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ELEC 7770 Advanced VLSI Design Spring 2007 Binary Decision Diagrams

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ELEC 7770Advanced VLSI DesignSpring 2007Binary Decision Diagrams

Vishwani D. Agrawal

James J. Danaher Professor

ECE Department, Auburn University

Auburn, AL 36849

http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr07

ELEC 7770: Advanced VLSI Design (Agrawal)

- Satisfiability algorithms
- ATPG methods
- Binary decision diagrams (BDD)

ELEC 7770: Advanced VLSI Design (Agrawal)

- C. E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” Trans. AIEE, vol. 57, pp. 713-723, 1938.
- Consider:
- Boolean variables, X1, X2, . . . , Xn
- Boolean function, F(X1, X2, . . . , Xn)

- Xi’ is complement of Xi
- Cofactors, F(Xi=j) = F(X1, X2, . . , Xi=j, . . , Xn), j = 0 or 1

ELEC 7770: Advanced VLSI Design (Agrawal)

(1)F = Xi F(Xi=1) + Xi’ F(Xi=0)∀ i=1,2,3, . . . n

(2)F = (Xi + F(Xi=0)) (Xi’ + F(Xi=1)) ∀ i=1,2,3, . . . n

F(Xi=0) F(Xi=1)

0 1

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F

ELEC 7770: Advanced VLSI Design (Agrawal)

- F = XiXj F(Xi=1, Xj=1) + XiXj’ F(Xi=1, Xj=0)
+ Xi’Xj F(Xi=0, Xj=1)

+ Xi’Xj’ F(Xi=0, Xj=0)

- In general, a Boolean function can be expanded about any number of input variables.
- Expansion about k variables will have 2k terms.

ELEC 7770: Advanced VLSI Design (Agrawal)

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Graph representation

of a Boolean function.

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Leaf nodes

ELEC 7770: Advanced VLSI Design (Agrawal)

- Binary decision diagram (BDD) is a graph representation of a Boolean function, directly derivable from Shannon’s expansion.
- References:
- C. Y. Lee, “Representation of Switching Circuits by Binary Decision Diagrams,” Bell Syst. Tech J., vol. 38, pp. 985-999, July 1959.
- S. Akers, “Binary Decision Diagrams,” IEEE Trans. Computers, vol. C-27, no. 6, pp. 509-516, June 1978.

- Ordered BDD (OBDD) and Reduced Order BDD (ROBDD).
- Reference:
- R. E. Bryant, “Graph-Based Algorithms for Boolean Function Manipulation,” IEEE Trans. Computers, vol. C-35, no. 8, pp. 677-691, August 1986.

ELEC 7770: Advanced VLSI Design (Agrawal)

- BDD of an n-variable Boolean function is a tree:
- Root node is any input variable.
- All nodes in a level are labeled by the same input variable.
- Each node has two outgoing edges, labeled as 0 and 1 indicating the state of the node variable.
- Leaf nodes carry fixed 0 and 1 labels.
- Levels from root to leaf nodes represent an ordering of input variables.
- If we trace a path from the root to any leaf, the label of the leaf gives the value of the Boolean function when inputs are assigned the values from the path.

ELEC 7770: Advanced VLSI Design (Agrawal)

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OBDD

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ELEC 7770: Advanced VLSI Design (Agrawal)

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ELEC 7770: Advanced VLSI Design (Agrawal)

- Start at leaf nodes and work toward the root – leaf node functions are 0 and 1.
- Function at a node with variable x is
f = x’.f(low) + x.f(high)

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ELEC 7770: Advanced VLSI Design (Agrawal)

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ELEC 7770: Advanced VLSI Design (Agrawal)

- Two OBDDs are isomorphic if there is one-to-one mapping between the vertex sets with respect to adjacency, labels and leaf values.
- Two isomorphic OBDDs represent the same function.
- Two identical circuits may not have identical OBDDs even when same variable ordering is used.
- Comparison is possible if:
- Same variable ordering is used.
- Any redundancies in graphs are removed.

ELEC 7770: Advanced VLSI Design (Agrawal)

- Directed acyclic graph (DAG) (*).
- Contains just two leaf nodes labeled 0 and 1.
- Variables are indexed, 1, 2, . . . n, such that the index of a node is greater than that of its child (*).
- A node has exactly two child nodes, low and high, that low ≠ high.
- Graph contains no pair of nodes such that subgraphs rooted in them are isomorphic.

* Properties common to OBDD.

ELEC 7770: Advanced VLSI Design (Agrawal)

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Isomorphic

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ELEC 7770: Advanced VLSI Design (Agrawal)

- Unique for given variable ordering – graph isomorphism verifies logic equivalence.
- Size (number of nodes) changes with variable ordering – worst-case size is exponential (e.g., integer multiplier).
- Other applications: logic synthesis, testing.
- For algorithms to derive ROBDD, see
- R. E. Bryant, “Graph-Based Algorithms for Boolean Function Manipulation,” IEEE Trans. Computers, vol. C-35, no. 8, pp. 677-691, August 1986.
- G. De Micheli, Synthesis and Optimization of Digital Circuits, New York: McGraw-Hill, 1994.
- S. Devadas, A. Ghosh, and K. Keutzer, Logic Synthesis, New York: McGraw-Hill, 1994.

ELEC 7770: Advanced VLSI Design (Agrawal)