1. Reducing Multi-Valued Algebraic Operations to Binary J.-H. Roland Jiang Alan Mishchenko Robert K. Brayton Dept. of EECS University of California, Berkeley

2. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

3. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

4. 06/03/2003DATE ’ 03 Motivation: MV optimization  In high-level design, system descriptions are inherently multi-valued.  Common sub-expressions should be extracted and preserved at a higher level.  Given an MV design, there are two paths of synthesis: 1.Encoding  binary optimization (SIS) – Optimizes design resting on the initial encoding – Explores optimization only on restricted design space 2.MV optimization  encoding  binary optimization (MVSIS) + Preserves common sub-expressions at a higher level + Explores larger design space – Was thought to be computationally much harder and time consuming

5. 06/03/2003DATE ’ 03 Motivation: Speed up MV operations  Previous MV algebraic methods can be slow on large examples.  Satisfiability matrix method  Graph matching method (2-cube divisors only)  Observations:  Many networks had many binary nodes.  Good speed-up can be obtained by gathering all these together and applying SIS fast_extract ( fx ) method.  Question:  Can we extend this to MV nodes besides binary ones ?

6. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

7. 06/03/2003DATE ’ 03 Preliminaries  Given a multi-valued function (a generalization of Boolean functions) f(a, b, … ): A × B × · · ·  F, we can express it in a sum-of-products (SOP) form.  E.g. f {0} (a,b) = a {2,3} b {0,1} + a {0,3} b {1,2} + a {1,2} b {0,3} + a {0,1} b {2,3}  Can think of this as a three level OR-AND-OR expression.

8. 06/03/2003DATE ’ 03 Definitions  Purely algebraic operations (used in SIS)  Operations which manipulate expressions like polynomials  Semi-algebraic operations (used in MVSIS)  Operations which include up to absorption rules  Boolean operations  Operations which include up to cube creation/annihilation  A fact:  Algebraic optimization in Approach 2 (semi-algebraic mv opt.  enc.  purely algebraic bin. opt.) may correspond to Boolean optimization in Approach 1 (enc.  Boolean opt.).

9. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

10. 06/03/2003DATE ’ 03 Problem formulation  Given an arbitrary multi-valued SOP expression E and an oracle  which optimally factors a binary SOP input, how can we take advantage of  to factor E ?  To do so, we have two criteria: 1.Transform E into E’ which “ looks like ” binary, and 2.Transform the resultant output of , say E’’, back to an expression that “ directly ” reflects an optimally factored form of E.

11. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

12. 06/03/2003DATE ’ 03 Some unsuccessful naïve trials  Apply binary encoding to the MV SOP expression  For example, using 1-hot code, negative 1- hot code or any logarithmic code  Satisfies the first criteria  But NOT for the second (i.e. doesn ’ t directly reflect a factored form)

13. 06/03/2003DATE ’ 03 The solution: co-singleton transform  Forward transformation: Input : an MV expression E Output : a disguised binary expression E’ Begin For each literal x S  E Replace it with  i  S x i End (x i  x {0,…, i –1, i +1,…, n } is a co-singleton literal)  Backward transformation: Input : a disguised binary expression E’ Output : an MV expression E Begin For each  i  T x i  E’ Replace it with x { j | j  T} End

14. 06/03/2003DATE ’ 03 Co-singleton transform  An example: Assume a, b are 4-valued variables E = a {2,3} b {0,1} + a {0,3} b {1,2} + a {1,2} b {0,3} + a {0,1} b {2,3}  (forward co-singleton transform) E’ = a 0 a 1 b 2 b 3 + a 1 a 2 b 0 b 3 + a 0 a 3 b 1 b 2 + a 2 a 3 b 0 b 1  (factoring in the binary domain) = (a 1 b 3 + a 1 b 3 ) (a 0 b 2 + a 2 b 0 )  (backward co-singleton transform) E = (a {0,2,3} b {0,1,2} + a {0,1,2} b {0,2,3} ) (a {1,2,3} b {0,1,3} + a {0,1,3} b {1,2,3} )

