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Lecture 8: Binary Decision Diagrams1. Classical Boolean expression representations2. If-then-else Normal Form (INF)3. Binary Decision Diagrams

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Today’s Program

• [13:00-13:55] Computation based representations– Boolean expressions and Boolean functions– Classical representations of Boolean expressions

• Truth tables• Two-level normal forms: CNF, DNF• Multi-level representations: circuits and formulas

• [14:05-15:00] Evaluation based representations– If-then-else normal form– Decision trees– Ordered Binary Decision Diagrams (OBDDs)– Reduced Ordered Binary Decision Diagrams

(ROBDDs or just BDDs)

2007 Rune M. Jensen

IT University of Copenhagen

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Classical Representations of Boolean Expressions

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Boolean Expressions• Boolean Expressions

t ::= x | 0 | 1 | t | t t | t t | t t | t t

• Literalsl ::= x | x

• Precedence, , , , (obs. two last swapped in HRA notes)

• Terminology– Boolean expression = Boolean formula, Propositional

formula, Sentence in propositional logic– Boolean variable = Propositional symbol/letter/variable

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Boolean functions

• DefinitionAn n-variable function f : Bn B is called a Boolean function if it is induced by a Boolean expression E.

• B = {0,1}• E represents the Boolean function

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• Order of arguments matterf(x,y) = x y f(y,x) = x y

• Several expressions may represent a function f(x,y) = x y = x y = (x y) (x x) = …

• Equalityf = g iff x . f(x) = g(x)

• Number of Boolean functions f : Bn B

Properties

2007 Rune M. Jensen

n22

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Desirable properties of a representation

• Compact• Equality check easy (true for canonical representations)

• Easy to evaluate the truth-value of an assignment• Boolean operations efficient• SAT check efficient• Tautology check efficient• Easy to implement

2007 Rune M. Jensen

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The curse of Boolean function representation

• Boolean functions in n variables… – How do we find a single compact representation for

all of them?

Ex. Boolean circuits : how many Boolean circuits of size g with n inputs and 1 output?

– Choice of gates: 16g

– Choice of connections: (2g+1)(n+g)

– Total: 16g (2g+1)(n+g)

n22

…

Out1

In1 In2 Inn

161 2

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The curse of Boolean function representation

• Assume g = O(nk)• Thus the fraction of Boolean functions of n

variables with a polynomial circuit size in n is

nnO

n

kk nnOknO

for 02

)1)(2(162

))(()(

Curse of Boolean function representation:

This problem exists for all representations we know!

2007 Rune M. Jensen

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Truth tables

• Compact table size 2n

• Equality check easy canonical

• Easy to evaluate the truth-value of an assignment linear

• Boolean operations efficient linear

• SAT check efficient linear

• Tautology check efficient linear

• Easy to implement e.g., DBs2007 Rune M. Jensen

x y z xyz

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

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• One level normal forms

• Two level normal Forms

Two Level Normal Forms

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OP1

Lit LitLit …

OP1

…

OP2

Lit Lit Lit

OP2

Lit Lit Lit

OP2

Lit Lit Lit… …

…

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Two-level normal forms: DNF CNF

• -monomials : x1 x2 x3 x4

• -monomials : x1 x2 x3 x4

• ( ,)-polynomials: -product of -monomials disjunctive normal form (DNF):

• Ex: x y x y

• (, )-polynomials: -product of -monomials conjunctive normal form (CNF):

• Ex: (x y) (x y)2007 Rune M. Jensen

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Two-level normal forms: DNF CNF• Is there a DNF and CNF of every expression?

• Given a truth table representation of a Boolean formula, can we easily define a DNF and CNF of the formula?

x y z xyz0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

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Two-level normal forms: DNF CNF

• Example DNF of xyz

x y z xyz0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

x y z

x y z

x y z

x y z

x y z

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Two-level normal forms: DNF CNF

• Example CNF of xyz

x y z xyz0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

( x y z)

(x y z)

(x y z)

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IT University of Copenhagen

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Two-level normal forms: DNF CNF

• Example CNF of xyz

x y z xyz0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

(x y z)

(x y z)

( x y z)

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Two-level normal forms: DNF CNF

• The special version DNF and CNF representations produced from on and off-tuples are canonical and called cDNF and cCNF

• Are cDNF and cCNF minimum size DNF and CNF representations?

