1. Model Checking of
Time Petri Nets
Marwa K. U. Al-Rikaby
Babylon University
IT College

2. Outlines:
 Introduction.
 Time Petri nets.
 Temporal Logics for time Petri nets.
 TPN state space abstraction.
 Model checking timed properties of TPN.

3. Introduction
Why time Petri nets?
TPN model is a good compromise between
modeling power and verification complexity of
concurrent systems with timed constraints
(real time systems).

4. Introduction
Why time Petri nets?
 TPN are able to model time constraints even if
the exact delays or durations of events aren't
known.
 TPN specify time constraints of real time
systems by giving worst case boundaries.

5. Introduction
How TPN differ from ordinary PN?
 A firing interval is associated with each
transition specifying the minimum and
maximum times it must be maintained
enabled, before its firing.
 Firing takes no time, but may lead to another
marking.
 In real world, events (firings) takes a time to
complete, but in Petri nets it is omitted for
simplicity.

6. Introduction
Model checking techniques of systems:
 Applied by:
 Representing the behavior of a system as a finite
state transition system (state space).
 Specifying properties of interest in a temporal logic
(LTL, CTL, CTL*, MTL, TCTL).
 Exploring the state space to determine whether they
hold or not.
 With TPN, an extra effort is required to abstract
their generally infinite state space.

7. Outlines:
 Introduction.
 Time Petri nets.
 Temporal Logics for time Petri nets.
 TPN state space abstraction.
 Model checking timed properties of TPN.

8. Time Petri nets
 Definition:
A TPN is PN with time intervals attached to its
transitions, it is a 6 tuple ϰ =(P, T, Pre, Post, m0,Is)
where:
 P and T are finite sets of places and transitions,
P∩T=Ø.
 Pre and Post are the backward and forward incidence
functions: P T N.
 m0 is the initial marking: P N.
 Is is a static firing interval associated with each
transition t, T Q+ (Q+ U {∞}).
 ↓Int is the lower bound.

9. Time Petri nets
 Example:
ϰ =(P, T, Pre, Post,
m0,Is)
P={ p0,p1,p2,p3}
T={t0,t1}
pre={(1,1),(1,1)}
post={(1,1),(1,1)}
m0={1, 0, 1, 3}
Is={[1,2],[1, ∞]}

10. Time Petri nets
 In ordinary PN, a transition t is said to be
enabled if there are enough tokens in its input
places.
 In TPN, that’s not enough, the time in which
the transition has the needed number of
tokens must not be less than the lower bound
nor exceeds the upper bound of the transition
interval.

11. Time Petri nets
Keep these notations in your mind:
 En(m) : is the set of all enabled transitions in a
marking m.
 s and s’: are two different states of TPN in the
state space.
 θ R+ is a number of time units.

12. Time Petri nets
The semantics of TPN
Defines the TPN state as a marking and a
function.
 Definition based on clocks:
associates with each transition t of the model a
clock to measure the time elapsed since t
became enabled most recently.
 Definition based on intervals:
associates a firing interval with each enabled

13. Time Petri nets
 Clocks based TPN:
 TPN clock state is a pair s=(m,v), where m is a
marking and v is a valuation function, v: En(m)
R+ .
 The initial clock state is: s0=(m0,v0)
 m0 is the initial marking.
 v0(t)=0 for all transitions in En(m).
 TPN state evolves either by time progression or
by firing transitions.

14. Time Petri nets
 Clocks based TPN:
 When t becomes enabled, its clock initialized to 0
and increases synchronously with time until t is
fired or disabled by another transition firing.
 t can fire if its clock value is inside its static firing
interval Is(t).
 If the clock reached ↑Is(t) then t must fire
immediately without any delay.

15. Time Petri nets

16. Time Petri nets

17. Time Petri nets
 Intervals based TPN
 The TPN interval state is a couple s=(m,I), where
m is a marking and I:En(m)  Q+ (Q+ U {∞}) is
an interval function.
 The initial interval state is s0=(m0,I0)
 m0 is the initial marking.
 I0(t)=is(t) for all t in En(m0).
 TPN state evolves either by time progression or
by firing transitions.

