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Functionally Linear Decomposition and Synthesis of Logic Circuits for FPGA

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Page 1: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Functionally Linear Decomposition and Synthesis of Logic Circuits for FPGA

Abstract

To present a novel logic synthesis method to reduce the area of XOR-based logic functions.

Idea: To exploit linear dependency between

logic sub- functions to create an implementation based on an XOR relationship with a lower area overhead.

Page 3: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Over ViewIntroductionBackgroundDecomposition and SynthesisVariable PartitioningBasic/ Selector OptimizationMulti-Output SynthesisExperiment ResultsRelated WorkConclusion and Future Work

Page 4: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Introduction

The XOR-based logic functions are an important type of functions, heavily used in arithmetic, error correcting and telecommunication circuits.

Focus on XOR-based logic functions and show that they exhibit a property that can be exploited for area reduction in this work.

Page 5: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

XOR Decomposition (early work)

Spectral decompositionLinear decompositionDavio expansion + help of BDDs (Reed-Muller logic equation) Look for x-dominators in a BDD that indicate

a presennce of an XOR gate Tabular methods based on AC decomposition

Page 6: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

FLDS

Look at a linear relationship between logic functions

Define functional linearity to be a decomposition of the form:

f( X ) = Σi g i ( Y ) hi ( X-Y ) where X and Y are sets of variables (Y is the

subset of X), while the summation represents an XOR gate.

Page 7: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

FLDS

Look at a linear relationship between logic functions

Define functional linearity to be a decomposition of the form:

f( X ) = Σi g i ( Y ) hi ( X-Y ) where X and Y are sets of variables (Y is the

subset of X), while the summation represents an XOR gate.

f is the weighted sum of gi

weighting factors

Page 8: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

FLDS

Look at a linear relationship between logic functions

Define functional linearity to be a decomposition of the form:

f( X ) = Σi g i ( Y ) hi ( X-Y ) where X and Y are sets of variables (Y is the

subset of X), while the summation represents an XOR gate.

f is the weighted sum of gi

weighting factors

Can synthesize XOR logic functions by

Davio and Shannon’s expansions

Page 9: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

FLDS

Look at a linear relationship between logic functions

Define functional linearity to be a decomposition of the form:

f( X ) = Σi g i ( Y ) hi ( X-Y ) where X and Y are sets of variables (Y is the

subset of X), while the summation represents an XOR gate.

basis vectors

selector vectors

XOR gates

Page 10: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Background

Galois Field of characteristic 2 (GF(2))Linear independenceVectors spacesGaussion Elimination

Page 11: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Field

Page 12: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Galois Field of characteristic 2 (GF(2))

Page 13: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Linear independence

Page 14: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Gaussion Elimination

Page 15: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Decomposition and Synthesis

Page 16: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Page 17: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Page 18: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Page 19: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Page 20: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Variable Partitioning

Page 21: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Basic/ Selector Optimization

Page 22: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Multi-Output Synthesis

Page 23: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Experiment Results

Page 24: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Related Work

Page 25: Abstract To present a novel logic synthesis method to reduce the area of XOR-based logic functions. Idea: To exploit linear dependency between logic sub-

Conclusion and Future Work

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