Talk at TACAS 2011 by Karsten Wolf

1. Harro Wimmel,
Karsten Wolf
Applying CEGAR to the Petri Net State Equation

2. CEGAR
Counterexample Guided
Abstraction Refinement 5
? -
-
+
1
2
4
?
3
-
x +
+
Karsten Wolf: CEGAR / PN State Equation

3. 1
s1 s3
t3
Petri Net
t1 t2 t5 s5
t4
s2 s4
1 –1 0 0 0
-1 1 0 0 0
C=
0 1 –1 0 1
0 -1 0 1 0
0 0 1 -1 -1
Incidence
Matrix
Karsten Wolf: CEGAR / PN State Equation

4. If m0 t t t t t
2 3 5 1 3 m then 1
m0 + t2 + t3 + t5 + t1 + t3 = m
m0 + (1•t1) + (1•t2) + (2•t3) + (0•t4) + (1•t5) = m
m0 + C • (1,1,2,0,1) = m
Parikh-vector of
t2 t3 t5 t1 t3
•If m0 w m then m0 + C • Parikh(w) = m
• Marking m is reachable only if
C • x = (m – m0) is feasible for natural x
The State Equation
Karsten Wolf: CEGAR / PN State Equation

5. 1
Example
t1 t3
t5
t2 t4
reachable marking: corresponding solutions
(1,0,0,0,0) ( 0 , 0 , 0 , 0 , 0), ( 1 , 0 , 1 , 0 , 1), ...
(0,1,0,0,1) ( 1 , 0 , 0 , 0 , 0), ...
(0,0,1,1,0) ( 0 , 1 , 0 , 0 , 0), ...
(0,0,0,1,1) ( 1 , 0 , 1 , 0 , 0), ( 0 , 1 , 0 , 1 , 0), ...
not reachable has solution, though! :-(
(0,1,1,0,0) ( 1 , 1 , 0 , 0 , 1), ...
Karsten Wolf: CEGAR / PN State Equation

6. 1
Solution Space
b1, b2, …, bj: base solutions: incomparable, minimal
i1, i2, …, ik: increments, solutions to C • x = 0
bi + n1 i1 + … nk ik
Karsten Wolf: CEGAR / PN State Equation

7. 1
The role of increments
t1 t2
u1 u2
t1 t2 unrealizable „to lend tokens“
u1 t1 t2 u2 realizable
Karsten Wolf: CEGAR / PN State Equation

8. CEGAR
Counterexample Guided
Abstraction Refinement 5
? -
-
+
1
2
4
?
3
-
x +
+
Karsten Wolf: CEGAR / PN State Equation

9. 2
Checking solutions
?
Given solution x, explore state space, but fire t at most x(t) times
-Apply partial order reduction technique of [Schmidt, Petri Nets `99]
.... does not insert invisible transitions
.... does not require invisible transitions
-Skip sequences that are covered by other solutions
Karsten Wolf: CEGAR / PN State Equation

10. CEGAR
Counterexample Guided
Abstraction Refinement 5
? -
-
+
1
2
4
?
3
-
x +
+
Karsten Wolf: CEGAR / PN State Equation

11. 3
Refinement
x
increment jump
Karsten Wolf: CEGAR / PN State Equation

12. 3
Problem: Increment after Jump
x
b jump
b' increment?
transform jump!
Karsten Wolf: CEGAR / PN State Equation

13. CEGAR
Counterexample Guided
Abstraction Refinement 5
? -
-
+
1
2
4
?
3
-
x +
+
Karsten Wolf: CEGAR / PN State Equation

14. Tool: Sara 4
(Structures for Automated Reachability Analysis)
http://service-technology.org/tools/download
Experiments:
-590 business processes (20-300 transitions)
(thanks to Jana Köhler, IBM)
checked for „relaxed soundness“ (510 are, 80 are not):
Sara: 198 sec LoLA: failed on 17 instances
24 min on the others
Hardest instance:
12278 calls to lp_solve, 24 sec
Karsten Wolf: CEGAR / PN State Equation

15. Tool: Sara 4
(Structures for Automated Reachability Analysis)
http://service-technology.org/tools/download
Experiments:
-4 models with context in verification of parameterized boolean
programs
(thanks to Daniel Kröning, Alexander Kaiser)
checked for coverability
Sara: 0 sec LoLA: failed on 1, 0 sec on the others
Karsten Wolf: CEGAR / PN State Equation

16. Tool: Sara 4
(Structures for Automated Reachability Analysis)
http://service-technology.org/tools/download
Experiments:
-1 challenge example from Petri Net mailing list
(thanks to Hubert Garavel)
776 transitions checked for quasi-liveness
Sara: 26 sec LoLA: 41 sec but tricks were needed on
2 instances
Witness paths:
Sara: <30 LoLA: up to 6000
Karsten Wolf: CEGAR / PN State Equation

17. CEGAR
Counterexample Guided
Abstraction Refinement 5
? -
-
+
1
2
4
?
3
-
x +
+
Karsten Wolf: CEGAR / PN State Equation

18. Diagnosing "no" 5
-
i
o
Karsten Wolf: CEGAR / PN State Equation

19. Conclusion
Applied CEGAR to a structural verification technique
+ performant
+ short witnesses
+ excellent behavior on negative instances
+ applicable to infinte state systems
+ diagnostic information in negative cases
- inherently incomplete ( reachability for Petri nets is EXSPACE hard)
Traverse solutions rather than states
Karsten Wolf: CEGAR / PN State Equation