Petri Net Theory and Its
Applications
Hsu-Chun Yen
Dept. of Electrical Engineering
National Taiwan University
[email protected]
http://www.ee.ntu.edu.tw/~yen
Instructor
Hsu-Chun Yen
Educational background:
Ph.d., 1986, Computer Science Dept.
Univ. of Texas at Austin, USA
Work Experiences:
Professor: EE Dept. National Taiwan Univ.
1991 – present
Assistant Prof.: CS Dept. Iowa State U. USA,
1986-1990
Research Interests:
Automata Theory, Petri net theory, Algorithms
Goals
 Introducing
various models of restricted
and extended Petri nets and their
related problems
 Understanding the analytical techniques
for reasoning about Petri net problems
 Analyzing the complexities of Petri net
problems
 Investigating applications of Petri net
theory in various CS related disciplines
Modeling and Analysis
System
Model
Revise
Petri net model
Analysis
Properties
of the
system
History

Model introduced by C. A. Petri (U. Darmstadt, W.
Germany) in 1962
– Ph.D. Thesis: “Communication with Automata”

Applications: distributed computing, manufacturing,
control, communication networks, transportation…

PNs describe explicitly and graphically:
– sequencing/causality
– conflict/non-deterministic choice
– concurrency
 For
recent developments, see
http://www.daimi.au.dk/PetriNets/
Applications











Membrane computing
Discrete event dynamic systems
Communication protocols
Performance evaluation
Asynchronous circuit design/synthesis
Database systems
Concurrent and parallel programs
Flexible manufacturing systems
Dataflow-computing systems
Fault-tolerant systems
…etc, etc, etc
Outline
 Definitions
of Petri nets (equiv. Vector
Addition Systems, VAS with states, …)
 Various PN problems
 Analytical techniques and complexity
results
 Extended PNs
 Conclusions and open problems
PART 1:
Definitions and Notations
of
Petri Nets
Petri Nets
In addition to their popularity in
modeling systems of concurrent
nature (such as communication
protocols, asynchronous circuits, and
more), Petri nets (C. Petri, 1962) also
pose many interesting and
challenging mathematical questions!
Petri Nets
place
P4
P1
t2
t1
toke
n
t3
t4
P3
transition
P2
P5
Transition Firing
P4
P1
t2
t1
t3
P3
P2
P5
t4
Transition
Transition
Firing
Firing
Goals
P4
P1
t2
t1
t3
t4
P3
P2
P5
Interleaving semantics
Formal definition
A classical Petri net is a four-tuple
(P,T,I,O) where:
P
is a finite set of places,
T is a finite set of transitions,
I : P x T  N is the input function, and
O : T x P  N is the output function.
: (P x T)  (T x P)  N
An example
P4
P1
I(p3,t3) = 3
t1
t2
3
t3
t4
O(t2, p3) = 2
I(t3,p5) = 1
…
2 P3
P2
P5
Marking
The marking of a Petri net (P,T,I,O) is
defined as follows:
: P  N, i.e., a function mapping
the set of places onto {0,1,2, … }.

Firing of an enabled transition
 A transition
t is enabled at a marking 
iff
(p)  I(p, t), p P
 The
firing of transition t results in
marking ’ such that
’ (p)= (p) + O(t, p) - I(p, t) , p P
Modeling FSMs
transition
state
State machines:
each transition
has exactly
one input and
one output
Allowing conflict, decision, choice, nondeterminism
Modeling concurrency
t2
t1
concurrency
t4
t3
Marked graph:
each place has
exactly one
incoming arc
and one
outgoing
arc.
Conventionally, interleaving semantics is assumed
Modeling communication
protocols
Asynchronous communication
ready
to send
ready
to receive
send
msg.
receive
msg.
wait
for ack.
proc.1
proc.2
msg.
received
receive
ack.
send
ack.
ack.
received
ack.
sent
Computation
P4
P1
t1
t2
Initial
marking
(1,0,0,1,0)
t3
t2
t4
(0,1,1,1,0)
P3
P2
t3
t1
P5
Reachable markings
(1,0,1,1,0)
{
t2
(0,1,2,1,0)
(0,1,0,0,1)
t3
t1
(1,0,0,0,1)

Let a marking be represented as a vector
(place 1, place 2, place 3, place 4, place 5)

The initial marking for the above Petri net is:
0=(1,0,0,1,0)

A path:
t1
t2
t2
(1,0,0,1,0)  (0,1,1,1,0)  (1,0,1,1,0)(0,1,2,1,0)
0

1
2
3
The markings 0, 1, 2 , 3 …are said to be
reachable.

