1. Rolf Drechlser’s slides used
Davio Gates and Kronecker Decision Diagrams
Lecture 5
Rolf Drechlser’s slides used

2. Absolutely Minimum Background on Binary Decision Diagrams (BDD) and Kronecker Functional Decision Diagrams
BDDs are based on recursive Shannon expansion
F = x Fx + x’ Fx’
Compact data structure for Boolean logic
can represents sets of objects (states) encoded as Boolean functions
Canonical representation
reduced ordered BDDs (ROBDD) are canonical
essential for simulation, analysis, synthesis and verification

3. BDD Construction Typically done using APPLY operator Reduction rules
remove duplicate terminals
merge duplicate nodes
(isomorphic subgraphs)
remove redundant nodes
Redundant nodes:
nodes with identical children a b f 1 b c

4. BDD Construction – your first BDD
Construction of a Reduced Ordered BDD
1 edge
0 edge
f = ac + bc 1 a b c f
a b c f
Truth table
Decision tree

5. BDD Construction – cont’d1 a b c f f 1 a b c
f = (a+b)c 1 a b c
1. Remove duplicate terminals
2. Merge duplicate nodes
3. Remove redundant nodes

6. Decomposition types
Decomposition types are associated to the variables in Xn with the help of a decomposition type list (DTL) d:=(d1,…,dn) where di { S, pD, nD}

7. KFDD
Definition

8. Three different reductions types
Type I :
Each node in a DD is a candidate for the application g f g f xi xi xi

9. Three different reductions types (cont’d)xi xj xj g f g f

10. Three different reductions types (cont’d)
Type D xi
This is used in functional diagrams such as KFDDs xj xj f f g g

11. Example for OKFDD 1 x1 F=

12. Example for OKFDD (cont’d)
This diagram below explains the expansions from previous slide
X1 S-node
X2 pD-node
X3 nD-node

13. Example for OKFDD’s with different DTL’s
You can save a lot of nodes by a good DTL

14. Complement Edges They can be applied for S-nodes
They can be applied for D-nodes
This is a powerful idea in more advanced diagrams – any operator on representation

15. Such recursive operations are a base of diagram creation and operation
XOR-operation
D-node
S-node
Such recursive operations are a base of diagram creation and operation

16. Such recursive operations are a base of diagram creation and operation

17. Restriction of Variables for S nodes is trivial
Xi=0
Xi=1 xi
S-node g f f g
Cofactor for Xi=1
Cofactor for Xi=0

18. Restriction of Variables for D nodes is more complex, requires XORing
Xi=1
Xi=0 xi
pD-node g f f
Please remember this

19. Restriction of Variables (cond’t)
Xi=1
Xi=0 xi
nD-node g f f

20. Optimization of OKFDD-Size
Exchange of Neighboring Variables
DTL Sifting

21. Experimental Results

22. Experimental Results (cont’d)

23. DIY PROBLEM
Use simulated annealing algorithm for choice of a better variable ordering of the OKFDD

26. End of Lecture 5 Simple solution Solution space Neighborhood structure
X1 X2 X3 X4
X3 X4 X1 X2
A solution
A solution
Neighborhood structure
X1 X2 X3 X4
X1 X3 X2 X4
change
End of Lecture 5