1 Petri Nets Ina Koch and Monika Heiner

2 Petri Nets(1962) Carl Adam Petri

3 Outline basic definition structural analysis biological network with Petri net

4 Petri net Definition : PN = (P, T, f, m 0 ) Two type nodes Set : P (Places) ; T (transitions) Edges Set : f (set of directed arcs) m 0 : initial marking(tokens)

5 Example Arcs only connect of different type

6 Firing rule Definition A transition t is enabled in a marking m m[t>,if ∀ p ∈ t : f (p, t) ≤ m(p). transition t,which is enabled in m, may fire. When t in m fires : m[t>m ’,with ∀ p ∈ P : m ’ (p) = m(p) − f (p, t) + f (t, p).

7 Example: firing 2NAD + + 2H 2 O → 2NADH + 2H + + O 2

8 Concurrent Firing actions partial order (r 1,r 2 ); (r 1,r 3 ); r 2,r 3 can fire independently

9 Behavioral Properties Reachability liveness, reversibility Boundedness others

10 Behavioral Properties : Reachability Marking M is reachable from marking M 0 if there exists a sequence of firings  M 0 t 1 M 1 t 2 M 2 … M that transforms M 0 to M. [M 0 > Set of marking M reachable from M 0

11 Behavioral Properties : Reachability [M 0 > = R(M 0 )={( ), ( ), ( ) }

12 Behavioral Properties : Liveness Definition: Liveness of transitions A transition t is dead in a marking m, if it is not enabled in every marking m ’ reachable from m:  ∃ m ’ ∈ [m> : m ’ [t>  m ’ ∈ [m> :  m ’ [t> A transition t is live, if it is not dead in any marking reachable from m 0

13 Behavioral Properties : Liveness Definition: Liveness of Petri net Deadlock-free (weakly live) : if there are no reachable dead markings. (marking m is dead if there is no transition enabled in m) Live (strongly live) : if each transition is live

14 Examples Weakly live

15 Behavioral Properties : Reversibility Definition : A Petri net is reversible: ∀ m ∈ [m 0 > : m 0 ∈ [m> Not reversible X

16 Behavioral Properties : Boundedness Definition : place p is k-bounded : if ∃ k ∈ postive integer : ∀ m ∈ [m 0 > : m(p) ≤ k Petri Net is k-bounded : if all its places are k-bounded

17 Unbounded

18 Unbounded

19 Unbounded

20 Unbounded

21 Structural Analysis ordinary : A Petri net is ordinary, if all arc weights are equal to 1 Pure: A Petri net is pure, if there are no two nodes, connected in both directions conservative : A Petri net is conservative, if all transitions fire token- preservingly

22 Structural Analysis (cont ’ s) connected : A Petri net is connected, if it holds for all pairs of nodes a and b that there is an undirected path from a to b. Strongly connected : A Petri net is strongly connected, if it holds for all pairs of nodes a and b that there is an directed path from a to b. free of boundary nodes : A Petri net is free of boundary nodes, if there are no transitions without pre-/postplaces and no places without pre/posttransitions

23 Structural Analysis (cont ’ s) free of static confilct : A Petri net is free of static conflicts, if there are no two transitions sharing a preplace..

24 Structural Analysis (cont ’ s) structural deadlock : D ⊆ P, D ⊆ D trap : Q ⊆ P, Q ⊆ Q

25 example structural deadlock : D ={A,B}, D ={r1,r2}, D ={r1,r2,r3} : D ⊆ D trap : Q ={C,D,E}, Q ={r4,r5}, Q ={r1,r3,r4,r5} : Q ⊆ Q

26 matrix representation matrix entry c ij : token change in place pi by firing of transition tj matrix column Δt j : vector describing the change of the whole marking by firing of t j

27 incidence (stoichiometric) matrix

28 T-invariants integer solutions y :( y is transition vector) C y= 0, y≠ 0, y ≥ 0 1y 1 −3y 2 +3y 3 = 0 2y 1 −2y 2 = 0 −2y 1 +3y 3 = 0 +2y 2 −3y 3 = 0

29 P-invariants integer solutions x : ( x is place vector) x C= 0, x≠ 0, x ≥ 0 x 1 +2x 2 −2x 3 = 0 −3x 1 −2x 2 +2x 4 +2x 5 = 0 3x 1 +3x 3 −3x 4 −3x 5 = 0

30 covered by invariants Definition : A Petri net is covered by p-invariants — CPI, if every place belongs to a p-invariant A Petri net is covered by t-invariants — CTI, if every transition belongs to a t-invariant.

31 example P-invar solutions (2, 0, 1, 0, 3)=>{A,C,E} (0, 1, 1, 0, 1)=>{B,C,E} (2, 0, 1, 3, 0)=>{A,C,D} (0, 1, 1, 1, 0)=>{B,C,D} T-invar solutions (3,3,2)=>{r1,r2,r3}

32 Reachability graph Let N = (P, T, f,m0) be a Petri net. The reachability graph of N is the graph RG(N) = (VN,EN ), where VN := [m0> is the set of nodes, EN := {(m, t,m ’ ) | m,m ’ ∈ [m0, t ∈ T : m[t>m ’ } is the set of arcs.

33 Reachability graph 1.k-bounded: iff there is no node in the reachability graph with a token number larger than k in any place. 2. reversible: iff the reachability graph is strongly connected. 3.deadlock-free: iff the reachability graph does not contain nodes without outgoing arcs.

34 Reachability graph

35 Different type of biological network metabolic networks signal transduction networks gene regulatory networks

36 Pathway vs Network Network cell behavior or the whole model of a cell Pathway represents functional subnetwork Network may consist of several pathways

37 Hypergraph to Petri Nets