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MVSIS Group

Optimization of Multi-Valued Multi-Level Networks. MVSIS Group Minxi Gao,, Jie-Hong Jiang, Yunjian Jiang, Yinghua Li, Alan Mishchenko 1 , Subarna Sinha, Tiziano Villa 2 , and Robert Brayton Dept. of Electrical Engineering and Computer Science University of California, Berkeley.

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MVSIS Group

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  1. Optimization of Multi-Valued Multi-Level Networks MVSIS Group Minxi Gao,, Jie-Hong Jiang, Yunjian Jiang, Yinghua Li, Alan Mishchenko1, Subarna Sinha, Tiziano Villa2, and Robert Brayton Dept. of Electrical Engineering and Computer Science University of California, Berkeley 1 Portland State Univ., Portland OR 2 Parades, Rome, Italy

  2. Outline • Motivations: From binary to multi-value • MV Networks & Design specification • MVSIS optimizations • Node simplification • Algebraic extraction • Pairing merging and encoding • Network manipulations • Demo • New capabilities • Conclusions

  3. Motivations • Synchronous binary hardware synthesis • Software synthesis from synchronous specifications • Asynchronous hardware synthesis • Multi-valued devices? • Current-mode CMOS devices • Optical logic circuits

  4. Verilog-MV Verilog vl2mv BLIF-MV Two-level MV-PLA synthesis R. Rudell, et al “Espresso-MV”, 1987 MV-Optimize FSM state encoding T. Villa, et al, “Nova”, 1990E. Goldberg, et al, “Minsk”, 1999 Opt-Encode Encode MVSIS 1.1 Multi-level FSM synthesis (single MV) L. Lavagno, et al “MIS-MV”, 1990 SIS Motivation – synchronous hardware • Design and synthesis from multi-valued logic • MV is natural method of specification • Larger design space

  5. EFSM’s POLIS F.Balerin, et al, “Synthesis of software programsfor embedded control applications”, TCAD 1999 MV-Optimize ESTEREL G.Berry, “The foundations of Esterel”, 2000 Code Gen POLISVCC MVSIS 1.1 MVSIS Y.Jiang, et al, “Logic optimization and code generationfor embedded control applications”, CODES 2000 C/Assembly Motivation – software synthesis • Synchronous programming of embedded systems • Esterel/Lustre/Signal • Interactive FSM semantics • Code generation from logic

  6. x m x+y vm y Motivation – multi-valued devices • Multi-valued current-mode MOS • signed digit arithmetic • High-speed, Low supply voltage Building blocks Iy Ix IT x y1 y2 T. Hanyu and M. Kameyama, “A 200 MHz pipelined multiplier using 1.5 V-supplymultiple-valued MOS current-mode circuits with dual-rail source-coupled logic”, IEEE Journal of Solid-Statee Circuits, 1995 A. Jain, R. Bolton and M. Abd El-Barr, “CMOS Multi-Valued Logic Design”, IEEE Trans. on Circuits and Systems, Aug. 1993.

  7. Each output value is called an i-set F(u,v,w): {0,1} x {0,1,2} x {0,1,2} {0,1,2} 0-set: F{0} = u{0} v{0} + u{0} v{1} w{0,1} 2-set: F{2} = u{1} v{1,2} + u{1} v{0} w{1,2} 1-set: F{1} = <default> Functional Semantics(MV-Network) F • Network of MV-nodes • Each variable xn has its own range{0, 1,…, |pn|-1} • Values are treated uniformly • MV-literal: x{0,2} • MV-cube: x{0,2}z{0,1} • MV-relation at each node (can be non-deterministic) Latch

  8. Design Specification blif-mv .model simple .inputs a b .outputs f .mv f 3 .mv x 3 .table x a b -> f .def 0 0 1 1 1 {1,2} 0 - 1 1 - 1 2 0 - 0 2 ... .reset x 0 .latch f x ... .exdc .inputs a b .outputs f .table a b -> f .def 0 0 0 1 .end • BLIF-MV subset • Single output MV nodes • Can be non-deterministic • Flat network, no hierarchy (yet) • Constant initial states (.reset) • Extensions • External don’t care networks (.exdc) • Can have an external don’t care specified for each output • Can specify datapaths

