IT University of Copenhagen Lecture 7: BDD Construction and Manipulation 1. BDD construction 2....

IT University of Copenhagen

Lecture 7: BDD Construction and Manipulation1. BDD construction2. Boolean operations on BDDs3. BDD-Based configuration


Today’s ProgramToday’s ProgramToday’s ProgramToday’s Program

• [12:00-13:15] – Unique table– Build(t)– Apply(op,u1,u2)– BDD-Based configuration (SJHAHM04)

• [13:20-14:00] Henrik Hulgaard, Configit A/S – A configuration technology company

2011 Rune Møller Jensen 2


BDD ConstructionBDD ConstructionBDD ConstructionBDD Construction

32011 Rune Møller Jensen


Last week:

1.Make a Decision Tree of formula

2.Keep reducing it until no further reductions are possible

This week:• Reduce the decision tree to a BDD while building

it

BDD constructionBDD constructionBDD constructionBDD construction



• Represent BDD by a table of unique nodes (UT) • Build BDDs recursively,

i.e. to add a new node u:1. Compute high(u) and low(u) and store them in UT

2. Maintain BDD reductions when adding u to UT:

a) Only extend UT with u if high(u) low(u) (non-redundancy test)

b) Only extend UT with u if u UT (uniqueness)

Reduce decision tree to BDD during constructionReduce decision tree to BDD during constructionReduce decision tree to BDD during constructionReduce decision tree to BDD during construction



Unique Table RepresentationUnique Table RepresentationUnique Table RepresentationUnique Table RepresentationNode Attributes

Represent Unique Table by two tables T and HT : u (i,l,h) H: (i,l,h) u


H is the inverse of T:T(u) = (i,l,h) H(i,l,h) = u

u unique node identifier {0,1,2,3,…} i = var(u) variable index {1,2,…,n,n+1}l = low(u) node identifier of lowh = high(u) node identifier of high

u

low(u) high(u)

var(u)


Primitive Operations on T and HPrimitive Operations on T and HPrimitive Operations on T and HPrimitive Operations on T and H



Unique Table Interface: MakeNode (MK)Unique Table Interface: MakeNode (MK)Unique Table Interface: MakeNode (MK)Unique Table Interface: MakeNode (MK)



Idea: Construct the BDD recursively using theShannon Expansion t = x t[1/x], t[0/x]

• Terminal casesBuild(0) = 0Build(1) = 1

• Recursive case

Build(t(xi,xi+1,…,xn)) = Mk( )

BuildBuildBuildBuild


Build(t(0,xi+1,…,xn)) Build(t(1,xi+1,…,xn))

xi


BuildBuildBuildBuild



BDD ManipulationBDD ManipulationBDD ManipulationBDD Manipulation



Multi-Rooted BDDMulti-Rooted BDDMulti-Rooted BDDMulti-Rooted BDD

2011 Rune Møller Jensen 12

5 6x1

x2

x3 2

3 4

1 0

Unique Table contains many BDDs

x1 x3 x1 x2

x2


ApplyApplyApplyApply

• Apply(op,u1,u2): computes the BDD of u1 op u2

where op : any of the 16 Boolean operatorsu1, u2: root nodes of BDDs

• Relies on the Shannon expansion properties:

(x t1,t0) op (x t’1,t’0) x (t1 op t’1),(t0 op t’0)

(x t1,t0) op t x (t1 op t),(t0 op t)132011 Rune Møller Jensen


• Terminal case: u {0,1} u’ {0,1}

App(u u’) = u u’

• Recursive case: u = xv u1, u0 u’ = xw u’1, u’0 App(u u’) = Mk(xv, App(u0 u’0), App(u1 u’1) ) if v = w

Mk(xv, App(u0 u’), App(u1 u’) ) if v < w

Mk(xw, App(u u’0), App(u u’1) ) if w < v

Apply with Apply with opop = = Apply with Apply with opop = =





Construct BDDs from expression treeConstruct BDDs from expression treeConstruct BDDs from expression treeConstruct BDDs from expression tree


x1 x2 x3

u5 u4

u2 u3

x1 x2 x3

u2 u3u4

u5

u6


Properties of ApplyProperties of ApplyProperties of ApplyProperties of Apply• Improvements?

– Early termination. E.g., no reason to keep recursing if the left side in a conjunction is 0

• Complexity : O(|u1||u2|) , due to dynamic programming

• So a BDD of any formula can be computed in poly time?



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



BDD-Based ConfigurationBDD-Based ConfigurationBDD-Based ConfigurationBDD-Based Configuration



A configuration problem C is a triple (Y,D,F)– Y is a set of variables y1, y2, … ,yn

– D is the Cartesian product of their finite domainsD = D1 D2 … Dn

– F = {f1,f2,…,fm} is a set of propositional formulas over atomic propositions yi = v, where v Di, specifying the conditions that the variable assignments must satisfy. Each formula is inductively defined by

f yi = v | f g | f g | f

Configuration ProblemsConfiguration ProblemsConfiguration ProblemsConfiguration Problems



• y1 {black, white, red, blue} : Color y2 {small, medium, large} : Sizey3 {“Men in black” - MIB, “Save the whales” -STW} : Print

• f1 (y3 = MIB) (y1 = black)f2 (y3 = STW) (y2 small)

• y1 y2 y3black small MIBwhite medium STWred largeblue

T-Shirt ExampleT-Shirt Example11T-Shirt ExampleT-Shirt Example11

212011 Rune Møller Jensen 1: Due to Erik van der Meer, now at Microsoft


























IPC(C)

1.R COMPILE(C)

2. while |R| > 1 do

3. choose (yi = v) VALIDASSIGNMENTS(R)

4. R R (yi = v)

Interactive Product ConfiguratorInteractive Product ConfiguratorInteractive Product ConfiguratorInteractive Product Configurator



Idea

1.Use a BDD to represent R

2.Use a polynomial-time BDD algorithm to compute VALIDASSIGNMENTS(R)

BDD-based configurationBDD-based configurationBDD-based configurationBDD-based configuration



1. Define domains in binary 00 : black, 01 : white, 10 : red, 11 : blue 00 : small, 01 : medium, 10 : large 0 : MIB, 1 : STW

2. Build a BDD of the rules

Represent Represent RR by a BDD by a BDDRepresent Represent RR by a BDD by a BDD

28

01

11 , xx

02

12 , xx03x


• Trace paths for each variable layer in the BDD

Compute VALIDASSIGNMENTS(Compute VALIDASSIGNMENTS(RR))Compute VALIDASSIGNMENTS(Compute VALIDASSIGNMENTS(RR))

29


Henrik HulgaardHenrik Hulgaardconfigit configit

Henrik HulgaardHenrik Hulgaardconfigit configit


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IT University of Copenhagen Lecture 7: BDD Construction and Manipulation 1. BDD construction 2....

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