Automatically Exploring Structural Symmetry in Symbolic Trajectory Evaluation

1. Automatically Exploring Structural Symmetry in Symbolic Trajectory Evaluation Yongjian Li The State Key Laboratory of Computer Science Chinese Academy of Sciences William N. N. Hung Synopsys Inc. Xiaoyu Song Portland 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

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 • Sucesfullyused for data-dominated circuits

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

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

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

7. A formal netlist model

8. Next state function Given a netlistnl, 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) Y: a next state function Y s c’ = s (a) & s( b ) …. nl: a netlist g1: gate c a b tab1 L2: Delay c’ c ….

9. Syntax and Semantics of trajectory formula

10. Syntax and Semantics of trajectoty assertion

11. Causal Subnetlist on an Assertion

12. Example

13. Evaluating an assertion in its causal subnetlist

14. Symmetric structures

15. Motivating Example nl0 nl1

16. Symmetry properties

17. Symmetry reduction

18. Motivating Example

19. Substitutions on Trajectory Formulas In forte, sometimes we need do substitution on Boolean guards in STE assertion in some context. After a substitution {ba1/ba0,bb0/bb1}

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

21. Overall strategy

22. Implementation

23. Codes –main body

24. Codes (1) –matching formulas

25. Codes (2) –Computing symmetry

26. Codes (2) –Computing symmetry

27. Case study on CAMs

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

29. Symmetry Reduction in CAMs Reduced to 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

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

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

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!)

34. Thank You ! Question & Answer