Multilevel Logic Synthesis
-- Introduction
Multilevel Logic Synthesis: Outline
 Introduction
 What is multilevel logic?
 Why we need it?
 Problems and challenges.
 Multilevel Logic Synthesis and Minimization
 Restructuring
 Technology Independent Local Optimization
 Technology Mapping
 Delay Analysis and Optimization (*)
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Multilevel Logic vs. Two-Level Logic
Example: Let F=a’b+ab’, define G and H as follows:
if F is true, then G=cd+a’d’, H=cd+a’d’+e(f+b),
else G=e(f+b), H=(cd+a’d’)e(f+b).
Multi-Level Implementation:
G=F(cd+a’d’)+F’e(f+b)
H=F(cd+a’d’+e(f+b))+F’(cd+a’d’)e(f+b)
4-level
5-level
c
d
a’
G
F
F’
e
d’
f
b
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Multilevel Logic vs. Two-Level Logic
Example: Let F=a’b+ab’, define G and H as follows:
if F is true, then G=cd+a’d’, H=cd+a’d’+e(f+b),
else G=e(f+b), H=(cd+a’d’)e(f+b).
G = F(cd+a’d’)+F’e(f+b) = (a’b + ab’)(cd + a’d’) + (a’b +
ab’)’(e)(f + b)
= a’bcd + a’bd’ + acdb’ + e(f+b)(a+b’)(a’ +b)
= a’bcd + a’bd’ + acdb’ + (efa + efb’ + eba)(a’+b)
= a’bcd + a’bd’ + acdb’ + efab + efb’a’ + eba
THIS IS MORE COMPLICATED TO IMPLEMENT
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Multilevel Logic vs. Two-Level Logic
Two-level:




At most two gates between a primary input and a primary output.
Real life circuits: PLA.
Exact optimization methods: well-developed, feasible.
Heuristics.
Multi-level:





Any number of gates between a primary input and a primary output.
Most circuits in real life are multilevel (e.g. standard cells, FPGA).
Smaller, less power, and (in many cases) faster.
Exact optimization methods: few, high complexity, impractical.
Heuristics.
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Optimization Objectives
1. Area: occupied by the logic gates and interconnect, e.g.
measured by “one literal = one transistor” in technology
independent optimization.
2. Delay: measured by the longest path (critical path) through
the logic.
3. Power Consumption
Optimization is performed while simultaneously
satisfying upper/lower bound constraints placed on
these physical quantities.
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Optimization Cost Criteria
(Figure source: Prof. Brayton’s lecture notes.)
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Technology Independent
 A design is technology independent when the formula
(function, system) has no connection with the building
blocks in the implementation.
 Advantage: cost can be simply estimated by the number of logic
symbols (I.e. one literal = one transistor), delay is the length of the
“critical path”, better for local optimization.
 Example: Full Adder




z(x,y,c) = XOR(x,y,c) = xyc+x’y’c+x’yc’+xy’c’
cnew(x,y,c) = Majority(x,y,c) = xy+(x+y)c
The literal count is 12+5=17
The critical path delay is 3: xx+y(x+y)ccnew
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Technology Dependent
 A design is technology
dependent if the formula
(function, circuit, system) is
implemented by one or more
logic gates in a pre-designed
set of gates (called technology
library or cell library).
 Advantage: gates in the cell library
have a highly optimized, pre-defined
path to silicon, so that the area and
delay parameters are known and
accurate.
 Example: Full Adder
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signal
formula
gate
transi
stors
g
(xy)’
NAND
4
a
g’
NOT
2
b
(x+y)’
NOR
4
e
(a+b)’
NOR
4
z
e’c+ec’
XOR
8
d
b’
NOT
2
h
(dc)’
NAND
4
f
h’
NOT
2
j
(a+f)’
NOR
4
cnew
j’
NOT
2
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Multilevel Logic Synthesis Problem
 Goal:
 Model the multilevel logic
 Optimize the logic based on the cost criteria
 Difficulty:
 Multilevel gives more design freedom and increases the design
complexity
 Technology dependent/independent views
 Models:
 Algebraic forms
 Boolean networks
 Optimization Techniques: HARD
 Exact method: few, exponential complexity, impractical
 Approximation method: heuristic algorithm, rules-based methods
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Sum of Products (SOP)
 Example:
 Advantages:
abc’+a’bd+b’d’+b’e’f
 Easy to manipulate and minimize.
 Algorithms available (e.g. AND, OR, TAUTOLOGY).
 Two-level theory applies.
 Disadvantages:
 Not representative of logic complexity. For example
f=ad+ae+bd+be+cd+ce
f’=a’b’c’+d’e’
which differ in their implementation by an inverter.
 Difficult to estimate logic and to estimate progress
during logic manipulation.
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Factored Forms: Overview
 Example:
 Advantages:
(ad+b’c)(c+d’(e+ac’))+(d+e)fg
 Good representative of logic complexity
f=ad+ae+bd+be+cd+ce=(a+b+c)(d+e)
f’=a’b’c’+d’e’
 Good estimator of logic implementation complexity
 Implicitly Imply Multi-Level Computation (Computation
using tree)
 Disadvantages:
 Not many algorithms available for manipulation
 Not canonical:
ab+c(a+b)=bc+a(b+c)=ac+b(a+c)
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Factored Forms: Definition
A factored form is defined recursively by the rules:
1. A product is either a single literal or a product of factored form.
2. A sum is either a single literal or a sum of factored forms.
3. A factored form is either a product or a sum
Another point of view: Example:






p l
p f*f
s l
s f+f
f p
f s
(1) (base case)
(2)
(3) (base case)
(4)
(5)
(6)
 Factored forms: x, y’, abc’,
ab+c’, ((a’+b)cd+e)(a+b’)+e’.
 However, (a’+b’)’c’ is not,
although it equals abc’. This
is because internal
complement in not allowed
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Factoring Tree
 Factored forms can be graphically represented
as labeled trees, called factoring trees, in which
each internal node including the root is labeled
with either + or ·, and each leaf has a label of
either a variable or its complement.
+
 Example:
+
·
+
a’
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e’
·
((a’+b)cd+e)(a+b’)+e’
+
e a
b’
c d
b
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Equivalent Factorizations
 Two factored forms are equivalent if they
represent the same logic function.
 Example:
a(b+c)+bc
and
ab+c(a+b)
 Two factored forms are syntactically equivalent if
their factoring tree are isomorphic. (WHAT IS
ISOMOSRPHIC?)
 Example:
(a+b)(c+d)e and
a(b+c)+bc and
(c+d)e(a+b)
ab+c(a+b)
+
+
·
·
a
+
b
b
c
·
·
c
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+
b c
a
b
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Maximal Factorizations
 A factored form is maximally factored if
 For every sum of products, there are no two
syntactically equivalent factors in the products;
 For every product of sums, there are no two
syntactically equivalent factors in the sums.
 Example:
 ab+ac is not maximally factored,
a(b+c) is.
 (a+b)(a+c) is not maximally factored,
a+bc is.
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+
·
·
a
b a
c
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Optimum Factored Forms
 The size of a factored form f, (f), is the number
of literals in the factored form.
 Example:
 ((a+b)ca’) = 4
 ((a+b+cd)(a’+b’)) = 6
 A factored form is optimum if no other equivalent
factored form has fewer literals.
 Example:
 ab+a’c+bc is not optimum (ab+a’c+bc = ab+a’c).
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Unate Factored Forms
 A factored form F is positive unate in x, if x
appears in F, but x’ does not. A factored form is
negative unate in x, if x’ appears in F, but x does
not. F is unate in either case, and is binate if it is
not unate.
 Example:
 (a+b’)c+a’ is (positive) unate in c, (negative) unate in
b, and binate in a.
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Cofactor of Factored Forms
 The cofactor of a factored form F with respect to a literal
x (or x’ ) is the factored form Fx= Fx=1(x) (or Fx’=Fx=0(x) )
obtained by:
 replacing all occurrences of x by 1, and x’ by 0
 simplifying the factored form using the Boolean
algebra identities
1y=y 1+y=1 0y=0 0+y=y
 after constant propagation (all constants are
removed), part of the factored form may appear as G
+ G. In general, G is another factored form, and the
G’s may have different factored forms.
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Cofactor of Factored Forms
 The cofactor of a factored form F with respect to
a cube c is a factored form obtained by
successively cofactoring F with each literal in c.
 Example:
 F = (x+y’+z)(x’u+z’y’(v+u’)) and c = vz’
Fz’ = (x+y’)(x’u+y’(v+u’))
Fz’ v = (x+y’)(x’u+y’)
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Algebraic and Boolean Expressions
 f is an algebraic expression if f is a set of cubes (SOP),
such that no single cube contains another (minimal with
respect to single cube containment). Otherwise, f is called
a Boolean expression.
 Example: a+bc is algebraic, a+ab is Boolean. (ab+a’c+bc???)
 Special Properties of Boolean Algebra:
 a+a = a·a = a
 a+bc = (a+b)(a+c)
 a+ab = a
 The support of an expression f, supp(f), is the set of
variables that f explicitly depends on. Two expressions f
and g are said to be orthogonal if supp(f)supp(g)=,
denoted by fg.
 Example: a+b  c+d
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Algebraic and Boolean Factored Forms
 A factored form f is said to be algebraic if the
SOP expression obtained by multiplying f out
directly (I.e. without using aa=a+a=a, aa’=0, a+ab=a)
is algebraic, otherwise it is Boolean.
 Example: abg+acg+adf+aef+afg+bd+ce+be+cd



(b+c)(d+e)+((d+e+g)f+(b+c)g)a
=(bd+be+cd+ce)+(df+ef+gf+bg+cg)a
= bd+ce+be+cd+abg+acg+adf+aef+afg
(b+c)(d+e+ag)+(d+e+g)af
(af+b+c)(ag+d+e)
=afag+afd+afe+bag+bd+be+cag+cd+ce
=afg+adf+aef+abg+acg+bd+be+cd+ce
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(algebraic)
(algebraic)
(Boolean)
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Value of a Factorization
 Given an algebraic factorization F=G1G2+R, its
factorization value is given by:
fact_val(F,G2) = lits(F)-(lits(G1)+lits(G2)+lits(R))
where lits(F) and cubes(F) are the number of literals and
cubes in SOP form of F respectively, G1,G2, and R are
algebraic expressions.
 Example: The algebraic expression
F = ae+af+ag+bce+bcf+bcg+bde+bdf+bdg
can be expressed in the form F = (a+b(c+d))(e+f+g), which requires 7
literals, rather than 24, a save of 17 literals.
If G1=(a+bc+bd) and G2=(e+f+g), then R= and fact_val(F,G2) =
23+25=16. Where the extra literal saving comes from?
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