ECE 697B (667) Spring 2006 Synthesis and Verification of Digital Systems

1ECE 667 - Synthesis & Verification - Lecture 14

ECE 697B (667)ECE 697B (667)Spring 2006Spring 2006

Synthesis and Verificationof Digital Systems

BDD-based Bi-decompositionBDD-based Bi-decomposition

BDS systemBDS system

ECE 667 - Synthesis & Verification - Lecture 14 2

OutlineOutline

• Review of current decomposition methods– Algebraic – Boolean

• Theory of BDD decomposition [C. Yang 1999]

– Bi-decomposition F = D Q– Boolean AND/OR Decomposition– Boolean XOR Decomposition– MUX Decomposition

• Logic optimization based on BDD decomposition


Functional Decomposition – previous workFunctional Decomposition – previous work

• Ashenhurst [1959], Curtis [1962]– Tabular method based on cut: bound/free variables– BDD implementation:

• Lai et al. [1993, 1996], Chang et al. [1996]

• Stanion et al. [1995]

• Roth, Karp [1962]– Similar to Ashenhurst, but using cubes, covers

– Also used by SIS

• Factorization based– SIS, algebraic factorization using cube notation

– Bertacco et al. [1997], BDD-based recursive bidecomp.


Drawbacks of Traditional Synthesis MethodsDrawbacks of Traditional Synthesis Methods

• Weak Boolean factorization capability.• Difficult to identify XOR and MUX decomposition.• Separate platforms for Boolean operations and

factorization.

• Our goal: use a common platform to carry out both Boolean operations and factorization: BDD’s


What is wrong with Algebraic Division?What is wrong with Algebraic Division?

• Divisor and quotient are orthogonal!!• Better factored form might be:

(q1+ q2+ …+qn) (d1+d2+…+dm)

– gi and dj may share same or opposite literals

• Example:

SOP form: F = abg + acg + adf + aef + afg + bd + ce + be + cd. (23 lits)

Algebraic: F = (b + c)(d + e + ag) + (d + e +g)af. (11 lits)

Boolean: F = (af + b + c)(ag + d + e). (8 lits)


First work, Karplus [1988]: 1-dominatorFirst work, Karplus [1988]: 1-dominator

d

10

ca

b

1 0

*

a + b c + d

• Definition: 1-dominator is a node that belongs to every path from root to terminal 1.

• 1-dominator defines algebraic conjunctive (AND) decomposition: F = (a+b)(c+d).

F

a

bc

d

10

1-dominator


Karplus: 0-dominatorKarplus: 0-dominator

a

b

0 1

d

10

c

a b c d

• Definition: 0-dominator is a node that belongs to every path from root to terminal 0.

• 0-dominator defines algebraic disjunctive (OR) decomposition: F = ab + cd.

a

bc

d

10

F

0-dominator


Bi-decomposition based on DominatorsBi-decomposition based on Dominators

• We can generalize the concept of algebraic decomposition and dominators to:• Generalized dominators• Boolean bi-decompositions (AND, OR, XOR)

• Bi-decomposition: F = D Q

• First, let’s review fundamental theorems for Boolean division and factoring.


Boolean DivisionBoolean Division

Definitions

• G is a Boolean divisor of F if there exist H and R such that F = G H + R, and G H 0.

• G is said to be a factor of F if, in addition, R=0, that is:F = G H.

where H is the quotient, R is the remainder.Note: H and R may not be unique.


Boolean Factor - Boolean Factor - TheoremTheorem

Theorem:

Boolean function G is a Boolean factor of Boolean function F iff F G , (i.e. FG’ = 0, or G’ F’).

FF

GGProof:: G is a Boolean factor of F. Then H s.t. F = GH;

Hence, F G (as well as F H).

: F G F = GF = G(F + R) = GH. (Here R is any function R G’.)

Notes:

- Given F and G, H is not unique. - To get a small H is the same as getting a small F + R.

Since RG = 0, this is the same as minimizing (simplifying) f with DC = G’.


Boolean Division - Boolean Division - TheoremTheorem

Theorem: G is a Boolean divisor of F if and only if F G 0.

GGFF

Proof:

: F = GH + R, GH 0 FG = GH + GR. Since GH 0, FG 0.

: Assume that FG 0. F = FG + FG’ = G(F + K) + FG’. (Here K G’.)

Then F = GH + R, with H = F + K, R = FG’. Since GH = FG 0, then GH 0.

Note:

f has many divisors. We are looking for a g such that f = gh + r, where g, h, r are simple functions. (simplify f with DC = g’)


Boolean DivisionBoolean Division

Goal : for a given F, find D and Q such that F = Q · D.

Boolean Space

FFD

FQ

F = e + bd, D = e + d, Q = e + b


Conjunctive (AND) DecompositionConjunctive (AND) Decomposition

• Conjunctive (AND) decomposition: F = D Q.

