Utilization of Modular Eight-Variable Karnaugh Maps in Digital Design | Chapter 9 | Novel Research Aspects in Mathematical and Computer Science Vol. 8

 The 8-variable Karnaugh map, which is a map with an extraordinarily high variable-handling capability, is introduced in this chapter in both a regular and modular form. Additionally, in the broader scenario of arbitrary n, the chapter analyses an n-bit comparator and visualises this analysis for n = 4 on the aforementioned map. On the six-variable, four-variable, and two-variable submaps of the original map, the situations n = 3, 2, and 1 are respectively exceptional cases. A tutorial exposition of several key concepts in switching theory, including implicants, prime implicants, essential prime implicants, minimal sum, complete sum, disjoint sum of products (or probability-ready expressions), Boole-Shannon Expansion, and unate and threshold functions, are provided in the evaluation.

Author(s) Details:

Ali Muhammad Ali Rushdi,
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O. Box-80200, Jeddah-21589, Saudi Arabia.

Sultan Sameer Zagzoog,
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O. Box-80200, Jeddah-21589, Saudi Arabia.

Please see the link here: https://stm.bookpi.org/NRAMCS-V8/article/view/8363

Map Visualization of Cause-Effect Relations in Qualitative Comparative Analysis | Chapter 1 | Novel Perspectives of Engineering Research Vol. 3

 This chapter investigates cause-and-effect interactions modeled by partially-defined Boolean functions (PDBF) using a recent regular and modular version of the eight-variable Karnaugh map (known also as incompletely-specified switching functions). First, we show how to use a simple Karnaugh-map test to determine whether a certain variable must be included in a list of supporting variables for a function, or whether it should be excluded from such a set. This exclusion is achieved through certain don't-care assignments that ensure the equivalence of the Boolean quotient with respect to the variable and its complement, i.e., the exact matching of the half map representing the internal region of the variable and the remaining half map representing the external region of the variable, in which case either of these two half maps replaces the original full map as a representation of the function. This type of variable exclusion could be repeated with respect to other variables until a minimal set of supporting variables is found. The work deals with obtaining all of a PDBF's minimal sets of supporting variables without explicitly using integer-programming techniques. Standard map methods for extracting prime implicants allow the development of all irredundant disjunctive forms for each of the minimal sets obtained (IDFs). We avoid the effort of drawing prime-implicant loops on the initial eight-variable map by following this approach of first finding a minimal set of supporting variables, and we postpone this task until the map is (typically drastically) reduced in size. The approach described here has significant implications for the recently founded field of Qualitative Comparative Analysis (QCA). The mainstream QCA community is unlikely to appreciate these repercussions, as they plainly show that the too-often impressive outcomes reported by QCA adherents must be reviewed and investigated. More observations, in our opinion, are needed to limit down the possibilities and reduce the number of possible IDFs.

Author(S) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

Raid Salih Badawi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

View Book:- https://stm.bookpi.org/NPER-V3/article/view/4752

Utilization of Eight-Variable Karnaugh Maps in the Digital Design of n-bit Comparators | Chapter 5 | Theory and Practice of Mathematics and Computer Science Vol. 8

 An n-bit comparator is a common combinational circuit that compares two n-bit inputs, Y and Z, and generates three orthonormal outputs: G (indicating that Y is strictly greater than Z), E (indicating that Y and Z are equal or equivalent), and L (indicating that Y and Z are equal or equivalent) (indicating that Y is strictly less than Z). The symbols 'G,' 'Equal to,' and 'Less than,' respectively, are chosen to represent the concepts of 'Greater than,' 'Equal to,' and 'Less than.' The study for n=4 on a standard and modular version of the 8-variable Karnaugh-map is visualised in this paper for an n-bit comparator in the general case of arbitrary n. On 6-variable, 4-variable, and 2-variable submaps of the original map, the cases n= 3, 2 and 1 appear as special cases. Many important concepts in switching theory are covered in this review, including implicants, prime implicants, critical prime implicants, irredundant disjunctive forms, minimal sums, the complete sum, and disjoint sums of products (or probability-ready expressions).

Author(s) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

Sultan Sameer Zagzoog
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

View Book :- https://stm.bookpi.org/TPMCS-V8/issue/view/50

Review of Methods for Conditional-Probability Computations in Medical Disciplines | Chapter 7 | Highlights on Medicine and Medical Research Vol. 1

There is a definite need for conditional probability representations and methods that increase comprehension, simplify calculations, promote intuition, and encourage reasoning. In a wide range of scientific disciplines, such representations and instruments are desirable and useful, but their usefulness in medical contexts and applications is emphasized here, in order to address a clinical audience rather than a mathematical audience. We use a plethora of time-tested pedagogical representations or conditional probability methods, including: (a) Visualization of Venn diagrams or Karnaugh maps, (b) Reformulation as natural frequencies, (c) Entity interrelationships through Signal Flow Graphs, (d) Specification of certain problem formats, such as Trinomial Graphs format, and (e) Use of a pair of Karnaugh-map-like or complementary Karnaugh-maps Most of the new representations or instruments have well-known records of pedagogical benefits, but in the particular realm of conditional probability they are yet to be evaluated. Further evaluation of the novel representations or instruments proposed herein is needed. - of these is to be taught to a group of students, and the traditional representation must be used to teach a control group of students. Detailed statistical analysis of the findings is warranted. Earlier, similar inquiries were conducted, but they were on a small scale. The need arises, therefore, for an exploratory analysis that exhausts all suggested representations or instruments.

Author (s) Details

Rufaidah Ali Rushdi
Department of Pediatrics, Kasr Al-Ainy Faculty of Medicine, Cairo University, Cairo, 11562, Arab Republic of Egypt.

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

Fayez Ahmed Talmees
Department of Electrical and Computer Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.

View Book :- https://stm.bookpi.org/HMMR-V1/issue/view/2