15. 06/03/2003DATE ’ 03 Co-singleton transform  Co-singleton transform produces a bijection between an MV expression and a “ binary ” expression.  Co-singleton transform is of time complexity linear in the size of the input expression.  Co-singleton transform is optimally compact.  Only n bits are used for an n-valued MV variable.  Co-singleton transform vs. negative 1-hot encoding  Co-singleton transform is NOT an encoding in the sense that its “ binary codes ” for the values of an MV variable are non- disjoint.  Co-singleton transform = negative 1-hot coding + some minimization w.r.t unused codes

16. 06/03/2003DATE ’ 03 Closure properties Let F be a factored form of an MV SOP expression E, and F’ and E’ be the co-singleton transformed versions of F and E respectively.  Theorem 1 If F is a purely algebraic factorization of E, then F’ is a purely algebraic factorization of E’.  Theorem 2 If F is a semi-algebraic factorization of E, then F’ can be derived from E’ using only semi-algebraic operations  Theorem 3 If F’ is a semi-algebraic factorization of E’, then F is a semi- algebraic factorization of E.  Corollary Semi-algebraic operations are closed under the co-singleton transform, but not for purely algebraic operations.

17. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

18. 06/03/2003DATE ’ 03 EBD operations  EBD: Execution in Binary-in-Disguise  Procedure:  Apply co-singleton transform  Apply binary operations  Apply inverse co-singleton transform  Binary operations:  Factorization and decomposition  Algebraic division  Common divisor extraction  (Non-algebraic operations)  In the following discussion, binary operations are meant to be those used in SIS, and therefore are purely algebraic.

19. 06/03/2003DATE ’ 03 EBD operations vs. MV operations  Different results  In MV cases, results are maximally lowered. In EBD cases, results are maximally raised.  Procedures exist to make them similar.  Due to the implementation, there are results that can be obtained by one method, but not by the other.

20. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

21. 06/03/2003DATE ’ 03 Experimental results  Constructed a script (MV-script, similar to script.rugged in SIS).  Replaced all MV algebraic operations with their equivalent EBD operation to obtain EBD- script.  Measured total time and final number of literals in all factored forms of MV network.

22. 06/03/2003DATE ’ 03 Experimental results circuit name EBD time EBD lits MV time MV lits vg22.9872.685 sse2.11282.2120 b122.4702.370 cht1.81631.9164 sqrt81.1671.256 clip5.31347.6129 duke210.749724.6488 sand23.654547.5525 f51m1.81082.497 sao22.41094110 circuit name EBD time EBD lits MV time MV lits term15.21476.2142 9sym3724.6120 alu212.526619.4278 sct1.983290 t48114.23663.940 ttt23.12334.4221 bw3.21944.4194 rd845.3879.5106 squar51.4581.458 z4ml1.2381.438

23. 06/03/2003DATE ’ 03 Experimental results circuit name EBD time EBD lits MV time MV lits C43246.318549.3195 planet24.760563.5611 vda32.376396777 cps93.31479364.11524 dk167.02489.3238 S95318.651029.6516 k2269.2142633511428 monk1t0.970.87 sleep36.63363.737 car1.2431.344 circuit name EBD time EBD lits MV time MV lits balance8.118284.1217 conv35c1.283172 employ11.7421.536 mm30.9230.823 mm55.31378.5130 pal3x41144.5100 aluack1.4911.476 iris1.3121.312 mm42752.360 monks2t1.2511.243

24. 06/03/2003DATE ’ 03 Experimental results  EBD method is significantly faster than MV method, especially for large examples.  EBD results sometimes have little loss in quality due to the fact that there is no semi- algebraic capabilities in the binary domain.

25. 06/03/2003DATE ’ 03 Outline  Motivation  Definitions  Problem formulation  Co-singleton transform  EBD operations  Experimental results  Conclusions

26. 06/03/2003DATE ’ 03 Conclusions  EBD algebraic operations do not give same results as MV algebraic operations  MV result is maximally “ lowered ” ; EBD result is maximally “ raised ”  Due to the fact that binary operations are purely algebraic  Binary semi-algebraic operations need to be revisited  EBD operations are significantly faster, especially on large examples.  When used in a full script, the quality of EBD results is comparable with that of MV results.