Symmetry Detection for Large Boolean Functions

1. Symmetry Detection for Large Boolean Functions Using Circuit Representation, Simulation and Satisfiability Jin S. Zhang, Portland State University Alan Mishchenko, UC Berkeley Robert Brayton, UC Berkeley Malgorzata Chrzanowska-Jeske, Portland State University Supported by: Maseeh Fellowship from PSU, NSF grant CCR-9988402 and CCR 0312676,SRC contract 1361.001, and by the California Micro program with our industrial sponsors, Altera, Intel, Magma, and Synplicity.

2. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

3. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

4. Importance of Classical Symmetries • Widely used in logic synthesis and verification • BDD minimization • Decomposition • Boolean matching • Constructive logic synthesis • Placement & routing • Formal verification • …

5. Prior Work on Symmetry Computation • BDD-based methods [Möller ICCAD93, Panda ICCAD94, Tsai TCAD96, Mishchenko TCAD03] • BDD construction takes majority of run time • Cannot be applied to large designs • Incomplete circuit-based methods performing structural analysis [Wang, ICCD03] • Detect classical and higher-order symmetry • Complete circuit-based method using simulation and ATPG [Pomeranz, TCAD94] • Does not rely on recent improvements in SAT

6. Circuit Representation State-of-art SAT Solver Structural Analysis Bit-Parallel Simulation Transitivity Analysis Our Contributions Compute all classical two-variable symmetries for large Boolean functions

7. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

8. Definitions • Completely specified Boolean function: • F (x1, …,xn) where x1, xn, F range over {0, 1}. • Supportof F: all the variables that F depends on. • Cofactors: • Cofactors w.r.t. variables xi and xj • F00 = F [xi 0, xj 0], • F01, F10, F11 are defined similarly.

9. x y y Representations of Boolean Functions • Binary Decision Diagram (BDD) • And-Inverter Graph (AIG)

10. F (…,x, y,…) = !F (…,y, x,…) • F01F10 = 1 Skew non-equivalent Skew equivalent Non-skew equivalent • F00F11 = 1 • F00F11 = 0 F (…,x, y,…) = !F (…,!y, !x,…) F (…,x, y,…) = F (…,!y, !x,…) Classical Symmetries For a pair of variables F (…,x, y,…) = F (…,y, x,…) • F01F10 = 0 Non-skew non-equivalent

11. F (…,x, y,…) = !F (…,y, x,…) • F01F10 = 1 Skew non-equivalent Skew equivalent Non-skew equivalent • F00F11 = 1 • F00F11 = 0 F (…,x, y,…) = !F (…,!y, !x,…) F (…,x, y,…) = F (…,!y, !x,…) Classical Symmetries For a pair of variables F (…,x, y,…) = F (…,y, x,…) • F01F10 = 0 Non-skew non-equivalent

12. S S S Transitivity of Symmetry Variables: a, b, c a b c Form larger symmetry group

13. Simulation • Computes the values of the internal signals and POs from the value of the PIs • Random / Guided • Control of simulation rounds • Static / Dynamic

14. Boolean Satisfiability (SAT) • Proves that a given Boolean formula has a satisfying assignment • F = (x+y)(y +z ), satisfying assignments: x = 1, y = 0 • SAT vs. BDDs • SAT involves search, heuristics can improve performance • BDDs “preprocess” the search space • By building canonical form of Boolean functions • Hard or impossible for large functions

15. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

16. Compute functional support SAT returns counter-example? Perform guided simulation Y Perform structural analysis N Perform random simulation N All variable pairs processed? All variable pairs processed? Y N returnall symmetric pairs Y returnall symmetric pairs Overview of the Algorithm Derivenetwork AIG Call SAT for a remaining pair

17. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

18. Structural Analysis: Detect Symmetric Variable Pairs • Convert AIGs to Implication Supergates (ISs) • Introduced in [Chang DAC00] • Compute local symmetries for each Implication Supergates • Propagate global symmetries from primary inputs to primary outputs

19. AIG e a d b c Represents 2-input AND gate AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached

20. AIG e a d b c Represents 2-input AND gate AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached

21. AIG e a d b c Represents 2-input AND gate AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached

22. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

23. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

24. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

25. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

26. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

27. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached AIG e a d b c Represents 2-input AND gate

28. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached Implication supergate AIG s1 e a b c e s2 a d b c Represents 2-input AND gate Represents multi-input AND gate

29. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached Implication supergate AIG s1 e a b c e s2 d a d b c Represents 2-input AND gate Represents multi-input AND gate

30. AIG  Implication Supergates • Expand the AND gate until a PI or complemented edge is reached • Recover lost structural symmetry in AIG Implication supergate AIG s1 e a b c e s2 s3 d a d b c b c Represents 2-input AND gate Represents multi-input AND gate

31. IS Positive PIs Negated PIs Other ISs Candidate Symmetry Kept Symmetry Final Symmetry s3 b,c (b,c) (b,c) s2 d s3 (b,c) s1 b,c,e a s2 (b,c) (b,e) (c,e) (b,c) (b,c) Structural Symmetry Detection IS • Divide the fanins of each IS into three categories: • - Positive PIs • - Negated PIs • - Other ISs s1 e a b c s2 d s3 b c

