Hardware and Petri nets

1. Hardwareand Petri nets Symbolic methods foranalysis and verification

2. Outline • Representing Boolean functions with BDDs • Symbolic traversal for reachability set calculation • State encoding • Structural methods for efficient encoding

3. 1 0 0 1 1 0 0 1 Representing Boolean functions a 1 0 b b 1 0 1 0 c c c c 1 0 1 0 1 0 1 0

4. 0 1 1 0 Representing Boolean functions a 1 0 b b 0 1 1 0 c c 1 0 1 0

5. 0 1 1 0 Representing Boolean functions a 1 0 b 0 1 c c 1 0 1 0

6. 0 1 Representing Boolean functions a 1 0 b 1 0 c c 0 0 1 1

7. 1 0 Binary Decision Diagrams • All variables appear in the same order • No isomorphic subgraphs • Canonical form • Efficient form for many functions • [Bryant,ACM Comp. Surveys,September 1992] a 1 0 b 1 0 c c 0 0 1 1 Reduced Ordered Binary Decision Diagram

8. p1 t2 t1 p2 p3 p4 p5 t3 t4 t7 t5 t6 p2 p7 p6 p3 p6 p5 p4 p7 t6 t4 t3 t5 p6 p7 Reachable markings p1 t1 t2 p4 p5 p2 p3 t6 t4 t5 t3 p7 p6 t7

9. 1000000 t2 t1 0110000 0001100 t3 t4 t7 t5 t6 0010010 0000110 0001001 0100001 t6 t4 t3 t5 Seven variables: p1 p2 p3 p4 p5 p6 p7 0000011 Boolean encoding p1 t1 t2 p4 p5 p2 p3 t3 t6 t4 t5 p7 p6 t7

10. 1000000 p1 p2 p3 (p4 p6) (p5 p7)  t2 t1 Enabled(t7) = p6 p7 0110000 0001100  t3 t4 t7 t5 t6 p1 p2 p3 p4 p5 p6 p7 0010010 0000110 0001001 0100001 (toggle p1, p6 , p7) t6 t4 t3 t5 Seven variables: p1 p2 p3 p4 p5 p6 p7 0000011 p1 p2 p3 p4 p5 p6 p7 Boolean encoding

11. Symbolic Traversal (BFS algorithm) Reached = From = {m0 }; repeat From [ T  To; New = To \ Reached; From = New; Reached = Reached  New; untilNew = Ø; # iterations: sequential depth of the net

12. New markings #iteration Reachability Set computation • Based on BFS Image computation S0 = M0 Si+1 = Si Image (Si) S3 S2 S1 S0 Monotonic increase until fix point Si+1 = Si

13. Sparse encoding: 1000000 000 Seven variables: p1 p2 p3 p4 p5 p6 p7 0110000 0001100 001 100 Optimal encoding: 011 110 010 101 0010010 0000110 0001001 0100001 Three variables log2 |RG|: v1 v2 v3 111 0000011 Boolean encoding t2 t1 Very easy to derive and use Less efficient in terms of BDDs t3 t4 t7 t5 t6 t6 t4 t3 t5 But the reachability graphshould be known a priori ...

14. p1 p2 p3 p4 p3 p4 p1 p2 Ø Encoding for safe PNs Observation: • Not all combinations of tokens are possible. • Find relations among places to reduce the number of variables!!!

