1. 7. Implementation

2. Plan
• Firing a transition
• Evaluating a state predicate
• Managing the state space
• Organizing search
• Detecting strongly connected components

3. Firing transitions
Marking changed via list of pre-, list of post-
places  effort does not depend on size of
net
After firing, only some transitions are checked
for enabledness
previously enabled transitions that lost
tokens
previously disabled transitions that gained
tokens
... managed through explicitly stored lists

4. Checking state predicates
• predicate = boolean combination of
• p {><=≤≥≠} k
• stored in negation-free normal form
φ
φ φ
φ φ


5. Managing the state space
1st state = bit vector
other states = bit vector +decision record

6. Managing the state space
find/insert a marking: one integrated process
dive down into decision tree
on mismatch:
at decision point: switch to next vector
at end: found, no insert
between decision points: insert at point of mismatch
decision records form tree

7. Organizing search
General remarks
Search consists of
- fire transitions ✔
- find/insert marking ✔
- backtracking: fire transition backwards
 only „constant“ time
search stack consists of reference to
transition +
list of enabled transitions
 state space is „write-only“ memory

8. Organizing search
b) Depth-first search: ability to detect SCC
c) Breadth-first search:
Simulated by bounded depth-first search with
incrementally increased bound
 Update of current marking, list of enabled
transitions, etc. through sequence of transition
occurrences

9. Detecting strongly connected
components
• Traditional approach: Tarjan‘s algorithm
4
4 6 21
5 
44
33 11
00

10. Detecting strongly connected
components
• LoLA approach: simplified lowlink
4
4 6 21
5 
41
31 11
00

11. Reduction techniques

12. Stubborn Sets
• Crucial: Core principle
• Simple method:
– If t enabled, add conflicting transitions
– If t disabled, add pre-transitions of some
unmarked pre-place
place pre-transitions
must be included
transition conflicting
updated at enabledness check

13. The sweep-line method
• constant change  successors lie in a small window of
progress values

14. Calculation of Symmetries
7 5 0 2 A1 ∪.... ∪ An = V
A1 B1
9 4 B1 ∪....∪ Bn = V
A2 1 4 3
2 6 B2 σ satisfies C iff
6 9 8
σ(Ai) = Bi (for all i)
07
3 7 B3 13
A3 0 29
5 35
48
..... 50
66
An Bn
8 1 74
81
92
C

15. Abstract Permutation – Examples

16. Abstract Permutation – Examples
PP all permutations that
TT respect node type

17. Abstract Permutation – Examples
PP all permutations that
TT respect node type
{p1}  {p1}
...... Elements of some orbit
{pi-1}  {pi-1} wrt. Ui in Ui-1
{pi}  {pk}
others1  others2

18. Abstract Permutation – Examples
PP all permutations that
TT respect node type
{p1}  {p1}
...... Elements of some orbit
{pi-1}  {pi-1} wrt. Ui in Ui-1
{pi}  {pk}
others1  others2
New problem: given.: abstract permutation C
compute an automorphism that satisfies C
... equivalent to graph isomorphism

19. REFINE
Choose A-B, A’-B’ and arc multiplicity c
# c-neighbors in A’ #c-neighbors in B’
3 2 1 0 0 1 2 3
A B
8 9 8 6
6 1
0 5 4 2
1 2 4 3 7 0
8 7
2 5 4 B’
A’ 1 0
3

20. REFINE
Choose A-B, A’-B’ and arc multiplicity c
# c-neighbors in A’ #c-neighbors in B’
3 2 1 0 0 1 2 3
A B
8 9 8 6
6 1
0 5 4 2
1 2 4 3 7 0
8 7
2 5 4 B’
A’ 1 0
3
Every automorphism that satisfies A-B, satisfies all resulting
constraints

21. DEFINE
Choose A-B,
p ∈A 2 5 1 2
3 8 4 5

22. DEFINE
Choose A-B,
p ∈A 2 5 1 2
3 8 4 5
2 5 1 2 2 5 1 2
3 8 4 5 3 8 4 5
2 5 1 2 2 5 1 2
3 8 4 5 3 8 4 5

23. DEFINE
Choose A-B,
p ∈A 2 5 1 2
3 8 4 5
2 5 1 2 2 5 1 2
3 8 4 5 3 8 4 5
2 5 1 2 2 5 1 2
3 8 4 5 3 8 4 5
Every automorphism tat satisfies A-B,
satisfies one of the new constraints

24. Computation tree
4 3
2
poly
1 8 1
6 7 8
5
a b f a
e c
R*
D
R* R* R*
D D
R* R* R* R* R*
2 3 2 3
3 8 3 8
c c c c
exp
= #Ai ! #Bi is rare. That is, often polynomial run time

25. Combination of techniques

26. More information
• LoLA: ICATPN 2000, Petri Nets 2007
– www.informatik.uni-rostock.de/tpp/
• Stubborn sets: ICATPN 1999, Fundamenta Informaticae 2000, FMSD 2006
• Symmetries: Acta Informatica 2000, TACAS 2000
• Linear Algebra: TACAS 2003
• Sweep-Line: TACAS 2004, STTT
• Coverability graphs: FMSD 1999
• Services: BPM 2005
• GALS: ASYNC Krstic et al, 2005
• Bio-chemistry: Talcott & Dill, 2005
• Workflow: BPM 2009