1. 1 Don´t Care Minimization of *BMDs: Complexity and Algorithms Christoph Scholl Marc Herbstritt Bernd Becker Institute of Computer Science Albert-Ludwigs-University 79110 Freiburg im Breisgau, Germany

2. 2 Overview Motivation Decision Diagrams: BDDs, BMDs and *BMDs More details on BMDs Problem Formulation Complexity Results Algorithms Experiments Conclusions

3. 3 Motivation Decision Diagrams (DDs): –function representation –used in: logic synthesis, testing, and verification Don´t Care conditions: –can occur between function modules –by system specification  Use Don´t Cares to minimize DD-representation –well-known for bit-level DDs (BDD, FDDs) –unknown for word-level DDs (BMD, *BMDs,...)

4. 4 Related Work Zilic, Radecka, IWLS, 1998: „Don´t Care FDD minimization by interpolation“ Problem: FDD represents boolean function, *BMD represents integer-valued function Varma, Trachtenberg, IEE Proceedings, 1991: „Computation of Reed-Muller expansions of incompletely specified boolean functions from reduced representations“ Problem: Spectral technique not applicable in case of integer-valued functions

5. 5 Decision Diagrams: BDDs Representation for boolean functions: Iterative Shannon-decomposition: Transformation into directed, acylic graph: Reduction: eliminate v iif low(v)=high(v) v low(v)high(v)

6. 6 Decision Diagrams: BMDs (1) Representation for integer-valued functions Iterative positive Davio decomposition: Transformation into directed, acyclic graph: Reduction: eliminate v iif high(v) = 0 v low(v)high(v)

7. 7 Decision Diagrams: BMDs (2) Example:

8. 8 Decision Diagrams: *BMDs More compact than BMDs: extract multiplicative factor of terminals into edge weights Example:

9. 9 – Special case: not labeled with Let, choose with: BMD-nodes reached by Node v labeled with variable low(v)=0-successor(v), high(v)=1-successor(v) Node reached by : –Follow starting at root node Go to node

10. 10 Example Reduced BMD:Non-reduced BMD: (0,1)-node: terminal 5 (1,0)-node: terminal 4 (1,1)-node: terminal 0 (0,0)-node: terminal 1

11. 11 Function represented by a BMD-node Let B be a BMD,,v is a -node. Funtion represented by v is: Modification of -node,, changes function represented by node v

12. 12 Example +k-k+k

13. 13 Problem Formulation Problem: DC*BMD Given: *BMD B, BDD C, Find: *BMD B *, such that and B * has minimal number of nodes. Problem: DCBMD Like DC*BMD, but BMD instead of *BMD.

14. 14 Complexity Theorem: DC*BMD and DCBMD are NP-complete. Proof: NP-hardness: Reduction from graph colorability problem. NP-membership: Using WLCDs.

15. 15 Method min_polynomial (1) Idea: Minimize size of polynomial extracted from BMD Polynomial contains term with iif node reached by is terminal Size of BMD B,, is less or equal to size of polynomial representing. Property holds also for *BMDs.

16. 16 Method min_polynomial (2) Try to change terminal c reached by to have value 0 by using don´t cares. If, change to : Important: Adjustment of all terminals

17. 17 Method min_polynomial (3) dc(0,0)=1 +1 dc(1,1)=1 -7

18. 18 *BMD min_polynomial( *BMD b, BDD dc) 1. if dc=1 return 0; if dc=0 return b; if b=constant return b 2. if (result,b,dc) in Computed Table return result 3. Let v be top variable of b and dc 4. b * low =min_polynomial(b low, dc low ) 5. b * high =min_polynomial(b high +(b low - b * low ), dc high ) 6. b * = b * low +v* b * high 7. if size(b * ) size(b) then b * = b 8. insert (b *,b,dc) into Computed Table 9. return b * Method min_polynomial (4)

19. 19 Method independent_dfs (1) Idea: Use don´t cares to assign high(v) to 0 application of Davio-reduction rule. At node v, to check if w=high(v) can be set to 0: 1. check if low(w) can be set to 0, 2. if 1. succeeds, check if high(w) can be set to 0.

20. 20 Method independent_dfs (2) dc(0,0)=1 dc(1,1)=1 +3-3 +3-11

21. 21 Method independent_dfs (3) *BMD independent_dfs( *BMD b, BDD dc) 1. if dc=1 return 0; if dc=0 return b; if b=constant return b 2. if (result,b,dc) in Computed Table return result 3. Let v be top variable of b and dc# 4. (success, b low-diff )=check_zero(b high, dc low, dc high ) 5. if success then b * =independent_dfs(b low +b low-diff,dc low *dc high ) else b * low =independent_dfs(b low,dc low ) b * high =independent_dfs(b high +(b low -b * low ),dc high ) b * = b * low +v* b * high 7. if size(b * ) size(b) then b * = b 8. insert (b *,b,dc) into Computed Table 9. return b *

22. 22 Experimental Results (1) Experimental setup: –Generated incompletely specified functions: Collapse benchmark to sum-of-products Select cubes with probability of 40% to be included in Don´t Care set Compute BDD C for Don´t Care Set –Build *BMD B for the outputs of the benchmark, where is: –Compute min_polynomial(B,C) and independent_dfs(B,C)

23. 23 Experimental Results (2) init: initial size of benchmark az: setting all don´t cares to 0 mp: min_polynomial dfs: independent_dfs

24. 24 Experimental Results (3) Initial *BMD:, minimized *BMD: Ratio:

25. 25 Conclusions We presented methods do minimize *BMD/BMDs using Don´t Care conditions Two methods: min_polynomial, independent_dfs –very effective in reducing the size of *BMDs –time efficient (most benchmarks finish in a few seconds)

26. 26 Future Work Modification of independent_dfs to independent_bfs: Idea: Don´t care assignments which could be made „at the top in the right half“ of the *BMD can become not applicable due to assignments made „at the bottom in the left half“ Extension of existing methods to K*BMDs: K*BMDs use several decomposition types: Shannon, positive and negative Davio Application of Don´t Care based minimization methods for verification of division circuits