18. Time Petri nets
 Interval based TPN
 When a transition t becomes enabled, its firing interval
is set to its static firing interval Is(t).
 The lower and upper bounds of this interval decrease
synchronously with time, until t is fired or disabled by
another firing.
 t can fire, if the lower bound of its firing interval
reaches 0, but must be fired, without any additional
delay, if the upper bound of its firing interval reaches
0.

19. Time Petri nets

21. Timed Petri nets

22. Timed Petri nets
 π(s) is the set of all execution paths starting from
state s.
 π(s0) is the set of all execution paths in the TPN
since it starts from s0.
 The TPN state space defines the branching
semantics of the TPN model, where as
defines its linear semantics.

23. Outlines:
 Introduction.
 Time Petri nets.
 Temporal Logics for time Petri
nets.
 TPN state space abstraction.
 Model checking timed properties of TPN.

24. Temporal logics of TPN
 Properties of timed systems are usually
specified using temporal logics, we introduce:
CTL* (computation tree logic star).
TCTL (timed computation tree logic).
 Markings are represented as atomic
propositions.

25. Temporal logics of TPN

34. Temporal logics of TPN
 TCTL
A time extension of CTL, in which a time interval
is associated with each temporal operator.
 Defined as:
 When interval I is omitted, its value is [0,∞] by
default.

35. Outlines:
 Introduction.
 Time Petri nets.
 Temporal Logics for time Petri nets.
 TPN state space
abstraction.
 Model checking timed properties of TPN.

36. TPN state space abstractions
 Aim to construct a finite contraction of the
model state space by removing irrelevant
details.
 Must preserve interested properties (markings,
linear and branching properties), which would
be verified using classical model checking
techniques later.
 The challenge is to construct a much coarser
abstraction with less resources (time and
space).

37. TPN state space abstractions
Abstraction process going into:
1. Abstract state space.
2. Abstract states.
3. Abstractions preserving linear properties.
4. Abstractions preserving branching properties.

43. TPN state space abstractions
2. Abstract states:
 Each transition enabled in m is represented
in f by a time variable with the same
name, Var(f)=En(m).
 All time variables are either clocks (clock
abstract state) or delays (interval abstract
state).

44. TPN state space abstractions
3. Abstractions preserving linear properties:
 Have exactly the same firing sequences as their
concrete state space.
 Three levels of abstractions:
 States reachable by time progression may either
represented or abstracted.
 States reachable by the same firing sequence
independently of their firing times are agglomerated
in the same node.
 The agglomerated states are then considered
modulo some relation of equivalence or
approximation.

45. TPN state space abstractions
3. Abstractions preserving linear properties:
t

46. TPN state space abstractions
4. Abstractions preserving branching properties:
 Can be done on two steps:
 Intermediate abstraction:
does not necessarily preserve branching
properties.
 Refinement process:
restore the condition AE (the resulting graph is
atomic).

47. TPN state space abstractions
4. Abstractions preserving branching properties:
Step1: Intermediate abstraction:
1. group abstract states whenever one of them includes all
the others or their union is convex.
2. replace the grouped states set by a new abstract state
representing their union.
3. all transitions between these abstract states become
loops for their union.
4. ingoing and outgoing transitions of the grouped abstract
states become ingoing and outgoing of their union.
5. if one of the grouped abstract states contains the initial
abstract state then their union become the initial abstract
state.

49. TPN state space abstractions
4. Abstractions preserving branching properties:
Step2: Refinement Process:
1. partition a into a set of convex subclasses so as
isolate the predecessors of a’ by t in a from those
are not.
2. replace a by its partition.
3. each subclass inherets all connections of a
according to condition EE.
4. repeat refinement process until condition AE is
established.
This step process generates a finite graph iff the

50. Outlines:
 Introduction.
 Time Petri nets.
 Temporal Logics for time Petri nets.
 TPN state space abstraction.
 Model checking timed
properties of TPN.