mihir choudhury, kartik mohanram (iccad’10 best paper nominee) presentor: abert liu

Bi-decomposition of large Boolean functions using blocking edge graphs

Mihir Choudhury, Kartik Mohanram(ICCAD’10 best paper nominee)

Presentor: ABert Liu

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Introduction Terminology Algorithm Illustration Experimental Result Conclusion

Outline

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Introduction

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Bi-decomposition is a special kind of functional

decomposition Functional decomposition

Break a large function into a network ofsmaller functions

Reduce circuit and communication complexityand thus simplify physical design

Bi-decomposition plays an important role in logic synthesis for restructuring Boolean networks

Introduction

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Problem

f

fA f B

h

XA XB XC

XBXC

Bi-decompose

XA

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Terminology

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Blocking Edge Graph (BEG):

It’s an undirected graph. Every vertex represent a variable. One graph represents only one non-

decomposability of and, or, or xor. The edges connected variable pair {i , j} means

no variable partition can decompose this variable pair.

BEG can extract variable partition.

Terminology

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( a, b) and ( b, d) are not blocked in the and BEG.

( c, d) and ( c, b) are not blocked in the or BEG. The xor BEG is complete graph.

BEG Example

ORAND

a b

dc

a b

dc

XOR

a b

dc

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Algorithm

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1.Construct BEG 2.Do variable partition on the BEG. 3.Compute the decomposed functions. 4.Recursively bi-decompose the decomposed

functions from 3. In 4. if the function is not decomposable it will

do some relaxation to make further decompose. (It’s not bi-decomposition)

Algorithm

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Illustration

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Giving a function f with variable set V. To construct BEG for f: 1.Choose a variable pair { i, j }. 2.Give an assignment c to V/{ i, j}. (Restricting the K-map of function to 2x2 squares.) (There are 2x2 K-maps for each pair { i, j}) 3.Test the blocking condition to add edges ( i,

j) to BEGs. 4. Go back to 1. until all the pairs have been

tested.

Construct BEG

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Blocking condition:

One square blocks the others.

Construct BEG

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K-map ij\k 0 1 00 1 1 01 1 0 11 0 1 10 1 0

Example

AND

i

kj

OR

i

kj

XOR

i

kj

For { i, j } pair : assign k=0 assign k=1 i\j 0 1 i\j 0 1 0 1 1 or 0 1 0 xor 1 1 0 square 1 0 1 squareBlocking all { i, j} in all BEGs.Other pair do the same thing to finish constructing the BEGs.

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K-map ij\k 0 1 00 1 0 01 1 0 11 0 0 10 0 0

Example

AND

i

kj

OR

i

kj

XOR

i

kj

For { i, j } pair : assign k=0 assign k=1 i\j 0 1 i\j 0 1 0 1 1 Literal 0 0 0 Zero 1 0 0 square 1 0 0 square

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We can compute whether the edge ( i, j)

should be added or not in and and or BEGs by compute x. If x is not constant 0 the edge will be added.

Construct BEG

i

j

k

On-SetOff-set

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The same for or and and:

Construct BEG

i

j

k

On-SetOff-set

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Whether the function is bi-decomposable or not?

Guarantee the existence of variable partition.

Variable Partition

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Who can be common variable?Vertex cut is necessary but not sufficient condition.

Variable Partition

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Vertex Cut: A set of vertices whose removal

renders a connected graph disconnected. Examples:

Variable Partition

a b

dc

Legal cut

Not a vertex cut

a b

dc

e Legal cut

Another legal cut

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Decomposed function for or and and: For an or bi-decomposition.

Decomposed Function

Off-set

On-set

An and bi-decomposition can be obtained in a similar manner by interchanging the off-set and the on-set of f.

It’s subset of off-set of f.

It’s obtained from expanding on-set of f and does not overlap with off-set of decomposed function.

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For xor :

Decomposed Function

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Example

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Experimental Result

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Experimental Result

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Experimental Result

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Conclusion

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The experimental result looks good. This work implemented in ABC using CUDD

package, it might have memory problem when function is large.

Solving problem with new graph structure might be a good idea. The BEG has global view of the decomposability of

every variable pairs to choose the best partition. Maybe we can improve our bi-decomposition

work.

Conclusion

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Thanks for Attention