State university of New York at New PaltzElectrical and Computer Engineering Department

Logic Synthesis OptimizationLect15: Heuristic Two Level Logic

Minimization

Dr. Yaser KhalifaElectrical and Computer Engineering Department

State University of New York at New Paltz

Unate Function

Definition :If, in the minimum sum-of-product form of a switching function, each variable appears either in its true form or its complemented form, but not both, then the function is called a unate function.

Ex) f1(x1 x2 x3)= x1 + x2 x3’ : unatef1(x1 x2 x3)= x1 + x1’ x2 x3’ : ≠ unate due to x1 & x1’

Th

A switching function is unate if and only if it can be expressed as a sum of essential prime cubes, all intersecting at a common subcube.

Heuristic Logic Minimization

Heuristic Logic Minimization (cube-based)

② Unate recursive paradigm : ESPRESSO(or in English; Divide-and-Conquer!)

i.e. Decomposition Make use of Shannon’s expansion to recursively operate on subsets of logic cover until cover has unate property.

Why looking for the unate property recursively?

Every prime of a unate function is essential.

Minimum form is given as the union of all EPIs.

Note that min form = EPIs + minimum PIs in general.

If F is a unate function, min form = EPIs : this is good to find min. form.

In ESPRESSO, used for complementation and tautology checking.

(b) Shannon Expansion

xjxj jjfxfxf

1

xx jjff

0x1xx jjjfff

where

Complementation

xjxjjj

fx)f(xf )(

xjxjjj

(f)x(f)x

How to determine if a function is unate :

The definition on prev. page requires the min. form of the function. But we are looking for it. Thus, we need another method.

A function is unate if it is unate in all its variables. A function is unate in xj if it is monotone in xj.

Thus, a function is unate if it is monotone in all its variables

Monotone in xj?

xj change causes all changing outputs to change in the same direction.

If xj of 0 1 causes all changing outputs to change

from 0 1 : monotone increasing in xj

from 1 0 : monotone decreasing in xj

A B C D F=C’D +ABD

F1=C’D +ABD+A’B’C’

0 0 0 0 1

0 0 0 1 1 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1 1 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1 1 1

1 0 1 0

1 0 1 1

1 1 0 0

1 1 0 1 1 1

1 1 1 0

1 1 1 1 1 1

Both OK for D: 0 1

A B D C F=C’D +ABD

F1=C’D +ABD+A’B’C’

0 0 0 0 1

0 0 0 1

0 0 1 0 1 1

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0 1 1

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0 1 1

1 0 1 1

1 1 0 0

1 1 0 1

1 1 1 0 1 1

1 1 1 1 1 1

Both OK for C: 0 1

A B C D F=C’D +ABD

F1=C’D +ABD+A’B’C’

0 0 0 0 1

1 0 0 0

0 0 0 1 1 1

1 0 0 1 1 1

0 0 1 0

1 0 1 0

0 0 1 1

1 0 1 1

0 1 0 0

1 1 0 0

0 1 0 1 1 1

1 1 0 1 1 1

0 1 1 0

1 1 1 0

0 1 1 1

1 1 1 1 1 1

F1 NOT OK for A: 0 1

Heuristic Minization Operators

• Expand– Make implicants prime– Remove covered implicants

• Reduce– Reduce size of each implicant while preserving cover

• Reshape– Modify implicant pairs: enlarge one implicant enabling

the reduction of another

• Irredendant– Make cover irredundant

Example

• On-set: 0000 10010 10100 10110 11000 11010 10101 10111 11001 11011 11101 1

• Prime implicants: | 0**0 1

| *0*0 1

| 01** 1

| 10** 1

| 1*01 1

| *101 1

Example

• Expand 0000 to = 0**0– Drop 0100, 0010. 0110 from the cover

• Expand 1000 to = *0*0– Drop 1010 from the cover

• Expand 0101 to = 01** – Drop 0111 from the cover

• Expand 1001 to = 10**– Drop 1011 from the cover

• Expand 1101 to = 1*01• Cover is { }

Example reduction

• Reduce 0**0 to nothing.

• Reduce = *0*0 to = 00*0

• Reduce = 1*01 to = 1101

• Cover is {}

Example reshape

• Reshape { to {• Where = 10*1

• Cover is {}

Example second expansion

• Expand = 10*1 = 10** • Expand = 1101 to = *101

• Cover is { }

Summary of Example

• Expansion:– Cover: { }– Prime, redundant, minimal w.r.t single cube containment

• Reduction: eliminated– B = *0*0 reduced to b = 00*0– E = 1*01 reduced to e = 1101– Cover: { }

• Reshape:– {} reshaped to {} where d = 10*1

• Second expansion – Cover: { }– Prime, irredundant (= minimal)