1. Logic Decomposition of Asynchronous Circuits Using STG Unfoldings Victor Khomenko School of Computing Science, Newcastle University, UK

2. 2 Asynchronous circuits  The traditional synchronous (clocked) designs lack flexibility to cope with contemporary design technology challenges Asynchronous circuits – no clocks: Low power consumption and EMI Tolerant of voltage, temperature and manufacturing process variations Modularity – no problems with the clock skew and related subtle issues [ITRS’09]: 22% of designs will be driven by ‘handshake clocking’ in 2013, and 40% in 2020  Synthesis algorithms are complicated  Computationally hard to synthesize efficient circuits

3. 3 Motivation Logic decomposition is one of the most difficult tasks in logic synthesis The quality of the resulting circuit (in terms of area and latency) depends to a large extent on the way logic decomposition was performed

4. 4 Speed-independency assumptions Gates are atomic (so no internal hazards) Gates’ delays are positive and unbounded (and perhaps variable) Wire delays are negligible (SI) or, alternatively, wire forks are isochronic (QDI) F instant evaluator delay …

5. 5 Speed-independent decomposition G … H1H1 HkHk … … delay F instant evaluator delay …

6. 6 VME Bus Controller Device VME Bus Controller lds ldtack d Data Transceiver Bus dsr dtack lds-d-ldtack-ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc-

7. 7 Complex-gate implementation Device d Data Transceiver Bus dsr dtack lds ldtack csc May be not in the gate library and has to be decomposed

8. 8 Naïve decomposition is hazardous d dsr dtack lds ldtack csc x lds-d-ldtack-ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc- Unexpected!

9. 9 Decompose at the PN level! d dsr dtack lds ldtack csc dec lds-d-ldtack- ldtack+ dsr-dtack+d+ dtack-dsr+lds+ csc+ csc- dec+ dec- Insert a new signal dec whose implementation is [dec] = ldtack + csc Multiway acknowledgement

10. 10 Latch utilisation d dsr dtack lds ldtack csc d dsr dtack lds ldtack csc C Only possible because there is no globally reachable state at which dsr=ldtack=0 and csc=1

11. 11 State Graphs: Relatively easy theory Many algorithms  Not visual  State space explosion problem State Graphs vs. Unfoldings

12. 12 State Graphs vs. Unfoldings Unfoldings: Alleviate the state space explosion problem More visual than state graphs Proven efficient for model checking  Quite complicated theory  Not sufficiently investigated  Relatively few algorithms

13. 13 Function-guided signal insertion forever do for all non-input signals x do S[x] ← ∅ for all G  {latches, gates} do S[x] ← S[x]  decompositions(x,G) bestH[x] ← best SI candidate in S[x] if for each x, bestH[x] is implementable Library matching stop if for each x, bestH[x]=UNDEFINED fail H ← the most complex bestH Insert a new signal z implementing H into the STG Logic decomposition algorithm [Cortadella et al, ’99]

14. 14 Function-guided signal insertion Problem: given a Boolean function F, insert a new signal dec (i.e. a set of new transitions labelled dec+ or dec-) with the implementation [dec]=F into the STG. Only unfolding prefix (rather than state graph) may be used.

15. 15 Previous work: Transformations [PN’07] Sequential pre-insertionSequential post-insertion Concurrent insertion

16. 16 Previous work: main results [PN’07] Validity criteria: safeness & bisimilarity  can be checked before the transformation is performed, i.e. on the original prefix (to avoid backtracking) Perform the insertion directly on the prefix  avoid re-unfolding  good for visualization (re-unfolding can dramatically change the look of the prefix)  Can transfer some information between the iterations of the algorithm The suite of transformations is good in practice for resolution of encoding conflicts

17. 17 Motivation for more transformations The suite of transformations is not sufficient for logic decomposition; intuitively:  only linear (in the PN size) number of sequential pre- and post-insertions (assuming that the pre- and postset sizes are bounded)  only quadratic (in the PN size) number of concurrent insertions  exponential number of ‘cuts’ in the PN where a Boolean expression can change its value

18. 18 Example: imec-sbuf-ram-write dec+ dec- dec Implementation of prbar: (csc2  req)  csc1  wsldin imec-sbuf-ram-write req precharged done wsldin wenin prbar wen wsen ack wsld

19. 19 Generalised transition insertion [ICGT’10] s1s1 s2s2 s3s3 d1d1 d2d2 All previously listed good points hold for GTIs as well Exponentially many GTIs can exist:  more likely that an appropriate transformation exists  no longer practical to enumerate them all   can enumerate only the ‘potentially useful’ (for logic decomposition) GTIs sources destinations

20. 20 Compatible insertions An insertion I is compatible with F if whenever an x  can fire and trigger I, F’ x =1, where F’ x = F x=0  F x=1 Intuitively, when x  fires, the value of F must change, as I becomes enabled. C xx I F=v F=  v

21. 21 Compatible insertions lds- d- ldtack- ldtack+ dsr- dtack+ d+ dtack- dsr+lds+ csc+ dsr+ csc+csc- F =  ldtack  csc F=1 F=0 F=1

22. 22 Find an optimal w.r.t. a heuristic cost function SAT assignment of the Boolean formula MUTEX  SA  CUTOFF  FUN depending on the variables I 1,..., I k corresponding to the compatible insertions, and conveying that:  no two insertions are non-commuting, or concurrent, or in auto-conflict, or one of them can trigger the other (MUTEX)  consistent assignment of signs is possible in the prefix (SA) and beyond cut-offs (CUTOFF)  F is a possible implementation of the newly inserted signal (FUN) Reduction to (incremental) SAT [ACSD’07]

23. 23 Cost function Parameterised by the user; takes into account: the delay introduced by the insertion the number of syntactic triggers of all non- input signals the number of inserted transitions of a signal the number of signals which are not locked with the newly inserted signal …

24. 24 Building FUN Let C be a configuration enabling some x , F’ x =1, and I be the set of compatible insertions such that: C xx I F=v F=  v Then the clause V I  I I is in FUN. One can build a Boolean formula FUNGEN depending on C and compatible insertions whose SAT assignments satisfy this condition.

25. 25 Building FUN (cont’d) Problem: it is infeasible to enumerate all configurations. Idea 1: The same clause can be generated by many different configurations, and hence once one such configuration is found, the others can be excluded from the search. Idea 2: Clauses subsumed by already generated ones can be excluded from the search. It is enough to add a clause V I  I  I to FUNGEN whenever a new clause V I  I I is computed.

26. 26 Building FUN (example) lds- d- ldtack- ldtack+ dsr- dtack+ d+ dtack- dsr+lds+ csc+ dsr+ csc+csc- F =  ldtack  csc F=1 F=0 F=1 C C

27. 27 Experimental results Implemented in MPSAT (library matching not implemented yet) and compared with PETRIFY Assorted small benchmarks:  Similar failure rates and the quality of circuits – structural insertions seem sufficient  The tests reflect the quality of heuristics in choosing the decomposition in each step rather than the quality of the signal insertion routine  Large benchmarks  Tend to be non-decomposable by both tools   Only one series (scalable pipelines) was useful – can be solved by a single insertion, hence minimizes the impact of heuristics and reflects the quality of the signal insertion routine – huge reachability graphs, so unfoldings win

28. 28 Conclusions Unfolding-based decomposition algorithm  alleviates state space explosion  completes the design cycle based fully on unfoldings (i.e. state graphs are never built) All advantages of state-based decomposition are retained:  multiway acknowledgement  latch utilisation  highly optimised circuits

29. 29 Thank you! Any questions?