  9. MVSIS Optimization • MVSIS optimizations • Node simplification • Kernel and cube extraction/decomposition • Pairing/Merging • Encoding • Network manipulations

  10. a b c d z 100 11 011 101 0 101 10 111 011 0 110 10 110 110 0 100 11 111 101 1 111 01 100 111 1 100 11 110 110 1 <default> 2 101 01 100 001 - 010 01 001 101 - DC MV-SOP Minimizers • For each i-set (MV-input, binary output) • Two-level: Espresso-MV • minimize an i-set with a don’t care. • a don’t care is an input for which the output can be any value. • Two-Level: ISOP (Minato) • Fast method to build a cover of cubes of an i-set from MDD of function and MDD of don’t cares (method of Minato extended to MV). • All i-sets at once • Quine-McCluskey type ND minimization • Given a ND relation, generate a cover of all i-sets such that the total number of cubes is minimum.

  11. Q1Q2...Qr Di i image P1P2...Pn DCi care set Node Simplification mvsis> simplify mvsis> fullsimp mvsis> reset_default Multi-level (using don’t cares - inputs for which output can be any value) • Compatible observability don’t cares(CODC) • Satisfiability don’t cares (SDC) • External don’t cares (XDC) • Generalization from binary case Reference: Y. Jiang et. al. “Compatible Observable Don’t Cares for MV Logic, ICCAD’01

  12. Node Simplification mvsis> complete_simplify -m [ISOP, ESP, QM] Using non-determinism • Derive Complete Flexibility at a node (an ND relation) • Minimize ND relation • Espresso • ISOP • QM • Automatically finds best default • Uses external specification (relation) Reference: A. Mishchenko and R. Brayton, “Simplification of Non-Deterministic MV Networks”, IWLS, June 2002.

  13. Z yj Yj X Computing Complete Flexibility at a node The complete flexibility at node j is

  14. P2 P1 P3 P0 minterms Quine-McCluskey type ND relation minimization Given an ND relation, e.g. the complete flexibility, its i-set is the set of input minterms that can produce output value i. • Generate for each i-set all its primes, Pi • Form covering table with one column for each pj in Pi for all i • One row for each minterm in the input space • Solve minimum covering problem • Primes chosen from each Piis the cover for each i-set.

  15. F Algebraic Decompositions Example: factoring/ decomposition/resub • Kernel extraction • Semi-algebraic division • Resubstitution • Factoring/Decomposition [-q]: Two-cube divisors [-g]: Best divisors • F = a{0,1,2} c{3} + b{1,2,3} c{3} • + a{0}b{1,2,3} c{0} +a{0} c{1} = (c{3} +a{0}c{0,1}) (a{0,1,2} c{1,3} +b{1,2,3} c{0,3}) mvsis> fx [-q] [-g] mvsis> resub mvsis> decomp mvsis> factor M. Gao and R. K. Brayton, “Multi-valued Multi-level Network Decomposition”, IWLS, June 2001.

  16. Example: EBD Algebraic Decompositions mvsis> ebd_fx mvsis> ebd_decomp • Stands for Encode, Binary, Decode • Use binary codes to encode multi-valued variable x, e.g • Operate with fast binary implementations imported from SIS • Convert (decode) back to multi-valued J-H Jiang, A. Mishchenko, R. Brayton, Reducing Multi-Valued Operations to Binary, IWLS’02 Results: Quality almost as good, but much faster

  17. a b c d x a b c d y 100 11 011 101 0 101 10 111 011 0 110 10 110 110 0 100 11 011 101 0 101 10 111 011 0 <default> 1 <default> 1 merge a b c d z x0y0 z0 100 11 011 101 0 101 10 111 011 0 x0y1 z1 100 11 111 101 1 111 01 100 111 1 110 10 110 110 1 x1y0 z2 x1y1 z3 <empty> 2 <default> 3 Pairing Merging and Encoding • Pair_decode/Merge • Combine two or more nodes into a single node with more values. • Explore different combinations • Encode • full and partial encode • Combine some i-sets • combine i-sets where those values always appear together in fanouts. mvsis> pair_decode mvsis> merge mvsis> encode