The set of reachable markings is known as
the reachability set: R(0 , P).
Reachability Tree Analysis
Initial
marking
(1,0,0,1,0)
Exhaustive search:
only applicable to
bounded Petri nets
of small size

t2
(0,1,1,1,0)
t3
t1
(1,0,1,1,0)
t2
(0,1,2,1,0)
(0,1,0,0,1)
t3
t1
(1,0,0,0,1)
Vector Addition Systems
addition vector
P4
P1
+ (-1,1,1,0,0)
t1
t2
t3
t4
(1,0,0,1,0)
Petri Nets
+ (1,-1,0,0,0)
(1,0,1,1,0)
P3
P2
(0,1,1,1,0)
P5
base vector
Vector Addition Systems
( v0; {a1, …, an} )
Equivalent Models
(-1,1,1,0,0)
(1,-1,0,0,0)
Petri Nets
(-2, -1,0,+3,0)
 Vector Addition Systems
 Vector Addition Systems with States
 Vector Replacement Systems
Equivalence between n-dim VASS
& (n+3)-dim VAS
Theorem [HP’79] n-dim VASS  (n+3)-dim VAS
(Proof) Let q1, q2,…, qk be states of VASS
(0, -ai, ak-i+1-bi, bk-i+1)
Let ai=i and bi = (k+1)(k+1-i)
-:j>i
VASS in state (qi , u)  VAS-- in (u, ai, bi:, j0)< i
To simulate VASS trans. q i  (qj, w)
(u, ai , bi , 0)  (u, 0, ak-i+1, bk-i+1 )  (u, bi, 0, ai )
--- (w, aj - bi, bj , - ai ) (u+w, aj, bj, 0)
NOTE: ai < ai+1, bi > bi+1, ai < bj, bi - bi+1 =k+1 > aj
Hence, (…) can be applied only at (… , bi, 0, ai ) because
am - ai < 0 (m < i); bm + aj - bi  aj – (k+1) < 0 (m > i)
Petri Net Properties
 Behavioral
marking
2
 Structural
marking
properties: depending on the initial
2
properties:
Bounded but not
structurally bounded
independent of the initial
Structurally
bounded
Behavioral properties (1)
 Reachability
– Marking n is reachable from initial marking 0 if
existss a sequence of firings that transform 0 into n
– reachability is decidable, but exponential-space hard
 Boundedness
– a PN is bounded if the number of tokens in each
place doesn’t exceed a finite number k for any
marking reachable from 0
– a PN is safe if it is 1-bounded
Behavioral properties (2)
 Liveness
– a PN is live if, no matter what marking has been
reached, it is possible to fire any transition with an
appropriate firing sequence
– equivalent to deadlock-free
– different levels of liveness are defined
 Reversibility
– a PN is reversible if, for each marking n reachable
from 0, 0 is reachable from n
– relaxed condition: a marking  is a home state if, for
each marking n reachable from 0,  is reachable
from n
Behavioral properties (3)
 Coverability
– a marking  is coverable if exists n reachable
from o s.t. 
n(p)  (p) for all places p
 Persistence
– a PN is persistent if, for every reachable marking
and any two enabled transitions, the firing of one
of them will not disable the other
– then, once a transition is enabled, it remains
enabled until it’s fired
Behavioral properties (4)
 Synchronic
distance
– maximum difference of times two transitions are fired
for any firing sequence
d12  max  (t1 )   (t2 )

– well defined metric for condition/event nets and
marked graphs
Number of occurrences of
 Fairness
t1 in σ
– bounded-fairness: the number of times one transition
can fire while the other is not firing is bounded
– unconditional(global)-fairness: every transition
appears infinitely often in a firing sequence
–…
More on liveness
A transition t in a Petri net (N, M0) is said to be:

Dead (L0-live) if t can never be fired in
any firing sequence in L(M0);

L1-live (potentially firable) if t can be fired at least
once in some firing sequence in L(M0);

L2-live if, given any positive integer k, t can be
fired at least k times in some firing sequence in
L(M0);

L3-live if t appears infinitely often in some firing
sequence in L(M0);

L4-live or live if t is L1-live for every marking M in
R(M0);
L4-liveness is the strongest => L3-liveness => L2-liveness
=> L1-liveness
The Classical Problems
 The
boundedness problem:
EXPSPACE-c
Is | R(0 , P)| < ? Lipton (Yale TR., 1976); Rackoff (TCS 1978)
 The
containment (equivalence)
Undecidable
problem:
Rabin (Baker, MIT TR. 1973)
Hack (TCS 1976)
Is R(0 , P)  R(’0 , P’) ?
(Is R(0 , P) = R(’0 , P’) ?)
 The
reachability problem:
Is   R(0 , P)?
Decidable
Mayr (SICOMP, 1984)
Petri Net Problems
Boundedness
Containment
Liveness
Model
Checking
Language
equivalence
Reachability
Bisimulation
equivalence
Regularity
...
Self-stabilization
Fairness
Controllability
Reachability is key to many of the Petri net problems
Complexity Hierarchy
Recursive
Boundary
between
decidability
/undecidabil
ity
Primitive recursive
EXPSPACE
EXPTIME
PSPACE
p2
Languages
expressible as
 x1 x2 … y1 y2 …
P(x1,x2,…,y1,y2,…)
p2
CoNP
NP
P
The study of Petri net
complexity/decidability
Since the original work by C.A. Petri in
1962, an outburst of theoretical work on
decidability/complexity of problem for
Petri nets was done in the late
seventies and early eighties.
Researchers include some well-known
computer scientists, like Rabin, Rackoff,
Lipton, Mayr, Meyer, Kosaraju, …just to
name a few.
PART 2:
Analytical Techniques
Analysis techniques
 State
Space Analysis techniques
– Reachability graph
– Coverability Tree
–…
– Normally for analyzing behavioral property
 Structural
analysis techniques
– Place & transition invariants
– Incidence matrix
–…
– Normally for analyzing structural property
 Simulation
…
Reachability graph
3
1
(1,0,1,0,1)
(0,1,1,0,0)
5
2
4
(1,0,0,1,0)
• Places 2 and 4 cannot
both contain tokens –
mutual exclusion
property
• PN is bounded (safe)
Reachability graph
Graph
containing a node for each
reachable marking.
Starting from the initial state, generate
all directly reachable markings.
Expensive technique.
Only feasible if finitely many markings
(otherwise use coverability graph).
Applicable to certain properties only.
Infinite reachability graph
3
1
5
2
6
4
Coverability Tree
 Coverability
tree
– tree representation of all possible
markings
root = M0 (initial marking)
nodes = markings reachable from M0
arcs = transition firings
– if net is unbounded, then tree is kept
finite by introducing the symbol 
PN is bounded iff  doesn’t
appear in any node
Coverability tree example
M0=(100)
t3
p2
t2
p1
t1
p3
t0
Coverability tree example
M0=(100)
t1
t3
p2
t2
p1
t1
p3
M1=(001)
“dead end”
t0
Coverability tree example
M0=(100)
t1
t3
p2
t2
p1
t1
p3
M1=(001)
“dead end”
t0
t3
M3=(110)
Coverability tree example
M0=(100)
t1
t3
p2
t2
p1
t1
p3
M1=(001)
“dead end”
t0
t3
M3=(10)
Coverability tree example
M0=(100)
t1
t3
p2
p1
t1
M1=(001)
“dead end”
t0
t3
M3=(10)
t1
M4=(01)
t2
p3
Coverability tree example
M0=(100)
t1
t3
p2
p1
t1
M1=(001)
“dead end”
t0
t3
M3=(10)
t1
M4=(01)
t2
p3
t3
M3=(10)
“old”
Coverability tree example
M0=(100)
t1
t3
p2
p1
t1
t3
M1=(001)
“dead end”
t0
M3=(10)
t1
M4=(01)
t2
p3
t2
M5=(01)
“old”
t3
M6=(10)
“old”
Coverability tree example
M0=(100)
100
t1
t1
t3
001
M1=(001)
“dead end”
10
t1
t2
t3
01
t3
M3=(10)
t1
M4=(01)
t2
M5=(01)
“old”
coverability graph
coverability tree
t3
M6=(10)
“old”
State Equation
1
t1
4
t2 3
t2 t1 t2
2
1
0
0
1
0
5
+
1 -1
-1 1
0 1
0 0
0 0
0
0
-1
-1
1
0
0
0
1
-1
State Equation:
1
2
0
0
=
0 + M  x = 
0
1
2
1
0
Condition for Reachability
State equation is a necessary but not
sufficient condition for reachability
i.e.
0 + M  x = 


