Logic Synthesis

1. Logic Synthesis Two-Level Minimization I

2. Quine-McCluskey Procedure (Exact) Given G’ and D (covers for F = (f,d,r) and d), find a minimum cover G of primes where: f  G  f+d (G is a prime cover of F ) Q-M Procedure: 1. Generate all the primes of F , {Pj} (i.e. primes of (f+d)=G’+D) 2. Generate all the minterms of f=G’D, {mi} 3. Build Boolean matrix B where Bij = 1 if mi Pj = 0 otherwise 4. Solve the minimum column covering problem for B (unate covering problem)

3. Difficulty Note:Can be • ~ 2n minterms • ~ 3n/n primes Thus O(2n) rows and O(3n/n ) columns AND minimum covering problem is NP-complete. (Hence can probably be double exponential in size of input, i.e. difficulty is O(23n) primes 3n/n 0 1 minterms 2n 0 0 0 1

4. y w xz xyzw xyzw xyzw xyzw Example Primes: y + w + xz Covering Table Solution: {1,2} y + w is minimum prime cover. (also w+ xz) xz y xy xy xy xy zw w Karnaugh map zw zw zw

5. y w xz xyzw xyzw xyzw xyzw Covering Table Primes of f+d Definition: An essential prime is any prime that uniquely covers a minterm of f. Minterms of f Row singleton (essential minterm) Essential prime

6. Covering Table Row Equality: In practice, many rows are identical. That is there exist minterms that are contained in the same set of primes. m1 0101101 m2 0101101 Any row can be associated with a cube -- called the signature cube. e.g. m1 m2  P2P4P5P7

7. Row and Column Dominance Definition:A row i1 whose set of primes is contained in the set of primes of row i2 is said to dominate i2. Example: i1 011010 i2 011110 i1 dominates i2 We can remove row i2, because we have to choose a prime to cover i1, and any such prime also covers i2. So i2 is automatically covered.

8. Row and Column Dominance Definition:A column j1 whose rows are a superset of another column j2 is said to dominate j2. Example: j1 dominates j2 We can remove column j2 since j1 covers all those rows and more. We would never choose j2 in a minimum cover since it can always be replaced by j1. j1 j2 1 0 0 0 1 1 0 0 1 1

9. Pruning the Covering Table 1. Remove all rows covered by essential primes (columns in row singletons). Put these primes in the cover G. 2. Group identical rows together and remove dominated rows. 3. Remove dominated columns. For equal columns, keep one prime to represent them. 4. Newly formed row singletons define n-ary essential primes. 5. Go to 1 if covering table decreased. The resulting reduced covering table is called the cyclic core. This has to be solved (unate covering problem). A minimum solution is added to G - the set of n-ary essential primes. The resulting G is a minimum cover.

10. Example Essential Prime and Column Dominance G=P1 n-ary Essential Prime and Column Dominance G=P1 + P3 Row dominance Cyclic Core

11. Solving the Cyclic Core Best known method (for unate covering) is branch and bound with some clever bounding heuristics. Independent Set Heuristic: Find a maximum set of “independent” rows I. Two rows Bi1 ,Bi2 are independent if j such that Bi1j =Bi2j=1. (They have no column in common) Example: Covering matrix B rearranged with independent sets first. 1 1 1 Independent set = I of rows B= 0 1 1 1 1 1 1 C A

12. Solving the Cyclic Core Lemma: |Solution of Covering|  |I| 1 1 1 0 1 1 1 1 1 1 C A

13. 1 1 1 0 1 1 1 1 1 1 C A Heuristic Let I={I1, I2, …, Ik} be the independent set of rows • choose j  Ii which covers the most rows of A. Put j  J • eliminate all rows covered by column j • I  I\{Ii} • go to 1 if |I|  0 • If B is empty, then done (in this case we have the guaranteed minimum solution -IMPORTANT) • If B is not empty, choose an independent set of B and go to 1

