Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah 21589, Saudi Arabia.

Waleed Ahmad
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah 21589, Saudi Arabia.

DOI https://dx.doi.org/10.33889/IJMEMS.2018.3.4-029

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Abstract

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.

Keywords- Digital design, Suppression of variables, ‘Big’ Boolean algebras, Boolean-equation solving, Parametric solutions.

Citation

Rushdi, A. M. A., & Ahmad, W. (2018). Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras. International Journal of Mathematical, Engineering and Management Sciences, 3(4), 404-428. https://dx.doi.org/10.33889/IJMEMS.2018.3.4-029.