Boolean Matching in Logic Synthesis

Boolean matching is a technique to detect
equivalence of two Boolean function
under permutation of the variables.
A permutation P over X is a one- to-one function
P : X -> X; X = {x1, x2, x3, …, xn }

Equivalence of two functions defined under
 Negation of input variables
 Permutation of input variables
 Negation of output

 X = {x1, x2, x3, …, xn }
 Set of Boolean input variables.
 f (X) and g (X)
 Denotes two Boolean functions of X.
  (X)
 Maps each xi to itself or its complement.
 If g (X) = f ( (X))
 Then we say f (X) and g (X) are equivalent functions.

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Example:
X = {x1, x2, x3}
f (X) = x1 + x2 + x3
g (X)= x1 + x2 + x3
 Mapping:
x1 x1
x2 x2
x3 x3
g (X) = f ( (X) ), so f (X) and g (X) are equivalent
functions.

  (X)
 Maps each xi to any xi.
  is a permutation of X.
 If g (X) = f ( (X))
 Then we say f (X) and g (X) are equivalent functions.
Example:
X = {x1, x2, x3}
f (X) = x1x3 + x2 + x3x1
g (X)= x3x2 + x1 + x2x3
 Mapping:
x1 x3
x2 x1
x3 x2
g (X) = f ( (X) ), so f (X) and g (X) are equivalent functions.

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 X = {x1, x2, x3,…, xn}
 Set of Boolean input variables.
 f (X) and g (X)
 Denotes two Boolean functions of X.
 If g (X) = f (X) or g (X) = f (X)
 Then we say f(X) and g(X) are equivalent functions.
Example:
X = {x1, x2, x3}
f (X) = x1x2x3
g (X)= x1 + x2 + x3
g (X) = f (X), so f (X) and g (X) are equivalent
functions.

 N-equivalent
 Equivalent under input Negation.
 P-equivalent
 Equivalent under input Permutation.
 NP-equivalent
 Equivalent under input Negation, input Permutation.
 NPN-equivalent
 Equivalent under input Negation, input Permutation,
output Negation.

Given functions f (X) and g (Y),
where,
X = { x1 , x2 ,…, xn}
Y = { y1 , y2 ,…, yn}
 : maps each xi to a unique yj or yj
g (Y) = f ( (X) ) or f ( (X) )
Total 2n  n!  2 mappings

Function f and g
Compute Signatures
of Functions
Equal?
Not
Matched
No
Compute Signatures
of variables
Yes
Equal?
Equivalence of
f and g ?
Matched
Not
Matched
Not
Matched
Yes
Yes
No
No

• The positive cofactor of function f (x1, ..,xn) with
respect to variable xi, denoted by fxi , is f (x1, ..,xi = 1,
...,xn).
▫ The negative cofactor denoted by fx′i, is f (x1, .., xi =0, ...,xn).
• A function f (x1, ...,xn) is positive unate in variable xi if
and only if
 the negative cofactor of f with respect to xi is covered by the
positive cofactor of f with respect to xi, i.e., fx′i⊆ fxi .
 Likewise, f is negative unate in variable xi if and only if
fxi ⊆ fx′i.
 f is called binate in xi if it is not unate in it.

A signature is defined as a necessary condition that
captures some key characteristic of a Boolean
function with respect to one or more of its input
variables.
3 types of key characteristic :
 Unateness property
 Symmetry property
 Size of on_set

 Matched functions having the same number of Unate
and Binate variables.
 If we denote by b the number of binate variables,
then b is a signature of the function.
 The number of Unate variables (n-b) is a signature.
 At most b! *(n – b)! variable permutations need to be
considered in the search for a match in the worst
case.

A function:
g has n = 7, 4 Unate variables, 3 Binate variables.
Consider another function with n=7.
• First, a necessary condition for f to match g is to also have 4
unate variables and 3 binate variables.
• Then, only 3!4! =144 variable orders and corresponding OBDDs
need to be considered in the worst case.

 Symmetry class : a set of variables that are
interchangeable without affecting the logic
functionality.
 A symmetry class is an ensemble of symmetry sets
with the same cardinality.
• A necessary condition for matching is to have
symmetry classes of the same cardinality for each
i = 1,2, . . . , n.

Consider,
• The support variables of f(x) can be partitioned into three
symmetry sets:{x1, x2, x3}, {x4, x5}, {x6, x7}.
• There are two non-void symmetry classes, namely:C2 ={{x4,
x5}, {x6, x7}} and C3 = {{x1, x2, x3}}.
• A signature is [0,2,1,0,0,0,0].
Consider, g1 = y1 + y2 y3 + y4 y5 + y6 y7
g2 = (y1’ + y2’)(y3’+ y4)( y5 + y6 +y7 ).
 The signatures of the cells are, [1,3,0,0,0,0,0] & [0,2,1,0,0,0,0].
• The signatures of f and g2 are equal, and indeed g2 is NPN
equivalent to f.

 Signatures can also be obtained by considering the
satisfy count of a function, which is the number of its
minterms.
 minterm is intended to imply it that each of the groups
in the expression takes on a value of 1 only for one of the
all possible combinations of the inputs and their inverses.
 The number of minterms for which a function f evaluates
to 1 (denoted as on_set(f) ).
 matched functions have the same size of on_set.
 computed on Ordered Binary Decision Diagram

 Cofactor signature
 Cofactor statistics and Component signature
 Single fault propagation weight signature (SFP)
and Partner patterns

 Cofactor signature of a variable xt for function f :
onsize (xt  fxt)
: 0-branch
: 1-branch
xt
f
xt  fxt
0 1

 A sorted list of onsize (xt xj  fxtxj) for all xt in the
input set
xt xj  fxtxj
f
xt
xj
0 1

( onsize (xt  fxt  fxt), onsize (xt  fxt  fxt), onsize (fxt  fxt) )
xt xt xt xt
0 1

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 Technology Mapping
 Matching of complex gates.
 Exploiting implicit don’t care.
 Logic Verification
 When input correspondence is not given.
 Verify correctness of a logic circuit.
 Logic Synthesis
 Cell library binding process.
23

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Boolean matching is based on exact equivalence
checking techniques that find a match whenever
possible. Various kinds of algorithms exist for
Boolean matching. An overview of Boolean
S i g n a t u r e b a s e d
algorithm is discussed.

Boolean Matching in Logic Synthesis

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