1. Binary Decision Diagrams (BDDs)
Daniel Kroening and Ofer Strichman

2. Part I Reminders - Deciding Propositional Logic What is Logic
Proofs by deduction
Proofs by enumeration
Decidability, Soundness and Completeness
some notes on Propositional Logic
Deciding Propositional Logic
SAT tools
BDDs      

3. Next: Binary Decision Diagrams
SAT looked for a satisfying solution to CNF
We will now examine a graph-based data structure called Binary Decision Diagrams.
It has several advantages and disadvantages comparing to SAT
Developed by Bryant [1986]. Next few slides are from the source …

4. Alternate Approach (A ∧ C) ∨ (C ∧ B) (A Ç B) Æ (C) A B C 1 A B C 1
Generate complete representation of function
Canonicity: functions are equal iff representations are identical
(A ∧ C) ∨ (C ∧ B) A B C 1
(A Ç B) Æ (C) A B C 1

5. Decision Structures Truth Table Decision Tree
Vertex represents decision
Follow green (dashed) line for value 0
Follow red (solid) line for value 1
Function value determined by leaf value.

6. Variable Ordering OK Not OK
Assign arbitrary total ordering to variables
e.g., x1 < x2 < x3
Variables must appear in ascending order along all paths OK
Not OK x x 3 1 x 2 x x3 1

7. Reduction Rule #1
Merge equivalent leaves a a a

8. Reduction Rule #2
Merge isomorphic nodes y x z y x z y x z

9. Reduction Rule #3
Eliminate Redundant Tests y x y

10. Example OBDD Initial Graph Reduced Graph
(x1 Ç x2) Æ x3
Canonical representation of Boolean functions
For a given variable ordering
Two functions are equivalent iff graphs are isomorphic
Can be tested in linear time
Desirable property: simplest form is canonical.

11. Satisfiability etc.
Constants
Unique unsatisfiable function
Unique tautology
Conclusion: given a BDD it takes constant time to check:
Validity
Contradiction
Satisfiability
Is this a free lunch ? …

12. Effect of Variable Ordering
Good Ordering
Bad Ordering
Linear Growth
Exponential Growth

13. Selecting Good Variable Ordering
Intractable Problem
Even when problem represented as OBDD
i.e., to find optimum improvement to current ordering
Application-Based Heuristics
Exploit characteristics of application
e.g., Ordering for functions of combinational circuit
Traverse circuit graph depth-first from outputs to inputs

14. Building BDDs ‘from below’
Starting from a binary decision tree is too hard for formulas with many variables.
Goal: construct the BDD ‘from below’.

15. Building BDDs ‘from below’
For this we will need a function called APPLY:
Given the BDDs for f1 and f2,
and a binary connective F 2 {Æ, Ç, !, $...} (any one of the 16 binary connectives),
Construct the BDD for f1 F f2,

16. Building BDDs ‘from below’
Def: a restriction of a function f to x=d, denoted f|x=d where x 2 vars(f), d 2 {0,1}, is equal to f after assigning x=d.
Given the BDD of f, deriving the BDD of f|x=0 is simple:
f: (x1 Ç x2) Æ x3
f|x2=1 1 x 3 x 3 1

17. Now, APPLY (1/3)
Let v1,v2 denote that root nodes of f1, f2, respectively, with var(v1) = x1 and var(v2)=x2.
If v1 and v2 are leafs, f1 F f2 is a leaf node with value val(v1) F val(v2) Ç 1 = 1 Æ 1 =

18. Now, APPLY (2/3)
If x1 = x2 = x, apply Shanon expansion: f1 F f2 = (:x Æ f1|x=0 F f2|x=0 Ç x Æ f1|x=1 F f2|x=1) x
BDD for
f1|x=1 Æ f2|x=1
f1|x=0 Æ f2|x=0 x x Æ =
BDD for
f1|x=0
BDD for
f1|x=1
BDD for
f2|x=0
BDD for
f2|x=1

19. Now, APPLY (3/3)
else, suppose x1 < x2 in the variable order. f1 F f2 = (:x1 Æ f1|x=0 F f2 Ç x1 Æ f1|x=1 F f2) x1 x2 x1 Æ =
BDD for
f1|x1=1
BDD for
f1|x=0 Æ f2
BDD for
f1|x=1 Æ f2
BDD for
f1|x1=0
BDD for
f2|x2=0
BDD for
f2|x2=1

20. BDDs from below: example.1 Ç x 2 1 x 1
BDD for
f1|x1=0 Ç f2
f1|x1=1 Ç f2 x2 1 x x 2 x 2 1 2 Ç = 1
f2: :x2
f1: x1 $ x2
BDD for
f1|x1=0 Ç f2 = = x 2 1
0 Ç 0 = 0
1 Ç 1 = 1

21. BDDs from below: example.1 x x 2 x 2 x 1
BDD for
f1|x=0 Ç f2
f1|x=1 Ç f2 2 Ç = 1 1
f2: :x2
f1: x1 $ x2
f1 Ç f2 = x1 Ç (:x1 Æ :x2) x 1 2 = x 1 x 2 = x 2 1

22. Comparing SAT to BDDs
BDD is a canonical data structure that represents the semantic of a function, i.e. all its solutions
Some applications (e.g. symbolic model checking) need canonicity to detect when two sets are equivalent.
Can require exponential space & time (highly sensitive to variable ordering)
SAT searches through CNF for a single solution
CNF is not canonical.
Poly-space algorithms exists. Time can be exponential.