Literature Review: A New Decomposition Algorithm for Threshold Synthesis and Generalization of Boolean Functions

Paper by José L. Subirats, José M. Jerez, and Leonardo FrancoPublished in IEEE Transactions on Circuits and Systems I in November, 2008

Thershold FunctionsWhat is a Threshold Function?

Threshold Functions

Threshold Functions

Threshold FunctionsWhat is the significance of a

threshold function?◦Model of a Neuron◦Important in Neural Networks

Selecting Output (Or/And)Determined by number of 1s in

the output of the truth table

When over half of the output of the truth table is 1, output function of final architecture is an OR

Otherwise, output function is an AND

Selecting Output (Or/And)

Unate Function

-Positive Unate variable:

-Negative Unate variable:

-Function is a Unate Function when all variables are either positive or negative unate

Unate FunctionWhat is the significance of the

function being unate?All threshold functions are unateChecking for a unate function

much quicker than checking for a threshold function

Can eliminate non-threshold functions more quickly, speeding up overall computation time

Function SplittingFirst, find variable with highest

influence

Influence defined as the number of input vectors where the change of the variable changes the value of the function

Function splittingSplit as follows, where xi is

function with highest influence for function F1

For case of OR representation:

Function SplittingModification necessary for use of

‘don’t cares’Again, for case of OR

representation:

ResultsComparisons to another

threshold function algorithm published in IEEE Trans. (2005)

Compared on number of gates, number of levels, and number of weights (interconnect) of generated circuits

Results

ResultsAlgorithm works with truth

vectors involving up to 21 inputsCan be applied to systems with

significantly more inputs with the addition of standard processing steps used in circuit design

Results

ResultsWith use of don’t cares,

comparisons are made to standard algorithms- C4.5 decision tree algorithm, feedforward neural networks, and implementation of nearest neighbour algorithm

For each function 60% of examples used for training, 40% used to test

Results compared on terms of generalization ability

Results

ConclusionsWithout preprocessing, significant

improvements to number of gates and levels for up to 21 inputs.

With preprocessing, some improvement to number of gates and significant improvement to number of levels

Increased amount of interconnect in both cases

Generalization ability comparable to existing standard algorithms