Variable-entered Karnaugh Map Research Articles

Reliability Evaluation of Multi-State Flow Networks Via Map Methods

This paper examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks. These two methods involve two Karnaugh map versions, namely the Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM). These two versions are crucial in providing not only the visual insight necessary to write better future software but also adequate means of verifying such software. We assess these two versions of map methods versus the exhaustive search method, which guarantees conceptual clarity at the expense of lack of computational efficiency. Our target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which we consider flow from a source node to a sink node in a capacitated network with a multistate capacity model for the links. Each network link has a varying capacity, which is assumed to exist in a mutually exclusive sense. The reliability of the system is wholly dependent on its ability to successfully transmit at least a certain required system flow from the source (transmitter) to the sink (receiver) station. The max-flow min-cut theorem is critical in obtaining all successful states. To demonstrate the proposed methods applicability, two demonstrative examples are given with ample details.

Boolean Curve Fitting with the Aid of Variable-Entered Karnaugh Maps

The Variable-Entered Karnaugh Map is utilized to grant a simpler view and a visual perspective to Boolean curve fitting (Boolean interpolation); a topic whose inherent complexity hinders its potential applications. We derive the function(s) through m points in the Boolean space B^(n+1) together with consistency and uniqueness conditions, where B is a general ‘big’ Boolean algebra of l≥1 generators, L atoms (2^(l-1)<L≤2^l) and 2^L elements. We highlight prominent cases in which the consistency condition reduces to the identity (0=0) with a unique solution or with multiple solutions. We conjecture that consistent (albeit not necessarily unique) curve fitting is possible if, and only if, m=2^n. This conjecture is a generalization of the fact that a Boolean function of n variables is fully and uniquely determined by its values in the {0,1}^n subdomain of its B^n domain. A few illustrative examples are used to clarify the pertinent concepts and techniques.

Handling Generalized Type-2 Problems of Digital Circuit Design via the Variable-Entered Karnaugh Map

This paper offers a novel treatment of the generalized Type-2 problem, a prominent fundamental problem of digital circuit design. We adapt the input-domain constraining technique via utilization of Variable-Entered Karnaugh Maps (VEKMs) together with careful employment of modern don’t-care notation. Our analysis covers the cases when an honest translator is possible or a sneaky translator is warranted, and is effective whether side inputs are absent or present, and for scalar or vectorial outputs.

Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis

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.

SWITCHING-ALGEBRAIC ANALYSIS OF RELATIONAL DATABASES

There is an established equivalence between relational database Functional Dependencies (FDs) and a fragment of switching algebra that is built typically of Horn clauses. This equivalence pertains to both concepts and procedures of the FD relational database domain and the switching algebraic domain. This study is an exposition of the use of switching-algebraic tools in solving problems typically encountered in the analysis and design of relational databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-entered Karnaugh map. The problems handled include; (a) the derivation of the closure of a Dependency Set (DS), (b) the derivation of the closure of a set of attributes, (c) the determination of all candidate keys and (d) the derivation of irredundant dependency sets equivalent to a given DS and consequently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates its pedagogical and computational merits over the traditional approach.

THE INVERSE PROBLEM FOR BOOLEAN EQUATIONS

The Forward Problem (FB) of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equation of the form f(X) = 0 where f(X): Bn → B and B is an arbitrary Boolean algebra. By contrast, the Inverse Problem (IB) of Boolean equations aims to reconstruct the equation f (X) = 0 given the set of solutions and hence to verify the correctness of this set. This study derives methods that handle this inverse problem for the main types of solutions of Boolean equations. These include: (a) Subsumptive general solutions, in which each of the variables is expressed as an interval by deriving successive conjunctive or disjunctive eliminants of the original function, (b) Parametric general solutions, in which each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra and (c) Particular solutions, each of which is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity. The reconstructed function f(X) in every case is set in a canonical form, such as the complete-sum form, to facilitate proving its equivalence to the original function. The methods presented herein are demonstrated with carefully-chosen illustrative examples over big Boolean algebras of various sizes. Among the methods utilized in handling the inverse problem for Boolean equations, the ones utilizing the variable-entered Karnaugh map offered pictorial insight and exhibited an efficient divide-and-conquer strategy.

