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

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

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+

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’

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

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

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

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

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

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

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)

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

Example: Re-unfolding  

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

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

Concurrent insertion Can introduce unsafeness Can introduce deadlocks

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

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

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

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

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

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

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 

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

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

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

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

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

Thank you! Any questions?