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Logic Synthesis 2

Outline Two-Level Logic Optimization ESPRESSO Goal Understand two-level optimization Understand ESPRESSO operation. Logic Synthesis 2. N-dimensional boolean space - 2 N points, each associated with a unique set of N literals e.g. entries in a Karnaugh map or truth table

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Logic Synthesis 2

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  1. Outline Two-Level Logic Optimization ESPRESSO Goal Understand two-level optimization Understand ESPRESSO operation Logic Synthesis 2

  2. N-dimensional boolean space - 2N points, each associated with a unique set of N literals e.g. entries in a Karnaugh map or truth table each point is a minterm e.g. abcd, ab’cd, in space <a:d> cube - conjunction (AND) of literals in N-dim boolean space points on N-dim hypercube that are 1 examples: a’bc, acd expression - disjunction (OR) of cubes, i.e. equation example: a’bc + def don’t cares - missing literals from cube example: abc in space of <a:d>, d is don’t care result is cube covering larger part of space abc = abcd’ + abcd Logic Optimization Definitions cube: a’ DC: b a’b ab space: <a:b> a’b’ ab’

  3. Approach find minimal set of cubes to cover ON-set (1 minterms) each cube = AND gate minimal cubes => minimal AND gates each expression = cubes + OR gate one expression (OR gate) per output exploit don’t cares to increase cube sizes each DC doubles cube size cube must only cover 1 or DC vertices or cover OFF-set (0 minterms) instead ON DC OFF Two-Level Logic Optimization a’b ab redundancy in cube cover a’ + b a’b’ ab’

  4. Minimal set of cubes minimum graph covering problem NP-complete - exponential in worst case must use heuristic search Complications solve simultaneously for each expression (output) minimize total number of unique cubes consider ON vs. OFF vs. DON’T CARE set Two-Level Logic Optimization ESPRESSO input ESPRESSO output .i 3 .o 3 .p 4 10x101 x01100 110110 11x010 .e .i 3 .o 3 .p 4 -01 100 11- 010 1-0 100 10- 001 .e x = b’c + ac’ y = ab z = ab’ x = ab’ + b’c + abc’ y = abc’ + ab z = ab’

  5. Approach minimize cover of ON-set of function ON-set is set of vertices for which expression is TRUE minimum set of cubes exploit don’t cares to increase cube sizes Algorithm start with cubes covering the ON-set this is just sum-of-products form iteratively expand, shrink, add, remove cubes remove redundant (covered) cubes result is irredundant cover Two-Level Logic Optimization x = a’ + b x = a’b + ab + a’b’ a’b ab a’b ab a’b’ ab’ a’b’ ab’

  6. ESPRESSO Algorithm • Forig = ON-set; /* vertices with expression TRUE */ • R = OFF-set; /* vertices with expression FALSE */ • D = DC-set; /* vertices with expression DC */ • F = expand(Forig, R); /* expand cubes against OFF-set */ • F = irredundant(F, D); /* remove redundant cubes */ • do { • do { • F = reduce(F, D); /* shrink cubes against ON-set */ • F = expand(F, R); • F = irredundant(F, D); • } until cost is “stable”; • /* perturb solution */ • G = reduce_gasp(F, D); /* add cubes that can be reduced */ • G = expand_gasp(G, R); /* expand cubes that cover another */ • F = irredundant(F+G, D); • } until time is up; • ok = verify(F, Forig, D); /* check that result is correct */

  7. Expand expand essentialcubes in F in decreasing size to a prime cube prime cube - fully expanded against OFF-set essential cube - contains essential vertex essential vertex - minterm no other cube covers remove any covered cubes ON 01 11 DC 00 10 OFF Cube Operations 01 11 Expand 00 10

  8. Irredundant find minimal cover with each cube containing an essential vertex find relatively essential cubes E removing them violates cover - keep them redundant cubes R = F - E can be individually removed totally redundant Rt - covered by E+D remove Rt partially redundant Rp - R - Rt new F = E + minimal set of Rp ON DC OFF Cube Operations E 01 11 01 11 Rp Rt Irredundant 00 10 00 10 Rp

  9. Reduce shrink cubes in descending order of size while maintaining cover smaller cubes can expand in more directions smaller cubes more likely to be covered by other cubes during expansion ON 01 11 01 11 DC 00 10 00 10 OFF Cube Operations Reduce

  10. Reduce Gasp for each cube add a subcube not covered by other cubes Expand Gasp expand subcubes and add them if they cover another cube later use Irredundant to discard redundant cubes this is a “last gasp” heuristic for exploration no ordering by cube size ON DC OFF Cube Operations 01 11 01 11 Reduce Gasp 00 10 00 10 01 11 01 11 Expand Gasp 00 10 00 10

  11. Example x = a’b + ab + a’b’ Expand a’b ab a’b ab a’b’ ab’ a’b’ ab’ Irredundant Reduce a’b ab a’b ab a’b’ ab’ a’b’ ab’ Expand x = a’ + b a’b ab Irredundant a’b ab a’b’ ab’ a’b’ ab’ Cost Stable

  12. Examples E Rp Rp E Prime & Irredundant Cover Essential and Redundant Cubes Initial Cover Reduce Expand in right direction

  13. Experimental Results ESPRESSO algorithm gets minimum or close to minimum cover where cover is known up to 10 000 input literals, 100 inputs, 100 outputs tested CPU time < 12 min on high-speed workstation Application PLA minimization use as subroutine in multi-level logic minimization minimize pieces of larger circuit Conclusions

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