  18. Other Commands • Network manipulations • mvsis> eliminate • mvsis> collapse • mvsis> sweep • IO interface • mvsis> read(write)_blifmv • mvsis> read(write)_blif • Verification • mvsis> validate -n # (uses simulation) • mvsis> verify • mvsis> gen_vec • mvsis> simulate • mvsis> qcheck (quick check for ND network) • Printing • mvsis> print • mvsis> print_stats • mvsis> print_factor • mvsis> print_range • mvsis> print_io • mvsis> print_value • Sequential • mvsis> extract_seq_dc

  19. Design Flow • Typical design flow mvsis> source mvsis.script (general MV script) mvsis> encode -i mvsis> source mvsis.scriptb (keeps all nodes binary)

  20. Example #1 Matrix multiplication (3 values) #2 X 2 matrix mult over the ring Z_3 .model matmul .inputs a11 a12 a21 a22 .inputs b11 b12 b21 b22 .outputs c11 c12 c21 c22 .mv a11, a12, a21, a22 3 .mv b11, b12, b21, b22 3 .mv c11, c12, c21, c22 3 .table a11 a12 b11 b21 c11 0 0 - - 0 0 1 - - =b21 0 2 - 0 0 0 2 - 1 2 0 2 - 2 1 1 0 - - =b11 1 1 0 0 0 1 1 0 1 1 1 1 0 2 2 1 1 1 0 1 … … .table a11 a12 b12 b22 c12 0 0 - - 0 0 1 - - =b22 0 2 - 0 0 0 2 - 1 2 0 2 - 2 1 1 0 - - =b12 1 1 0 0 0 1 1 0 1 1 1 1 0 2 2 … … .table a21 a22 b11 b21 c21 0 0 - - 0 0 1 - - =b21 0 2 - 0 0 0 2 - 1 2 0 2 - 2 1 1 0 - - =b11 1 1 0 0 0 1 1 0 1 1 1 1 0 2 2 1 1 1 0 1 1 1 1 1 2 1 1 1 2 0 1 1 2 0 2 1 1 2 1 0 1 1 2 2 1 1 2 0 0 0 1 2 0 1 2 1 2 0 2 1 1 2 1 0 1 1 2 1 1 0 … … .table a21 a22 b12 b22 c22 0 0 - - 0 0 1 - - =b22 0 2 - 0 0 0 2 - 1 2 0 2 - 2 1 1 0 - - =b12 1 1 0 0 0 1 1 0 1 1 1 1 0 2 2 1 1 1 0 1 … … .end

  21. Demo • simulated • live

  22. [cadntws11:/home/wjiang/mvsis/examples/bob] mvsis UC Berkeley, MVSIS mvsis> help alias chng_name collapse decomp delete echo elim_part eliminate encode extract_seq_dc factor fullsimp fx gen_vec help history merge pair_decode print print_altname print_factor print_io print_level print_part_value print_range print_stats print_value qcheck quit read_blif read_blifmv reset_default reset_name resub runtime set simplify simulate source sweep unalias undo unset usage validate write_blifmv mvsis> mvsis> read_blifmv matmul-c mvsis> mvsis> chng_name changing to short-name mode mvsis> print_stats matmul: 4 nodes, 4 POs, 128 cubes(sop), 480 lits(sop) mvsis>

  23. mvsis> print_io primary inputs: a b c d e f g h primary outputs: {i} {j} {k} {l} mvsis> mvsis> set autoexec pfs matmul: 4 nodes, 4 POs, 128 cubes(sop), 480 lits(sop), 216 lits(fact.) mvsis> mvsis> print_range {i}: 3 {j}: 3 {k}: 3 {l}: 3 a: 3 b: 3 c: 3 d: 3 e: 3 f: 3 g: 3 h: 3 matmul: 4 nodes, 4 POs, 128 cubes(sop), 480 lits(sop), 216 lits(fact.) mvsis> mvsis> simplify matmul: 4 nodes, 4 POs, 96 cubes(sop), 320 lits(sop), 160 lits(fact.) mvsis> mvsis> reset_default matmul: 4 nodes, 4 POs, 96 cubes(sop), 320 lits(sop), 160 lits(fact.) mvsis>

