ECE 3060
VLSI and Advanced Digital Design
Lecture 10
Two Level Logic Minimization
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Two Level Logic Minimization in VLSI and Advanced Digital Design - Prof. Vincent Mooney, Study notes of Electrical and Electronics Engineering
A lecture note for ece 3060, which discusses modern techniques for manipulating and minimizing boolean functions in two level logic minimization. It covers exact methods, heuristic methods, representation of boolean expressions, manipulation of realistic multilevel networks, and definitions such as binary space, operations or(+), and(.), not, single output, multiple output, implicant, minterm, and product or cube. It also explains tabular representations, cube representation, prime definitions, and logic minimization using exact methods and heuristic methods.
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ECE 3060
VLSI and Advanced Digital Design
Lecture 10
Two Level Logic Minimization
ECE 3060
Lecture 10–
Motivation
•^
We will study modern techniques for manipulating andminimizing boolean functions
-^
Issue: Tractibility of minimization problem for largenumber of variables
-^
Exact methods
-^
Heuristic methods
•^
Issue: Representation of boolean expressions in aform conducive to boolean operations
-^
Implicant tables
-^
Binary decision diagrams
•^
Issue: Manipulation of realistic multilevel networks
-^
Graph representations
-^
multilevel minimization
-^
Technology mapping
ECE 3060
Lecture 10–
Cube Representation
•^
can be represented by a binary n-cube, i.e. an n- dimensional binary hypercube
-^
As usual, literals may be replaced with binary values,i.e.
-^
B Adjacent minterms (vertices) differ in only one vari-able similar to K-map
n a a
b b
b b
a a
a a
bac bac
b
b
a
a c
b c
b
a
a c
c
bac bac
a
b c
abc
ECE 3060
Lecture 10–
Definitions
•^
Boolean
variable:
•^
Boolean
literal:
or
•^
Product or
cube: product of literals
•^
Implicant: product term implying a value of a function(usually TRUE)
-^
binary hypercube in the boolean space
•^
Minterm: product using all input variables implying avalue of a function (usually TRUE)
-^
vertex in the boolean space
a^
B
a^
a
ECE 3060
Lecture 10–
Cube Representation
• •^
c
a b
F
abc
abc
abc
abc
abc
+^
+^
+^
F
ab
bc
ac
ab
ECE 3060
Lecture 10–
Prime Definitions
•^
Prime implicant
-^
implicant not contained by any other implicant
•^
Prime cover
-^
cover of prime implicants
•^
Essential Prime Implicant (EPI):
-^
there is at least one minterm covered by EPI and not covered by anyother prime implicant
ECE 3060
Lecture 10–
Minimal or Irredundant Cover
•^
Cover of the function that is not a proper superset ofanother cover
-^
no implicant can be dropped
-^
local optimum
c
a b
ECE 3060
Lecture 10–
Minimal Cover with respect tosingle-implicant containment
-^
no implicant is contained by any other implicant
-^
weak local optimum
c
a b
ECE 3060
Lecture 10–
Exact Logic Minimization
•^
Quine’s theorem:
-^
There is a minimum cover that is prime
-^
Consequently, the search for minimum cover can be restricted toprime implicants
•^
Quine-McCluskey method:
compute prime implicants
determine minimum cover via branching
•^
Petrick’s method
compute prime implicants
determine minimum cover via covering clause
ECE 3060
Lecture 10–
Computing Prime Implicants
•^
The
Hamming weight of a minterm is the number of
ones in that minterm.
-^
Start with list of minterms sorted by Hamming weight.
Combine all possible implicants (minterms) using
. Note that this
algebraic reduction specifies two implicants with Hamming weights that differby one.
Group resulting implicants by Hamming weight.
Repeat 1. and 2. on the resulting implicants until no further factoring is possible(i.e. all implicants are prime)
•^
Example:
α y
α y +^
α
f^
abcd
abcd
abcd
abcd
abcd
abcd
abcd
abcd
abcd
abcd
+^
+^
+^
+^
ECE 3060
Lecture 10–
Example
•^
Function:
-^
Choose cover by selecting a set of implicants whichcover all minterms.
f^
abc
abc
abc
abc
abc
•^
Primes:
•^
Implicant Table:
Label
PIs 00011111
α β γ δ
Minterms
Primes 1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
α^
β^
γ^
δ
abc abc abcabc abc
ECE 3060
Lecture 10–
Cube Representation
c
a b
(a) prime implicants
(b) minimum cover
ECE 3060
Lecture 10–
Exact Two-level Logic Minimization
•^
Matrix representation
-^
Covering problem
-^
Reduction strategies
-^
Branch and bound covering algorithm
ECE 3060
Lecture 10–
Matrix representation
•^
View implicant table of some function
as Boolean
matrix:
-^
the ith minterm is covered by the jth prime
implicant
•^
The (Boolean) selection vector
selects which prime
implicants will be in the cover.
-^
To cover
, find an
which satisfies
-^
i.e. select enough columns to cover all rows
•^
To find a minimum cover, minimize cardinality of
, i.e.
the number of nonzero entries of
f
A
a^ ij
(^
x
f^
x
y^ i
i ∀ ≥^
Ax
y
x
x