Binary Decision Diagrams

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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
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BDD is defined as Directed Acyclic Graph
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Binary Decision Diagrams
Short described a cyclic structure for a BDD
variant (1960).
We suggest cycles are a general phenomenon.
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Binary Decision Diagrams
Short described a cyclic structure for a BDD
variant (1960).
We suggest cycles are a general phenomenon.
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Binary Decision Diagrams
Short described a cyclic structure for a BDD
variant (1960).
We suggest cycles are a general phenomenon.
0
0
1
0
1
1
6
Binary Decision Diagrams
Short described a cyclic structure for a BDD
variant (1960).
We suggest cycles are a general phenomenon.
0
0
1
0
1
1
Future research awaits...
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Analysis of Biochemcial Systems
Result of computation is a certain token
distribution.
time t
time (t?t)
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Analysis of Biochemcial Systems
Colored tokens (multi-valued logic).
time t
time (t?t)
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
1 bit of memory
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Balancer

• Asynchronous token routing device

inputs
outputs
balanced token counts
1 bit of memory
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Balancer

• Snapshot

inputs
outputs
x
y
1 bit of memory
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Counting Network

• Data structure for multiprocessor coordination

c
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d
a
a
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c
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d
g
g
step sequence
a
b
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f
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f
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a
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f
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Counting Network
Execution trace token counts on all wires
step sequence
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Counting Network
Tokens are assigned value based on the output
wire number.
4, 8, 12, 16, . . .
3, 7, 11, 15, . . .
Counting Network
2, 6, 10, 14, . . .
1, 5, 9, 13, . . .