Fast Spectral Transforms and Logic Synthesis

1. Fast Spectral Transforms and Logic Synthesis DoRon Motter August 2, 2001

2. Introduction • Truth Table Representation • Value provides complete information for one combination of input variables • Provides no information about other combinations • Spectral Representation • Value provides some information about the behavior of the function at multiple points • Does not contain complete information about any single point

3. Spectral Transformation • Synthesis • Many algorithms proposed leveraging fast transformation • Verification • Correctness may be checked more efficiently using a spectral representation.

4. Review – Linear Algebra • Let M be a real-valued square matrix. • The transposed matrix Mt is found by interchanging rows and columns • M is orthogonal if MMt = MtM = I • M is orthogonal up to the constant k if MMt = MtM = kI • M-1 is the inverse of M if MM-1 = M-1M = I • M has its inverse iff the column vectors of M are linearly independent

5. Review – Linear Algebra • Let A and B be (nn) square matrices • Define the Kronecker product of A and B as

6. Spectral Transform • General Transform Idea • Consider the 2n output values of f as a column vector F • Find some transformation matrix T(n) and possibly its inverse T-1(n) • Produce RF, the spectrum of F, as a column vector by • RF = T(n)F • F = T-1(n)RF

7. Walsh-Hadamard Transform • The Walsh-Hadamard Transform is defined:

8. Walsh-Hadamard Matrices

9. Walsh-Hadamard Example

10. Walsh-Hadamard Example 3

11. Walsh-Hadamard Example

12. Walsh-Hadamard Example

13. Walsh-Hadamard Transform • Why is it useful? • Each row vector of T(n) has a ‘meaning’ 1 x1 x2 x1x2 • Since we take the dot product of the row vector with F we find the correlation between the two

14. Walsh-Hadamard Transform Constant, call it x0 x1 x2 x1 x2 x3 x1 x3 x2 x3 x1 x2 x3

15. Walsh-Hadamard Transform • Alternate Definition • Recursive Kronecker Structure gives rise to DD/Graph Algorithm

16. Walsh-Hadamard Transform • Alternate Definition • Note, T is orthogonal up to the constant 2:

17. Decision Diagrams • A Decision Diagram has an implicit transformation in its function expansion • Suppose • This mapping defines an expansion of f

18. Binary Decision Diagrams • To understand this expansion better, consider the identity transformation • Symbolically,

19. Binary Decision Diagrams • The expansion of f defines the node semantics • By using the identity transform, we get standard BDDs • What happens if we use the Walsh Transform?

20. Walsh Transform DDs

21. Walsh Transform DDs 1

22. Walsh Transform DDs • It is possible to convert a BDD into a WTDD via a graph traversal. • The algorithm essentially does a DFS

23. Applications: Synthesis • Several different approaches (all very promising) use spectral techniques • SPECTRE – Spectral Translation • Using Spectral Information as a heuristic • Iterative Synthesis based on Spectral Transforms

24. Thornton’s Method • M. Thornton developed an iterative technique for combination logic synthesis • Technique is based on finding correlation with constituent functions • Needs a more arbitrary transformation than Walsh-Hadamard • This is still possible quickly with DD’s

25. Thornton’s Method • Constituent Functions • Boolean functions whose output vectors are the rows of the transformation matrix • If we use XOR as the primitive, we get the rows for the Walsh-Hadamard matrix • Other functions are also permissible

26. Thornton’s Method • Convert the truth table F from {0, 1} to {1, -1} • Compute transformation matrix T using constituent functions {Fc(x)} • Constituent functions are implied via gate library • Compute spectral coefficients • Choose largest magnitude coefficient • Realize constituent function Fc(x) corresponding to this coefficient

27. Thornton’s Method • Compute the error e(x) = Fc(x) F with respect to some operator,  • If e(x) indicates w or fewer errors, continue to 8. Otherwise iterate by synthesizing e(x) • Combine intermediate realizations of chosen {Fc(x)} using  and directly realize e(x) for the remaining w or fewer errors

28. Thornton’s Method • Guaranteed to converge • Creates completely fan-out free circuits • Essentially a repeated correlation analysis • Extends easily to multiple gate libraries

29. Thornton’s Method: Example • Step 1: Create the truth table using {-1, 1}

30. Thornton’s Method: Example • Step 2: Compute transformation matrix T using constituent functions. • In this case, we’ll use AND, OR, XOR

31. Thornton’s Method: Example 1 x1 x2 x3 x1 x2 x1 x3 x2  x3 x1  x2 x3 x1 + x2 x1 + x3 x2 + x3 x1 + x2 + x3 x1x2 x1x3 x2x3 x1x2x3

32. Thornton’s Method: Example • Step 3: Compute spectral coefficients • S[Fc(x)]= [-2, -2, 2, -2, 2, -2, 2, -6, 2, -2, 2, 2, -2, -2, -2, -4] • Step 4: Choose largest magnitude coefficient. • In this case, it corresponds to x1  x2 x3

33. Thornton’s Method: Example • Step 5: Realize the constituent function Fc • In this case we use XNOR since the coefficient is negative

34. Thornton’s Method: Example • Step 6: Compute the error function e(x) • Use XOR as a robust error operator

35. Thornton’s Method: Example • Step 7: Since there is only a single error, we can stop, and realize the final term directly

36. Conclusion • The combination of spectral transforms and implicit representation has many applications • Many ways to leverage the spectral information for synthesis