1. A Semi-Canonical Form for Sequential AIGs
Alan Mishchenko, Niklas Een, Robert Brayton UC Berkeley
Michael Case, Pankaj Chauhan, Nikhil Sharma
Calypto Design Systems

2. Motivation Logic networks often contain duplicate sub-circuits
Leads to redundant work
Synthesis and verification tools re-analyze the same sub-circuit
Verification tools waste time on duplicate proof obligations

3. Motivation Key idea: identify duplicate logic regions
Ideal solution: isomorphism
Our approximate solution: semi-canonical mapping
Use the circuit structure to classify each sub-circuit

4. Example Fanout count  {b}, {m} are unique
Complemented outputs  {c} is unique
Fanin level  {F}, {G} are unique

5. Example Sometimes nodes cannot be distinguished
After semi-canonicization: {F, F’}, {G, G’}
Isomorphisms: F ≈ F’ , G ≈ G’ F’ G’

6. Implementation Implemented within ABC as an internal routine
Puts the netlist in semi-canonical form
Node order reflects both topological order & signatures
Application 1: “write_aiger –u”
Writes the netlist in semi-canonical form
Application 2: “&iso”
Discard isomorphic POs

7. Algorithm Overview Compute signatures for each node
Signatures indicate which class a node is in
Singleton nodes: nodes that have unique signatures
Goal: assign unique signatures for as many nodes as possible
Initially all signatures are 0

8. Algorithm Overview Edge value: reflects the structure around an edge
edge_value := hash(driver_signature, edge_is_complemented) 4 1 3 5 6 8 7 2

9. Algorithm Overview Function: PropagateSignaturesForward()
Assign the same random signature to all inputs
For each node, signature := hash(old_signature, fanin edge_values) 71 42 1 2 15 3 19 8 4 5 6 7 12

10. Algorithm Overview Function: PropagateSignaturesBackward()
Recompute edge values
Assign the same random signature to all outputs
For each node, signature := hash(old_signature, fanout edge_values) 71 42 15 19 12 12 12 12

11. Algorithm Overview
Sometimes we cannot distinguish nodes  tie breaking
Choose the equiv class with the largest level
Assign unique signatures to the class nodes
Propagate signatures to other nodes until convergence
If there some equiv classes are left, go to Step 1 71 42 15

12. Implementation Implemented within ABC as an internal routine
Puts the netlist in semi-canonical form
Node order reflects both topological order & signatures
Application 1: “write_aiger –u”
Writes the netlist in semi-canonical form
Application 2: “&iso”
Discard isomorphic POs

13. Implementation Application 1: “write_aiger –u”
Writes the netlist in semi-canonical form
Useful for quickly comparing AIGER netlists
Netlist N
write_aiger -u
N.aig
diff
Netlist N’
write_aiger -u
Nprime.aig

14. Implementation Application 2: “&iso”
Convert each PO to semi-canonical form (separately)
Discard POs that have duplicate semi-canonical forms
i.e. “drop isomorphic proof obligations” F’ G’ a' b’ c’ d’
Counterexamples/invariants on F/G can be re-mapped to F’/G’

15. Experimental Results With industrial verification benchmarks
Remove isomorphic POs
Apply synthesis, remove proved POs
Remove isomorphic POs (again)
Example
FFs
ANDs
POs
POs after iso
iso (sec)
POs after synthesis
SDCU
2442
14418
834
727
3.2
361
353
.09
TPC_P
1619
1270
948
795
16.00
394
393
.28
TCP_Oh
3838
36890
598
553
4.78
545
541
.55
PC_T
2565
20101
274
258
129
.01
SDXIA
7822
78858
600
490
32.04
245
.63 DU
10864
84397
946
518
198.14
458
432
.80
Initial AIG Size

16. Conclusion Previous work:
computing functional symmetries and automorphisms
simplifying reachability and SAT instances with symmetries
First work on semi-canonical labeling of nodes in a sequential AIG
Allows for caching intermediate AIGs in EDA tools
Future work may include:
Speeding up propagation of node signatures
Generalizing the algorithm to work for logic networks other than the traditional AIGs

17. THANK YOU
Public implementation is available in ABC