Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras
A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of t...
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| Format: | Article |
| Language: | English |
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Ram Arti Publishers
2018-12-01
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| Series: | International Journal of Mathematical, Engineering and Management Sciences |
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| Online Access: | https://www.ijmems.in/assets//29-ijmems-17-012-vol.-3%2c-no.-4%2c-404%E2%80%93428%2c-2018.pdf |
| _version_ | 1828312854028091392 |
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| author | Ali Muhammad Ali Rushdi Waleed Ahmad |
| author_facet | Ali Muhammad Ali Rushdi Waleed Ahmad |
| author_sort | Ali Muhammad Ali Rushdi |
| collection | DOAJ |
| description | A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design. |
| first_indexed | 2024-04-13T16:22:27Z |
| format | Article |
| id | doaj.art-f15690486e504f0fb7af9c27095384d1 |
| institution | Directory Open Access Journal |
| issn | 2455-7749 2455-7749 |
| language | English |
| last_indexed | 2024-04-13T16:22:27Z |
| publishDate | 2018-12-01 |
| publisher | Ram Arti Publishers |
| record_format | Article |
| series | International Journal of Mathematical, Engineering and Management Sciences |
| spelling | doaj.art-f15690486e504f0fb7af9c27095384d12022-12-22T02:39:54ZengRam Arti PublishersInternational Journal of Mathematical, Engineering and Management Sciences2455-77492455-77492018-12-013440442810.33889/IJMEMS.2018.3.4-029Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean AlgebrasAli Muhammad Ali Rushdi0Waleed Ahmad1Department of Electrical and Computer Engineering , King Abdulaziz University, P. O. Box 80204, Jeddah 21589, Saudi ArabiaDepartment of Electrical and Computer Engineering , King Abdulaziz University, P. O. Box 80204, Jeddah 21589, Saudi ArabiaA task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.https://www.ijmems.in/assets//29-ijmems-17-012-vol.-3%2c-no.-4%2c-404%E2%80%93428%2c-2018.pdfDigital designSuppression of variables‘Big’ Boolean algebrasBoolean-equation solvingParametric solutions |
| spellingShingle | Ali Muhammad Ali Rushdi Waleed Ahmad Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras International Journal of Mathematical, Engineering and Management Sciences Digital design Suppression of variables ‘Big’ Boolean algebras Boolean-equation solving Parametric solutions |
| title | Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras |
| title_full | Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras |
| title_fullStr | Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras |
| title_full_unstemmed | Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras |
| title_short | Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras |
| title_sort | digital circuit design utilizing equation solving over big boolean algebras |
| topic | Digital design Suppression of variables ‘Big’ Boolean algebras Boolean-equation solving Parametric solutions |
| url | https://www.ijmems.in/assets//29-ijmems-17-012-vol.-3%2c-no.-4%2c-404%E2%80%93428%2c-2018.pdf |
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