Technical Note] Purely-Algebraic versus VEKM Methods for Solving Big Boolean Equations

Manual methods of solving big Boolean equations include purely-algebraic methods and methods utilizing the variable-entered Karnaugh map (VEKM). The VEKM techniques are demonstrated to be at least competitive with (and occasionally superior to) purelyalgebraic techniques, since the VEKM techniques (a) include purelyalgebraic methods as special cases, and (b) have a better control on the minimality of the pertinent function representations, and hence are more capable of producing more compact general solutions.

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

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

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general ‘big’ Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of ‘big’ Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.

Selective Deduction with the Aid of the Variable-Entered Karnaugh Maps

An important class of logical reasoning problems involves selective deduction from given hypotheses, which is deduction with the knowledge of certain information, or the lack thereof, about some of the pertinent variables. This note solves selective deduction problems via a powerful manual pictorial tool, viz., the variable-entered Karnaugh map (VEKM). The VEKM can be utilized to obtain the set of all prime implicants of the underlying function and then apply the knowledge available to select a subset thereof. Moreover, it allows a skipping of the tedious step of obtaining the complete sum in its general form by incorporating the available knowledge right from the outset. This is a time-saving short cut indeed, since no implicants are generated except those that are ultimately retained. An illustrative example serves to explain the steps of the method proposed herein and to demonstrate its superiority to the conventional method.

Variable-entered Karnaugh Map Procedures for Obtaining the Irredundant Disjunctive Forms of a Switching Function from Its Complete Sum

The covering problem, a classical problem in switching theory, involves the selection of some prime implicants irredundantly to cover the asserted part of an incompletely specified switching function. This paper advocates the use of the variable-entered Karnaugh map (VEKM) to solve this problem. To set the stage for achieving this purpose, a novel tutorial exposition of the VEKM-related variable-entered cover matrix is presented. Subsequently, the paper offers two VEKM solutions of the covering problem. The first solution relies on the new concept of VEKM loops. By contrast to a loop on a conventional Karnaugh map which is characterized by a single product, a VEKM loop is identified by two products, namely loop coverage (which depends only on map variables) and loop depth (which depends only on entered variables). In the second solution, the VEKM is used directly as a cover map in such a way that it is not littered with unnecessary details or too many overlapping loops. All solution procedures are demonstrated via illustrative examples that help reveal their visual merits as well as their applicability to relatively large problems.

Derivation of the Complete Sum of a Switching Function with the Aid of the Variable-entered Karnaugh Map

The complete sum (CS) of a switching function f is defined as a sum-of-products formula whose products constitute all, and nothing but, the prime implicants of f It has many useful applications including the simplification and minimization of switching functions, proving equivalence or independence, solution of Boolean equations, and transient hazard analysis. This paper presents two novel techniques for deriving the CS with the aid of the variable-entered Karnaugh map (VEKM); a map that enjoys several pictorial advantages and a doubled variable-handling capability, and hence is recommended when the number of variables ranges from 7 to 12 or even more. Only completely specified switching functions (CSSFs) are considered herein, simply because the CS of an incompletely specified function f is that of the CSSF that represents the upper bound for f. Our first technique uses the VEKM for obtaining any product-of-sums expression for the function which can be multiplied out to produce the CS after deletion of any absorbable terms. The second technique starts with a VEKM of CS entries, and after repeated folding of the VEKM ends up with the required CS provided necessary absorptions are implemented after each folding. Both techniques gain much from the use of a novel multiplication matrix that restricts the number of term comparisons needed for implementing absorptions. This matrix can be used to advantage also with some purely algebraic techniques such as Tison method. However, algebraic techniques, even after improvement, might remain inferior to VEKM techniques, obviously since the latter can combine most merits of map and algebra. Dual versions of the VEKM techniques considered can be used to obtain the dual of the complete sum, viz., the complete product.

Further Improved Variable-Entered Karnaugh Map Procedures for Obtaining the Irredundant Forms of an Incompletely-Specified Switching Function

