Binary Decision Diagrams
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1 Binary Decision Diagrams An Introduction and Some Applications Manas Thakur PACE Lab, IIT Madras Manas Thakur (IIT Madras) BDDs 1 / 25
2 Motivating Example Binary decision tree for a truth table Manas Thakur (IIT Madras) BDDs 2 / 25
3 Motivating Example Binary decision tree for a truth table Manas Thakur (IIT Madras) BDDs 3 / 25
4 Motivating Example Collapse redundant nodes Manas Thakur (IIT Madras) BDDs 4 / 25
5 Motivating Example Collapse redundant nodes Manas Thakur (IIT Madras) BDDs 5 / 25
6 Motivating Example Collapse redundant nodes Manas Thakur (IIT Madras) BDDs 6 / 25
7 Motivating Example Collapse redundant nodes Manas Thakur (IIT Madras) BDDs 7 / 25
8 Motivating Example Eliminate unnecessary nodes Manas Thakur (IIT Madras) BDDs 8 / 25
9 Motivating Example We got an ROBDD!! Manas Thakur (IIT Madras) BDDs 9 / 25
10 Introduction Overview 1 Motivating Example 2 Introduction 3 Constructing ROBDDs 4 Applications 5 Conclusion Manas Thakur (IIT Madras) BDDs 10 / 25
11 Introduction Binary Decision Diagrams Definition A Binary Decision Diagram is a rooted DAG with One or two terminal nodes of out-degree zero labeled 0 or 1 A set of variable nodes of out-degree two Manas Thakur (IIT Madras) BDDs 11 / 25
12 Introduction Ordered Binary Decision Diagrams (OBDDs) A BDD is ordered if on all paths through the graph, the variables respect a given linear order. b 1 < b 2 <... < b n Manas Thakur (IIT Madras) BDDs 12 / 25
13 Introduction Ordered Binary Decision Diagrams (OBDDs) A BDD is ordered if on all paths through the graph, the variables respect a given linear order. An unordered BDD b 1 < b 2 <... < b n Manas Thakur (IIT Madras) BDDs 12 / 25
14 Introduction Ordered Binary Decision Diagrams (OBDDs) The size of a BDD depends on the variable ordering The problem of finding the best variable ordering in OBDDs is NP-Complete Manas Thakur (IIT Madras) BDDs 13 / 25
15 Introduction Reduced Ordered Binary Decision Diagrams (ROBDDs) Definition An OBDD is reduced if it satisfies the following properties: Uniqueness low(u) = low(v) and high(u) = high(v) implies u = v Non-redundant tests low(u) high(u) We already saw an example of ROBDDs!! See it again Manas Thakur (IIT Madras) BDDs 14 / 25
16 Introduction Properties of ROBDDs Size is correlated to amount of redundancy, NOT size of relation Insight: As the relation gets larger, the number of dont-care bits increases, leading to fewer necessary nodes (usually) Canonicity: For every Boolean function, there is exactly one ROBDD representing it Hence, satisfiability, tautology-check, and equivalence can be tested in deterministic time For Boolean expressions, this problem is NP-Complete Manas Thakur (IIT Madras) BDDs 15 / 25
17 Constructing ROBDDs Normal forms for Boolean expressions Disjunctive Normal Form (DNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Satisfiability: easy; Tautology check: hard Manas Thakur (IIT Madras) BDDs 16 / 25
18 Constructing ROBDDs Normal forms for Boolean expressions Disjunctive Normal Form (DNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Satisfiability: easy; Tautology check: hard Conjunctive Normal Form (CNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Tautology check: easy; Satisfiability: hard Manas Thakur (IIT Madras) BDDs 16 / 25
19 Constructing ROBDDs Normal forms for Boolean expressions Disjunctive Normal Form (DNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Satisfiability: easy; Tautology check: hard Conjunctive Normal Form (CNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Tautology check: easy; Satisfiability: hard Reduction? Manas Thakur (IIT Madras) BDDs 16 / 25
20 Constructing ROBDDs Normal forms for Boolean expressions Disjunctive Normal Form (DNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Satisfiability: easy; Tautology check: hard Conjunctive Normal Form (CNF) (a 1 a 2... a n )... (a 1 a 2... a n ) Tautology check: easy; Satisfiability: hard Reduction? No hopes since conversion between CNF and DNF is exponential Manas Thakur (IIT Madras) BDDs 16 / 25
21 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. x y 0, y 1 = (x y 0 ) ( x y 1 ) Manas Thakur (IIT Madras) BDDs 17 / 25
22 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. More examples x y 0, y 1 = (x y 0 ) ( x y 1 ) x = x (1, 0) Manas Thakur (IIT Madras) BDDs 17 / 25
23 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. More examples x y 0, y 1 = (x y 0 ) ( x y 1 ) x = x (1, 0) x = (x 0, 1) Manas Thakur (IIT Madras) BDDs 17 / 25
24 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. More examples x y 0, y 1 = (x y 0 ) ( x y 1 ) x = x (1, 0) x = (x 0, 1) x y = (x 1, y) Manas Thakur (IIT Madras) BDDs 17 / 25
25 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. More examples x y 0, y 1 = (x y 0 ) ( x y 1 ) x = x (1, 0) x = (x 0, 1) x y = (x 1, y) x y = (x y, 1) Manas Thakur (IIT Madras) BDDs 17 / 25
