Binary Decision Diagram

Binary Decision Diagram (BDD) is an effective way to represent the Switching function. It is a Data-Structure used to represent a Boolean Function and can be represented as a compressed form of sets or relations.

Definition:

A BDD is a set of an Acyclic, Rooted Graph with Directed Edges(Top-to-Bottom), which consist of Decision Node and 2-Terminal Nodes and have a special Node Called ROOT.

A binary decision diagram is a rooted, directed, acyclic graph. Nonterminal nodes in such a graph are called decision nodes; each decision node is labeled by a Boolean variable and has two child nodes, referred to as low child and high child.

BDD is a Shannon cofactor tree:

• f = v f_{v +}v’ f_v’  (Shannon expansion)
• Vertices represent decision nodes (i.e. multiplexers) controlled by variables
• leaves are constants “0” and “1”

Representation of BDD:

Decision and Child Node

• Each Decision Node is labeled by a Boolean variable and has two child nodes called low child and high child.
• The edges from a node to a low child represent, an assignment of variable 0 and for high as 1.

Representation of BDD

• Every Decision Node with a variable a and has two child nodes.

A BDD is said to be Ordered if different variables appear in the same order on all paths from the root. 

A three-variable function is represented by Truth-table and BDD.

BDD of Three-variable function

 Variable ordering restriction and reduction rules make (ROBDD) representation canonical

Example:

f = ac+bc+a'b'c

 Solution-    

Step -1: Decomposing the function with respect to variable a

for a’ put a=0 & for a put a=1 in function;

Now, we get-

f =  a' (b'c'+bc) + a(c+bc)    
f =  a'(b'c'+bc) + a(c)

Step – 2: Decomposing the functions with respect to variable b-

• decompose (b’c’+bc) w.r.to b = b'(c’)+b(c)
• decompose (c) w.r.To b = b'(c)+b(c)

Step – 3: Decomposing the functions with respect to variable c-

• decompose c’ w.r.to c = c'(1)
• decompose c w.r.to c = c(0)
• decompose c’ w.r.to c’ = c'(0)
• decompose c w.r.to c’ = c(1)

We get:

c'(1)+c(0)      c'(0)+c(1)      c'(0)+c(1)      c'(0)+c(1)

Note: Here order of variables is a, b, c. But we can take variables in any order.

BDD of f=ac+bc+a’b’c

Logical Operation in BDD

Conjunction(∧) – AND

BDD of ( a∧ b)

In mathematical notation, we usually write a ∧ b.

In the diagram above for a ∧ b, we indicate the 0 (left) branch with a dashed line and the 1 (right) branch with a solid line. This convention will make it easier to draw diagrams without a lot of intersecting lines.

Disjunction( ∨ ) – OR

BDD of  ( a ∨  b)

In mathematical notation, we usually write a ∨ b.

Negation(⊕ )– NOT

BDD of ( a⊕ b )

Exclusive or, which is mathematically often written as a⊕ b.

Variable Ordering in BDD:

The size of BDD is determined both by the function being represented and the chosen ordering of the variables.

 For some functions, the size of a BDD may vary between a linear to an exponential range depending upon the ordering of the variables.

Example:

f(x1,.......,x2n) = x1x2+ x3x4+.......+x2n-1x2n

Variable Ordering: x₁ < x₃ <……<x_2n-1< x₂ < x₄<……< x_2n

BDD requires 2ⁿ⁺¹ nodes to represent the function.

Variable Ordering: x₁<x₂<x₃<x₄<x₅<……..<x_2n-1<x_2n

BDD requires 2n nodes to represent the function.

Application of BDD:

• BDDs are extensively used in CAD software to synthesize circuits (logic synthesis) and in formal verification.
• In the analysis of Digital Circuit.
• There are several lesser-known applications of BDD, including fault tree analysis, Bayesian reasoning, product configuration, and private information retrieval.