1. Behaviour-Preserving Transition Insertions in Unfolding Prefixes
Victor Khomenko
University of Newcastle upon Tyne

2. Motivation
Some design methods based on Petri nets repeatedly execute the following steps:
Analyze the original PN spec
Modify the PN by behaviour-preserving transition insertion

3. Example: VME Bus Controller
Device
VME Bus
Controller
lds
ldtack d
Data Transceiver
Bus
dsr
dtack
lds- d-
ldtack-
ldtack+
dsr-
dtack+ d+
dtack-
dsr+
lds+

4. Example: Encoding Conflict
dtack-
dsr+
00100
ldtack-
00000
10000
lds-
01100
01000
11000
lds+
ldtack+ d+
dtack+
dsr- d-
01110
01010
11010
01111
11111
11011
10010
M’’ M’

5. State Graphs vs. Unfoldings
Relatively easy theory
Many efficient algorithms
Not visual
State space explosion problem

6. State Graphs vs. Unfoldings
Alleviate the state space explosion problem
More visual than state graphs
Proven efficient for model checking
Quite complicated theory
Not sufficiently investigated
Relatively few algorithms

7. Example: Encoding Conflict
dtack-
dsr+ e1 e2 e3 e4 e5 e6 e7
e12
dsr+
lds+
ldtack+ d+
dtack+
dsr- d-
lds+
Code(conf’)=10110
Code(conf’’)=10110
lds-
ldtack- e9
e11

8. Example: Resolving the conflict
lds- d-
ldtack-
ldtack+
dsr-
dtack+ d+
dtack-
dsr+
lds+
csc+
csc-

9. Example: Resolving the conflict
dtack-
dsr+
csc+
001000
000000
100000
100001
lds+
ldtack-
ldtack-
ldtack-
dtack-
dsr+
011000
100101
010000
110000
ldtack+
lds-
lds-
lds-
dtack-
dsr+
M’’ M’
011100
110101
010100
110100 d+ d-
csc-
dsr-
dtack+
011110
011111
111111
110111

10. Example: Resulting Circuit
Data Transceiver
Device
Bus d
lds
dtack
dsr
csc
ldtack

11. Motivation: validity Need to check the validity of the transformation
safeness
bisimulation
The validity should be checked before the transformation is performed, i.e. on the original prefix (to avoid backtracking)

12. Motivation: avoid re-unfolding
Perform the transformation directly on the prefix to avoid re-unfolding
Re-unfolding is time-consuming
Good for visualization (re-unfolding can dramatically change the look of the prefix)
Can transfer information (e.g. encoding conflicts) between the iterations of the algorithm

13. Example: Re-unfolding 

14. Sequential pre-insertion
Preserves safeness
Preserves traces
Can introduce deadlocks: need to check that the new transition never ‘steals’ tokens from any other enabled transition
simple state property
can be checked on the original prefix

15. Sequential post-insertion
Preserves safeness
Yields a bisimular PN
Nothing to check!

16. Concurrent insertion
Can introduce unsafeness
Can introduce deadlocks

17. Place insertion: token
If the place insertion is valid and t’ or t’’ is not dead then p contains token iff there is a t’’-labelled event in the prefix which does not have t’-labelled predecessor

18. Place insertion: validity
Tokens(C)=n + #t’C – #t’’C
The transformation is valid if:
for all instances e of t’ and t’’ of the prefix, Tokens([e]){0,1}, and
for all cut-offs e with a corresponding configuration C, Tokens([e])=Tokens(C)
If a valid transformation is rejected by this criterion then t’ and t’’ are not live

19. Pre-insertion in the prefix
Naïve splitting can yield an incomplete prefix!

20. Pre-insertion in the prefix
Naïve splitting can yield an object which is not a branching process!

21. Pre-insertion in the prefix
Find all possible extensions of the prefix by the new transition
Amend the instances of the split transitions
Amend the cut-off corresponding configurations

22. Post-insertion in the prefix
Naïve splitting can yield an incomplete prefix!  

23. Post-insertion in the prefix
Definition: a configuration is extendible if in the modified prefix it can be extended by an instance of the new transition
If there is a cut-off event e with a corresponding configuration C such that [e] is extendible and C is not extendible then terminate unsuccessfully
Amend the instances of the split transition
Amend the cut-off corresponding configurations 

24. Place insertion in the prefix
Assumption: the place insertion has passed the validity check
If n = 1 then create a new (causally minimal) instance cmin of p
For each instance e of t′ (including cut-offs), create a new instance of p and connect it to e
For each instance e of t′′ (including cut-offs): connect e to cmin if e has no t′-labelled predecessor and to the instance of p in the postset of the (unique) maximal t′-labelled predecessor of e otherwise

25. Concurrent insertion in the prefix
Perform the corresponding place insertion
Perform the sequential pre-insertion
This two steps can easily be combined p t’
t’’ n

26. Equivalent insertions
Equivalence is easy to check
Fewer transformations to consider
Can convert to ‘canonical form’, e.g. pre-insertions – good for unfolding
No need to check validity – post-insertions are always valid

27. Commutative insertions
Definition: two transition insertions commute if they can be performed in any order
concurrent insertions commute with any other insertions
pre-insertions commute with post-insertions
two pre/post-insertions commute iff they split different transitions or the sets of split off places do not overlap
A valid insertion remains valid if another valid commutative insertion is applied first, i.e. the validity needs to be checked only once

28. Summary Rigorous validity criteria developed
can be checked on the original prefix – no backtracking
Algorithms for performing transformations directly on the prefix
avoids re-unfolding, good for performance and visualization
proofs of correctness
Optimisation
equivalent transformations
commutative transformations

29. Thank you!
Any questions?