1. Heuristic Minimization of Two Level Circuits
Sungho Kang
Yonsei University

2. Outline Local Search Checking for Equivalence and Tautology
Choosing the Right Direction
Identifying Essential Implicants

3. Local Search Local search algorithms
The algorithms that start from an initial solution and try to find a better one by applying successive modifications
One property of the solution of a local search algorithm is local optimality
A solution is locally optimal if its neighborhood does not contain any solution of lower cost
In order to guarantee local optimality, it is sufficient to use the following stopping criterion
Stop when there is no cheaper solution in the neighborhood of the current solution

4. Local Search Applied to Logic Minimization
Starting point is a cover for a multi-output function and the cost is the number of cubes with the number of literals as a tie-breaker
The neighborhood is defined as the set of covers that are obtained from the original cover by adding or removing exactly one literal to one of the cubes
A new cover obtained is a feasible point if it is equivalent to the original one i.e. if it represents the same function
The solutions with more cubes than the initial cover are not part of the search space

5. Local Search Applied to Logic Minimizationx y z f 1 - x y z f - 1
change into 0-0 1
Expanding input parts and removing the cube from the cover

6. Local Search Applied to Logic Minimization
Expand output part and remove the cube x y z f g - 1 x y z f g - 1 x y z f g - 1 x y z f g 1 - x y z f g 1 -

7. Simple Local Search Algorithm
F = EXPAND(F,D);
F = IRREDUNDANT(F,D);
do {
cost = |F|;
F = REDUCE(F,D);
} while (|F| < cost);
F = MAKE_SPARSE(F,D);

8. MAKE_SPARSE Example x y z f g 1 -
Local Search x y z f g 1 -
All cubes are maximally expanded in their input and output parts The input term xz of f is redundant MAKE_SPRSE eliminates it x y z f g 1 -

9. Checking for Equivalence
Equivalence and Tautology
Two problems
One is the choice of the best move when many are possible
The other is checking a move for validity
When a cube is expanded new minterms are added to it
Check that none of these minterms belong to the OFF-set
When a cube is reduced minterms are removed from it
Make sure that these minterms are either covered by some other cubes or belong to the DC-set

10. Checking for Equivalence
Equivalence and Tautology
The expansion or reduction is valid if and only if ci*  ( F-{ci} )  D
F={cj} is the current cover for ON-set and D is the cover for DC-set
ci is the cube of F that is being added or reduced
ci* is the cube that contains the newly added or removed minterms
check the validity of the expansion of to 0-0 1
ci is and ci* is 010 1
must check the containment of in x y z f 1 - x y z
( F-{ci} )  D 1 -

11. Checking for Tautology
Equivalence and Tautology
A possible simplistic approach is to examine all the minterms of ci* for containment in ( F-{ci} )  D
This is simple but there could be too many minterms to examine
Improvement
Initially check for single cube containment of ci*
If that fails, we break ci* into halves and check the two halves separately
If both are contained, we are done
If only one is contained, the other is split again, and so on
If eventually we find no minterms that are not contained, the containment test has succeeded

12. Checking for Tautology
Equivalence and Tautology
The cofactor of a function f(x1, x2, …, xn) with respect to literal x1, written fx1, is f(1, x2, …, xn)
The cofactor of a function f(x1, x2, …, xn) with respect to literal x1’, written fx1’, is f(0, x2, …, xn)
The cofactor of a function f(x1, x2, …, xn) with respect to cube c is the successive cofactoring of f with respect to all the literals in c
A function that is identically 1 is called a tautology
For a function f and a cube c c  f  fc  1
(c  f → cc  fc  1 fc  fc  1)

13. Checking for Tautology
Equivalence and Tautology
Advantages of [ ( F-{ci} )  D ] ci*
Cofactoring may greatly simplify the function by reducing the number of variables and cubes
Tautology
The cofactor is taken by selecting the rows such that
The first and the third input columns are not 0
The first output column is a 1

14. Checking for Tautology
Equivalence and Tautology
General rule
Eliminate the rows that disagree with the input part of ci*
A disagreement means that the row has a 1 in an input column where ci* has a 0 or vice versa
Eliminate the rows that do not have at least a 1 in an output column where ci* has a one

15. Unate Functions
Equivalence and Tautology
A function f(x1, x2, …, xn) is monotonically increasing in variable x1 if and only if f(0, x2, …, xn )  f(1, x2, …, xn)
A function f(x1, x2, …, xn) is monotonically decreasing in variable x1 if and only if f(0, x2, …, xn )  f(1, x2, …, xn)
If neither of the two above is true, f is non monotonic in x1
A function f(x1, x2, …, xn) is unate in variable x1 if and only if it is either monotonic increasing or monotonic decreasing in x1
It is unate if it is unate in all of its variables

16. Unate Functions
Equivalence and Tautology
A cover f is monotonically increasing in variable x1 if and only if x1 never appears complemented in the terms of f
It is monotonically decreasing in variable x1 if and only if x1 never appears uncomplemented in the terms of f
If neither of the two above is true, f is non monotonic in x1
If function ia unate in variable x1 then there exists a cover of f unate in x1
If a cover in F is unate in x1 then the function F represents is also unate in x1
A unate cover F is a tautology if and only if it contains the constant term 1

17. Unate Functions Non-unate cover xyz f yz fx yz fx’
Equivalence and Tautology
Non-unate cover
xyz f yz fx yz fx’
1-1 1 after splitting
00- 1
0-0 1
It is sufficient to examine one of them to see that the function is not a tautology

