1. Variable-Time-Frame Gate-Level Abstraction Alan Mishchenko Niklas Een Robert Brayton Alan Mishchenko Niklas Een Robert Brayton UC Berkeley UC Berkeley Jason Baumgartner Hari Mony Pradeep Nalla IBM IBM

2. 2 Overview Introduction Introduction Motivation Motivation Algorithm Algorithm Experimental results Experimental results Conclusion Conclusion

3. Abstraction Finding a subset of logic gates of the miter, large enough to complete the proof

4. Taxonomy of Abstraction Methods Automatic vs. manual SAT-based vs. BDD-based vs. other Proof-based vs. CEX-based vs. hybrid Flop-level vs. gate-level Fixed time-frame vs. variable time-frame

5. The Proposed Approach is… Automatic SAT-based Hybrid Gate-level Variable time-frame

6. 6 Previous Work Flop-level abstraction Flop-level abstraction N. Een, A. Mishchenko, and N. Amla, "A single-instance incremental SAT formulation of proof- and counterexample- based abstraction", Proc. FMCAD'10. N. Een, A. Mishchenko, and N. Amla, "A single-instance incremental SAT formulation of proof- and counterexample- based abstraction", Proc. FMCAD'10. Gate-level abstraction Gate-level abstraction J. Baumgartner and H. Mony, “Maximal Input Reduction of Sequential Netlists via Synergistic Reparameterization and Localization Strategies”. Proc. CHARME’05, pp. 222-237. J. Baumgartner and H. Mony, “Maximal Input Reduction of Sequential Netlists via Synergistic Reparameterization and Localization Strategies”. Proc. CHARME’05, pp. 222-237.

7. 7 Motivation Flop-level abstraction is too crude Flop-level abstraction is too crude Adds too much logic to the abstracted model Adds too much logic to the abstracted model (but refinement with external CEXes is easier…) (but refinement with external CEXes is easier…) Gate-level abstraction is also too crude Gate-level abstraction is also too crude Includes all abstracted logic in each time-frame Includes all abstracted logic in each time-frame Solution: “Variable-time-frame” gate-level abstraction Solution: “Variable-time-frame” gate-level abstraction Adds logic to each time-frames on demand (a gate may be added in one time-frame but not in others) Adds logic to each time-frames on demand (a gate may be added in one time-frame but not in others)

8. Improved BMC In the classical BMC, in each timeframe, we add the complete “tent” (bounded cone-of-influence) experiments show that a small fraction of this logic (typically, 5-20%) is enough to prove the problem UNSAT This motivates a smarter approach add logic on-demand This may reduce the SAT solver size substantially, resulting in a faster and more scalable BMC Frame 0 Frame 1 Frame 2 Frame 3

9. Deciding What Logic to Add It is enough to add only logic in the UNSAT cores But we do not know what is the next UNSAT core We use previous cores: Lift K previous UNSAT cores to the given level If the problem is still SAT, refine it by selectively adding gates to time-frames Use the rollback feature of SAT solver to include the minimal amount of logic UNSAT core of Frame 0 UNSAT core of Frame 1 UNSAT core of Frame 2 UNSAT core of Frame 3

10. Improved Gate-Level Abstraction Use the variable-time-frame approach to BMC Then, build a gate-level abstraction, by taking the union of all gates, present in any time-frame

11. Improved Interpolation Interpolation-based model checking can benefit from the variable-time-frame approach to BMC When the transition relation is unrolled, there is no need to add all logic in the COI of the property The proposed approach can be used to decide what logic to include As a result The SAT problem becomes simpler The intermediate interpolants becomes smaller

12. 12 Experimental Results abc 01> read ex1.aig; ps ex1: i/o = 1570/ 1 lat = 3113 and = 16745 lev = 31 abc 02> pdr Invariant F[29] : 5033 clauses with 734 flops (out of 3113) Property proved. Time = 808.01 sec abc 03> read ex1.aig; ps ex1: i/o = 1570/ 1 lat = 3113 and = 16745 lev = 31 abc 04> &vta -S 5 -P 2 -F 45 -v Solver UNSAT = 1.49 sec ( 14.50 %) Solver SAT = 2.57 sec ( 24.94 %) Refinement = 5.37 sec ( 52.17 %) Other = 0.86 sec ( 8.37 %) TOTAL = 10.29 sec (100.00 %) SAT vars = 36976. Clauses = 92646. Confs = 5074. Used 0.75 Mb for proof-logging. abc 05> &vta_gla; &ps; &gla_derive; &put; pdr Gate-level abstraction: PI = 1 PPI = 66 FF = 143 (4.59 %) AND = 505 (3.02 %) Invariant F[22] : 545 clauses with 114 flops (out of 143) Property proved. Time = 3.92 sec

13. abc 02> &r ex1.aig; &ps abc 02> &vta -S 5 -P 2 -F 45 -v Frame Confl One Cex All 0 : 0 7 0 6 1 : 0 11 0 11 2 : 0 66 0 80 3 : 0 73 0 31 4 : 0 84 0 135 5 : 0 90 0 61 6 : 0 90 0 71 7 : 0 96 0 100 8 : 0 96 0 116 9 : 0 104 0 152 10 : 0 104 0 174 11 : 0 112 0 219 12 : 0 112 0 249 13 : 0 139 3 323 14 : 0 139 0 360 15 : 0 150 100 555 16 : 0 150 0 572 17 : 0 150 0 674 18 : 0 150 0 692 19 : 0 150 0 831 20 : 0 150 0 849 21 : 16 536 131 2112 22 : 0 536 2 2131 23 : 51 602 36 3057 24 : 2 602 0 3080 25 : 147 617 9 3783 26 : 2 617 0 3806 27 : 118 628 22 4581 28 : 2 628 0 4602 29 : 144 629 1 5259 30 : 2 629 0 5290 31 : 125 635 7 5851 32 : 2 635 0 5929 33 : 160 640 1 6549 34 : 3 640 0 6570 35 : 212 650 11 7274 36 : 2 650 0 7295 37 : 217 650 0 7931 38 : 3 650 0 7952 39 : 229 650 5 8519 40 : 2 650 0 8540 41 : 295 650 0 9087 42 : 3 650 0 9109 43 : 296 650 0 9694 44 : 2 650 0 9715 SAT completed 45 frames. Time = 10.28 sec Solver UNSAT = 1.49 sec ( 14.50 %) Solver SAT = 2.57 sec ( 24.94 %) Refinement = 5.37 sec ( 52.17 %) Other = 0.86 sec ( 8.37 %) TOTAL = 10.29 sec (100.00 %) SAT vars = 36976. Clauses = 92646. Confs = 5074. Used 0.75 Mb for proof-logging.

14. ABC’s &vta vs. IBM’s SixthSense Tried two SixthSense configurations: Config2: automatic, SAT-based, counter-example- based, gate-level, fixed time-frame Config5: automatic, SAT-based, hybrid, gate-level, fixed time-frame Used a suite of 58 model checking benchmarks submitted to HWMCC’11 by IBM Result 1: Config5 produces abstractions that are 20% (16%) smaller in terms of gates (flops) Result 2: Config2 completed more timeframes in 5 minutes for 75% of benchmarks

15. 15 Conclusions Reviewed abstraction algorithms Reviewed abstraction algorithms Motivated an improvement to BMC Motivated an improvement to BMC Connected it with gate-level abstraction Connected it with gate-level abstraction Showed preliminary experimental results Showed preliminary experimental results

16. Future Work Using coarser objects to abstract, refine, and derive CNF Adopting min-cut heuristics to decide what gates to add to the abstraction Performing the initialized unrolling with proof-logging