Courtesy RK Brayton (UCB) and A Kuehlmann (Cadence) 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, {P j } (i.e. primes of (f+d)=G’+D) 2. Generate all the minterms of f=G’  D, {m i } 3. Build Boolean matrix B where B ij = 1 if m i  P j = 0 otherwise 4. Solve the minimum column covering problem for B (unate covering problem)

3 Difficulty Note: Can be ~ 2 n minterms ~ 3 n /n primes Thus O(2 n ) rows and O(3 n /n ) columns AND minimum covering problem is NP-complete. (Hence can probably be double exponential in size of input, i.e. difficulty is O(2 3 n ) n /n minterms primes 2n2n

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

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

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. m m Any row can be associated with a cube -- called the signature cube. e.g. m 1  m 2  P 2 P 4 P 5 P 7

7 Row and Column Dominance Definition: A row i 1 whose set of primes is contained in the set of primes of row i 2 is said to dominate i 2. Example: i i i 1 dominates i 2 We can remove row i 2, because we have to choose a prime to cover i 1, and any such prime also covers i 2. So i 2 is automatically covered.

8 Row and Column Dominance Definition: A column j 1 whose rows are a superset of another column j 2 is said to dominate j 2. Example: j 1 dominates j 2 We can remove column j 2 since j 1 covers all those rows and more. We would never choose j 2 in a minimum cover since it can always be replaced by j 1. j 1 j

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

A 1 C 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 B i 1,B i 2 are independent if  j such that B i 1j =B i 2j =1. (They have no column in common) Example: Covering matrix B rearranged with independent sets first. Independent set = I of rows B=

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

13 Heuristic Let I={I 1, I 2, …, I k } be the independent set of rows choose j  I i which covers the most rows of A. Put j  J eliminate all rows covered by column j I  I\{I i } 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 A 1 C

14 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 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  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 x’ y’ x’ z’ F = x’ y’ + w x y + x’ y z’ + w y’ z Courtesy: Maciej Ciesielski, UMASS

15 Generating Primes – multiple outputs Procedure similar to single-output function, except: –include also the primes of the products of individual functions Example: x y z 0 – – 1 f 1 f Can also represent it as: x y z 0 – – – – 1 f 1 f x y z f1f1 f2f2 101

16 Generating Primes - example Modification (w.r.t single output function): –When two adjacent implicants are merged, the output parts are intersected 000 | 01  010 | 01  011 | | 10  111 | 10  0 – 0 | – | 01 – 1 1 | 10 1 – 1 | 10 x y z 0 – – 1 f 1 f There are five primes listed for this two-output function. - What is the min cover ? f2f f1f1 101

17 Minimize multiple-output cover - example List multiple-output primes f1f1 f2f p 1 = | 11 p 2 = 0 – 0 | 01 p 3 = 0 1 – | 01 p 4 = – 1 1 | 10 p 5 = 1 – 1 | | | | | | | 10 listed twice p 1 p 2 p 3 p 4 p 5 Min cover has 3 primes: F = { p 1, p 2, p 5 } Create a covering table, solveCreate a covering table, solve

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 = x i q, q is a prime of, P = x i r, 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=x i abc. Suppose r=ab is a prime of. Then is an implicant of f Thus abc and x i ab are implicants, so x i abc is not prime. Note: abc is prime because if not, ab  f x (or ac or bc) contradicting abc prime of Note: x i ab is prime, since if not then either ab  f or x i a  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 primes 2. Make G irredundant (in optimum way) Note: Q-M is exact i.e. it gives an exact minimum Heuristic Methods: 1.Generate (somehow) a cover of  using some of the primes 2.Make G irredundant (maybe not optimally) 3.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=x i. Shannon cofactor is Generalized cofactor with respect to g=x i is Note that In fact f x i is the unique cover of co(f, x i ) independent of the variable x i.

23 Shannon Cofactor off on Don’t care

24 Shannon Cofactor (Cont.) So

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