1. Boolean Functions and their Representations
ECE 697B (667) Spring Synthesis and Verification of Digital Circuits
Boolean Functions
and their Representations
Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003
ECE 667

2. The Boolean Space Bn B = { 0,1}, B2 = {0,1} X {0,1} = {00, 01, 10, 11}
Karnaugh Maps:
Boolean Cubes: B0 B1 B2 B3 B4
ECE Synthesis & Verification - Lecture 9a

3. Boolean Functions x2 x1
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4. Boolean Functions
Literal x1 represents the logic function f, where f = {x| x1 = 1}
Literal x1 represents the logic function g where g = {x| x1 = 0} x1 x3 x2
f = x1 x1 x2 x3
f = x1
Notation: x’ = x
ECE Synthesis & Verification - Lecture 9a

5. Set of Boolean Functions
Truth Table or Function Table:
x1x2x3
 1 x3 x2 x1
There are 2n vertices in input space Bn
There are 22n distinct logic functions.
Each subset of vertices is a distinct logic function: f  Bn
ECE Synthesis & Verification - Lecture 9a

6. Boolean Operations - AND, OR, COMPLEMENT
Given two Boolean functions:
f : Bn  B
g : Bn  B
AND operation
f × g = {x | f(x)=1 Ù g(x)=1}
The OR operation
f + g = {x | f(x)=1 Ú g(x)=1}
The COMPLEMENT operation (^f or f’ )
f’ = {x | f(x) = 0}
ECE Synthesis & Verification - Lecture 9a

7. Cofactor and Quantification
Given a Boolean function:
f : Bn  B, with the input variables (x1,x2,…,xi,…,xn)
Positive Cofactor of function f w.r.t variable xi
fxi = {x | f(x1,x2,…,1,…,xn)=1}
Negative Cofactor of f w.r.t variable xi
fxi’ = {x | f(x1,x2,…,0,…,xn)=1}
Existential Quantification of function f w.r.t variable xi,
$xi f = {x | f(x1,x2,…,0,…,xn)=1 Ú f(x1,x2,…,1,…,xn)=1}
Universal Quantification of function f w.r.t variable xi,
"xi f = {x | f(x1,x2,…,0,…,xn)=1 Ù f(x1,x2,…,1,…,xn)=1}
ECE Synthesis & Verification - Lecture 9a

8. Representations of Boolean Functions
We need representations for Boolean Functions for two reasons:
to represent and manipulate the actual circuit we are “synthesizing”
as mechanism to do efficient Boolean reasoning
Forms to represent Boolean Functions
Truth table
List of cubes: Sum of Products, Disjunctive Normal Form (DNF)
List of conjuncts: Product of Sums, Conjunctive Normal Form (CNF)
Boolean formula
Binary Decision Tree, Binary Decision Diagram
Circuit (network of Boolean primitives)
Canonicity – which forms are canonical?
ECE Synthesis & Verification - Lecture 9a

9. Truth Table Truth table (Function Table): ab’c’d + ab’cd + abc’d +
The truth table of a function f : Bn  B is a tabulation of
its values at each of the 2n vertices of Bn. (all mintems)
Example: f = a’b’c’d + a’b’cd + a’bc’d +
ab’c’d + ab’cd + abc’d +
abcd’ + abcd
(Notation for complement: a’ = a )
The truth table representation is
- intractable for large n
- canonical
Canonical means that if two functions are the
same, then the canonical representations of each
are isomorphic (identical).
abcd f
ECE Synthesis & Verification - Lecture 9a

10. Boolean Formula
A Boolean formula is defined as an expression with the following syntax:
formula ::= ‘(‘ formula ‘)’
| <variable>
| formula “+” formula (OR operator)
| formula “×” formula (AND operator)
| ^ formula (complement)
Example:
f = (x1×x2) + (x3) + ^^(x4 × (^x1))
typically the “×” is omitted and the ‘(‘ and ‘^’ are simply reduced by priority, e.g.
f = x1x2 + x3 + x4^x1
ECE Synthesis & Verification - Lecture 9a

