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This chapter attempts to enhance the interplay between the ecology and reliability fields by employing Boolean-based reliability language and techniques to quantify ecological metrics related to connectivity and redundancy. We emphasize the question of connectivity in models of probabilistic networks as a common area of interest for both fields. The chapter borrows techniques from mainstream reliability theory to treat a prominent problem of ecology, namely that of survivability (of a species), defined here as the probability of successful migration of a certain organism escaping from critical source habitat patches and seeking refuge in specific destination habitat patches via heterogeneous deletable ecological corridors, possibly with uninhabitable stepping stones en route. This problem might be reformulated in contexts other than that of migration, including those of (a) dynamics of metapopulations, colonization, or invasion, (b) gene flow, (c) spread of infectious diseases, epidemics, or pandemics, and (d) energy transfer within food webs. Indicators of network connectivity in classical reliability theory are probabilities that might be designated according to the set of source nodes and the set of destination nodes as one to one, one to many, many to many, or all to all. Our present notion of survivability (of a species) is also a probability of connectivity, now measured from any node (among many nodes) to any node (among many nodes). We explore methods for computing the survivability (of a species) by adapting switching-algebraic techniques that are usually employed in the reliability field. In addition to this survivability metric, we comment on some other connectivity indicators that are currently used in ecology. We stress two recent contributions to the ecology literature, one employing analogy with electric circuit theory, and another concerning the most reliable (or minimum-lag) dispersal paths.

This paper advocate and demonstrates the utility of the Karnaugh map, as a pictorial manual tool of Boolean algebra, in the exploration of medical problems as exemplified herein by the problem of Fetal Malnutrition (FM). The paper briefly introduces the FM problem, and specifies four metrics or tests used frequently in its study. Clinical data collected about these metrics (as continuous variables or dichotomized versions thereof) are conventionally treated via statistical methods. The Karnaugh map serves as a convenient way for aggregating the set of clinical data available into a pseudo-Boolean function. The map can be used to produce a two-by-two contingency matrix (confusion matrix or frequency matrix) that relates an assessed test or metric to a reference or standard one. Each of these two metrics can be any of the map variables or a function of some or all of these variables. While the map serves in this capacity as a supplement or aid to statistical methods, it is also useful for certain non-statistical methods (specifically Boolean ones). The paper shows how the map entries can be dichotomized via an appropriate threshold for use in Boolean Analysis (BA), which can be conducted despite the lack of a gold standard. The map also implements Qualitative Comparative Analysis (QCA) for the given clinical data. The map variable-handling capability does not pose as a shortcoming for either BA or QCA, since the number of variables involved (not only herein but in other typical medical problems as well) is relatively small. The concepts and methods introduced herein are demonstrated through application to the aforementioned set of clinical data for the FM problem, and can be extended to a wide variety of medical problems.

Modern digital circuits, especially those based on large-scale integration devices employ DE flip flops, which are an extension of the D type with the capacity to store an input value only upon request or enabling. The DE flip flop could possibly be described algebraically by its characteristic equation or tabularly by its next-state table (used for analysis purposes) and its excitation table (used for synthesis purposes). This paper explores two novel characterizations of DE flip flops. First, equational and implicational descriptions are presented, and the Modern Syllogistic Method is utilized to produce complete statements of all propositions that are true for a general DE flip flop. Next, methods of Boolean-equation solving are employed to find all possible ways to express the excitations in terms of the present state and next state. The concept of Boolean quotient plays a crucial role in exposing the pertinent concepts and implementing the various desired derivations. This paper is expected to be of an immediate pedagogical benefit, and to facilitate the analysis and synthesis of contemporary sequential digital circuits.

The paper utilizes the Karnaugh map in a pedagogical calculation of the Shapley-Shubik index of voting power, which is perhaps the most prominent such index for managerial, political, and engineering voting systems. The Karnaugh map has features that allow it to readily implement this calculation, namely: (a) it can conveniently represent pseudo-switching functions of the form g= B_2^n →R, (b) it can immediately detect the number of orderings in going from one of its configurations (cells) to another, and (c) it visually identifies a pair of configurations that is critical w. r. t. a certain voter, which are a primitive winning coalition of which this voter is a member and a corresponding primitive losing coalition that results when this voter defects. The paper offers a detailed description of these three features and of the associated calculations using the celebrated system of the European Economic Community as a running example.