• Theorem:

Boolean function F has conjunctive decomposition iff F D. For a given choice of D, the quotient Q must satisfy: F Q F + D’.

DDQQ F

• For a given pair (F,D), this provides a recipe for Q .


Boolean Division Boolean Division AND decompositionAND decomposition

FFD

FQ

F = e + bd, Q = e + b F + D’

Boolean Space

D = e + d,

Given function F and divisor D F, find Q such that:

F Q F + D’.

D’


AND DecompositionAND Decomposition ( (F = D Q): Example Example

• Recall: given (F,D), quotient Q must satisfy: F Q F + D’.

d

e

b

0 1

F = e + bde

b

0 11

De

b

0 1

D = e + b,

d

e

b

0 1

DC

Q

d

e

0 1

Q = e + d

DC

DC

ebd

00 01 11 10

0

1

1 1

111

Q


Disjunctive (OR) DecompositionDisjunctive (OR) Decomposition

• Disjunctive (OR) decomposition: F = G + H.

• Theorem:

Boolean function F has disjunctive decomposition iff F G. For a given choice of G, the term H must satisfy: F’ H’ F’ + G.

• For a given (F,G), this provides a recipe for H .

GGHH

F

Dual to conjunctive

decomposition.


Boolean AND/OR Bi-decompositionsBoolean AND/OR Bi-decompositions

• Conjunctive (AND) decomposition

If D F, F = F D = Q D.

• Disjunctive (Disjunctive (OROR) decomposition) decomposition

If If G F, F = F + G = H + G.

• D, G = generalized dominators


Generalized DominatorGeneralized Dominator D D

af

bb

c c

0 0

D

g

d

e

a

f

bb

c c

F

1 00

1

Boolean divisor

F = D Q


Generalized DominatorGeneralized Dominator G G

af

bb

c c

0 0

G

g

d

e

a

f

bb

c c

F

1 00

0

Boolean “subtractor”

F = G + H


Boolean Division Based on Boolean Division Based on Generalized DominatorGeneralized Dominator

a

f

bb

c c

0 0

D

g

d

e

a

f

bb

c c

F

1 00

g

d

e

a

f

bb

c c

Q

1

DC

0

DC

minimizeg

d

e

a

Q

10

Q = ag + d + e

1

reduce

a

f

b

c

0

D

D = af + b + c

1


Special Case: Special Case: 1-dominator1-dominator

a

b

1 0

a + b

d

10

c

*

c + dFa

b

c

d

10

1-dominator

F = (a+b)(c+d)


Special Case: Special Case: 0-dominators0-dominators

a

b

0 1

d

10

c

a b c da

b

c

d

10

F

0-dominator

F = ab + cd


Algebraic Algebraic XOR XOR DecompositionDecomposition

h

F

f

h

F

f’f

+= h

f

0

hf

h

f’

0

h’f’

= complement edge = 1-edge edge

= 0-edge edge


AlgebraicAlgebraic XOR XOR Decomposition: Decomposition: x-dominatorsx-dominators

F

1

a

d

b

c

x-dominator a+b

1

a

b

c+d

1

d

c


Boolean Boolean XORXOR Decomposition: Decomposition: Generalized x-dominatorsGeneralized x-dominators

Given F and G, there exists H : F = G H; H = F G.

F

1

a

d

b

c c

Generalized x-dominator

c

d

1

G

1

H

a

d

b

c


MUX MUX DecompositionDecomposition

f g

F

v 0 1

f g

F

v

f g

h

F

0 1

f gh

F

• Simple MUX decomposition

• Complex MUX

decomposition


FunctionalFunctional MUX MUX Decomposition - example Decomposition - example

F

h

f g

F

h

f g

0 1

1

0 1

b

a

0

F

dc

F

1

d

b

a

c

0


Synthesis FlowSynthesis FlowBoolean Network

Construct Global BDDs

Variable Reorder

Decompose BDD

Factoring Tree Processing

Technology Mapping

Construct Factoring Trees


Factoring Tree Processing: Factoring Tree Processing:


A Complete Synthesis ExampleA Complete Synthesis Example


A Complete Synthesis Example (Decompose function A Complete Synthesis Example (Decompose function gg))


A Complete Synthesis Example (Decompose function A Complete Synthesis Example (Decompose function hh))


A Complete Synthesis Example (Sharing Extraction)A Complete Synthesis Example (Sharing Extraction)


ConclusionsConclusions

• BDD-based bi-decomposition is a good alternative to traditional, algebraic logic optimization– Produces Boolean decomposition– Several types: AND, OR, XOR, MUX

• BDD decomposition-based logic optimization is fast.

• Stand-alone BDD decomposition scheme is not amenable to large circuits– Global BDD too large– Must partition into network of BDDs (local BDDs)

ECE 697B (667) Spring 2006 Synthesis and Verification of Digital Systems

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