1. Multi-Valued Logic Network Optimizations EE219B Project Presentation Jason Shamberger May 16, 2000

2. Outline Input Variable Pairing –results Input Variable Encoding –results Conclusions

3. Input Variable Pairing By pairing input variables, we can make the cover smaller Example:

4. Input Variable Pairing Trade-off: pairing variables increases input width If input variables have m, n values, pairing them increases the width from m+n to m*n For some applications we wish to limit the total input width to 32 bits

5. Input Variable Pairing Simple Heuristic: –Count the original number of cubes –Eliminate the variables to pair –Count the new number of cubes –The difference gives an approximation of the savings The real savings could be more than this, since further simplifications could be made by calling Espresso

6. Input Variable Pairing Results Number of cubes: Original network mv_simpvar_pair mv_simp test1416375467422 test2785740930871 test31617149719651818 test413768126811578913297 test524715220582882126275

7. Input Variable Pairing Results Possible explanations? –Heuristic is not good enough, can try calling espresso on each node, each pair of variables to get an accurate count of savings Another problem: demux nodes

8. Input Variable Pairing We pair variables by inserting demux nodes

9. Input Variable Pairing Results Original network mv_simpvar_pair mv_simp All nodes Excluding demux nodes All nodesExcluding demux nodes test1416375467409422364 test2785740930772871713 test3161714971965157318181426 test413768126811578913297 11870 test5247152205828821232552627519992

10. Input Variable Pairing The new node we insert must have at least one cube for each value of the MV variable we generate Solutions: –Set the threshold equal to the number of cubes in demux node Result: no pairing

11. Input Encoding The opposite of variable pairing: Take one MV input and encode it with smaller variables Choosing the encoding affects the size of the cover (see class notes)

12. Input Encoding Implementation: –Find which variables to encode –Generate the constraints –Generate the dichotomies –Generate the prime dichotomies –Find the minimum column cover –Translate this into an encoding scheme

13. Input Encoding Results The optimal method is not fully implemented –Currently generate dichotomies, then find a cover which satisfies all of them Encounter the same problem with insertion of new nodes for muxing –Insert one node for each variable to encode with –Each node has one cube for each value of the encoding variable

14. Input Encoding Results Original network mv_simpinput_encode mv_simp All nodes Excluding mux nodes All nodesExcluding mux nodes test1416375793393481375 test27857401371759896740 test3161714973201154119191497 test4137681268129200128761716212428 In all test cases, input size at each node decreased or remained the same

15. Conclusions Original thought: alternately applying variable pairing and input encoding could lead to a small cube cover and small input width (similar to extraction and elimination) –This does not work, just end up inserting more and more nodes into the network

16. Conclusions

17. For better results look at the network level Instead of pairing variables, pair nodes (combine two sibling nodes into one) –Will increase the input width of the paired nodes, so make sure two nodes share common inputs AND outputs

18. Conclusions

19. Instead of encoding the inputs, encode at the outputs of previous nodes –Requires multiple outputs from a single node

20. References Class Notes “Satisfaction of Input and Output Encoding Constraints”, Saldanha, Villa, Brayton, Sangiovanni-Vincentelli “Multiple-Valued Logic and Optimization of Programmable Logic Arrays”, Sasao “On the Optimal Design of Multiple-Valued PLA’s”, Sasao