Neue boolesche orthogonalisierende operative methoden und gleichungen
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- Tez No: 509110
- Danışmanlar: Prof. GEORG FISCHER, Prof. DIETMAR FEY
- Tez Türü: Doktora
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2016
- Dil: Almanca
- Üniversite: Friedrich-Alexander-Universität Erlangen-Nürnberg
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 213
Özet
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Özet (Çeviri)
Orthogonality is a special property of Boolean functions because an orthogonal function can be transformed in another form and so it simplifies the handling for further calculations as the Boolean Differential Calculus. It applies, the orthogonal form of a disjunctive normalform is equal to the orthogonal form of an antivalent normalform consisting of the same terms, DNForth = ANForth. Thus, the orthogonalization of a Boolean function simplifies the transformation in another equivalent form. With the use of the Ternary-Vector-List to represent a Boolean function the Boolean Differential Calculus on orthogonal Ternary-Vector-Lists will be facilitated. Furthermore, the advantages in terms of computation times and memory usage will be used. In this work two logical operation methods called 'orthogonalizing difference-building ' and 'orthogonalizing OR-ing _g' are newly presented. The orthogonalizing differencebuilding is used to calculate the difference of two product terms respectively of two Ternary-Vectors or of two functions respectively of two Ternary-Vector-Lists of the disjunctive normal form which is provided in orthogonal form. The orthogonalizing difference-building also enables the calculation of the complement of a function as well as the EXOR and EXNOR of two product terms respectively Ternary-Vectors or two functions respectively two Ternary-Vector-Lists with a result in orthogonal form. Two calculation steps - difference-building derived out of the set theory and the subsequent orthogonalization - are performed in one step by the use of this new operation method . On the basis of orthogonalizing difference-building a further logical operation method called orthogonalizing OR-ing _gis going to be introduced. Orthogonal OR-ing is used to calculate the orthogonal union of two product terms respectively of two Ternary-Vectors. It also enables the calculation of the orthogonalizing OR-ing of two orthogonal Boolean functions respectively of two orthogonal Ternary-Vector-Lists of disjunctive normal form to get orthogonal results. Two calculation steps - OR-ing and the subsequent orthogonalization - are performed in one step by the use of _g. The advantages of both operation methods are their orthogonal results that have an essential advantage for succeeding calculations. As an orthogonal disjunctive normal form is equivalent to the orthogonal antivalence normal form with the same product terms, the application of the Boolean Differential Calculus will be simplified in TVLarithmetic. Additionally, the implemented algorithms of both new operation methods has faster computing times in comparison to the according compositions. By reducing of two calculation steps to one step the requested memory space is reduced. Further calculation steps in the Ternary-Vector-List arithmetic is facilitated by the inherent orthogonalization. Furthermore, two new Boolean equations for the orthogonalization of Boolean functions respectively of Ternary-Vector-Lists of the disjunctive and also conjunctive normal form based on these new methods are set up. They provide the mathematical solution of orthogonalization for the first time. In addition, the new equations can be used as a part in the calculation procedure of getting suitable test patterns for combinatorial circuits for verifying feasible logical faults in the Ternary-Vector-Lists arithmetic by the method of [61]. Computation time and memory request are important criteria in the application of algorithms. Due to the exponential growth of the complexity of integrated circuits the test methods and test design will fall through in the future. Therefore, the consideration to implement better algorithms for the calculation of test pattern is enormous important. That means algorithms with smaller complexity. Algorithms which provides minor computation times and generates minimized test pattern are of high relevance for the field of test. Accordingly, the implemented algorithms ORTH[ ] and ORTH[_g] based on the new equations are analyzed in computation time and memory request in compare to other methods known from the literature. The algorithms ORTH[ ] and ORTH[_g] reduce the computation time to a factor of approximately 2.5 with increasing dimension to 50 and increasing length of Ternary-Vectors-List of 25 Ternary-Vectors. For higher dimensions an acceleration of computation time is theoretically expected. Further advantage is the smaller number of the product terms respectively of the Ternary-Vectors in the orthogonalized result which reduces the number of further calculation steps. ORTH[ ] and ORTH[_g] reduce the number of Ternary- Vectors by approximately 50% and enable faster calculation of the Boolean Differential Calculus due to the fewer number of operations. Thus, the additional computation time and the memory usage will be reduced. With this reduction further reductions of Ternary-Vectors is continued in the calculation procedures until the determining of test pattern . Thereby, smaller set of suitable test pattern will be provided for combinatorial circuits for verifying feasible logical faults. The minor set of test pattern and the algorithms providing faster computation time are important factors when the test time and the resulting test costs should be minimized. The scope of the new mathematical methods of orthogonalization are not limited by the area of determining of test pattern. Similar advantages can be expected for the applications in cryptology, but this is not scope of this thesis. Possibly the new methods can be used in the area of the application of Boolean functions and their orthogonalization, such as in the logic, the Boolean algebra, the game theory and the combinatorics.
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