26 Constructing ROBDDs If-then-else Normal Form (INF) An If-then-else Normal Form (INF) is a Boolean expression built from the if-then-else operator and the constants 0 and 1, such that all tests are performed only on variables. More examples x y 0, y 1 = (x y 0 ) ( x y 1 ) x = x (1, 0) x = (x 0, 1) x y = (x 1, y) x y = (x y, 1) x y = x (y 1, 0), (y 0, 1) Manas Thakur (IIT Madras) BDDs 17 / 25
27 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 Manas Thakur (IIT Madras) BDDs 18 / 25
28 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 Manas Thakur (IIT Madras) BDDs 18 / 25
29 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 Manas Thakur (IIT Madras) BDDs 18 / 25
30 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 Manas Thakur (IIT Madras) BDDs 18 / 25
31 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 Manas Thakur (IIT Madras) BDDs 18 / 25
32 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 Manas Thakur (IIT Madras) BDDs 18 / 25
33 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 Manas Thakur (IIT Madras) BDDs 18 / 25
34 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 t 110 = y 2 0, 1 Manas Thakur (IIT Madras) BDDs 18 / 25
35 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 t 110 = y 2 0, 1 t 111 = y 2 1, 0 Manas Thakur (IIT Madras) BDDs 18 / 25
36 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 t 110 = y 2 0, 1 t 111 = y 2 1, 0 Manas Thakur (IIT Madras) BDDs 18 / 25
37 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 t 110 = y 2 0, 1 t 111 = y 2 1, 0 Manas Thakur (IIT Madras) BDDs 19 / 25
38 Constructing ROBDDs Example: t = (x 1 y 1 ) (x 2 y 2 ) t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 11, 0 t 00 = x 2 t 001, t 000 t 11 = x 2 t 111, t 110 t 000 = y 2 0, 1 t 001 = y 2 1, 0 t 110 = y 2 0, 1 t 111 = y 2 1, 0 t = x 1 t 1, t 0 t 0 = y 1 0, t 00 t 1 = y 1 t 00, 0 t 00 = x 2 t 001, t 000 t 000 = y 2 0, 1 t 001 = y 2 1, 0 Manas Thakur (IIT Madras) BDDs 19 / 25
39 Applications Let s move ahead 1 Motivating Example 2 Introduction 3 Constructing ROBDDs 4 Applications 5 Conclusion Manas Thakur (IIT Madras) BDDs 20 / 25
40 Applications Applications of BDDs Correctness of Combinational Circuits Equivalence of Combinational Circuits Model Checking And yes, Program Analysis! Manas Thakur (IIT Madras) BDDs 21 / 25
41 Applications BDDs for representing Points-to relation Points-to analysis using BDDs. Berndl et al. PLDI 03. Let a,b,c be reference variables and A,B,C be object references. The points-to relation (a,a),(a,b),(b,a),(b,b),(c,a),(c,b),(c,c) is represented as: Manas Thakur (IIT Madras) BDDs 22 / 25
42 Applications BDDs for representing Points-to relation Points-to analysis using BDDs. Berndl et al. PLDI 03. Let a,b,c be reference variables and A,B,C be object references. The points-to relation (a,a),(a,b),(b,a),(b,b),(c,a),(c,b),(c,c) is represented as: On the board. Manas Thakur (IIT Madras) BDDs 22 / 25
43 Applications bddbddb Cloning-Based Context-Sensitive Pointer Alias Analysis Using Binary Decision Diagrams. John Whaley and Monica S. Lam. PLDI 04. Manas Thakur (IIT Madras) BDDs 23 / 25
44 Applications bddbddb Cloning-Based Context-Sensitive Pointer Alias Analysis Using Binary Decision Diagrams. John Whaley and Monica S. Lam. PLDI 04. On the Board 1 Represent program information using relations Express relations as BDDs Manas Thakur (IIT Madras) BDDs 23 / 25
45 Applications bddbddb Cloning-Based Context-Sensitive Pointer Alias Analysis Using Binary Decision Diagrams. John Whaley and Monica S. Lam. PLDI 04. On the Board 1 Represent program information using relations Express relations as BDDs 2 Write program analyses as Datalog queries Express queries as BDD operations Manas Thakur (IIT Madras) BDDs 23 / 25
46 Applications bddbddb Cloning-Based Context-Sensitive Pointer Alias Analysis Using Binary Decision Diagrams. John Whaley and Monica S. Lam. PLDI 04. On the Board 1 Represent program information using relations Express relations as BDDs 2 Write program analyses as Datalog queries Express queries as BDD operations 3 Get solutions! Perform operations on BDDs Manas Thakur (IIT Madras) BDDs 23 / 25
47 Applications Pointers for the enthusiast An Introduction to Binary Decision Diagrams. Tutorial by Henrik Reif Andersen. Program Analysis using Binary Decision Diagrams. Ondrej Lhotak s PhD Thesis (2006). Context-Sensitive Pointer Analysis using Binary Decision Diagrams. John Whaley s PhD Thesis (2007). Fun with Binary Decision Diagrams. Video lecture by Donald Knuth. Manas Thakur (IIT Madras) BDDs 24 / 25
48 Conclusion Conclusion Manas Thakur (IIT Madras) BDDs 25 / 25
49 Conclusion Conclusion BDDs are very interesting and useful! Manas Thakur (IIT Madras) BDDs 25 / 25
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