14. w’ x’ y’  w’ x’ z’  x’ y’ z’  x’ y’ z  x’ y z’  w x’ y’  w x’ z’  w y’ z w y z’ w x y w x z w’ x’ y’ z’  w’ x’ y’ z  w’ x’ y z’  w x’ y’ z’  w x’ y’ z  w x’ y z’  w x y z’  w x y’ z  w x y z  x’ y’ x’ z’ Generating Primes - single output func. Tabular method (based on consensus operation): • Start with all minterm canonical form of F • Group pairs of adjacent minterms into cubes • Repeat merging cubes until no more merging possible; mark () + remove all covered cubes. • Result: set of primes of f. Example: F= x’ y’ + w x y + x’ y z’ + w y’ z F= x’ y’ + w x y + x’ y z’ + w y’ z Courtesy: Maciej Ciesielski, UMASS

15. x y z 0 – 0 0 1 1 1 – 1 f1 f2 0 1 1 1 1 0 f2 011 111 f1 011 111 010 010 101 101 010 110 010 110 000 100 x y z 0 – 0 0 1 – – 1 1 1 – 1 f1 f2 0 1 0 1 1 0 1 0 000 100 y x z Can also represent it as: Generating Primes – multiple outputs • Procedure similar to single-output function, except: • include also the primes of the products of individual functions Example:

16. x y z 0 – 0 0 1 1 1 – 1 f1 f2 0 1 1 1 1 0 000 | 01  010 | 01  011|11 101 | 10  111 | 10  0 – 0|01 0 1–|01 – 1 1|10 1 – 1|10 f2 f1 011 111 011 111 010 010 101 101 010 010 110 110 000 100 000 100 Generating Primes - example • Modification (w.r.t single output function): • When two adjacent implicants are merged, the output parts are intersected There are five primes listed for this two-output function. - What is the min cover ?

17. p1 p2 p3 p4 p5 000 | 01 010 | 01 011 | 01 011 | 10 101 | 10 111 | 10 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 listed twice f1 f2 011 111 011 111 010 010 101 101 010 010 110 110 000 100 000 100 Minimize multiple-output cover - example • List multiple-output primes • Create a covering table, solve p1 = 0 1 1 | 11 p2 = 0 – 0 | 01 p3 = 0 1 – | 01 p4 = – 1 1 | 10 p5 = 1 – 1 | 10 Min cover has 3 primes: F = { p1, p2, p5 }

18. Generating Primes We use the unate recursive paradigm. The following is how the merge step is done. (Assumption: we have just generated all primes of and .) Theorem: p is a prime of f iff p is maximal among the set consisting of • P = xiq, q is a prime of , • P = xir, r is a prime of , • P = qr, q is a prime of , r is a prime of

19. Generating Primes Example: Assume q = abc is a prime of . Form p=xiabc. Suppose r=ab is a prime of . Then is an implicant of f Thus abc and xiab are implicants, so xiabc is not prime. Note: abc is prime because if not, ab  fx(or ac or bc) contradicting abc prime of Note: xiab is prime, since if not then either ab  f or xia  f . The first contradicts abc prime of and the second contradicts ab prime of

20. Quine-McCluskey Summary Q-M: 1. Generate cover of all primes2. Make G irredundant (in optimum way) Note: Q-M is exact i.e. it gives an exact minimum Heuristic Methods: • Generate (somehow) a cover of  using some of the primes • Make G irredundant (maybe not optimally) • Keep best result - try again (i.e. go to 1)

21. Generalized Cofactor Definition: Let f, g be completely specified functions. The generalized cofactor of f with respect to g is the incompletely specified function: Definition: Let  = (f, d, r) and g be given. Then

22. Relation with Shannon Cofactor • Let g=xi . Shannon cofactor is • Generalized cofactor with respect to g=xi is • Note that In fact fxi is the unique cover of co(f, xi )independent of the variable xi .

23. Shannon Cofactor on off Don’t care

24. Shannon Cofactor (Cont.) So

25. Properties of Generalized Cofactor Shannon Cofactor Generalized Cofactor We will get back to use of generalized cofactor later