  24. mvsis> fullsimp matmul: 4 nodes, 4 POs, 96 cubes(sop), 320 lits(sop), 160 lits(fact.) mvsis> mvsis> pair_decode 1 m{0} = a{0}e{2} + e{0} m{1} = a{0}e{1} m{3} = a{1}e{2} + a{2}e{1} n{0} = a{0}f{2} + f{0} n{1} = a{0}f{1} n{3} = a{1}f{2} + a{2}f{1} o{0} = e{0}c{2} + c{0} o{1} = e{0}c{1} o{3} = e{1}c{2} + e{2}c{1} p{0} = f{0}c{2} + c{0} p{1} = f{0}c{1} p{3} = f{1}c{2} + f{2}c{1} q{0} = b{0}g{2} + g{0} q{1} = b{0}g{1} q{3} = b{1}g{2} + b{2}g{1} r{0} = b{0}h{2} + h{0} r{1} = b{0}h{1} r{3} = b{1}h{2} + b{2}h{1} s{0} = g{0}d{2} + d{0} s{1} = g{0}d{1} s{3} = g{1}d{2} + g{2}d{1} t{0} = h{0}d{2} + d{0} t{1} = h{0}d{1} t{3} = h{1}d{2} + h{2}d{1} matmul: 12 nodes, 4 POs, 64 cubes(sop), 184 lits(sop), 160 lits(fact.) mvsis>

  25. mvsis> simplify matmul: 12 nodes, 4 POs, 56 cubes(sop), 96 lits(sop), 96 lits(fact.) mvsis> mvsis> reset_default matmul: 12 nodes, 4 POs, 56 cubes(sop), 96 lits(sop), 96 lits(fact.) mvsis> mvsis> fullsimp matmul: 12 nodes, 4 POs, 56 cubes(sop), 96 lits(sop), 96 lits(fact.) mvsis>

  26. mvsis> print_factor {i}{1} = m{2}q{2} + m{1}q{0} + m{0}q{1} {i}{2} = m{2}q{0} + m{1}q{1} + m{0}q{2} {j}{1} = n{2}r{2} + n{1}r{0} + n{0}r{1} {j}{2} = n{2}r{0} + n{1}r{1} + n{0}r{2} {k}{1} = o{2}s{2} + o{1}s{0} + o{0}s{1} {k}{2} = o{2}s{0} + o{1}s{1} + o{0}s{2} {l}{1} = p{2}t{2} + p{1}t{0} + p{0}t{1} {l}{2} = p{2}t{0} + p{1}t{1} + p{0}t{2} m{0} = a{0} + e{0} m{2} = a{2}e{1} + a{1}e{2} n{0} = a{0} + f{0} n{2} = a{2}f{1} + a{1}f{2} o{0} = c{0} + e{0} o{2} = c{2}e{1} + c{1}e{2} p{0} = c{0} + f{0} p{2} = c{2}f{1} + c{1}f{2} q{0} = b{0} + g{0} q{2} = b{2}g{1} + b{1}g{2} r{0} = b{0} + h{0} r{2} = b{2}h{1} + b{1}h{2} s{0} = d{0} + g{0} s{2} = d{2}g{1} + d{1}g{2} t{0} = d{0} + h{0} t{2} = d{2}h{1} + d{1}h{2} matmul: 12 nodes, 4 POs, 56 cubes(sop), 96 lits(sop), 96 lits(fact.) mvsis>

  27. mvsis> validate -m mdd matmul-c Networks are combinationally equivalent according to MDD method. matmul: 12 nodes, 4 POs, 56 cubes(sop), 96 lits(sop), 96 lits(fact.) mvsis>

  28. [cadntws11:/home/wjiang/mvsis/examples/bob] mvsis UC Berkeley, MVSIS mvsis> mvsis> read_blifmv red-add.mv mvsis> mvsis> chng_name changing to short-name mode mvsis> mvsis> print_io primary inputs: a b c d e primary outputs: {f} {g} {h} mvsis> mvsis> set autoexec pfs red_adder: 3 nodes, 3 POs, 48 cubes(sop), 240 lits(sop), 69 lits(fact.) mvsis> mvsis> reset_default red_adder: 3 nodes, 3 POs, 48 cubes(sop), 240 lits(sop), 69 lits(fact.) mvsis> mvsis> simplify red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis> mvsis> fullsimp red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis>

