1. Rewiring – Review, Quantitative Analysis and Applications Matthew Tang Wai Chung CUHK CSE MPhil 10/11/2003

2. Agenda Review Introduction RAMBO, REWIRE RAMFIRE GBAW Quantitative Analysis Rewiring power Runtime Coverage Applications Circuit Partitioning FPGA LUT mapping

3. Introduction – What is Rewiring? Replacing a target wire by an alternative wire without changing the functionality of the circuit. An alternative wire is a redundant wire to the circuit. Its addition would make the target wire become redundant hence removable. Let’s check out the classic example.

4. Classic Example

5. Classic Example – Resultant Circuit # gates: 9  6; # interconnect: 15  13

6. How can we find AWs? RAMBO / REWIRE ATPG: Mandatory assignments to find candidate nodes for AWs. RAMFIRE FIRE: Redundancy identification. ATPG: connection type. GBAW Graph-based pattern matching.

7. Automatic Test Pattern Generation (ATPG) We consider stuck-at-fault model. A wire is called stuck-at-1/0 (s-t-1/0) if the value at the wire is always 1/0. A wire is called redundant if its addition or removal to the circuit does not affect the functions at output. In ATPG, some stuck-at-faults for some wires are not testable  these wires are redundant.

8. ATPG by Example s-t-1 0/1 1 1 0 0 0 1 1 1 0 0 0 0 0 g6  g7 s-t-1 fault testable  g6  g7 not redundant

9. RAMBO – Basic Notion Consider TW ( t s, t d ) & AW ( a s, a d ) RAMBO considers absolute dominators t d of as candidates for a d. Create conflicts in M.A. by choosing suitable a s. e.g. if dominator is a AND gate, choose with node M.A. = 0.

10. RAMBO - Algorithm foreach wire w in the network N { perform stuck-at-fault test on w; build candidate set C for w; Verify each wire in C by redundancy test; } C can be very large  a lot of redundancy test needed.

11. REWIRE Observation that some MA is forced, which means conflicts these would result in untestable stuck-at-fault. (observability MA) Some fast filters are proposed to prune the candidate set to avoid unnecessary time- costy implication. The flow of REWIRE is similar to that of RAMBO.

12. REWIRE - Algorithm foreach node n d in the network N { find OMA of n d ; construct array of wires with n d ; foreach wire w t of the array { do stuck-at-fault test on w t ; build candidate set C; apply filters to C; Verify each wire in C; }

13. RAMFIRE Inefficiency in finding redundancy is a big problem with RAMBO and REWIRE. FIRE: a fast redundancy identification algorithm. No brute-force testing is needed, instead apply FIRE to get a set of alternative wires directly.

14. FIRE - Example 11 * * 1 Uncontrollability Unobservability 1111 * * * 0 * 0

15. RAMFIRE - Algorithm foreach wire w in the network N { perform stuck-at-fault test on w (get MA); perform FIRE (1’ & 0’) on w; Build a set of redundant wire R for w; Form AW in R by creating conflicts in MA; } Only three logic implications are used for each wire.

16. GBAW Based on pre-configured graph patterns. These patterns are extracted from result of RAMBO. There are around 20 – 25 patterns in the library. Each pattern correspond to a known TW / AW pair. Notion: Pattern matching on the network to find AWs.

17. GBAW – Pattern Example Local 13: a champion pattern Actual logic transformation: DeMorgan’s law Backward Forward

18. Agenda Review Introduction RAMBO, REWIRE RAMFIRE GBAW Quantitative Analysis Rewiring power Runtime Coverage Applications Circuit Partitioning FPGA LUT mapping

19. Analysis Basis Operation 1: Given a target wire, find a set of alternative wires. Operation 2: Given a redundant wire to be added, find a set of wires becoming redundant after this addition. We compare the algorithms in handling operation 1. The algorithms of RAMBO and REWIRE are modified accordingly. All the experimental results are obtained on the same PC running Redhat Linux 8.0. (Duron 1.3 GHz, 256 MB)

20. Comparing Rewiring Power

22. Comparing Runtime (Theoretical) Consider a circuit n nodes and w wires. Let c and c’ be the sizes of the candidate set in RAMBO & REWIRE respectively. Recursive Learning (r max = 1): O(w 2 ) TechniqueTime Complexity GBAWO(kw) = O(w) RAMBOO(cw 3 ) REWIREO(c’w 3 ) RAMFIREO(3w 3 )

23. Comparing Runtime (Experimental)

24. Comparing GBAW & RAMBO (coverage) Forward Pattern: Backward Pattern: PI side  higher depth  lower depth  PO side

25. Comparing GBAW & RAMBO (coverage) RAMBO is able to locate all forward patterns by GBAW. Why? Forward patterns are actually extracted from results of RAMBO. RAMBO can’t find any backward patterns by GBAW. Why? The candidate destination node of the AW must be an absolute dominator in RAMBO.

26. Comments GBAW & RAMFIRE: fast, strong in locating backward wires RAMBO & REWIRE: much stronger rewiring power Tradeoff: CPU runtime  Rewiring Power GBAW can be an efficient pre-process filter to skip time-costy redundancy test.

27. Agenda Review Introduction RAMBO, REWIRE RAMFIRE GBAW Quantitative Analysis Rewiring power Runtime Coverage Applications Circuit Partitioning FPGA LUT mapping

28. Circuit Partitioning Rewiring can be used in reducing the cut size. Consider the wires on the cut as target wires, we try to find alternative wires which can be placed within clusters  smaller cut size. Experimental Results: 9% reduction in cut size with small penalty on the # of literals. (5-way partition)

29. FPGA LUT Mapping Given a Boolean network, map the nodes into fixed size lookup tables (LUT). The number of inputs to a LUT is fixed, say k. A k -LUT can implement any Boolean function with k variables. Objective: Minimize the total number of LUT used. Minimize the depth of the mapping solution.

30. LUT-based mapping Example k = 3

31. Motivation Most LUT-based mapping algorithms do not consider modifying the Boolean network during mapping. We would like to explore the possibility to apply logic transformation during mapping. Logic transformation: our favourite – rewiring We have picked Flowmap to work with due to its solution’s depth optimality.

32. Intuition – Utilizing LUTs We found that quite a lot LUTs are not fully utilized, i.e. input used <= k. e.g. pcler8: k = 5; 12 / 47 nodes has 5 inputs. Using rewiring, we can find backward wire to fill the unused input in LUTs. When some LUT is utilized further, we may save some LUTs in the whole network.

33. Example

34. Example (Cont’) GBAW: Local 111 g6  g7 replaced by g6  g3

35. Future Work Verify the intuition through more examples or experiments. Observe and find out more intuitions. Purpose a more complete scheme. THANK YOU