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.
Over ViewIntroductionBackgroundDecomposition and SynthesisVariable PartitioningBasic/ Selector OptimizationMulti-Output SynthesisExperiment ResultsRelated WorkConclusion and Future Work
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.
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
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.
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
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
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
Background
Galois Field of characteristic 2 (GF(2))Linear independenceVectors spacesGaussion Elimination
Field
Galois Field of characteristic 2 (GF(2))
Linear independence
Gaussion Elimination
Decomposition and Synthesis
Variable Partitioning
Basic/ Selector Optimization
Multi-Output Synthesis
Experiment Results
Related Work
Conclusion and Future Work