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EE4271 VLSI Design
Logic Synthesis
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Logic Synthesis
Starts from RTL description in HDL or Boolean
expressions
Outputs a standard cell netlist
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Boolean Function
f: Bm
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
Yn
B = {‘0’, ‘1’}, Y = {‘0’, ‘1’, ‘-’}
The function is incompletely specified, don’t care ‘-’
For each output, space of Bm can be partitioned into



on-set: all inputs leading to output ‘1’
off-set: all inputs leading to output ‘0’
dc-set: all inputs leading to output ‘-’
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Points in Input Space
Assigning ‘1’ or ‘0’ to each of the m Boolean variables
x1, x2, …, xm specifies a point in input space Bm
Example
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(‘1’, ‘0’, ‘1’) in B3
x1=‘1’ Λ x2=‘0’ Λ x3=‘1’
x1•x2•x3
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Terminology
Literal: a Boolean variable or its complement
Minterm: a product of all input variables or their
complements – a point in Bm
Cube: a product of input variables or their
complements

The fewer number of variables, the bigger space covered
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Boolean Function Representation
A Boolean function can be specified by a sum of minterms
The expression has a minterm for each point in the on-set
f  x1  x 2  x3  x1  x 2  x3  x1  x 2  x3  x1  x 2  x3  x1  x 2  x3  x1  x 2  x3
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Implicant and Cover
An implicant is a cube whose points are either in the on-set or
the dc-set
A prime implicant is an implicant that is not included in any other
implicant
A set of prime implicants that together cover all points in the onset (and some or all points of the dc-set) is called a prime cover
f  x1  x 2  x1  x3  x 2  x3  x1  x 2  x 2  x3  x1  x3
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Irredundant Cover
An prime cover is irredundant when none of its prime implicants
can be removed from the cover
An irredundant prime cover is minimal when the cover has the
minimal number of prime implicants
f  x1  x3  x 2  x3  x1  x 2
f  x1  x 2  x 2  x3  x1  x3
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Goal of Logic Synthesis
Find a minimum irredundant prime cover
An irredundant prime cover is not necessarily a
minimum
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Quine-McCluskey Algorithm
Calculate all prime implicants of the union of the onset and dc-set, omitting prime implicants that only
cover points of dc-set
Finds the minimum cost cover of all minterms in the
on-set by the obtained prime implicants
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Set Covering Problem
Given a universe U={e1, …, en}, a collection of subsets
{S1, …, Sm} where each subset contains some
elements in universe
cost wi is associated with each subset Si
find a subcollection C (cover) such that C covers the
entire universe
Famous NP-complete problem
On-set U, a prime implicant an S
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Example of Set Cover
U={1,2,3,4}
S1={1,2}, w=2
S2={1,3,4}, w=3
S3={3}, w=1
S4={2,4}, w=2
S5={2,3}, w=2
S6={1,2,4}, w=3
C={S3,S6} is the optimal solution
S3 is redundant in C={S1,S2,S3}, and
C={S1,S2} is not optimal
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Greedy Algorithm
C = empty [the cover]
E= empty [record elements which have been covered]
While there is uncovered element
Find the subset Si which is most cost effective,
that is, the Si with smallest w(Si)/|Si-E|. [weight
of subset over the elements in the subset but not
covered so far]
For each e in Si-E, set price(e)=w(Si)/|Si-E|
Put all e in Si to E
Put Si in the current partial cover C
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Running Example (Iter 1)
U={1,2,3,4}
S1={1,2}, w=10
S2={1,3,4}, w=9
S3={3}, w=7
S4={2,4}, w=4
S5={2,3}, w=2
Iteration 1, E = empty
w(S1)/|S1-E|=10/2=5
w(S2)/|S2-E|=9/3=3
w(S3)/|S3-E|=7/1=7
w(S4)/|S4-E|=4/2=2
w(S5)/|S5-E|=2/2=1
Pick S5
E = {2,3}
C = {S5}
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Running Example (Iter 2)
U={1,2,3,4}
S1={1,2}, w=10
S2={1,3,4}, w=9
S3={3}, w=7
S4={2,4}, w=4
S5={2,3}, w=2
Iteration 2, E={2,3}, C = {S5}
w(S1)/|S1-E|=10/1=10
w(S2)/|S2-E|=9/2=4.5
w(S3)/|S3-E|=7/0=+infty
w(S4)/|S4-E|=4/1=4
Pick S4
E = {2,3,4}
C = {S5,S4}
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Running Example (Iter 3)
U={1,2,3,4}
S1={1,2}, w=10
S2={1,3,4}, w=9
S3={3}, w=7
S4={2,4}, w=4
S5={2,3}, w=2
Iteration 3, E={2,3,4}, C={S5,S4}
w(S1)/|S1-E|=10/1=10
w(S2)/|S2-E|=9/1=9
w(S3)/|S3-E|=7/0=+infty
Pick S2
E = {2,3,4,1}
C = {S5,S4,S2}
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The Other Example
n elements and a collection of m=n+1 subsets
U={1,2,…n}
S1={1}, w(S1)=1/n
S2={2}, w(S2)=1/(n-1)
S3={3}, w(S3)=1/(n-2)
…
Sn={n}, w(Sn)=1
Sn+1={1,2,…,n}, w(Sn+1)=1.0001
Our solution {S1,S2,…,Sn}, weight = ln n
Optimal solution {Sn+1}, weight = 1.0001
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Summary
Logic synthesis
Primary implicant
Minimum cost irredundant prime cover
Set Cover
Greedy Algorithm
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