This document compares Petri nets and state spaces for modeling and verification. It discusses that state spaces allow modeling global state changes over time, while Petri nets consider asynchronous components and causality of events. The document also describes techniques for efficient state space generation from Petri nets, such as checking enabled transitions with constant time, firing transitions with constant effort, backtracking transitions, and storing markings in a set. Reduction techniques like linear algebra, sweep-line methods, symmetries, and stubborn sets are also covered to reduce the state space.

1. Petri Net (versus) State Spaces
Karsten Wolf
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

2. My experience with
state spaces
-INA Integrated Net Analyzer
-LoLA A Low Level Analyzer
-The service-technology.org tool family
Case studies and applications:
-Finding hazards in a GALS wrapper
-Integration into Pathway Logic Assistent
-Soundness check for 700+ industrial business process models in (avg) 2 msec
-Verification of web service choreographies
-Verification of parameterized Boolean programs
-Solving AI planning challenges
-Integration into BP related tools like ProM, Oryx
-Integration into model checking platforms (MC Kit, PEP, CPN-AMI,…)
-….To be continued
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

3. Why state spaces? Why Petri nets?
Verification based on state space
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

4. -Consider asynchronously communicating
components rather than global state changes
Why state spaces?
-Consider causality of events rather than their
ordering in time!
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

5. Petri net principles
Presence or absence of ressources
Monotonicity of firing rather than reading / writing variables
Linearity of firing rule -Consider asynchronously communicating
components rather than global state changes
Locality
-Consider causality of events rather than their
Partially ordered ordering in time!
event structures
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

6. Petri net specific
verification
Monotonicity of firing
Coverability graphs
Siphons / traps
Linearity of firing rule
invariants
Locality
Net reduction
Partially ordered
event structures
Branching prefixes
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

7. State space generation
1. Checking enabledness
2. Firing a transition
3. Backtracking
4. Managing the visited states
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

8. State space generation
1. Checking enabledness
it y
oto n ic
Mon
After firing, only check:
previously enabled transitions which have lost tokens
previously disabled transitions which have gained tokens
... managed through explicitly stored lists
… typical: reduction from linear to constant time
li ty
L oca
2. Firing a transition
3. Backtracking
4. Managing the visited states
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

9. State space generation
1. Checking enabledness
2. Firing a transition
Marking changed via list of pre-, list of post-places
- effort does not depend on size of net
- Typically: constant effort
li ty
L oca
3. Backtracking
4. Managing the visited states
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

10. State space generation
1. Checking enabledness
2. Firing a transition
3. Backtracking
ri ty
Li nea
In depth-first search: fire transition backwards
In breadth-first search: implemented as incremental depth-first search
li ty
L oca
4. Managing the visited states
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

11. Consequence:
„write-only“ storage of markings
current marking m
-fire old/new
-fire backwards
Set of visited markings
Search
stack t1
t2
t3
...
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

12. 4. Managing the visited states
only performed actions: search, insert
p1 p2 p3 p4 p5 p6 p7 p8
in e arity
L
a1 p1 + a2 p2 + a3 p3 = const.
Place
invariants
b2 p2 + b4 p4 + b6 p6 = const.
c3 b3 + c7 p7 + c8 p8 = const.
30-60% less memory
preprocessing <1sec
run time gain: 30-60%
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

13. Reduction techniques
1. Linear Algebra
2. The Sweep-Line Method
3. Symmetries
4. Stubborn Sets
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

14. 1. Linear algebra
• The invariant calculus
– originally invented for replacing state spaces
– in LoLA: used for optimizing state spaces
Already seen: place invariants
Transition invariant: firing vector of a potential cycle
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

15. Transition invariants
ty
L in eari
for termination sufficient: store one state per cycle of occurrence graph
implementation in LoLA:
transition invariants
- set of transitions that occur in every cycle
- store states where those transitions enabled
saves space, if applied in connection with stubborn sets, costs time
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

16. 2. The sweep-line method
• Relies on progress measure
ty
L in eari
LoLA computes measure automatically:
m4: p4
t3
transition invariant
p2=p1+Δt1 m3: p3
p3 = p2+Δt2 t2
... m2: p2
t1
m1: p1
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

17. 3. The symmetry method
li ty
L oca
LoLA: A symmetry = a graph automorphism of the PT-Net
All graph automorphisms = a group (up to exponentially many members)
- stored in LoLA: polynomial generating set
A marking class: all markings that can be transformed into each other by a
symmetry
- executed in LoLA: polynomial time approximation
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

18. Example
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19. … as derived from a program
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

20. 4. Stubborn set method
• Dedicated method for each supported property
li ty
traditional LTL-preserving method: L o ca
- one enabled transition
- the basic stubborness principal
- only invisible transitions
- at least once, on every cycle, all enabled transitions
n
Tra d i t io
LoLA: PN
- can avoid some of the criteria, depending on property
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK

21. Conclusion
Why state spaces? Why Petri nets?
That‘s why
Further reading:
• Tools: www.service-technology.org
• Group / Papers: www.informatik.uni-rostock.de/tpp/
UNIVERSITÄT ROSTOCK | FAKULTÄT INFORMATIK UND ELEKTROTECHNIK | INSTITUT FÜR INFORMATIK