This paper deals with one of the most prominent problems in industrial prognostics, namely the estimation of the Remaining Useful Life (RUL) of the most popular industrial battery, viz., the lithium-ion battery. The paper presents a state-space model of the battery, and then estimates the dynamic behavior of seven of its process variables and two of its sensor variables. The estimation is achieved via two well known estimators, the Unscented Kalman Filter (UKF) and the Particle Filter (PF) when noise of various levels and types is injected. Numerical and chart comparisons of these two computing estimators are reported and discussed.

This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM). In the classical method, the number of parameters used is minimized and compact solutions are obtained. However, the parameters belong to the underlying big Boolean algebra. By contrast, the novel method does not attempt to minimize the number of parameters used, as it uses independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom in the Boole-Shannon expansion of the formula ݃(ࢄ .(Though the method produces non-compact expressions, it is much quicker in generating particular solutions. The two methods are demonstrated via two detailed examples.

Internet Service Providers (ISPs) are expected to continuously provide services without interruptions. However, planning for this activity requires a number of tasks such as forecasting clients' demand that needs to be considered in fulfilling their objectives particularly in face of calamity and traffic load variations. We present a Software-Defined Networking (SDN) load distribution technique for a typical ISP. We have implemented an optimization method employing linear programming to achieve this goal. A multi-objective model has been developed for this purpose. The multiple objectives are maximization of link utilization, maximization of link availability and minimization of average round-trip time for the top visited websites by users. Furthermore, an automated load distribution system has been developed for carrying out these optimization related activities. In addition, an example is presented to illustrate traffic load distribution under different operational modes (normal, critical, and catastrophic). The findings of this study should be of great help to the service provider to better plan for load distribution between international links, and help in approving down times and maintenance windows requested by cable providers.

Natural and designed ecological corridors are key elements for the survival of a species, as they allow the species to avoid local extinction by migrating to more suitable habitat patches. This paper studies various reliability metrics for the process of migration in a metapopulation landscape network from a critical habitat patch to destination habitat patches via perfect stepping stones and imperfect (deletable) corridors. The work presented herein generalizes earlier work on the application of reliability theory in ecology by allowing corridors to be heterogeneous (of non-identical unreliabilities). The paper is a tutorial exposition of modern reliability techniques, which formulate a problem in the Boolean domain, manipulate formulas to achieve disjointness of logically added subexpressions and retain statistical independence of logically multiplied ones, and finally reach a probability-ready expression that is directly transformed back to the probability domain. Several metrics are covered including system unreliability, life expectancy (MTTF), and component importance measures. An interesting finding is that the life expectancy of a classical landscape network is more than double that of a single corridor. Extensions to quantification of uncertainty in the above metrics and to evaluation of more sophisticated metrics of landscape connectivity are also pointed out.

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premisesallthat can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.

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

Any system of ‘big’ Boolean equations can be reduced to a single Boolean equation . We propose a novel method for producing a general parametric solution for such a Boolean equation without attempting to minimize the number of parameters used, but instead using independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom that appears in the discriminants of the function . We sacrifice minimality of parameters and algebraic expressions for ease, compactness and efficiency in listing all particular solutions. These solutions are given by additive formulas expressing a weighted sum of the asserted atoms of , with the weight of every atom (called its contribution) having a number of alternative possible values equal to the number of appearances of the atom in the discriminants of . This allows listing a huge number of particular solutions within a very small space and the possibility of constructing solutions of desirable features. The new method is demonstrated via three examples over the ‘big’ Boolean algebras, , , and , respectively. The examples demonstrate a variety of pertinent issues such as complementation, algebra collapse, incremental solution, and handling of equations separately or jointly.

A prominent reliability model is that of the partially-redundant (k-out-of-n) system. We use algebraic as well as signal-flow-graph methods to explore and expose the AR algorithm for computing k-out-of-n reliability. We demonstrate that the AR algorithm is, in fact, both a recursive and an iterative implementation of the strategy of Reduced Ordered Binary Decision Diagrams (ROBDDs). The underlying ROBDD for the AR recursive algorithm is represented by a compact Signal Flow Graph (SFG) that is used to deduce AR iterative algorithms of quadratic temporal complexity and linear spatial complexity. Extensions of the AR algorithm for (single or scalar) threshold, double-threshold, vector-threshold, and k-to-l-out-of-n systems have similar ROBDD interpretations.

