1. ECE 667 Synthesis and Verification of Digital Systems
Heuristic Logic Optimization (Espresso)

2. Outline Two-level logic minimization Unate recursive paradigm
Heuristic logic optimizer Espresso
Unate recursive paradigm
Unateness
Recursive optimization
Single output optimization
Application of containment check
Multiple output optimization
Generalization from single-output optimization
Concept of a characteristic function
ECE 667- Espresso logic minimizer

3. ESPRESSO – two-level logic optimizer
ECE 667- Espresso logic minimizer

4. Shannon Expansion f : Bn  B Shannon Expansion:
Theorem: F is a cover of f. Then
F = x Fx + x’i Fx ‘
We say that f (F) is expanded about xi.
xi is called the splitting variable.
ECE 667- Espresso logic minimizer

5. List of Cubes (Cover Matrix)
We often use matrix notation to represent a cover:
Example: F = ac + c’d + bcd’
a b c d a b c d
a c
c’d or
bcd’
Each row represents a cube
1 means that the positive literal appears in the cube
0 means that the negative literal appears in the cube
The 2 (or -, don’t care) means that the variable does not appear in the cube.
Finding factors from matrix representation is easy.
ECE 667- Espresso logic minimizer

6. Cover Matrix
ECE 667- Espresso logic minimizer

7. Fundamental Theorem Theorem: Let c be a cube and f a function.
Then c  f  fc  1.
Proof. We use the fact that x fx = x f, and fx is independent of x.
If : Suppose fc  1. Then cf =fcc = c. Thus, c  f. f c
ECE 667- Espresso logic minimizer

8. Some Special Functions
Definition: A function f : Bn  B is symmetric with respect to variables xi and xj iff
f(x1,…,xi,…,xj,…,xn) = f(x1,…,xj,…,xi,…,xn)
Definition: A function f : Bn  B is totally symmetric iff any permutation of the variables in f does not change the function
Symmetry can be exploited in searching the binary decision tree
because:
- That means we can skip one of four sub-cases
- used in automatic variable ordering for BDDs
ECE 667- Espresso logic minimizer

9. Unate Functions ECE 667- Espresso logic minimizer
Definition: A function f : Bn  B is positive unate in variable xi iff
This is equivalent to monotone increasing in xi:
for all minterm pairs (m-, m+) where
For example, m-3=1001, m+3=1011 (where i =3)
ECE 667- Espresso logic minimizer

10. Unate Functions ECE 667- Espresso logic minimizer
Similarly for negative unate
monotone decreasing:
A function is unate in xi if it is either positive unate or negative unate in xi.
Definition: A function is unate if it is unate in each variable.
Definition: A cover F is positive unate in xi iff xi  cj for all cubes cjF
ECE 667- Espresso logic minimizer

11. Example - unateness ECE 667- Espresso logic minimizer
f is positive unate in a,b :
f(ma+)  f(ma-)
f(mb+)  f(mb-)
and negative unate in c:
f(mc-) = 1  f(mc+) = 0
mc+ c b a
off on
mc-
Minterms associated with c
mc- = (010) = 1
mc+ = (011) = 0
ECE 667- Espresso logic minimizer

12. The Unate Recursive Paradigm
Key pruning technique based on exploiting the properties of unate functions
based on the fact that unate leaf cases can be solved efficiently
New case splitting heuristic
splitting variable is chosen so that the functions at lower nodes of the recursion tree become unate
ECE 667- Espresso logic minimizer

13. The Unate Recursive Paradigm
Unate covers F have many extraordinary properties:
If a cover F is minimal with respect to single-cube containment, all of its cubes are essential primes.
In this case F is the unique minimum cube representation of its logic function.
A unate cover represents the tautology iff it contains a cube with no literals (constant 1).
Positive unate: f = x fx + fx’
Negative unate: f = fx + x’fx’
This type of implicit enumeration applies to many sub-problems (prime generation, reduction, complementation, etc.).
ECE 667- Espresso logic minimizer

14. Unate Recursive Paradigm
Create cofactoring tree stopping at unate covers
choose, at each node, the “most binate” variable for splitting
recurse until no binate variable left (unate leaf)
“Operate” on the unate cover at each leaf to obtain the result for that leaf. Return the result
At each non-leaf node, merge (appropriately) the results of the two children.
Main idea: Operation on unate leaf is computationally less complex
Operations: complement, simplify, tautology, generate-primes,...etc. a c b
merge
ECE 667- Espresso logic minimizer

15. Two Useful Theorems - Tautology
Checking for tautology is simplified for unate functions
Positive unate (f = x fx + fx’ ) f  1  fx’ = 1
Negative unate (f = fx + x’fx’) f  1  fx = 1
Theorem 2: Let A be a unate cover matrix.
Then A  1 if and only if A has a row of all “-”s.
Proof:
If. A row of all “-”s is the tautology cube.
Only if. Assume no row of all “-”s. Without loss of generality, suppose function is positive unate. Then each row has at least one “1” in it. Consider the point (0,0,…,0). This is not contained in any row of A. Hence A1.
ECE 667- Espresso logic minimizer

