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 itsminterms.

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

xt fxt

0 1

f

A sorted list of onsize (xt xj fxtxj) for all xt in the input set

xt xj fxtxj

0 1

fxt

xj

( onsize (xt fxt fxt), onsize (xt fxt fxt), onsize (fxt fxt) )

xt xt xt xt

0 1

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

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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.

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Boolean matching is based on exact equivalencechecking techniques that find a match wheneverpossible. Various kinds of algorithms exist forBoolean matching. An overview of BooleanS i g n a t u r e b a s e d

algorithm is discussed.