32. IS Positive PIs Negated PIs Other ISs Candidate Symmetry Kept Symmetry Final Symmetry s3 b,c (b,c) (b,c) s2 d s3 (b,c) s1 b,c,e a s2 (b,c) (b,e) (c,e) (b,c) (b,c) Structural Symmetry Detection IS 2. Identify initial candidate symmetric pairs for each IS: - Pair up variables in positive and negative PIs separately s1 e a b c s2 d s3 b c

33. IS Positive PIs Negated PIs Other ISs Candidate Symmetry Kept Symmetry Final Symmetry s3 b,c (b,c) (b,c) s2 d s3 (b,c) s1 b,c,e a s2 (b,c) (b,e) (c,e) (b,c) (b,c) Structural Symmetry Detection IS 3. Keep a symmetric pair if the variables: - not in the support of other IS or - symmetric for each fanin in other IS s1 e a b c s2 d s3 b c

34. IS Positive PIs Negated PIs Other ISs Candidate Symmetry Kept Symmetry Final Symmetry s3 b,c (b,c) (b,c) s2 d s3 (b,c) s1 b,c,e a s2 (b,c) (b,e) (c,e) (b,c) (b,c) Structural Symmetry Detection IS 4. Propagate the symmetric pairs to its parent IS if: - the symmetry holds for at least one fanin of the other IS, and - the variables are not in the support for other fanins s1 e a b c s2 d s3 b c

35. Summary: Structural Analysis • Detect symmetry in one sweep over the circuit • Discover most easy structural symmetries • Simpler, faster and more scalable than [Wang ICCD03] • We only compute classical symmetries

36. Outline • Motivations • Background • Our approach to symmetry computation • Structural analysis • Simulation • Transitivity analysis • SAT • Experimental results • Conclusions

37. Simulation: Detect Non-symmetric Variable Pairs • Random & guided simulation • Performance random simulation until saturation • Guided simulation uses SAT counter examples and their distance-1 patterns • E.g. Distance-1 patterns of “011” are “111”, “001”, “010”

38. Why Distance-1 Patterns? • For a realistic and sparse function • Realistic: appear in practical circuit • Sparse: many 0s and fewer 1s or vice versa

39. Why Distance-1 Patterns? • For a realistic and sparse function • Random simulation often fails

40. Why Distance-1 Patterns? • For a realistic and sparse function • Random simulation often fails • SAT counterexamples reach unlikely values

41. Why Distance-1 Patterns? • For a realistic and sparse function • Random simulation often fails • SAT counterexamples reach unlikely values • Distance-1 patterns represent its neighborhood, contain other unlikely values

42. 0101 1101 Simulation:Detect Non-symmetric Variable Pairs • Random & guided simulation • Bit-Parallel computation • Simulating one AIG node involves bit-wise operations • Simulate 32 or 64 bits simultaneously 1010

43. Simulation:Detect Non-symmetric Variable Pairs • Random & guided simulation • Bit-Parallel computation • One vector targets all variable pairs simultaneously

44. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 ab cd 00 01 11 10 00 0 0 1 0 01 0 0 1 1 (4) Simulate the above patterns 0 11 1 1 0 0 10 0 1 0 • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

45. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab cd 00 01 11 10 00 0 0 1 0 01 0 0 1 1 (4) Simulate the above patterns 0 11 1 1 0 0 10 0 1 0 • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

46. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab 1 cd 00 01 11 10 00 0 0 1 0 01 0 0 1 1 (4) Simulate the above patterns 0 11 1 1 0 0 10 0 1 0 • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

47. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab 1 cd 00 01 11 10 1 00 0 0 1 0 01 0 0 1 1 (4) Simulate the above patterns 0 11 1 1 0 0 10 0 1 0 • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

48. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab 1 cd 00 01 11 10 1 00 0 0 1 0 0 01 0 0 1 1 (4) Simulate the above patterns R = 0110 0 11 1 1 0 0 10 0 1 0 • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

49. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab 1 cd 00 01 11 10 1 00 0 0 1 0 0 01 0 0 1 1 (4) Simulate the above patterns R = 0110 0 11 1 1 0 0 10 0 1 0 (5) Check variable pairs with the same polarity in P: {a, c}, {a, d}, {c, d} • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1

50. Simulation Example F(a, b, c, d) = ab + d (ac+ bc) (3) Flip the bits on the diagonal: 0011 1111 1001 1010 0 ab 1 cd 00 01 11 10 1 00 0 0 1 0 0 01 0 0 1 1 (4) Simulate the above patterns R = 0110 0 11 1 1 0 0 10 0 1 0 (5) Check variable pairs with the same polarity in P: {a, c}, {a, d}, {c, d} • Suppose the vector P is 1011. (2) Create the bit matrix: a b c d 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 (6) non-symmetric pairs: {a, c}, {c, d}