11. Cubes
A cube is defined as the product (AND) of a set of literal functions (“conjunction” of literals).
Example:
C = x1x’2x3
represents the following function
f = (x1=1)(x2=0)(x3=1) x1 x2 x3
c = x1
f = x1x2
f = x1x2x3
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12. Cubes If C  f, C a cube, then C is an implicant of f.
If C  Bn, and C has k literals, then |C| covers 2n-k vertices.
Example:
C = xy  B3
k = 2 , n = 3 => |C| = 2 = 23-2.
C = {100, 101}
In an n-dimensional Boolean space Bn, an implicant with n literals is a minterm.
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13. List of Cubes Sum of Products (SOP) F = {ab, ac, bc}
A function can be represented by a sum of cubes (products):
f = ab + ac + bc
Since each cube is a product of literals, this is a “sum of products” (SOP) representation
A SOP can be thought of as a set of cubes F
F = {ab, ac, bc}
A set of cubes that represents f is called a cover of f.
F1={ab, ac, bc} and F2={abc, abc, abc, abc}
are covers of
f = ab + ac + bc.
ECE Synthesis & Verification - Lecture 9a

14. Binary Decision Diagram (BDD)
f = ab+a’c+a’bd
Graph representation of a Boolean function
- vertices represent decision nodes for variables
- two children represent the two subfunctions
f(x = 0) and f(x = 1) (cofactors)
- restrictions on ordering and reduction rules
can make a BDD representation canonical c a b d 1
c+bd
root
node
c+d 1
ECE Synthesis & Verification - Lecture 9a

15. Boolean Networks Used for two main purposes
as representation for Boolean reasoning engine
as target structure for logic implementation which gets restructured in a series of logic synthesis steps until result is acceptable
Efficient representation for most Boolean problems
memory complexity is same as the size of circuits we are actually building
Close to input representation and output representation in logic synthesis
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16. Definitions – fanin, fanout, support
A Boolean circuit is a directed graph C(G,N) where G are the gates and N Í G´G is the set of directed edges (nets) connecting the gates.
Some of the vertices are designated:
Inputs: I Í G
Outputs: O Í G, I Ç O = Æ
Each gate g is assigned a Boolean function fg which computes the output of the gate in terms of its inputs.
ECE Synthesis & Verification - Lecture 9a

17. Definitions – fanin, fanout, support
The fanin FI(g) of a gate g are all predecessor vertices of g:
FI(g) = {g’ | (g’,g) Î N}
The fanout FO(g) of a gate g are all successor vertices of g:
FO(g) = {g’ | (g,g’) Î N}
The cone CONE(g) of a gate g is the transitive fanin of g and g itself.
The support SUPPORT(g) of a gate g are all inputs in its cone:
SUPPORT(g) = CONE(g) Ç I
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18. Example – Boolean Network6 1 5 3 4 7 8 9 2 O I
FI(6) = {2,4}
FO(6) = {7,9}
CONE(6) = {1,2,4,6}
SUPPORT(6) = {1,2}
Nodes = logic functions of arbitrary complexity
ECE Synthesis & Verification - Lecture 9a

19. Boolean Network Representations
Vertices can have arbitrary number of inputs and outputs
typically single-output functions are used
Vertices can represent any Boolean function stored in different ways, such as:
other circuits (hierarchical representation)
truth tables or cube representation
Boolean expressions read from a library description
BDDs, AIGs, etc.
Data structure allow very general mechanisms for insertion and deletion of vertices, pins (connections to vertices), and nets
ECE Synthesis & Verification - Lecture 9a

20. AND-INVERTER Circuits
Base data structure uses two-input AND function for vertices and INVERTER attributes at the edges (individual bit)
use De’Morgan’s law to convert OR operation etc.
Hash table to identify and reuse structurally isomorphic circuits f g g f
Means complement
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21. Data Representation Vertex: Edge:
pointers (integer indices) to left and right child and fanout vertices
collision chain pointer
other data
Edge:
pointer or index into array
one bit to represent inversion
Global hash table holds each vertex to identify isomorphic structures
Garbage collection to regularly free un-referenced vertices
ECE Synthesis & Verification - Lecture 9a

22. Data Representation Hash Table one 8456 6423 …. ... …. 0455 0456 0456
Constant
One Vertex
zero
left
0457 1
right
...
1345 ….
next
7463 ….
fanout
hash value
left pointer
complement bits
right pointer
0456
left
next in collision chain
right
array of fanout pointers
next
fanout
ECE Synthesis & Verification - Lecture 9a