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Paper by José L. Subirats , José M. Jerez, and Leonardo Franco

Literature Review: A New Decomposition Algorithm for Threshold Synthesis and Generalization of Boolean Functions. Paper by José L. Subirats , José M. Jerez, and Leonardo Franco Published in IEEE Transactions on Circuits and Systems I in November, 2008. Thershold Functions.

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Paper by José L. Subirats , José M. Jerez, and Leonardo Franco

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  1. Literature Review: A New Decomposition Algorithm for Threshold Synthesis and Generalization of Boolean Functions Paper by José L. Subirats, José M. Jerez, and Leonardo Franco Published in IEEE Transactions on Circuits and Systems I in November, 2008

  2. Thershold Functions • What is a Threshold Function?

  3. Threshold Functions

  4. Threshold Functions

  5. Threshold Functions • What is the significance of a threshold function? • Model of a Neuron • Important in Neural Networks

  6. 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

  7. Selecting Output (Or/And)

  8. Unate Function -Positive Unate variable: -Negative Unate variable: -Function is a Unate Function when all variables are either positive or negative unate

  9. Unate Function • What is the significance of the function being unate? • All threshold functions are unate • Checking for a unate function much quicker than checking for a threshold function • Can eliminate non-threshold functions more quickly, speeding up overall computation time

  10. Function Splitting • First, 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

  11. Function splitting • Split as follows, where xi is function with highest influence for function F1 • For case of OR representation:

  12. Function Splitting • Modification necessary for use of ‘don’t cares’ • Again, for case of OR representation:

  13. Results • Comparisons 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

  14. Results

  15. Results • Algorithm works with truth vectors involving up to 21 inputs • Can be applied to systems with significantly more inputs with the addition of standard processing steps used in circuit design

  16. Results

  17. Results • With 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

  18. Results

  19. Conclusions • Without 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

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