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Petri NetsIna Koch and Monika Heiner

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Petri Nets(1962)

Carl Adam Petri

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Outline

basic definition structural analysis biological network with Petri

net

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Petri net

Definition : PN = (P, T, f, m0) Two type nodes Set : P (Places) ; T (transition

s) Edges Set : f (set of directed arcs) m0 : initial marking(tokens)

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Example

p1 p2

p3

t1

t2

PN = ( P,T,F,M0 )P = { p1,p2,p3 }T = { t1,t2}F = {(p1,t1),(p2,t1),(t1,p3), (p3,t2),(t2,p1),(t2,p2)}M0(p1) = 1M0(p2) = 1M0(p3) = 0

Arcs only connect of different type

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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).

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Example:

firing

2NAD+ + 2H2O → 2NADH + 2H+ + O2

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Concurrent Firing actions

partial order

(r1 ,r2); (r1 ,r3);

r2,r3 can fire independently

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Behavioral Properties

Reachability liveness, reversibility Boundedness others

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Behavioral Properties :Reachability Marking M is reachable from

marking M0 if there exists a sequence of firings

M0 t1 M1 t2 M2… M that transforms M0 to M.

[M0> Set of marking M reachable from M0

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Behavioral Properties :Reachability

p1 p2

p3

t1

t2

p4

p1 p2

p3

t1

t2

p4

p1 p2

p3

t1

t2

p4

[M0> = R(M0)={(1 1 0 0) , (0 0 1 0) , (1 0 0 1) }

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Behavioral Properties :Liveness

Definition: Liveness of transitions A transition t is dead in a marking m, if it is not e

nabled 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 m0

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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

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Examples

Weakly live

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Behavioral Properties :Reversibility

Definition : A Petri net is reversible:

∀m ∈ [m0> : m0 ∈ [m>

dead nonlive live

Not reversible

X

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Behavioral Properties :Boundedness

Definition : place p is k-bounded :

if ∃k ∈ postive integer : ∀m ∈ [m0> : m(p) ≤ k Petri Net is k-bounded :

if all its places are k-bounded

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Unbounded

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Unbounded

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Unbounded

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Unbounded

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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-pr

eservingly

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Structural Analysis (cont’s)

connected : A Petri net is connected, if it holds for all pairs of node

s 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 tr

ansitions without pre-/postplaces and no places without pre/posttransitions

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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..

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Structural Analysis (cont’s) structural deadlock :

D ⊆ P , •D ⊆ D •

trap : Q ⊆ P , Q • ⊆ • Q

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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

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matrix representation

• matrix entry cij :

token change in place pi by firing of transition tj• matrix column Δtj:

vector describing the change of the whole marking by firing of tj

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incidence (stoichiometric) matrix

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T-invariants integer solutions y :( y is transition

vector) C • y= 0, y≠ 0, y ≥ 0

1y1 −3y2 +3y3 = 02y1 −2y2 = 0−2y1 +3y3 = 0+2y2 −3y3 = 0+2y2 −3y3 = 0

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P-invariants integer solutions x : ( x is place vector)

x • C= 0, x≠ 0, x ≥ 0

x1 +2x2 −2x3 = 0−3x1 −2x2 +2x4 +2x5 = 03x1 +3x3 −3x4 −3x5 = 0

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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.

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exampleP-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}

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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.

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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 nod

es without outgoing arcs.

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Reachability graph

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Different type of biological network

metabolic networks

signal transduction networks

gene regulatory networks

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Pathway vs Network Network

cell behavior or the whole model of a cell Pathway

represents functional subnetwork Network may consist of several pathways

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Hypergraph to Petri Nets