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# GAP 4 Package Thelma

## THreshold ELements, Modeling and Applications

1.0.2

2018

Victor Bovdi
Email: vbovdi@gmail.com
Address:
Department of Mathematical Sciences
UAEU
Al Ain, United Arab Emirates

Vasyl Laver
Email: vasyl.laver@uzhnu.edu.ua
Address:
Department of Mathematical Sciences
UAEU
Al Ain, United Arab Emirates

### Copyright

© 2018 by the authors

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.

### Contents

1 Introduction
1.1 Overview over this manual
1.2 Installation
1.3 Feedback
2 Boolean Functions
2.1 Basic Operations
2.1-1 LogicFunction
2.1-2 IsLogicFunction
2.1-3 PolynomialToBooleanFunction
2.1-4 IsUnateInVariable
2.1-5 IsUnateBooleanFunction
2.1-6 InfluenceOfVariable
2.1-7 SelfDualExtensionOfBooleanFunction
2.1-8 SplitBooleanFunction
2.1-9 KernelOfBooleanFunction
2.1-10 ReducedKernelOfBooleanFunction
3 Threshold Elements
3.1 Basic Operations
3.1-1 ThresholdElement
3.1-2 IsThresholdElement
3.1-3 OutputOfThresholdElement
3.1-4 StructureOfThresholdElement
3.1-5 RandomThresholdElement
3.1-6 Comparison of Threshold Elements
3.2 Single Threshold Element Realizability
3.2-1 CharacteristicVectorOfFunction
3.2-2 IsCharacteristicVectorOfSTE
3.2-3 IsInverseInKernel
3.2-4 IsKernelContainingPrecedingVectors
3.2-5 IsRKernelBiggerOfCombSum
3.2-6 BooleanFunctionBySTE
3.2-7 PDBooleanFunctionBySTE
3.3 Iterative Training Methods
3.3-1 ThresholdElementTraining
3.3-2 ThresholdElementBatchTraining
3.3-3 WinnowAlgorithm
3.3-4 Winnow2Algorithm
3.3-5 STESynthesis
4 Networks of Threshold Elements
4.1 Basic Operations
4.1-1 NeuralNetwork
4.1-2 IsNeuralNetwork
4.1-3 OutputOfNeuralNetwork
4.2 Networks of Threshold Elements
4.2-1 BooleanFunctionByNeuralNetwork
4.2-2 BooleanFunctionByNeuralNetworkDASG
5 Multi-Valued Threshold Elements
5.1 Threshold Elements over GF(p^k)
5.1-1 MVThresholdElement
5.1-2 IsMVThresholdElement
5.1-3 OutputOfMVThresholdElement
5.1-4 StructureOfMVThresholdElement
5.1-5 MVBooleanFunctionBySTE
References
Index
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