C A C = ˆ - A C ˆ A C ⊂ A C 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 = ˆ ˆ O O O ⊂ O A C ⊂ ˆ A C A C = ˆ P ⊆ ˆ P - A C ˆ A C ⊂ A C 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 A C = ˆ A C . 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 = ˆ ˆ O O O ⊂ O A C ⊂ ˆ A C A C = ˆ P ⊆ ˆ ˆ P - P ⊆ P A C ˆ A C ⊂ A C 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 A C = ˆ A C . 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 = ˆ ˆ O O O ⊂ O A C ⊂ ˆ A C ? ? A C = ˆ P ⊆ ˆ ˆ P - P ⊆ P A C ˆ A C ⊂ A C 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 A C ⊂ ˆ A C . 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 = ˆ ˆ O O O ⊂ O A C ⊂ ˆ A C ? ? A C = ˆ P ⊆ ˆ ˆ P - P ⊆ P A C ˆ A C ⊂ A C ? ? 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 A C ⊂ ˆ A C . Then, in general, neither P ⊆ ˆ P nor ˆ P ⊆ P follows. J. Wegener December 06, 2012 12 / 15
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