Every Boolean formula has a DNF and CNF representation

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Two-level normal forms: DNF CNF

• Properties of DNF and CNF

• Problem: conversion between CNFs and DNFs is exponential

SAT Tautology

CNF NP complete Polynomial (exercise)

DNF Polynomial (exercise)

Co-NP complete

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Two-level normal forms: DNF CNF

• Example– CNF

– Corresponding DNF

nn xxxxxx 1021

20

11

10

nn

nn

nn

nn

xxxx

xxxx

xxxx

xxxx

11

121

11

01

121

11

11

020

10

01

020

10

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Two-level normal forms: DNF CNF • Compact Translating a formula to either CNF or DNF may cause an exponential

blow-up in expression size

• Equality check easy ”compact” CNF and DNF formulas are not canonical

• Easy to evaluate the truth-value of an assignment linear

• Boolean operations efficient disjunction e.g. efficient for DNF

• SAT check efficient Not for CNF

• Tautology check efficient Not for DNF

• Easy to implement programmable logic arrays (PLAs)2007 Rune M. Jensen

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Circuits

• Circuits are multi-level representations • CNF and DNF are special circuits with depth 3• Thus, circuits are not canonical

Out

x y z

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Circuits

• Compact But still suffers from the curse of Booelan function representation.

• Equality check easy Not canonical

• Easy to evaluate the truth-value of an assignment linear

• Boolean operations efficient constant

• SAT check efficient (NP-Hard)

• Tautology check efficient Not canonical

• Easy to implement Integrated circuits2007 Rune M. Jensen

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Formulas

• Formulas are circuits where each node has out-degree 1

• Multiple use of intermediate functions is not allowed

((x y) (z y)) ((x y) (z x))

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Formulas • Compact At least as large as a circuit

• Equality check easy Not canonical

• Easy to evaluate the truth-value of an assignment linear

• Boolean operations efficient constant

• SAT check efficient (Circuit-SAT ≤p Formula-SAT)

• Tautology check efficient Not canonical

• Easy to implement Simpler algorithmic handling than circuits2007 Rune M. Jensen

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Classical Representations• Observations

– There is a trade off between the size of a representation and the efficiency of operations and checks

– Truthtables: very large, but all checks and Boolean operations can be carried out efficiently

– Circuits: quite compact, but SAT and equality checks are very hard

• Next: Binary decision diagrams, probably the best known balance between size and efficiency!

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Binary Decision Diagrams

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Computation and Evaluation Based Representation

• Computation Process Representation– Representation gives a way to compute the

expression value– Truth-tables, DNF, CNF, circuits and formulas

• Evaluation Process Representation– Representation gives a way to classify the expression

value by means of tests– Decision trees

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If-then-else operator

• The if-then-else Boolean operator is defined by xy1 , y0 = ( x y1 ) ( x y0 )

• We have

(xy1 , y0) [1/x] = ( 1 y1 ) ( 0 y0 ) = y1

(xy1 , y0) [0/x] = ( 0 y1 ) ( 1 y0 ) = y0

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If-then-else operator• All operators can be expressed using

– only operators and the constants 0 and 1– tests on un-negated variables

• What are if-then-else expressions for– x, x– x y, x y– x y

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INF normal form

• Proposition: any Boolean expression t is equivalent to an expression in INFProof:

t = x t[1/x], t[0/x] (Shannon expansion of t)

Apply the Shannon expansion recursively on t. The recursion must terminate in 0 or 1, since the number of variables is finite

An if-then-else Normal Form (INF) is a Boolean expression build entirely from the if-then-else operator and the constants 0 and 1 such that all test are performed only on variables

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Example

• Example: t = (x1 y1) (x2 y2)

• Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

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Example

• Example: t = (x1 y1) (x2 y2)

• Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

t0 = y1 t01, t00

t1 = y1 1, t10

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Example• Example: t = (x1 y1) (x2 y2)

• Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

t0 = y1 t01, t00

t1 = y1 1, t10

t01 = x2 t011, 0

t00 = x2 t001, 0

t10 = x2 t101, 0

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IT University of Copenhagen

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Example• Example: t = (x1 y1) (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

t0 = y1 t01, t00

t1 = y1 1, t10

t01 = x2 t011, 0

t00 = x2 t001, 0

t10 = x2 t101, 0 t011 = y2 1,0 t001 = y2 1,0 t101 = y2 1,0

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Decision tree• Example: t = (x1 y1) (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

t0 = y1 t01, t00

t1 = y1 1, t10

t01 = x2 t011, 0

t00 = x2 t001, 0

t10 = x2 t101, 0 t011 = y2 1,0 t001 = y2 1,0 t101 = y2 1,0

1

x1

y1 y1

x2 x2 x2

y2 y2

t

t0 t1

t00t01 t10

t001 t011 t101

0 0 y2 0

0 0 0 1 1 12007 Rune M. Jensen

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Reduction I: substitute identical subtrees

• Example: t = (x1 y1) (x2 y2)

• Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t0

t0 = y1 t01, t01

t1 = y1 1, t01

t01 = x2 t011, 0

t011 = y2 1,0

x1

y1 y1

x2

y2

t

t0 t1

t01

t011

0 1

Result: an Ordered Binary Decision Diagram (OBDD)

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Reduction II: remove redundant tests

• Example: t = (x1 y1) (x2 y2)

• Shannon expansion of t in order x1,y1,x2,y2

t = x1 t1, t01

t1 = y1 1, t01

t01 = x2 t011, 0

t011 = y2 1,0

x1

y1

x2

y2

t

t1

t01

t011

0 1

Result: a Reduced Ordered Binary Decision Diagram (ROBDD) [often called a BDD]

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Reductions

x

Uniqueness requirement

Non-redundant tests requirement

x x

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Another reduction examplex

z

y y

z z

0 1

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Another reduction examplex

z

y y

z z

0 1

Equal subtrees

Redundant test

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Another reduction examplex

z

y y

z

0 1

Equal subtrees

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Another reduction examplex

z

y y

0 1

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Another reduction examplex

z

y y

0 1

Equal subtrees

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Another reduction examplex

z

y

0 1

Redundant test

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Another reduction example

z

y

0 1

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Another reduction example

z

y

0 1

y z

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Definition of ROBDDs

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Canonicity of ROBDDs

• Canonicity Lemma: for any function f : Bn B there is exactly one ROBDD u with a variable ordering x1 < x2 < … < xn such that fu = f(x1,…,xn)

Proof (by induction on n)

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Canonicity of ROBDDs

• The canonicity of ROBDDs simplifies– Equality check– Tautology check– Satisfiability checkWhy?

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• What are the ROBDDs of– 1– 0– x y order x, y– (x y) z order x, y, z

Practice

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Size of ROBDDs

• ROBDDs of many practically important Boolean functions are small – E.g., Adders have polynomial sized ROBDDs

• Have all functions polynomial ROBDD size? NO– ROBDDs do not escape the curse of Boolean function

representation– E.g., Multipliers have exponential ROBDD size

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Size of ROBDDs• The set of nodes of a ROBDD of f is equal to the set of

different subfunctions of f (in an INF expansion)

• Example: 4 subfunctions = 4 nodes in the ROBDD

f(x,y,z) = y zf(0,y,z) = y zf(1,y,z) = y zf(0,0,z) = zf(0,1,z) = 1f(1,0,z) = zf(1,1,z) = 1

f(0,0,0) = 0f(0,0,1) = 1f(0,1,0) = 1f(0,1,1) = 1f(1,0,0) = 0f(1,0,1) = 1f(1,1,0) = 1f(1,1,1) = 1

z

y

0 1

y z

z

0 1

This property can be used to find upper bounds on the ROBDD size 2007 Rune M. Jensen

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Size Of ROBDDs

• The size of an ROBDD depends heavily on the variable ordering

• Example: t = (x1 y1) (x2 y2)

• Build ROBDD of t in order x1,x2,y1,y2

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Size of ROBDDs• The size of an ROBDD depends heavily on the

variable ordering(x1 y1) (x2 y2) … (xn yn)

x1 < y1 < x2 < y2 < … < xn < yn x1 < x2 < … < xn < y1 < x2 < …< yn

2007 Rune M. Jensen