reachable from 0
Example
1
(0,0,1) is not reachable even though
it satisfies the state equation
3
2
Structural analysis techniques
To
avoid state-explosion phenomenon.
Properties
independent of initial state.
We
only consider place and transition
invariants.
Invariants
can be computed using linear
algebraic techniques.
Place invariant
man
Assigns
x
z
couple
marriage
divorce
y
woman
1x+1y+2z
a weight to
each place.
The weight of a
token depends on
the weight of the
place.
The weighted
token sum is
invariant, i.e., no
transition can
change it
Other invariants
man
z
couple
marriage
divorce
y
woman
1x +0y+1z
 2x+3y+5z
 -2 x + 3 y + z
 x–y
 y–x
(Any linear combination of
invariants is an
invariant.)

x
Transition invariant
man
 Assigns
a weight
to each transition.
couple
x
ydivorce
marriage
woman
2 x+2y
 Firing
the number
of times indicated
bring the system
back in the initial
state.
Other invariants
1
man
couple
x
y
marriage
divorce
woman
x+ 1y
 20 x + 20 y
Any linear combination of
invariants is an invariant,
but transition invariants
with negative weights
have no obvious
meaning.
Two ways of calculating
invariants
“Trial-and-error":
Formulate the
property that you think holds and
verify it.
"Linear-algebraic
way": Solve a
system of linear equations.
Incidence matrix of a Petri net
y
 Each
row
corresponds to a
place.
 Each column
corresponds to a
transition.
man
m
couple
x
m
 1 1 


N   1 1 
 1 1
x
z
d
marriage
divorce
woman
y
z
d
Place invariant
Let
N be the incidence matrix of a net
with n places and m transitions
Any solution of the equation XN = 0 is a
transition invariant
– X is a row vector (i.e., 1 x n matrix)
– O is a row vector (i.e., 1 x m matrix)
Note
that (0,0,... 0) is always a place
invariant.
Basis can be calculated in polynomial
time.






Example
 1 1 


( x, y, z )  1 1   (0, 0)
 1 1
man
x
z
couple
marriage
divorce
y
woman
Solutions:
 (0,0,0)
 (1,0,1)
 (0,1,1)
 (1,1,2)
 (1,-1,0)
Transition invariant
Let
N be the incidence matrix of a net
with n places and m transitions
Any solution of the equation NX = 0 is a
place invariant
– X is a column vector (i.e., m x 1 matrix)
– 0 is a column vector (i.e., n x 1 matrix)
that (0,0,... 0)T is always a place
invariant.
Basis can be calculated in polynomial
time.
Note
Example
 1 1 
0
 1 1   m   0 

 d   


 1 1
0 
man
m
couple
x
z
d
marriage
divorce
woman
y
Solutions:
(0,0)T
(1,1)T
(32,32)T
Applications of S- and T-invariants
 T-invariant
– Cyclic schedules: need to return to the
initial state
 S-invariant
– Structural Boundedness: bounded for any
finite initial marking
– Existence of a positive S-invariant is CS
for structural boundedness
 initial
marking is finite
 weighted token count does not change
Petri Net Transformations
Preserving boundedness, liveness, …