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Unit-11: Binary Decision Diagrams (BDDs)
- B. Srivathsan
Chennai Mathematical Institute
NPTEL-course July - November 2015
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Module 1: Introduction to BDDs
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Unit-11: Binary Decision Diagrams (BDDs) B. Srivathsan Chennai - - PowerPoint PPT Presentation
Unit-11: Binary Decision Diagrams (BDDs) B. Srivathsan Chennai - - PowerPoint PPT Presentation
Unit-11: Binary Decision Diagrams (BDDs) B. Srivathsan Chennai Mathematical Institute NPTEL-course July - November 2015 1 / 24 Module 1: Introduction to BDDs 2 / 24 Model-checking Transition Systems + Properties + NuSMV State-space Bchi
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Model-checking
Transition Systems + Properties + NuSMV
Automata
Unit: 4
Büchi Automata
Unit: 5,6
LTL properties
Unit: 7,8
CTL properties
Unit: 9
State-space explosion
Unit: 10
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In this unit: An efficient data structure for representing transition systems
◮ Module 1: Introduction to the data structure: Binary Decision
Diagrams (BDDs)
◮ Module 2: Operations on BDDs ◮ Module 3: Using BDDs in the model-checking process
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In this unit: An efficient data structure for representing transition systems
◮ Module 1: Introduction to the data structure: Binary Decision
Diagrams (BDDs)
◮ Module 2: Operations on BDDs ◮ Module 3: Using BDDs in the model-checking process
Reference: Logic in Computer Science, 2nd edition, by Huth and Ryan, Section 6.1 - 6.3
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Boolean functions
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x,y : Boolean variables
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x,y : Boolean variables Boolean function f (x,y) = x + y
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x,y : Boolean variables Boolean function f (x,y) = x + y 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1
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x,y : Boolean variables Boolean function f (x,y) = x + y 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 Logical OR
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x,y : Boolean variables Boolean function f (x,y) = x + y 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 Logical OR
1 1 1 1 1 1 1 x y f (x,y)
Truth table
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x,y : Boolean variables
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x,y : Boolean variables Boolean function f (x,y) = x · y
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x,y : Boolean variables Boolean function f (x,y) = x · y 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1
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x,y : Boolean variables Boolean function f (x,y) = x · y 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1 Logical AND
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x,y : Boolean variables Boolean function f (x,y) = x · y 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1 Logical AND
1 1 1 1 1 x y f (x,y)
Truth table
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x : Boolean variable
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x : Boolean variable Boolean function f (x) = x
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x : Boolean variable Boolean function f (x) = x 0 = 1 1 = 0
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x : Boolean variable Boolean function f (x) = x 0 = 1 1 = 0 Logical NOT
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x : Boolean variable Boolean function f (x) = x 0 = 1 1 = 0 Logical NOT
1 1 x f (x)
Truth table
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x1,x2,...,xn: Boolean variables f : {x1,x2,...,xn} → {0,1} Boolean function
+
· Boolean operations
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x1,x2,...,xn: Boolean variables f : {x1,x2,...,xn} → {0,1} Boolean function
+
· Boolean operations Examples: f1(x,y) = x + y, f2(x,y,z) = x · y + y · z, f3(x,y,z) = x + y · z
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Representing boolean functions
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f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Truth table
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f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y y z z z z 1 1 1 1
Truth table
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f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y y z z z z 1 1 1 1
Truth table Binary Decision Tree
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Operations on truth tables
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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g(x,y,z) = f (x,y,z) = x · y + y · z
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z
1 1 1 1 1 1 1 1 1 1 1 1 14/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 43
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 46
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 47
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 48
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 49
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
SLIDE 50
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x · y h(x,y,z) = f (x,y,z) + g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z
1 1 1 1 1 1 1 1 1 1 1 1 15/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
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x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 57
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 58
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 59
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 60
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 61
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 15/24
SLIDE 62
x y z f
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x y z g
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f (x,y,z) = x · y + y · z g(x,y,z) = x h(x,y,z) = f (x,y,z) · g(x,y,z)
x y z h
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15/24
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Truth table representation for boolean functions
◮ Space: For n variables, needs to store 2n · (n + 1) bits ◮ Operations: Visit each entry of truth table
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Truth table representation for boolean functions
◮ Space: For n variables, needs to store 2n · (n + 1) bits ◮ Operations: Visit each entry of truth table ◮ Sequential circuits can be modeled using boolean functions
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Truth table representation for boolean functions
◮ Space: For n variables, needs to store 2n · (n + 1) bits ◮ Operations: Visit each entry of truth table ◮ Sequential circuits can be modeled using boolean functions
If boolean functions are represented using truth tables, a circuit with 100 variables needs more than 2100 bits!
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Coming next: Efficient representation for Boolean formulas
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x · y
x y y 1
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x · y
x y y 1 x y y 1
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x · y
x y y 1 x y y 1
unnecessary
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x · y
x y y 1 x y y 1
unnecessary
x y 1
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x · y
x y y 1 x y y 1
unnecessary
x y 1
Binary Decision Diagram
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x y y z z z z 1 1 1 1
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x y y z z z z 1 1 1 1 x y y z z z z 1
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SLIDE 74
x y y z z z z 1 1 1 1 x y y z z z z 1 x y y z z 1
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x y y z z z z 1 1 1 1 x y y z z z z 1 x y y z z 1 x y y z 1
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Binary Decision Diagrams
x y y z z z z 1 x y y z 1 x y y 1 x y 1
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Binary Decision Diagrams
x y y z z z z 1 x y y z 1 x y y 1 x y 1
Reduced BDDs
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Reduction rules for BDDs
◮ C1: Removal of duplicate leaves ◮ C2: Removal of redundant tests ◮ C3: Removal of duplicate sub-trees
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x y y z z z z 1 1 1 1 x y y z z z z 1 x y y z z 1 x y y z 1
C1 C2 C3
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x · y
x y y 1 x y y 1 x y 1
C1 C2
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Representing boolean functions
BDDs Reduced BDDs
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