  29. mvsis> print_range {f}: 2 {g}: 2 {h}: 2 a: 8 b: 8 c: 8 d: 8 e: 8 red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis> mvsis> encode red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis> mvsis> print_range {f}: 2 o: 2 {g}: 2 p: 2 {h}: 2 q: 2 i: 2 r: 2 j: 2 s: 2 k: 2 t: 2 l: 2 u: 2 m: 2 v: 2 n: 2 w: 2 red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis>

  30. mvsis> simplify red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis> mvsis> fullsimp red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis> mvsis> print_io primary inputs: i j k l m n o p q r s t u v w primary outputs: {f} {g} {h} red_adder: 3 nodes, 3 POs, 15 cubes(sop), 44 lits(sop), 28 lits(fact.) mvsis>

  31. mvsis> read_blifmv red-add.mv red_adder: 3 nodes, 3 POs, 48 cubes(sop), 240 lits(sop), 69 lits(fact.) mvsis> mvsis> help encode Feb 16, 2001 MVSIS(1) encode [-i] [-n] [-s] Encode the whole network into a binary one, considering both output and input constraints. For sequential networks, a latch is encoded with constraints generated from both its inputs and outputs -i keep primary inputs and outputs as multi-valued; add interface nodes between the internal encoded binary network and PI/POs. This option allows validation of the result. -n use natural code -s use NO_COMP rather than ESPRESSO as the intermediate minimization method. The difference is only in performance. Ordinary users should not be concerned with this option. red_adder: 3 nodes, 3 POs, 48 cubes(sop), 240 lits(sop), 69 lits(fact.) mvsis> mvsis> encode -n red_adder: 3 nodes, 3 POs, 1251 cubes(sop), 9432 lits(sop), 577 lits(fact.) mvsis>

  32. mvsis> simplify -t 1000 red_adder: 3 nodes, 3 POs, 315 cubes(sop), 2010 lits(sop), 156 lits(fact.) mvsis> mvsis> simplify -t 1000 -m exact red_adder: 3 nodes, 3 POs, 300 cubes(sop), 1989 lits(sop), 130 lits(fact.) mvsis> mvsis> validate -m mdd red-add-bin.mv Networks differ on (at least) primary output s1 i-set 0 Incorrect input is: 0 x1_b0 1 x1_b1 1 x1_b2 0 x0_b0 0 x0_b1 0 x0_b2 0 y1_b0 0 y1_b1 0 y1_b2 0 y0_b0 0 y0_b1 0 y0_b2 0 cin_b0 0 cin_b1 0 cin_b2 Networks are NOT combinationally equivalent. red_adder: 3 nodes, 3 POs, 300 cubes(sop), 1989 lits(sop), 130 lits(fact.) mvsis>

  33. New Capabilities • Non-deterministic MV Networks • Post Networks • Delay Insensitive Asynchronous Synthesis

  34. Conclusions • MV logic networks important in various applications • Presented MVSIS, an multi-valued logic synthesis software infrastructure • Release 1.1 on Linux and Windows platforms (as of May, 2002) • Support registers • External and sequential don’t cares • Verification based on MDD representations • software generation from Esterel • use of complete flexibility • non-determinism http://www-cad.eecs.berkeley.edu/Respep/Research/mvsis free download (binary versions available on Windows, Unix)

  35. Multi-Valued Logic Optimization on Post Logic Networks submitted to ICCAD 2002 Don’t Care Computations in Minimizing Extended Finite State Machines with Presburger Arithmetic IWLS 2002 Software Synthesis from Synchronous Specifications Using Logic Simulation Techniques DAC 2002 Simplification of Non-Deterministic Multi-Valued Networks IWLS 2002 A Boolean Paradigm in Multi-Valued Logic Synthesis IWLS 2002 Some Recent Publications

  36. Thank You

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