Resubstitution for Logic Optimization

1. Invariant-Strengthened Elimination of Dependent State Elements Michael L Case †‡ Alan Mishchenko † Robert K Brayton † Jason Baumgartner ‡ Hari Mony ‡ † University of California, Berkeley ‡ IBM Systems and Technology, Austin, TX

2. Resubstitution Basics • Working in an RTL level design • Let g1, g2, …be “basis functions” • Let X be a “target function” • To resubstitute is to find a dependency function f() s.t. X = f(g1, g2, …) Mike Case, FMCAD 2008

3. Why Resubstitute? • For Synthesis: • Increases logic sharing • Decreases total area • For Verification: • Can be tailored to reduce the number of latches • #latches more important than #gates • Traditionally performed with BDDs • BDDs don’t scale to many/large g’s • SAT-based approach first proposed by Lee, Jiang et. al., ICCAD 2007 Mike Case, FMCAD 2008

4. = = = Replace X with I Resubstitution with Interpolation 1 • Interpolant I has the properties: • I is a function of the g’s only • onset → I → ¬ offset X’s onset X’s offset (1) (0) X X g1 g3 g2 g1 g3 g2 Circuit copy 2 Circuit copy 1 Mike Case, FMCAD 2008

5. Our Objectives • Proposed approach is a good foundation, but missing many practical details • Improve SAT-based Resub • Guard against interpolant logic bloat • Improve speed and scalability • Provide details necessary for industrial use • Extend to sequential synthesis • Apply to Verification • Reduce the registers as much as possible Mike Case, FMCAD 2008

6. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

7. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

8. time == 0 S InitialState S T1 T2 T1 T2 (c) Eliminate S by separating time 0 and time > 0 behavior (b) Resubstitute next(S) in terms of the other next-state function 0 1 Register Elimination S T1 T2 (a) Original Circuit Mike Case, FMCAD 2008

9. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

10. = = = = = = 1 1 X X X X g1 g3 g2 g1 g3 g1 g3 g1 g3 g2 g2 g2 Dependency Function Minimization • Interpolant logic may be bloated • Note: Test function is symmetric • Extract both dependency functions and choose the “best” one Mike Case, FMCAD 2008

11. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

12. Dynamic Basis • Large basis set can slow resubstitution basisSet :=  while (1) { switch (resubsitute(X, basisSet)) { case Counterexample: Find new basis elements to block the cex. if (! new basis elements) return “resub failed” case Unsat: Interpolate, simplify circuit return “resub succeeded” } } Mike Case, FMCAD 2008

13. 0 1 X 1 1 0 0 0 X 0 0 X X X X X Dynamic Basis • Heuristic method with risks • May add too many basis functions • Might miss a basis function • If tuned properly, 80x speedup Resubstitution Test Function • For a particular state var S, next(S1)=1 & next(S2)=0 • For every other state var T, next(T1) = next(T2) next(S1) next(S2) 1 0 1 1 0 X Next StateFunctions Next StateFunctions Copy 1 Copy 2 Inputs / Current State Inputs / Current State Mike Case, FMCAD 2008

14. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

15. One-Hot Encoding: Valid resubstitutions: R1 R2 R3 R4 R1= R2 R3  R4 1 0 0 0 R2= R1 R3  R4 0 1 0 0 0 0 1 0 Optimize Logic: 0 0 0 1 R1= R1 R3  R4 Compatible Dependencies • Not all resubstitutions are compatible • Use of many dependency functions may lead to combinational cycles Mike Case, FMCAD 2008

16. Compatible Dependencies candidates := order(foundResubs) accepted :=  foreach C in candidates { if (! isCyclic(C accepted)) { accepted += C } } • Need to discard 24% of the found resubstitutions, on average • Ordering is very important for keeping the most valuable resubs Mike Case, FMCAD 2008

17. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

18. Invariants Invariants Strengthening with Invariants • Leverage invariants Resubstitution Test Function • For a particular state var S, next(S1)=1 & next(S2)=0 • For every other state var T, next(T1) = next(T2) • Invariants = 1 Next StateFunctions Next StateFunctions Copy 1 Copy 2 Inputs / Current State Inputs / Current State • Makes this a sequential transformation • Allows us to find resubstitutions post-retiming Mike Case, FMCAD 2008

19. Invariant Generation Techniques • Over-approximate reachability • Conjunction of many small local invariants gives a reachability approximation • Simple algorithm: • Use simulation to guess candidate invariants • Use induction to prove the candidates • Many types of invariants in our code: • Constants • Equivalences • Implications • Prove invariants in batches and vary the invariant families to take advantage of their orthogonal strengths • Invariants from circuit cuts • Random clauses Mike Case, FMCAD 2008

20. Candidate Invariant C Amount by which Cstrengthens Iis C I Reachable State Set If C is proved,I := C I Simulation Filtering • Simultation allows us to focus on the “best” candidate invariants • Want a tight reachability approximation Reachable State Set Proved Invariants I Mike Case, FMCAD 2008

21. Typical Invariant Run Few candidate implications Mike Case, FMCAD 2008

22. Outline • Resubstitution Enhancements • Direct register elimination • Dependency function minimization • Dynamic Basis • Compatible Dependencies • Invariant Strengthening • Results • Register reductions obtained • So what? The effect on verification. Mike Case, FMCAD 2008

23. Results: Reg Reductions • Heavily synthesized designs • Invariants very important • 12% register reduction on average • Usually a net gain in ANDs Mike Case, FMCAD 2008

24. Results: Effect on Verification • Run verification flow (SixthSense) on 267 IBM designs • 141 have remaining properties • Resubstitute, run verification flow again • Additional properties proven in 10 designs • Register reductions helpful in: interpolation, BMC, induction Mike Case, FMCAD 2008

25. Summary • Extended previous work: • Direct register elimination • Address potential logic bloat • Scalability improvements • Compatibility issues explored • Invariant strengthening • Results demonstrated to: • Reduce design size • Simplify verification efforts • Questions? Mike Case, FMCAD 2008