Example: Verification

1
Example Verification
e.g., input/output specification of multiplier
e.g., multi-level logic representation
2
Binary Decision Diagrams
Graph-based Representation of Boolean Functions

• Introduced by Lee (1959).
• Popularized by Bryant (1986).

• compact (functions of 50 variables)
• efficient (linear time manipluation)

Widely used has had a significant impact on the
CAD industry.
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Binary Decision Diagrams
Graph-based Representation of Boolean Functions
1
0
BDD is defined as Directed Acyclic Graph
0
0
1
1
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Analysis of Digital Circuits
Large domain, small range.
inputs
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Analysis of Digital Circuits
Large domain, small range.
inputs
2m possibilities
2 possibilities
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Data Structures
Truth Tables
Example
4 rows
2 variables
8 rows
3 variables
264 rows
64 variables
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Data Structures
Decision Diagrams
Example
x1
x2
x3
f
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
1
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Data Structures
Decision Diagrams
Optimize by merging nodes
x3
x3
x3
x3
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Data Structures
Decision Diagrams
S
Optimize by merging nodes
x1
1
0
x2
1
x3
x3
x3
0
1
0
1
1
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Data Structures
Decision Diagrams
Optimize by merging nodes
x2
x2
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Data Structures
Decision Diagrams
S
Optimize by merging nodes
x1
0
x2
1
0
x3
1
0
12
Data Structures
Logic Operations

AND
13
Decision Diagrams
Properties

• Canonical unique up to variable ordering
• Compact represent functions of up to 1000
  variables
• Efficient perform logic operations in
  linear-time

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Ordered Binary Decision Diagrams
(a.k.a. Branching Programs)
Example
Directed Acyclic Graph variables are inspected
in order.
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Reducing OBDDs
Terminal Rule eliminate duplicate terminals.
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Reducing OBDDs
Elimination Rule eliminate a node if its 0 and
1 edges lead to the same node.
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Reducing OBDDs
Merging Rule merge two nodes that reference
the same variable and point to the same
successors.
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Reduced OBDDs
Question
What is the optimal strategy for eliminate and
merge operations?
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Reduced OBDDs
Strategy eliminate and merge nodes repeatedly,
in any order, until no further simplifications
are possible.
Is the result unique?
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Uniqueness (proof)
By induction on the number of variables.
Induction Hypothesis Assume that any two ROBDDs
for a function with k 1 variables, k gt 0, are
isomorphic.
Inductive Step Show that any two ROBDDs for a
function with k variables are isomorphic.
21
Inductive Step
22
Inductive Step
Show that this mapping is well-defined and
one-to-one.
23
Inductive Step
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Mapping Well-Defined
low(v)
high(v)
Counter Example (Unreduced BDDs)
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Mapping One-To-One
Counter Example (Unreduced BDDs)
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Logic Operations
27
Logic Operations
28
Logic Operations
Compute
0
1
0
1
29
Logic Operations
30
Logic Operations
31
Logic Operations
32
Logic Operations
33
Logic Operations
0
1
34
Logic Operations
0
1
35
Logic Operations
0
1
36
Logic Operations
0
1
37
Logic Operations
Simplify
0
1
38
Logic Operations
Simplify
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Variable Ordering