18. Unate Functions
Equivalence and Tautology
It is sufficient to test the negative cofactor for tautology to get the answer for the complete functions
If a function is negative unate in a variable, it is sufficient to test the positive cofactor for tautology
wxyz f
All minterms with w=1 and x=0 cause f to be 0
After simplification, no row is left
The fucntion is not tautologous

19. Speed-up for Tautology Checking
Equivalence and Tautology
A row of all - in the input part
If such a row is found, the function is tautologous in all the outputs that have a 1 in that row
This condition is always sufficient and is also necessary for unate functions
An input column of all 1s or all 0s
If such a column is found, the function is not a tautology
If the number of inputs is less than 8, the truth table is generated and the tautology question is answered by inspection
The rationale is that if the number of inputs is small enough, generating the truth table is faster than recurring
If the vertex count of the cover is insufficient, the function is not a tautology
The vertex count of the cover is the sum of the vertex counts of the cubes

20. Speed-up for Tautology Checking
Equivalence and Tautology
Partitioning a search problem is normally very advantageous
If a cover F can be written as F = G + H where G and H have disjoint support (no variables in common), then F is tautologous if and only if either G and H are tautologous
Let x1, …xk be the variables on which G depends and xk+1, … xn be the variables on which H depends
If G(x1*, …xk*)=0 and H(xk+1*, … xn*)=0, then F(x1*, …xk*, xk+1*, … xn*)=0

21. Speed-up for Tautology Checking
Equivalence and Tautology
Finding a bipartition
The set of columns of the first block C1, is initialized to empty
One then picks a cube of the cover
All the columns where the cube is 0 or 1 are added to C1
Then all the cubes that intersect any of the column in C1 are selected and their columns added to C1
The process is repeated until no addition to C1 is possible
There are two possible outcomes
C1 includes all the columns of the cover
There is no bipartition
At least one column is left over
The leftover columns and rows identify the second block of the partition

22. Speed-up for Tautology Checking
Equivalence and Tautology
Partitioning example
12345
C1={1,3,4} and C2={2,5}
Neither subproblem is tautologous

23. Tautology Checks : Example
Equivalence and Tautology
Not a tautology
wxyz f cover is unate in w and x
cofactor wrt to w’ and x yz f

24. Tautology Checks : Example
Equivalence and Tautology
A multiple output cover is tautologous if all the outputs are tautologous
wxyz fg cover is unate in w and x
cofactor yz fg
can drop y
g is not a tautology
Not a tautology

25. Choosing the Right Direction
The choice of one direction over another may influence the quality of the solution
Having an explicit representation of the OFF-set of the function helps

26. Recursive Complementation
Right Direction
For a Boolean function f f’ = xf’x +x’f’x’
Proof
Let g = xf’x +x’f’x’ ( f = xfx +x’fx’ )
f + g = x(fx + f’x) + x’(fx’ + f’x’) = x + x’ = 1
fg = (xfx +x’fx’)(xf’x +x’f’x’) = 0

27. Recursive Complementation Example
Right Direction
Positive cofactor wrt x
Positive cofactor wrt x
f’xy’ = 0 x y z f 1 - y z fx - 1 z
fxy 1 z
f’xy 1 z
fxy’ - 1 z
f’xy’ y z
f’x’ - 1 y z
fx’ - 1 y z
f’x 1 x y z f’ 1 -

28. Recursive Complementation
Right Direction
For f monotonic increasing in x f’ = fx’ +x’ fx’’
For f monotonic decreasing in x f’ = x fx’ + fx’’

29. Recursive Complementation
Right Direction
Example
wxyz If we select w, a row of all dashes will appear in fw
Complement of the tautology is trivial to compute
fw’ = 0
compute fw’’ cofactoring f xyz
11-
111
fw’’ = x’+G’ xyz final result
wxyz
0 - 00

30. Using the OFF-set in the Expansion
Right Direction
Build a blocking matrix, that has a row for each cube in the OFF-set and a column for each variable
A row of the blocking matrix tells us that it is OK to expand the cube, as long as at least one of the directions corresponding to one of its ones is not expanded
It is not difficult to see that finding a maximum expansion corresponds to finding a minimum subset of the columns of the blocking matrix that covers all the rows

31. Identifying Essential Primes
Let F be a cover composed of prime implicants
Let e be one of the primes in F and let G be the cover composed of the remaining primes
Then e is an essential prime if and only if it is not covered by the union of ;
1. The consensus terms of e and each term of G
2. The intersections of e and each term of G

32. Identifying Essential Primes
Consider y’z’+xy’+xz and test y’z’
This prime intersects xy’ (the intersection is xy’z’)
It also has a consensus term with xz (xy’)
To see whether y’z’ is essential, check y’z’  xy’ + xy’z’
Since xy’z’ xy’, need to check y’z’  xy’
The answer is no; hence y’z’ is essential
If we consider xy’, need to check whether xy’ is covered by xy’z’ + xy’z
The answer is positive and xy’ is not essential

33. Identifying Essential Primes
Consider f = x’z’+x’y+xz
The cover is not a complete sum
Therefore it is not immediate to see whether a given prime is essential
Suppose we want to test x’y
Its intersection with x’z’ is x’yz’, while its consensus term with xz is yz
Hence we test x’y  x’yz’ + yz
Cofactoring the right hand side, we get z’+x, which is clearly a tautology
Hence x’y is not essential