1. A New Type of Behaviour- Preserving Transition Insertions in Unfolding Prefixes Victor Khomenko

2. 2 Motivation Some design methods based on PNs repeatedly execute the following steps:  Analyze the PN  Modify the PN by inserting a new transition in a behaviour-preserving way: – bisimilar – safe E.g: synthesis of asynchronous circuits:  Resolution of encoding conflicts  Logic decomposition of complex gates

3. 3 Unfolding prefixes Using unfolding prefixes for analysing the PN  alleviate state space explosion  efficient analysis and synthesis algorithms  good for visualisation It is desirable to avoid re-unfolding after each transformation, i.e. to mirror the PN transformation by local transformations in the unfolding prefix  re-unfolding is time-consuming  good for visualisation (re-unfolding can dramatically change the look of the prefix)  can transfer information (e.g. encoding conflicts) between the iterations of the algorithm  the difficult bit and the main point of the paper

4. 4 Example: VME Bus Controller lds-d-ldtack-ldtack+ dsr-dtack+d+ dtack-dsr+lds+ Device VME Bus Controller lds ldtack d Data Transceiver Bus dsr dtack

5. 5 Example: Encoding Conflict dtack-dsr+ dtack-dsr+ dtack-dsr+ 00100 ldtack- 00000 10000 lds- 01100 01000 11000 lds+ ldtack+ d+ dtack+dsr- d- 01110 01010 11010 011111111111011 11010 10010 M’’M’

6. 6 Example: Resolving the conflict lds-d-ldtack-ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc-

7. 7 Example: Resolving the conflict dtack-dsr+ dtack-dsr+ dtack-dsr+ 001000 ldtack- 000000 100000 lds- 011000 010000 110000 lds+ ldtack+ d+ dtack+dsr- d- 011100 010100 110100 011111111111110111 110101 100101 011110 csc+ csc- 100001 M’’M’

8. 8 Complex-gate implementation Device d Data Transceiver Bus dsr dtack lds ldtack csc May be not in the gate library and has to be decomposed

9. 9 Naïve decomposition is hazardous d dsr dtack lds ldtack csc x lds-d-ldtack-ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc- Unexpected!

10. 10 Decompose at the PN level! d dsr dtack lds ldtack csc dec lds-d-ldtack- ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc- dec+ dec- Insert a new signal dec whose implementation is [dec] = ldtack + csc

11. 11 Previous work: Transformations Sequential pre-insertionSequential post-insertion Concurrent insertion

12. 12 Previous work: main results Validity criteria  safeness  bisimilarity The validity can be checked before the transformation is performed, i.e. on the original prefix (to avoid backtracking) Avoiding re-unfolding The suite of transformations is good in practice for resolution of encoding conflicts

13. 13 Motivation for more transformations The suite of transformations is not sufficient for logic decomposition; intuitively:  only linear (in the PN size) number of sequential pre- and post-insertions (assuming that the pre- and postset sizes are bounded)  only quadratic (in the PN size) number of concurrent insertions  exponential number of ‘cuts’ in the PN where a Boolean expression can change its value

14. 14 Example: imec-sbuf-ram-write dec+ dec- dec Implementation of prbar: (csc2  req)  csc1  wsldin imec-sbuf-ram-write req precharged done wsldin wenin prbar wen wsen ack wsld

15. 15 Lock relation Transitions t and t’ are in the lock relation, denoted t  t’, if in every execution of the PN:  their occurrences alternate  the first occurrence of t precedes that of t’ A conservative approximation of  can be efficiently computed from the prefix The approximation is exact in the live case

16. 16 Generalised transition insertion (GTI) s1s1 s2s2 s3s3 d1d1 d2d2 For each s i : either s i  d j for each d j or d j  s i for each d j GTIs preserve safeness and yield bisimilar PNs It is possible to mirror a GTI by local transformations in the unfolding prefix sources destinations

17. 17 Computing useful GTIs There can be exponentially many GTIs (in the size of the PN):  more likely that an appropriate transformation exists  no longer practical to enumerate them all  How to enumerate only the ‘potentially useful’ (for a particular application) GTIs? Can be done in two steps:  computing sources (application-dependent, but the main idea is quite general)  computing the destinations for given sources (relatively independent on the application)

18. 18 Computing sources (logic deco) Given a Boolean expression E, find a position where E changes its value, i.e. a configuration C whose causally maximal events e correspond to the support of E and are such that E(C\{e})  E(C) Reducible to SAT When a solution is found:  add a clause removing all the solutions yielding the same sources  run the solver again  repeat until all possible sets of sources are computed Incremental SAT

19. 19 Computing destinations Given a set of sources S, compute all possible sets of destinations yielding a valid GTI  W.l.o.g: no two transitions in S are locked Compute the set L S = {d |  s  S : s  d  d  s} of transitions locked with each source Build the graph on L S : d 1 and d 2 are connected by an edge iff they are not locked with each other and are consistently locked with each source (i.e. s  d 1 iff s  d 2 ) Enumerate the non-empty cliques of this graph

20. 20 Summary of contributions A method for computing the approximated lock relation using an unfolding prefix; it is conservative, and exact in the live case A new kind of transition insertion preserving safeness and yielding a bisimilar PN An algorithm for applying a GTI on the existing prefix, avoiding thus (expensive) re-unfolding A method for computing ‘potentially useful’ GTIs in the context of logic decomposition; some parts of this method are relatively independent on the application These contributions form a complete framework for efficient use of GTIs together with the transformations developed earlier

21. 21 Thank you! Any questions?