Versatile SATbased Remapping for Standard Cells Alan Mishchenko

Versatile SAT-based Remapping for Standard Cells Alan Mishchenko and Robert Brayton UC Berkeley Thierry Besson, Sriram Govindarajan, Harm Arts and Paul van Besouw Mentor Graphics

Overview l l l Motivation Resynthesis framework SAT-based formulation SAT vs. other methods in logic synthesis Experimental results Conclusion and future work 2

Motivation l Improving existing circuit structure is an important task in logic synthesis l An efficient, resource-aware, versatile engine to perform this task is very useful l Applications l l l AIG (MIG, XOR-MIG, MUX-G, etc) optimization Resynthesis for mapped networks Translation of AIG into CNF

Previous Work l V. N. Kravets and P. Kudva, “Implicit enumeration of structural changes in circuit optimization”. Proc. DAC’ 04, pp. 438 -441 l l A. Mishchenko and R. Brayton, "SAT-based complete don't-care computation for network optimization", Proc. DATE '05, pp. 418 -423. l l Similar approach, based on BDDs Early-stage technology-independent optimization A. Mishchenko, R. Brayton, J. -H. R. Jiang, and S. Jang. "Scalable don't-care-based logic optimization and resynthesis", ACM Trans. Reconfigurable Technology and Systems (TRETS), Vol. 4(4), April 2011, Article 34. l Similar approach, specialized to LUT-based mapping

Resynthesis Framework

Constructing Boolean Relation of the Node and Candidate Divisors 1 Construction steps: l Collect candidate divisors di of node n l l l Use the same output variables Add inverter for node n in one copy Create comparator for the outputs Set the comparator to 1 l l … Duplicate the window of node n l l Divisors are not in the TFO of n Their support is a subset of that of node n This is the care set of node n Convert all gates to CNF d 1 n d 2 n How the relation is used: l l Function n = F(d 1, d 2, …) belongs to the relation iff n can be implemented as a gate with function F in terms of divisors d 1, d 2, … SAT solver is used to recursively cofactor the relation using different variable orders, resulting in several qualifying functions F X

SAT-based Relation Solving l l Given Boolean relation in CNF, find contained Boolean function(s) Pseudo-code function Solve. BR_rec( ordered set of divisors D, assigned divisors A, cnf C ) { if ( F=1 is UNSAT under assumptions in A ) return 0; if ( F=0 is UNSAT under assumptions in A ) return 1; if ( F=di or F=!di for a divisor di in D under assumptions in A ) return di or !di; d = the topmost variable in D; return d ? Solve. BR_rec( D/d, A d=1, C ) : Solve. BR_rec( D/d, A d=0, C ); } functions Solve. BR( number of functions N, set of divisors D, cnf C ) { result = ; for ( i = 0; i < N; i++ ) { find a new ordering of divisors in D; result = result Solve. BR_rec( D, , C ); } return result; }

SAT vs. Other Methods l The presented computations can be implemented with l l The implementations differ in terms of l l Complexity, resource usage, quality of results, scalability Based on these metrics, SAT is rated highly l l l SAT, ATPG, BDDs, SOPs, truth tables, etc Relatively easy to implement Relatively inexpensive (both memory and runtime are reasonable) Good quality of result (typically, there is no clear win) The most scalable among known engines (the main advantage!) The main reason why SAT is more scalable than BDDs l l BDDs require construction of a canonical form before they can be used SAT can start working on the problem right away l As a result, SAT has more chances to solve a hard instance

Experimental Results l l The results explore scalability of the mapper in the context of area-optimization Using 10 large combinational logic cones abc 01> r ex 02. aig; amap; ps; mfs 3 -aev -I 4 -O 2; ps; time; echo ex 02 : i/o =25237/18422 lat = 0 nd = 34817 edge = 110193 area =34817. 00 delay =308. 95 lev = 13 Library processing: Var = 6. Cell = 71. Fun = 38660. Obj = 38660. Ave = 1. 00. Skip = 0. Rem = 0. Time = 0. 07 sec Remapping parameters: TFO = 2. TFI = 4. Fan. Max = 10. Mffc. Min = 1. Mffc. Max = 3. Dec. Max = 1. 0 -cost = no. Effort = yes. Sim = no. Node = 34817. Try = 34817. Change = 8764. Const 0 = 0. Const 1 = 0. Buf = 44. Inv = 7655. Gate = 1065. And. Or = 0. Effort = 962. Max. Div = 111. Max. Win = 136. Ave. Div = 8. Ave. Win = 11. Calls = 5442733. (Sat = 2698400. Unsat = 2744333. ) Over = 0. T/O = 0. Lib = 0. 07 sec ( 0. 62 %) Win = 0. 12 sec ( 1. 06 %) Cnf = 0. 10 sec ( 0. 88 %) Sat = 10. 98 sec ( 97. 08 %) Sat = 8. 19 sec ( 72. 41 %) Unsat = 1. 73 sec ( 15. 30 %) Eval = 0. 00 sec ( 0. 00 %) Timing = 0. 00 sec ( 0. 00 %) Other = 0. 04 sec ( 0. 35 %) ALL = 11. 31 sec (100. 00 %) Cone sizes: 1=7699 2=5 3=164 4=34 5=861 6=1 Gate sizes: 1=7867 2=897 Reduction: Nodes 1210 out of 34817 ( 3. 48 %) Edges 1702 out of 110193 ( 1. 54 %) ex 02 : i/o =25237/18422 lat = 0 nd = 33607 edge = 108491 area =33607. 00 delay =308. 95 lev = 13

Experimental Results Area-only mapping was performed using a unit-area library. Parameters of mfs 3 were selected to match the runtime of amap and &nf. Each of these commands took about 3 min for all benchmarks listed.

Conclusion l l Introduced SAT-based network optimization engine Discussed the key difference between SAT and BDDs l And why SAT replaced BDDs in many application domains l Discussed experimental results l Future work l Improving implementation l l Using cubes rather than minterms to improve quality Using assumption_push/assumption_pop to reduce runtime Using bookmark/rollback to reduce runtime Comparing SAT-based vs QBF-based solutions