1. Yongjian Li The State Key Laboratory of Computer Science Chinese Academy of Sciences William N. N. HungSynopsys Inc. Xiaoyu SongPortland State University Presented by Yongjian Li

2. Outline Introduction A formal netlist model Syntax and semantics of trajectory logic Symmetry reduction Applying symmetry reduction automatically Case study on CAMs Conclusion and future work 2

3. STE overview X value + symbolic simulation Provides a theoretical foundation for symbolic evaluation of partially ordered state space Used in Intel, Motorola etc Directly using EXLIF netlist as circuit model Specification is as Impoverished temporal logical specification Poweful capacity Sucesfully used for data-dominated circuits 3

4. Related work Classical semantic work in STE literature (Carl Seger et al., 1995; Mark D. Aagaard et al., 2002. ) usually assuming a next state function Y core techniques: symbolic indexing and parametric representation combining with theorem proving rather cumbersome to reason about combinational parts of a circuit A closure semantics on a netlist model(Roorda and Classen, 2005,2006) a closure function from the structure of a netlist, which can be seen as a special next state function convenient in reasoning about combinational parts sat-based refinement 4

5. Related work(cont.) Symmetry reduction in STE (Pandey 1997 ) use sub-graph isomorphism detecting symmetry manually did not answer why symmetry in circuit netlist structure implies symmetry in next state function Symmetry reduction in STE (Darbari 2006) propose a structured model -- a high level modeling language recording the symmetry of a circuit make a connection from the model to STE : proving the symmetry in the structured model derives symmetry in the corresponding next-state function 5

6. Our contribution A formal BLIF netlist model in Isabelle Formally define the structural symmetry A soundness theorem guaranteeing the correctness of symmetry reduction Applying symmetry reduction automatically as a tactic in Forte 6

7. A formal netlist model 7

8. Next state function Given a netlist nl, a next state function Y can be formally induced, which is a closure function, i.e., Y is monotonic. Y is idempotent. Y is extensive. (See Rooda et al, 2005 and Li et al, 2009 for the detail) 8 nl: a netlist g 1 : gate c a b tab 1 L 2 : Delay c’ c …. Y: a next state function Y s c’ = s (a) & s( b ) ….

9. Syntax and Semantics of trajectory formula 9

10. Syntax and Semantics of trajectoty assertion 10

11. Causal Subnetlist on an Assertion 11

12. Example 12

13. Evaluating an assertion in its causal subnetlist 13

14. Symmetric structures 14

15. Motivating Example 15 nl 0 nl 1

16. Symmetry properties 16

17. Symmetry reduction 17

18. Motivating Example 18

19. Substitutions on Trajectory Formulas 19 In forte, sometimes we need do substitution on Boolean guards in STE assertion in some context. After a substitution {ba 1 /ba 0,bb 0 /bb 1 }

20. Applying Symmetry Reduction Automatically--problem Have a lemma: Try to prove another lemma: 20

21. Overall strategy 21

22. Implementation 22

23. Codes –main body 23

24. Codes (1) –matching formulas 24

25. Codes (2) –Computing symmetry 25

26. Codes (2) –Computing symmetry 26

27. Case study on CAMs 27

28. Case study on CAMs 28 A fully-encoding style specification:

29. Symmetry Reduction in CAMs 29 Reduced to

30. Discussion For n-t-d CAMs (n – entries, t- tag width, d- data width) in full encoding style Need theorem proving techniques to make assertions can be applied by symmetry reduction Structure symmetry between two bits of a bit- vector such as match, dout in CAMS 30 No Sym reductionSym reduction For property on hitn*t2*t For property on dout(t+d)*n+t2t+d

31. Discussion (continued) Exploring structure symmetries needs time especially, when the properties checked involves many Next operators Compared and related with symbolic indexing Need human guidance to make assertions to be applied by symmetry reduction in ours, then can be done automatically Need human guidance to make an index symbolic indexing assertion, then fully automatic ally checked by running STE One indexing case is symmetric to each other 31

32. Conclusion and future work Theoretical result of symmetry reduction Formalize the correspondence between structure symmetry and property symmetry reformulated in a netlist based closure semantics framework Automatic symmetry reduction works on a netlist model, automatically checking structure symmetry online use heuristics inherited in verification problem itself 32

33. Conclusion and future work Link Isabelle with Forte? take both the advantage of Isabelle' strong theorem proving and Forte's symbolic simulation features difficulty: interface between each other Need more detail of Forte's document (but Forte can not be downloaded now!) 33

34. Question & Answer Thank You ! 34