scholarly journals Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis

2021 ◽  
pp. 57-84
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Boolean Function ◽  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Programming Approach ◽  
Minimal Set ◽  
First Case ◽  
Desirable Feature ◽  
Karnaugh Map ◽  
Variable Problem

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According  to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and  postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

scholarly journals Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

2021 ◽  
pp. 117-137
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Systematic Investigation ◽  
Exact Matching ◽  
Minimal Set ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Switching Functions ◽  
Programming Techniques ◽  
Karnaugh Map

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. 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 any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

scholarly journals Quality of Minimal Sets of Prime Implicants of Boolean Functions

2017 ◽  
Vol 63 (2) ◽  
pp. 165-169
Author(s):  
V. C. Prasad
Keyword(s):  
Power Consumption ◽  
Boolean Function ◽  
Boolean Functions ◽  
New Technique ◽  
Minimal Set ◽  
Reduce Power Consumption ◽  
Minimal Sets ◽  
Prime Implicants

Abstract Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

scholarly journals Using Variable-Entered Karnaugh Maps in Determining Dependent and Independent Sets of Boolean Functions

2012 ◽  
Vol 1 (2) ◽  
pp. 45-67
Author(s):  
Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Independent Sets ◽  
Solution Procedure ◽  
Black Box ◽  
Important Class ◽  
Time Saving ◽  
Prime Implicants ◽  
Simultaneous Processing ◽  
Karnaugh Map

An important class for Boolean reasoning problems involves interdependence among the members of a set T of Boolean functions. Two notable problems among this class are (a) to establish whether a given subset of T is dependent, and (b) to produce economical representations for the complementary families of all dependent subsets and independent subsets of T. This paper solves these two problems via a powerful manual pictorial tool, namely, the variableentered Karnaugh map (VEKM). The VEKM is utilized in executing a Label-and-Eliminate procedure for producing certain prime implicants or consequents used in tackling the two aforementioned problems. The VEKM procedure is a time-saving short cut indeed, since it efficiently handles the three basic tasks demanded by the solution procedure, which are: (a) To combine several Boolean relations into a single one, (b) to compute conjunctive eliminants of a Boolean function, and (c) to derive the complete sum (CS) of a Boolean function. The VEKM procedure significantly reduces the complexities of these tasks by introducing useful shortcuts and allowing simultaneous processing. The VEKM procedure is described in detail, and then demonstrated via two illustrative examples, which previously had only black-box computer solutions as they were thought to be not amenable to manual solution. The first example deals with switching or bivalent functions while the second handles 'big' Boolean functions. Both examples indicate that the VEKM procedure proposed herein enjoys the merits of insightfulness, simplicity and efficiency

scholarly journals Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

2021 ◽  
pp. 26-42
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Qualitative Comparative Analysis ◽  
Digital Design ◽  
Boolean Formula ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Boolean Minimization ◽  
Minimization Technique ◽  
Context Specific ◽  
Karnaugh Map

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

scholarly journals Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis

2018 ◽  
Vol 3 (1) ◽  
pp. 28-46 ◽  
Author(s):  
Ali Muhammad Ali Rushdi
Keyword(s):  
Comparative Analysis ◽  
Wide Spectrum ◽  
Qualitative Comparative Analysis ◽  
Software Tools ◽  
Sub Saharan Africa ◽  
Common Belief ◽  
Recent Debate ◽  
Input Domain ◽  
Sub Saharan ◽  
Karnaugh Map

A recent debate in the literature of Qualitative Comparative Analysis (QCA) concerns the potentials and pitfalls of the multi-value variant (mvQCA) in comparison with the more established crisp-set QCA (csQCA) and fuzzy-set QCA (fsQCA) variants. So far, the mvQCA methodology has been implemented either algebraically or via specific software tools such as TOSMANA. The main goal of this paper is to enhance the mvQCA methodology through the utilization of several varieties of Karnaugh maps including (a) the Conventional Karnaugh Map (CKM), (b) the Multi-Valued Karnaugh Map (MVKM), and (c) the Variable-Entered Karnaugh Map (VEKM). The paper offers a tutorial exposition of each of these maps in terms of two recently-published problems concerning the legal provision (introduction) and implementation of party bans in sub-Saharan Africa. Results obtained via various map techniques agree exactly among themselves, and are generally more compact than those obtained earlier via elementary algebraic manipulations, or even via software tools. We show, by way of example, that coding multi-valued variables by binary ones has a harmful primary effect of increasing the input domain. This effect is partially counterbalanced by a (contrarily to common belief) beneficial secondary effect of introducing genuine don’t-care configurations. We also address the issue of unresolved contradictory configurations, and propose two strategies to cope with them. The maps used tackle seven binary variables (or their equivalent), a number beyond the typical map limit of six variables. They are used to produce not only the minimal sum of a Boolean function but the complete sum as well. Though this paper is basically intended as a contribution to mvQCA methodology, it is also of significant utility in any field that demands the use of the Karnaugh map. It serves as a unification/exposition of three fundamental variants of the map, and has a definite pedagogical advantage for the wide spectrum of map users.

Impact of Globalisation on Pandemics: A comparative study on SARS and Covid-19 (Preprint)

2020 ◽  
Author(s):  
Johann Johann And Devika
Keyword(s):  
Comparative Analysis ◽  
Comparative Study ◽  
Descriptive Analysis ◽  
Qualitative Comparative Analysis ◽  
Global Scale ◽  
Global Level ◽  
Economic Stability ◽  
The Past ◽  
The World ◽  
Close Relationship

BACKGROUND Since November 2019, Covid - 19 has spread across the globe costing people their lives and countries their economic stability. The world has become more interconnected over the past few decades owing to globalisation and such pandemics as the Covid -19 are cons of that. This paper attempts to gain deeper understanding into the correlation between globalisation and pandemics. It is a descriptive analysis on how one of the factors that was responsible for the spread of this virus on a global scale is globalisation. OBJECTIVE - To understand the close relationship that globalisation and pandemics share. - To understand the scale of the spread of viruses on a global scale though a comparison between SARS and Covid -19. - To understand the sale of globalisation present during SARS and Covid - 19. METHODS A descriptive qualitative comparative analysis was used throughout this research. RESULTS Globalisation does play a significant role in the spread of pandemics on a global level. CONCLUSIONS - SARS and Covid - 19 were varied in terms of severity and spread. - The scale of globalisation was different during the time of SARS and Covid - 19. - Globalisation can be the reason for the faster spread in Pandemics.

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