16. Recursive Tautology – termination rules
ECE 667- Espresso logic minimizer

17. Recursive Tautology - example
ECE 667- Espresso logic minimizer

18. Recursive Complement Operation
Theorem:
Proof:
ECE 667- Espresso logic minimizer

19. COMPLEMENT Operation ECE 667- Espresso logic minimizer
Algorithm COMPLEMENT(List_of_Cubes C) {
if(C contains single cube c) {
Cres = complement_cube(c) // generate one cube per
return Cres // literal l in c with ^l }
else {
xi = SELECT_VARIABLE(C)
C0 = COMPLEMENT(COFACTOR(C,^xi)) Ù ^xi
C1 = COMPLEMENT(COFACTOR(C,xi)) Ù xi
return OR(C0,C1)
ECE 667- Espresso logic minimizer

20. Recursive Complement – termination rules
ECE 667- Espresso logic minimizer

21. Recursive Complement – example (split)
ECE 667- Espresso logic minimizer

22. Recursive Complement – example (merge)
ECE 667- Espresso logic minimizer

23. Incompletely Specified Boolean Functions
F = (f, d, r) : Bn  {0, 1, *}
where * represents a don’t care.
f = onset function - f(x)=1  F(x)=1
r = offset function - r(x)=1  F(x)=0
d = don’t care function - d(x)=1  F(x)=*
(f,d,r) forms a partition of Bn, i.e.
f + d + r = Bn
fd = fr = dr =  (pairwise disjoint)
ECE 667- Espresso logic minimizer

24. Incompletely Specified Boolean Functions
A completely specified Boolean function g is a cover for F = (f,d,r) if
f  g  f+d
Note:
g r = 
if xd , then g(x) = 0 or 1 (don’t care)
if xf , then g(x)=1
if xr , then g(x)=0.
Also: r = f’d’  g’  f’ f d r
ECE 667- Espresso logic minimizer

25. ESPRESSO – 2-level logic optimizer
ECE 667- Espresso logic minimizer

26. Example: Logic Minimization (single output)
Consider F(a,b,c)=(f,d,r), where f={abc, abc, abc} and d ={abc, abc}, and the sequence of covers illustrated below:
F1= abc + abc+ abc
Expand abc a
off on
F2= a+abc + abc
Don’t care
abc is redundant
a is prime
F3= a+abc
Expand abc  bc c b a
F4= a+bc
ECE 667- Espresso logic minimizer

27. Two-level minimization (multiple-outputs)
Initial representation:
a b c
0 – 0
0 1 –
– 1 1
1 – 1
f1 f2
0 1
1 0
000
100
110
010
111
011
001 f1 f2
101
a b c
0 – 0
0 1 1
1 – 1
f1 f2
0 1
1 1
1 0
Minimized cover: f1 f2
000
100
110
010
111
011
001
101 c b a
ECE 667- Espresso logic minimizer

28. ESPRESSO Illustrated ECE 667- Espresso logic minimizer minimum Local

29. REDUCE ECE 667- Espresso logic minimizer
Problem: Given a cover F and c  F, find the smallest cube c  c such that
F\{ c } + { c }
is still a cover. Cube c is called the maximally reduced cube of c.
REDUCE is
order dependent on
off
Don’t care
ECE 667- Espresso logic minimizer

30. Expand and Reduce Fundamental procedures of Espresso
ECE 667- Espresso logic minimizer

31. Validity of expansion ECE 667- Espresso logic minimizer
Containment check: is the expanded cube contained in F ?
ECE 667- Espresso logic minimizer

32. Expand - example ECE 667- Espresso logic minimizer
Validity of expansion
ECE 667- Espresso logic minimizer

33. Reduce Transform a cover of prime implicants:
Replace each prime implicant p, wherever possible, with a smaller, non-prime implicant contained by p.
Purpose of Reduce: iterative improvement
Moves function away from local minimum
The subsequent Expand may be able to find a better set of primes
Similar to Expand, uses the same containment check
ECE 667- Espresso logic minimizer

34. Single output function
Expand input part:
(0 0 0 | 1)  (0 - 0 | 1)
Check if (f – a’b’c’)a'’bc’ = 1
2. Reduce output part:
(0 1 - | 1)  (0 1 - | 0)
(remove the cube)
Check if (f – a’b)a'’b = 1
ECE 667- Espresso logic minimizer

35. Multiple output function - expand
Expand output part:
(0 - 1 | 1 0)  (0 - 1 | 1 1)
C = 0-1| 10, C~= 0-1|01
Check if (f – {0-1|10})0-1|01 = 1
ECE 667- Espresso logic minimizer

36. Multiple output function – remove redundancy
Remove redundant cube:
(0 0 - | 0 1)  (0 0 - | 0 0)
C = 00-|01 , C~= 00-|00
Check if (f – {00-|01})00-|01 = 1
ECE 667- Espresso logic minimizer

37. Multiple output function – summary
ECE 667- Espresso logic minimizer

38. How to Expand and Reduce
ECE 667- Espresso logic minimizer