Out of the potpourri of methods available for traditional minimization of switching functions, the method of the Karnaugh map is distinguished as a quick manual method that provides the user with pictorial insight. An advanced version of this map, viz., the variable- entered Karnaugh map (VEKM) doubles the variable-handling capa- bility of the map and allows its use for big Boolean algebras. The present paper offers a novel exposition of the essential features and properties of the VEKM, many of which are published for the first time. It also presents a simple and further improved VEKM procedure that obtains one of the irredundant disjunctive forms (IDFs) of an in- completely specified switching function (ISSF). Duality concepts are used to convert the present procedure into a dual one that obtains an irredundant conjunctive form for an ISSF. These procedures differ from their predecessors in two respects. First, the present procedures are rather advanced ones equipped with an explicitly stated set of rules that are clearer, though more powerful, than those of the preced- ing procedures. Second, the present procedures are more precise in handling the contributions of a map entered term, or alterm, and hence are more likely to capture minor details in the intrinsic structure of the ISSF under consideration. Therefore, the present procedures, if fol- lowed strictly, are more likely to achieve exact minimality, and even if not, the resulting expressions from them are always guaranteed to be almost minimal. Many detailed examples are given to demonstrate the essential features and properties of the map and to illustrate the rules and steps of the new procedures.

Using variable-entered karnaugh maps to solve boolean equations

A new method for obtaining a compact subsumptive general solution of a system of Boolean equations is presented. The method relies on the use of the variable-entered Karnaugh map (VEKM) to achieve successive elimination through successive map folding. It is superior in efficiency and simplicity to methods employing Marquand diagrams or Conventional Karnaugh maps; it requires the construction of significantly smaller maps and produces such maps in a minimization-ready form. Moreover, the method is applicable to general Boolean equations and is not restricted to the two-valued case. Details of the method are carefully explained and then illustrated via a classical example.

A Boolean minimization procedure using the variable-entered Karnaugh map and the generalized consensus concept

Both the conventional Karnaugh map (CKM) and the variable-entered Karnaugh map (VEKM) are powerful manual tools that have many pictorial and pedagogical advantages. However, while the CKM is suitable only for small-sized problems, the VEKM can handle more variables and hence can be used for a variety of mediumsized applications. This paper presents an advanced and further improved VEKM procedure that aims at obtaining most, if not all, of the irredundant disjunctive forms of an incompletely-specified switching function (ISSF). This procedure differs from previous ones in two respects. First, it is a rather elaborate one employing the concept of the generalized consensus of a set of terms. Second, it uses a more systematic and more precise two-pass strategy that is more likely to capture minor details in the intrinsic structure of the ISSF under consideration. This procedure is intended as a supplement rather than a replacement of more powerful algorithms that have computer implementations but are too complicated to have some graphical insight for the user. Two detailed examples are given to illustrate the rules and steps of the new procedure and to demonstrate its efficiency and power.

Improved variable-entered Karnaugh map procedures

An improved variable-entered Karnaugh map (VEKM) procedure for obtaining the minimal disjunctive form of a switching function and a dual procedure for obtaining its minimal conjunctive form are presented. These procedures apply to any general switching function of moderate complexity that can be incompletely specified with respect to all its variables. Existing VEKM procedures are compared to the new ones and are shown to be special cases of them. Various examples are given to illustrate the details of the new procedures and to demonstrate their efficiency and power.

Map differentiation of switching functions

An important problem in reliability analysis and in testing of digital circuits is the problem of deriving the partial derivative (Boolean difference) of a swithching (Boolean) function. This problem is tackled herein by a divide and conquer technique that relies on the use of a recently developed minimization procedure via the variable-entered Karnaugh map (VEKM). This technique is simple, systematic, and can handle switching functions of moderate complexity. Two detailed examples are given to illustrate the derivation of non conventional Boolean differences as well as single and multiple conventional ones.

Map derivation of the minimal sum of a switching function from that of its complement

An important problem in reliability is the problem of inverting the minimal sum representation of a switching (Boolean) function to obtain that of its complement. This problem is tackled by a divide-and-conquer inversion procedure that relies on the use of the variable-entered Karnaugh map (VEKM). An improved VEKM minimization procedure is presented, and then utilized as a part of the inversion procedure. Both procedures are illustrated by simple examples that cover the noncoherent as well as the coherent case.

Symbolic Reliability Analysis with the Aid of Variable-Entered Karnaugh Maps

A method for finding the symbolic reliability of a moderately complex system is presented. The method uses Bayesian decomposition to reduce the system into simpler series-parallel subsystems. The successes or failures of these subsystems are found by inspection and then recast into disjoint sum-of-product forms with the aid of the Karnaugh map. Subsequently, an almost minimal disjoint expression for the system success or failure is obtained with the aid of a variable-entered Karnaugh map (VEKM). This VEKM method is illustrated by applying it to some examples recently solved in the literature. The main advantage of the method is the pictorial insight it provides to the reliability analyst.