In a recent seminal paper, Rudeanu derived necessary and sufficient conditions for the most general form of the subsumptive general solution of a Boolean equation. The present paper investigates four prominent classes of the aforementioned most general form. These include the conventional (eliminants) class, the don't-care class, the Rudeanu class, and the complete-sum (CS) class. We show that each of these classes satisfies the Rudeanu conditions, and hence fits into the frame of the most general form of the subsumptive solution. The four classes are of variable but comparable computational complexities. The conventional (eliminants) class is the least likely to produce a compact solution, while the don't-care class is the most likely to produce such a solution. Three examples are presented to demonstrate solutions produced by the four classes and compare them form the point of view of compactness of solutions, which determines the ease of generating the tree of particular solutions. A remaining open question that has yet to be settled is whether these four classes are exhaustive, or in other words, whether there are general solution classes other than them. We demonstrate that the don't-care class is never inferior (and occasionally superior) to the other classes. In fact, this class seeks global minimality and has a better control on the pertinent variables and algebra generators. Both the CS class and the Rudeanu class have an advantage over the eliminants method. The CS class utilizes a canonical representation that explicitly shows all information in the simplest form. The Rudeanu class enhances the eliminants class by incorporating the consensus (Boolean resolution) concept in a way similar to that of the CS approach.

There is an established equivalence between relatio nal database Functional Dependencies (FDs) and a fragment of switching algebra that is built typical ly of Horn clauses. This equivalence pertains to bo th concepts and procedures of the FD relational databa se 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 relationa l databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-ente red Karnaugh map. The problems handled include; (a) the derivation of the closure of a Dep endency Set (DS), (b) the derivation of the closure of a set of attributes, (c) the determination of all c andidate keys and (d) the derivation of irredundant dependency sets equivalent to a given DS and conseq uently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates i ts pedagogical and computational merits over the tradi tional approach.

Boolean-equation solving permeates many diverse areas of modern science. To solve a system of Boolean equations, one usually combines them into an equiva lent single Boolean equation {f (X) 0} = � whose set of solutions is exactly the same as that of the origin al system of equations. One of the general classes of solutions for Boolean equations is the subsumptive general solution, in which each variable is express ed as an interval decided by a double inequality in terms of the succeeding variables. The solution validity depends on the satisfaction of a required consisten cy condition. In this study, we introduce a novel m ethod (henceforth called the CS method) for producing sub sumptive Boolean-equation solutions based on deriving the complete sum (CS(f (X)) �� of the pertinent Boolean function f (X) � . The complete sum CS(f (X)) � is a disjunction of all prime implicants of f (X) �� and nothing else. It explicitly shows all informat ion about f (X) � in the most compact form. We demonstrate the proposed CS solutions in terms of four examples, covering Boolean algebras of different sizes and using two p rominent methods for deriving CS(f (X)) � . Occasionally, the consistency condition results in a collapse of the underlying Boolean algebra into a smaller subal gebra. We also illustrate how an expansion tree (typically reduced to an acyclic graph) can be used to deduce a complete list of all particular solutions from the subsumptive solution. The present CS method yields correct solutions, since it fits into the frame of the most general subsumptive solution. Among competing subsumptive methods, the CS method strikes a reasonable tradeoff between the conflicting requirements of less computational cost and more compact form for t he solution obtained. In fact, it is the second bes t known method from both criteria of efficiency and c ompactness of solution.

This chapter deals with the paradigm of handling system reliability analysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain. This paradigm is well-established within the academic circles of reliability theory and engineering, albeit virtually unknown outside these circles. The chapter lists and explains arguments in favor of this paradigm for systems described by verbal statements, fault trees, block diagrams, and network graphs. This is followed by a detailed exposition of the pertinent concept of the Real or Probability Transform of a switching (two-valued Boolean) function, and that of a Probability-Ready Expression (PRE). Some of the important rules used in generating a PRE are presented, occasionally along with succinct proofs. These include rules to achieve disjointness (orthogonality) of ORed formulas, and to preserve statistical independence, as much as possible, among ANDed formulas. Recursive relations in the Boolean domain are also discussed, with an application to the four versions of the AR algorithm for evaluating the reliability and unreliability of the k-out-of-n:G and the k-out-of-n:F systems. These four versions of the algorithm are explained in terms of signal flow graphs that are compact, regular, and acyclic, in addition to being isomorphic to the Reduced Ordered Binary Decision Diagram (ROBDD). An appendix explains some important properties of the concept of Boolean quotient, whose expectation in Boolean-based probability is the counterpart of conditional probability in event-based probability.

