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

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.

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Quality of Minimal Sets of Prime Implicants of Boolean Functions

International Journal of Electronics and Telecommunications ◽

10.1515/eletel-2017-0022 ◽

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.

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

Asian Journal of Research in Computer Science ◽

10.9734/ajrcos/2021/v8i230199 ◽

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.

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

Asian Journal of Education and Social Studies ◽

10.9734/ajess/2021/v17i230418 ◽

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.

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

Journal of King Abdulaziz University-Computing and Information Technology Sciences ◽

10.4197/comp.1-2.3 ◽

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

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Efficient Minimisation of Boolean Functions

International Journal of Electrical Engineering Education ◽

10.7227/ijeee.45.4.5 ◽

2008 ◽

Vol 45 (4) ◽

pp. 321-326 ◽

Cited By ~ 1

Author(s):

V. C. Prasad

Keyword(s):

Boolean Functions ◽

Minimal Set ◽

Undergraduate Level ◽

Popular Method ◽

Prime Implicants

Quine Mc Cluskey's (QM) method is a popular method for minimisation of Boolean functions. This method is widely taught at undergraduate level. In this paper simple modifications are suggested to make it more efficient. They allow us to avoid repetitions in the QM method. Further, a minimal set of prime implicants is easily obtained.

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Minimal Sets on Graphs and Dendrites

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007576 ◽

2003 ◽

Vol 13 (07) ◽

pp. 1721-1725 ◽

Cited By ~ 28

Author(s):

Francisco Balibrea ◽

Roman Hric ◽

L'ubomír Snoha

Keyword(s):

Topological Structure ◽

Partial Characterization ◽

Minimal Set ◽

Full Characterization ◽

Minimal Sets ◽

Continuous Maps

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

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Pseudoprojective strongly minimal sets are locally projective

Journal of Symbolic Logic ◽

10.2307/2275467 ◽

1991 ◽

Vol 56 (4) ◽

pp. 1184-1194 ◽

Cited By ~ 9

Author(s):

Steven Buechler

Keyword(s):

Predicate Symbol ◽

Infinite Dimension ◽

Elementary Extension ◽

Morley Rank ◽

Minimal Set ◽

Unary Predicate ◽

Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

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One theorem of Zil′ber's on strongly minimal sets

Journal of Symbolic Logic ◽

10.2307/2273990 ◽

1985 ◽

Vol 50 (4) ◽

pp. 1054-1061 ◽

Cited By ~ 4

Author(s):

Steven Buechler

Keyword(s):

Main Part ◽

Minimal Set ◽

Minimal Sets ◽

Image Position

AbstractSuppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,We prove the following theorems.Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

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