15. Encoding for safe PNs PN structure: Place Invariant • Set of places with a constant weighted sum of tokens k1p1+ k2p2+…..+ knpn = B • Specially efficient for safe PNs (State Machines) p1+ p2+…..+ pn = 1 • Computed by linear programming techniques

16. p1 p1 p1 p1 p1 t1 t1 t1 t2 t2 t2 t1 t2 t1 t2 p4 p4 p4 p4 p5 p5 p5 p2 p2 p2 p3 p3 p3 p2 p5 p3 t3 t3 t3 t6 t6 t6 t3 t4 t4 t4 t5 t5 t5 t5 t6 t4 p7 p7 p7 p6 p6 p6 p6 p7 t7 t7 t7 t7 t7 State Machine Components

17. 00 00 p1 p1 t1 t2 t1 t2 10 01 10 01 p4 p2 p5 p3 t3 t5 t6 t4 11 11 p6 p7 t7 t7 Encoding for safe PNs Two additional variables: v3 v4 Two variables: v1 v2

18. 0000 p1 0000 t2 t1 t1 t2 10 - - - - 10 1010 0101 p4 p5 p2 p3 - - 01 01 - - t3 t4 t7 t5 t6 t3 t6 t4 t5 1110 1101 0111 1011 p7 p6 t6 t4 t3 t5 11 - - - - 11 t7 1111 Encoding for safe PNs Four variables: v1 v2 v3 v4

19. t3 t2 p2 2010 3002 2104 2 t1 t2 t1 t3 t1 t3 t1 t2 p1 2 0120 1112 0214 1206 2 p4 p3 t3 t3 t2 t3 Dense encoding: 3 variables Sparse encoding: 10 variables max(p1)=3 max(p3)=2 max(p2)=3 max(p4)=8 Invariants of the PN: 010 000 001 I1: 2p1 + 4p2 - p4 = 4 I2: p1 + p2 + p3 = 3 111 100 110 011 Encoding for bounded PNs

20. Dense encoding: 3 variables Invariants of the PN: I1: 2p1 + 4p2 - p4 = 4 I2: p1 + p2 + p3 = 3 Encoding for bounded PNs Invariants characterize the set of all potentially reachable markings (an overestimation of the reachable markings) t3 t2 2010 3002 2104 0308 t2 t1 t3 t1 t3 t1 t2 0120 1112 0214 1206 t3 t2 t3 Dense encoding: 3 variables

21. p3 p4 p2 p1 p2 p2 p2 p4 p4 p4 p2 p4 p4 p4 p3 p3 p2 p4 p1 p4 p4 0 3 3 1 2 2 1 2 3 1 2 0 1 2 0 1 0 1 0 0 4 8 2 6 0 4 8 2 6 1 0 0 8 6 7 4 6 5 9 5 3 0 7 2 1 Encoding for bounded PNs 2p1 + 4p2 - p4 = 4 (8 vars) p1 + p2 + p3 = 3 3 variables 4 variables

22. p1 + p2 + p3 = 3 M(p1)+M(p2) p3 p1 p2 p2 p3 p2 p3 p3 p3 p3 p3 p2 p3 p3 p3 p3 1 2 3 0 1 2 3 2 1 0 1 2 3 1 2 0 1 0 2 1 0 1 0 1 0 0 0 1 3 2 2 4 0 7 4 3 1 Encoding for bounded PNs

23. T F T F x1 x2 x2 x3 x3 x3 x3 x2 x2 x3 x3 x1 x3 x3 T F T F T F T F T F T F T F T F T F T F T F T F p1 p3 2 0 2 1 2 3 0 0 3 0 1 2 1 1 1 0 0 0 0 1 0 1 1 2 p2 Encoding for bounded PNs 2p1 + 4p2 - p4 = 4 p3 = 3 - p1 - p2

24. F x1 x3 x4 x4 x3 x2 x3 x4 x4 x3 x4 x2 x4 x4 x4 x1 x2 x3 T F T T F T F T F T F T F T F T F T F F T T T T T T T T T T T T F T F T F T F T F T F T F T F F F F F F F Upper bounds for the state space 2p1 + 4p2 - p4 = 4 Characteristic function for potentially reachable states

25. Experimental results

26. Conclusions • Formal verification and synthesis oftensuffer from the state explosion problem • Symbolic techniques can be used to efficientlyrepresent the state space • Structural techniques are crucial to overestimateand encode the state space • Try to resist the temptation of using BDDs fromthe very beginning. Use them rationally and onlyif desperate.