The tasks of guard placement or sensor deployment in an art gallery, a museum or in the corridors of public and security buildings pose the same problem, which requires placing the guards or sensors so as to cover a specified set of nodes with a minimum number of sensors or guards, thereby reducing the overall cost of the system as well assist power consumption. Generally, minimization can be done using optimization techniques such as linear programming, but in case of sensor deployment or guard placement there is a need either to place or not to place the sensor or guard, and hence only Boolean or binary values are used. Therefore, in order to optimize such a problem, we use the special case of linear integer programming known as Boolean integer linear programming (0-1 ILP). Other algorithms like Pseudo-Boolean SAT Solvers can also be used for the minimization purpose. In this paper, we introduce these minimization algorithms for the sensor deployment problem. We also contribute a greed-based heuristic, which utilizes the fact that the pertinent propositional formulas have variables of purely un-complemented literals. This heuristic has much less computational cost compared to those of 0-1 ILP and the Pseudo-Boolean SAT Solvers.

The tasks of guard placement or sensor deployment in an art gallery, a museum or in the corridors of public and security buildings pose the same problem, which requires placing the guards or sensors so as to cover a specified set of nodes with a minimum number of sensors or guards, thereby reducing the overall cost of the system as well assist power consumption. Generally, minimization can be done using optimization techniques such as linear programming, but in case of sensor deployment or guard placement there is a need either to place or not to place the sensor or guard, and hence only Boolean or binary values are used. Therefore, in order to optimize such a problem, we use the special case of linear integer programming known as Boolean integer linear programming (0-1 ILP). Other algorithms like Pseudo-Boolean SAT Solvers can also be used for the minimization purpose. In this paper, we introduce these minimization algorithms for the sensor deployment problem. We also contribute a greed-based heuristic, which utilizes the fact that the pertinent propositional formulas have variables of purely un-complemented literals. This heuristic has much less computational cost compared to those of 0-1 ILP and the Pseudo-Boolean SAT Solvers.

An important class of square binary matrices over t he simplest finite or Galois Field GF(2) is the cla ss of involutory or self-inverse (SI) matrices. These mat rices are of significant utility in prominent engin eering applications such as the study of the Preparata Tra nsformation or the analysis of synchronous Boolean Networks. Therefore, it is essential to devise appr opriate methods, not only for understanding the pro perties of these matrices, but also for characterizing and constructing them. We survey square binary matrices of orders 1, 2 and 3 to identify primitive SI matrices among them. Larger SI matrices are constructed as (a) the direct sum, or (b) the Kronecker product, of smalle r ones. Illustrative examples are given to demonstr ate the construction and properties of binary SI matrices. The intersection of the sets of SI and permutation binary matrices is studied. We also study higher-order SI binary matrices and describe them via recursive rel ations or Kronecker products. Our work culminates in an exposition of the two most common representations of Boolean functions via two types of Boolean SI matri ces. A better understanding of the properties and methods of constructing SI binary matrices over GF (2) is achieved. A clearer picture is attained abou t the utility of binary matrices in the representation of Boolean functions.

A total description of a synchronous Boolean networ k is typically achieved by a matrix recurrence relation. A simpler alternative is to use a scalar equation which is a possibly nonlinear equation tha t involves two or more instances of a single scalar variable and some Bool ean operator(s). Further simplification is possible in terms of a linear reduced scalar equation which is the simplest two-term scalar equa tion that includes no Boolean operators and equates the value of a scalar variabl e at a latter instance t 2 to its value at an earlier instance t 1. This equation remains valid when the times t 1 and t2 are both augmented by any integral multiple of the underlying time period. In other words, there are i nfinitely many versions of a reduced scalar equatio n, any of which is useful for deducing information about the cyclic behavior of the network. However, to obtain correct information about the transient behavior of the network, one must find the true red uced scalar equation for which instances t1 and t 2 are minimal. This study investigates the nature, d erivation and utilization of reduced scalar equations. It relies on Boolean-algebraic ma nipulations for the derivation of such equations an d suggests that this derivation can be facilitated by seeking certain orthogonality relations among certain successive (albeit not necessarily consecutive) instances of t he same scalar variable. We demonstrate, contrary t o previously published assumptions or assertions, tha t there is typically no common reduced scalar equat ion for all the scalar variables. Each variable usually sat isfies its own distinct reduced scalar equation. We also demonstrate that the derivation of a reduced scalar equation is achieved not only by proving it but also by disproving an immediately preceding version of it when such a version might exist. We also demonstrate that, despite the useful insight supplied by the reduced scalar equations, they do not provide a total solut ion like the one offered by matrix methods and therefore they ne ed be supported by other techniques of mathematical reasoning. We present three classical examples to i llustrate our techniques. Two examples are tutorial s on the necessary Boolean-algebraic techniques. They present corrections of previously published results and r efute purported claims of discrepancies between scalar an d matrix methods. The third example illustrates how the reduced scalar equations can be supplemented by tec hniques of number theory, Diophantine equations and Boolean equations in making subtle inferences about Boolean networks. We achieved a better understanding of the nature of reduced scalar equations, demonstr ated Boolean-algebraic techniques for deriving them, presented other mathematical tools for utilizing th em and finally reconciled them with the more encompassing but more complex and less insightful matrix methods.

The Forward Problem (FB) of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equati on of the form f(X) = 0 where f(X): B n → B and B is an arbitrary Boolean algebra. By contrast, the Inve rse 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 t his set. This study derives methods that handle this in verse 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 con junctive or disjunctive eliminants of the original function, (b) Parametric general solutions, in whic h 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 identit y. The reconstructed function f(X) in every case i s set in a canonical form, such as the complete-sum f orm, 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 uti lizing the variable-entered Karnaugh map offered pictorial insight and exhibited an efficient divide -and-conquer strategy.

This paper utilizes concepts and tools of switching algebra to describe an algorithm for computing all candidate keys of a relational database. Our algorithm starts with a superkey comprising all attributes, and then uses logical inference to reduce it to the set of candidate keys. The essential step in this inference is the construction of the complete sum of a pertinent switching function. Some computational saving is possible through the use of Boolean quotients if essential and/or superfluous attributes are recognized.

Weighted voting systems play a crucial role in the investigation and modeling of manyengineering structures and political and socio-economic phenomena. There is an urgentneed to describe these systems in a simplified powerful mathematical way that can begeneralized to systems of any size. An elegant description of voting systems is presentedin terms of threshold Boolean functions. This description benefits considerably fromthe wealth of information about these functions, and of the potpourri of algebraic andmap techniques for handling them. The paper demonstrates that the prime implicantsof the system threshold function are its Minimal Winning Coalitions (MWC). Thepaper discusses the Boolean derivative (Boolean difference) of the system thresholdfunction with respect to each of its member components. The prime implicants of thisBoolean difference can be used to deduce the winning coalitions (WC) in which thepertinent member cannot be dispensed with. Each of the minterms of this Booleandifference is a winning coalition in which this member plays a pivotal role. However,the coalition ceases to be winning if the member defects from it. Hence, the numberof these minterms is identified as the Banzhaf index of voting power. The conceptsintroduced are illustrated with detailed demonstrative examples that also exhibit someof the known paradoxes of voting- system theory. Finally, the paper stresses the utilityof threshold Boolean functions in the understanding, study, analysis, and design ofweighted voting systems irrespective of size.

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.

This paper reviews earlier work claiming that water-resource inputs closely follow a self-similar model, which is a manifestation of the more general multi-fractal behavior. Such claims have been repeatedly verified for wet terrains, but are still to be tested for arid regions. The paper presents a tutorial introduction to the concepts and phenomena of self similarity and multi-fractality for water-resource inputs. It also discusses the important implications of these phenomena for water consumption and storage. While conservative water consumption is an all-time virtue, it must be stressed and strictly imposed during periods of water shortages or droughts, which are more likely to be extended than to be brief. Periods of heavy rainfalls and floods should be optimally utilized to improve existing water storage levels. A mandatory policy for water conservation is strongly advocated by this paper, which contributes a scientific confirmation for the intuitively observed self-similarity phenomenon in arid regions. The paper presents a pilot and pioneering statistical study of a rainfall data set in a desert arid region. This study clearly indicates that this data set has a multi-fractal behavior, and that a mono-fractal model does not suit it. To the best of our knowledge, this result is the first published decisive confirmation for the self-similarity and multifractality phenomenon for rainfall data in arid regions.

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.